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488A64eOf6	The method requires sampling in order to approximate the new parametric distribution (via the WIS algorithm). The runtime of this algorithm, and hence the additional computational complexity incurred by this decoding method, is not discussed. Further, there is an additional sampling + renormalization step that must happen during decoding (algorithm 2). The lack of discussion of these costs is especially pertinent given that the authors point to the tuning required by other decoding methods as one of their downsides “But unlike those previous decoding methods that require heavy manual hyper-parameter tuning for trade-off among different metrics”	LANGUAGE MODEL DECODING AS DIRECT METRICS OPTIMIZATION Haozhe Ji Pei Ke∗ Hongning Wang Minlie Huang∗ The CoAI Group, DCST, BNRist, Tsinghua University, Beijing 100084, China jihaozhe@gmail.com aihuang@tsinghua.edu.cn ABSTRACT Despite the remarkable advances in language modeling, current mainstream decoding methods still struggle to generate texts that align with human texts across different aspects. In particular, sampling-based methods produce less-repetitive texts which are often disjunctive in discourse, while search-based methods maintain topic coherence at the cost of increased repetition. Overall, these methods fall short in achieving holistic alignment across a broad range of aspects. In this work, we frame decoding from a language model as an optimization problem with the goal of strictly matching the expected performance with human texts measured by multiple metrics of desired aspects simultaneously. The resulting decoding distribution enjoys an analytical solution that scales the input language model distribution via a sequence-level energy function defined by these metrics. And most importantly, we prove that this induced distribution is guaranteed to improve the perplexity on human texts, which suggests a better approximation to the underlying distribution of human texts. To facilitate tractable sampling from this globally normalized distribution, we adopt the Sampling-Importance-Resampling technique. Experiments on various domains and model scales demonstrate the superiority of our method in metrics alignment with human texts and human evaluation over strong baselines. 1 INTRODUCTION Although pre-trained on large corpora of human texts with scaled up sizes, existing auto-regressive language models (LMs) (Radford et al., 2019; Brown et al., 2020; Zhang et al., 2022) are still struggling to produce human-like texts measured in various aspects, such as repetition, coherence, and consistency (Pillutla et al., 2021; Dou et al., 2022). Existing decoding methods are mainly driven to address two main mis-specifications of an LM’s distribution: (i) The long tail of the distribution is unreliable (Holtzman et al., 2020), such that sampling from these low-probability regions often produces low-quality contents that are incoherent. (ii) The mode of the distribution is degenerated (Welleck et al., 2020), where samples with high probabilities exhibit low diversity with repetitive patterns. As a result, sampling-based decoding methods (Fan et al., 2018; Holtzman et al., 2020; Meister et al., 2022) use various truncation strategies to avoid sampling from the unreliable long tail of the distribution, while recent search-based methods (Li et al., 2022; Su et al., 2022) incorporate additional contrastive objectives to avoid the collapse of degenerated repetitions. Since these two mis-specifications reside at opposing extremes of the probability spectrum, current decoding methods inevitably concentrate on ∗Corresponding Author. Figure 1: The decoding distribution $p_{\theta,\mu}$ induced by DAEMON scales the input LM distribution $p_\theta$ with a sequence-level energy function $E_\mu$, which leads to a more accurate recovery of the underlying data distribution $p_d$. just one of them which addresses only a limited subset of aspects. Although heuristic designs and sophisticated hyper-parameter tuning allow trade-offs, these approaches usually cannot effectively align with human texts with respect to a broad range of critical aspects simultaneously. Attempts have been made to fix the mis-specification issue of LM distribution by directly augmenting the standard Maximum Likelihood Estimation (MLE) with auxiliary training objectives (Welleck et al., 2020; Su et al., 2022; Xu et al., 2022). However, exposure bias (Chiang & Chen, 2021; Arora et al., 2022) prevents the effectiveness of such attempts. Specifically, since during training the autoregressive LM is conditioned on the ground-truth context, it is not guaranteed that the imposed properties in these training objectives would be preserved during decoding time, where the context is progressively generated by the LM itself. On the other hand, approaches based on Reinforcement Learning (RL) (Ranzato et al., 2016; Yu et al., 2017) address the exposure bias issue, but often encounter challenges to maintain proximity to the distribution of human texts (characterized by a low perplexity) (Caccia et al., 2020). Overall, these methods do not guarantee a general enhancement over the standard training paradigm, owing to the potential conflicts between their designated objectives and MLE (Lin et al., 2021b). More related work discussion is provided in Appendix B. In this work, we focus on the decoding route and present a novel framework, Decoding As Direct Metrics Optimization (DAEMON) that explicitly targets at aligning desired aspects with human texts. DAEMON frames decoding from a language model as an optimization problem with the goal of locating the optimal decoding distribution where sampled texts can strictly match with human texts in multiple evaluation metrics simultaneously. Formally, given the input LM distribution $p_\theta$ learned on the human text distribution $p_d$, DAEMON searches for the decoding distribution $q$ that minimizes the reverse Kullback-Leibler divergence (KL), $D_{KL}(q\|\|p_\theta)$, subject to the constraints of matching the expected evaluation metric scores under $g$ and $p_d$. We choose the reverse KL to induce the decoding distribution $q$, as it forces $q$ to recover the major probability masses within the support of $p_\theta$ (Huszár, 2015; Malinin & Gales, 2019), which contains mostly high-quality samples. Moreover, besides directly enforcing alignment on chosen metrics, we also rigorously prove that the optimization problem guarantees an improvement of the solution over the input LM in perplexity, which indicates a more general gain in aligning with human texts. In addition to the theoretical guarantee, the decoding distribution induced by DAEMON also enjoys an analytical solution denoted as $p_{\theta,\mu}$. It scales the locally normalized LM distribution $p_\theta$ with a sequence-level energy function $E_\mu$ which depicts the underlying distribution $p_d$ from various perspectives by satisfying the corresponding constraints. In Figure 1, we visualize $p_{\theta,\mu}$ in an illustrative example where the energy captures the disjoint regions of modes in $p_d$, which empowers the input LM distribution $p_\theta$ to facilitate a better approximation of $p_d$. To enable tractable sampling from $p_{\theta,\mu}$, which is globally normalized over the space of all possible sequences, we adopt the Sampling-Importance-Resampling (SIR) technique (DB, 1988; Smith & Gelfand, 1992) that first samples candidates from $p_\theta$ and then resamples based on the importance weight defined by the energy function. We empirically demonstrate the effectiveness of DAEMON in open-ended text generation by considering a wide range of critical aspects including repetition, coherence, diversity, and information content across different model scales and data domains. Experimental results show that DAEMON outperforms strong decoding baselines in both automatic evaluation of metrics alignment with human texts and human evaluation. ## Method: Decoding as Direct Metrics Optimization We consider conditional language generation from a pre-trained language model specified by the distribution $p_\theta$, where the model is provided with a relatively short prefix $x_{<t_0} = \{x_i\}_{i=1}^{t_0}$ of length $t_0$ and required to generate a continuation that results in a full text $\hat{x}_{\leq T} = \{\hat{x}_i\}_{i=1}^{T}$ of total length $T$. In the following, the subscript of $x_{<T}$ is omitted for convenience. Instead of directly sampling from $p_\theta$, we look for a decoding distribution induced from $p_\theta$ to produce human-like texts measured by a set of chosen metrics. For example, in the canonical top-$k$ sampling (Fan et al., 2018), the decoding distribution is obtained by truncating the conditional distribution $p_\theta(x_t\|x_{<t-1})$ to keep the top-$k$ candidates at every decoding step, so as to improve the reliability of generated content. Ideally, a perfect decoding distribution $q_{opt}$ assigns an arbitrary text sample $x$ with the probability equals to $p_d(x)$, where $p_d$ is the underlying distribution of human texts. In practice, this is infeasible since we only have samples from $p_d$, rather than $p_d$ itself. However, given a text evaluation metric we are interested in (such as repetition and coherence), formally $ f : \mathcal{X} \to \mathbb{R} $ that maps $ x $ in the text space $ \mathcal{X} $ to a real value, an alternative criterion for measuring the closeness from $ q_{\text{opt}} $ to $ p_d $ can be achieved by matching the expectation of $ f $ under $ q_{\text{opt}} $ and $ p_d $, i.e., $ \left\| \mathbb{E}_{x \sim q_{\text{opt}}} [f(x)] - \mathbb{E}_{x \sim p_d} [f(x)] \right\| $. This expectation-matching criterion, commonly employed in prior studies (Holtzman et al., 2020; Meister et al., 2022; Su et al., 2022) as an empirical evaluation of the resemblance of generated texts against human texts. This forms the basis of our proposed optimization-based decoding framework that directly aligns the generated texts with human texts against the set of chosen text evaluation metrics. ### 2.1 Formulation of the Optimization Problem At the core of our proposed decoding framework, we look for the optimal solution $ q_{\text{opt}} $ of the following constrained optimization problem which searches for the decoding distribution $ q $ closest to the given LM distribution $ p_\theta $ and strictly matching the expectations on the generated texts with that of human texts measured by a set of chosen evaluation metrics: \[ q_{\text{opt}} = \arg\min_{q \in \mathcal{P}} D_{\text{KL}}(q \\| p_\theta) \] subject to \[ \mathbb{E}_{x \sim q} [f_k(x)] = \mathbb{E}_{x \sim p_d} [f_k(x)], \quad k \in \{1, \cdots, K\}, \] where $ f = \{f_k\}_{k=1}^K $ is a set of evaluation metrics we concern, and $ \mathcal{P} $ is the set of all probability densities in the input space $ \mathcal{X} $. The formulation of our proposed optimization problem hinges on our key insight of constructing a decoding distribution from a language model to acquire samples that closely resemble human texts. The constraints, as defined to match the performance of evaluated metrics on generations with those obtained on human texts, explicitly ensure this goal in expectation. The reverse KL divergence in the optimization objective, i.e., $ D_{\text{KL}}(q \\| p_\theta) $, restricts the decoding distribution $ q $ to deviate minimally from the LM distribution $ p_\theta $ by encouraging mode-seeking behavior, which satisfies the quality-demanding nature of decoding. Although the forward KL is extensively employed as an optimization objective in learning data-driven probabilistic models (Radford et al., 2019), its induced distribution is shown to mismatch with the quality assessment of human (Pang & He, 2021) by overestimating the long tail of the target distribution (Ji et al., 2023) due to its mean-seeking behavior. More discussion is provided in Appendix C. We believe the learning and decoding phases posit different goals: the former is to capture all modes in the data, while the latter is to decode high-quality ones. Hence, we require the decoding distribution to only explore within the support of the given LM distribution, which is naturally realized by minimizing the reverse KL. Existing truncation-based sampling (Fan et al., 2018; Welleck et al., 2020; Meister et al., 2022) can be deemed as a heuristic that shares the same spirit of ours in maintaining a finite reverse KL, since the support of the truncated distribution is always the strict subset of the support of the given LM distribution. The formulation of our optimization problem is also known as information projection in previous literature of information geometry (Csiszár & Mátus, 2000; Nielsen, 2020), which can be deemed as finding the projection of $ p_\theta $ on the manifold of distributions that constrains $ p_d $. In the following proposition, we show that it actually leads to a nice analytical solution. The full proof of Proposition 1 is provided in Appendix A.1. Proposition 1. The distribution that solves the optimization problem (1) is in the form of: \[ p_{\theta,\mu}(x) \propto p_\theta(x) \exp \left[ - E_\mu(x) \right], \quad \forall x \in S(p_{\theta,\mu}) \] where $ E_\mu(x) = \mu^\top f(x) $ and $ S(p) = \{x : p(x) > 0\} $ is the support of distribution $ p $. $ \mu \in \mathbb{R}^K $ is determined by the constraints in (1). The unnormalized form of $ p_{\theta,\mu}(x) $, also known as the Energy-Based Model (EBM) (Rosenfeld et al., 2001; Hinton, 2002; LeCun et al., 2006), takes advantage from both the given LM distribution $ p_\theta $ and the energy function $ E_\mu(x) $ that serves as a sequence-level assessment about the satisfaction of constraints measured by the evaluation metrics. The contribution of individual metrics to the overall alignment performance is characterized by the derived coefficients $ \mu = \{\mu_k\}_{k=1}^K $. Decoding from Eq. (2) requires determining $ \mu $ and tractable sampling from the normalized density, which will be discussed in §2.3. In the next subsection, we take a step further and demonstrate that the optimal solution of the problem (1) guarantees a theoretical improvement in perplexity of human texts. 2.2 Theoretical Improvement in Perplexity Although explicitly driving the generation to align with human texts under the chosen evaluation metrics is appealing, we are still confronted with the question of whether the resulting decoding distribution is generally a better approximation to the underlying distribution of human texts. For most existing heuristic decoding methods, a distribution-level evaluation (e.g., perplexity) is infeasible because of their ad-hoc treatments on the input LM distribution. For example, distribution truncation (Pan et al., 2018; Welleck et al., 2020; Meister et al., 2022) leads to a sparse support which is smaller than the underlying distribution of human texts, while heuristic searching algorithms (Li et al., 2022; Su et al., 2022) such as beam search do not have a parametric decoding distribution. Martins et al. (2020) proposed a variant of the standard perplexity, $ \epsilon $-perplexity by smoothing a sparse distribution, which still cannot faithfully reflect the true perplexity of the truncated distribution. For the decoding distribution derived from the proposed optimization problem, we show that not only is the perplexity feasible to compute, but it also improves the perplexity of human texts against the original LM distribution. The full proof is provided in Appendix A.2. Proposition 2. The optimal solution $ q_{\text{opt}} $ of the optimization problem (1) satisfies: 1. $ S(q_{\text{opt}}) \supseteq S(p_d) $, where $ S(p) = \{ x : p(x) > 0 \} $. 2. $ H(p_d, q_{\text{opt}}) = H(p_d, p_\theta) - D_{\text{KL}}(q_{\text{opt}} \\| p_\theta) $, where $ H(p, q) = - \sum_x p(x) \log q(x) $. Proof sketch. The proof starts with the convexity of the set $ C $ of distributions that satisfy the constraints in Eq. (1). We then consider $ p_\alpha = (1 - \alpha)q_{\text{opt}} + \alpha p_d \in C $, for $ \alpha \in [0, 1] $. The key insight is the following observation: \[ \frac{\partial}{\partial \alpha} D_{\text{KL}}(p_\alpha \\| p_\theta) \bigg\|_{\alpha=0} = H(p_d, p_\theta) - H(p_d, q_{\text{opt}}) - D_{\text{KL}}(q_{\text{opt}} \\| p_\theta). \] $ \partial D_{\text{KL}}(p_\alpha \\| p_\theta)/\partial \alpha $ can also be written as the limit of $ [D_{\text{KL}}(p_\alpha \\| p_\theta) - D_{\text{KL}}(q_{\text{opt}} \\| p_\theta)]/\alpha $ which is non-negative when $ \alpha \to 0^+ $ due to the optimality of $ q_{\text{opt}} $. Therefore, for Eq. (3) to be non-negative, we must have $ q_{\text{opt}}(x) \neq 0 $ for any $ x \in S(p_d) $ (otherwise it converges to $-\infty$), which proves the first claim. Next, given $ S(q_{\text{opt}}) \supseteq S(p_d) $, there exists some $ \alpha' < 0 $ such that $ p_{\alpha'} $ is a probability density function, which by definition also belongs to $ C $. Therefore, $ [D_{\text{KL}}(p_{\alpha'} \\| p_\theta) - D_{\text{KL}}(q_{\text{opt}} \\| p_\theta)]/\alpha' $ is non-positive when $ \alpha' \to 0^- $, leading to $ \partial D_{\text{KL}}(p_\alpha \\| p_\theta)/\partial \alpha \big\|_{\alpha'=0} = 0 $, which proves the second claim. The first outcome of Proposition 2 establishes the feasibility of computing perplexity under $ p_{\theta,\mu} $ when evaluated using the underlying human text distribution $ p_d $. And the second result reveals the perplexity improvement over $ p_\theta $: $ 2^{H(p_d,q_{\text{opt}})} < 2^{H(p_d,p_\theta)} $, due to the non-negativity of $ D_{\text{KL}}(q_{\text{opt}} \\| p_\theta) $. Note that the perplexity of $ q $ is defined as $ 2^{H(p_d,q)} $. Intuitively, more powerful constraints in the optimization problem that better measure the alignment with human texts cause a larger deviation from the input LM distribution, which in turn leads to a better approximation of underlying human text distribution, and thus a lower perplexity. 2.3 Decoding from the Optimal Solution In this section, we describe the method to decode from the sampling distribution derived from the optimization problem (1). First, we describe our method to estimate the coefficients $ \mu $ by satisfying the constraints with a conditional proposal distribution. Then we introduce a tractable sampling method to obtain samples from the decoding distribution defined by the EBM. 2.3.1 Coefficients Estimation The only degrees of freedom in the analytical solution of the optimal decoding distribution $ p_{\theta,\mu} $ are the coefficients $ \mu = \{\mu_k\}_{k=1}^K $ in the energy function $ E_\mu(x) $, whose optimal values $ \mu_{\text{opt}} $ can be estimated by first calculating $ F = \mathbb{E}_{x \sim p_{\theta,\mu}}[f(x)] $ and then approximating the target expectation $ F = \mathbb{E}_{x \sim p_d}[f(x)] $ to satisfy the constraints with iterative gradient updates. Note that this procedure is done on a small development set once for all before the inference stage. First, $\hat{F}$ can be estimated by Weighted Importance Sampling (WIS) (Geweke [1989]; Hesterberg [1995]) which first obtains $N$ i.i.d. trajectories $\{\hat{x}^i\}_{i=1}^N \sim p_\theta$, and then computes the weighted sum of $f(\hat{x}^i)$ with importance weight proportional to $\exp(-E_\mu(\hat{x}^i))$ normalized over all trajectories. As the asymptotic bias and variance of $\hat{F}$ estimated by WIS are both proportional to $N^{-1}$ (Hesterberg [1995]), the target expectation can be approximated with required estimation error by drawing enough samples from the proposal. Detailed derivation of WIS is provided in Appendix A.4.1. Next, given $\hat{F}$ as a parametric function of the variable $\mu$, we propose to approximate the target expectation $F$ by minimizing the Root Mean Squared Relative Error (Shcherbakov et al. [2013]), $$\sqrt{\frac{1}{K} \\|1 - \hat{F}/F\\|_2^2},$$ where the estimation error of each $f_k$ is normalized to the same scale. Then the optimal coefficient $\mu_{\text{opt}}$ is obtained by iteratively updating $\mu$ until convergence, i.e., reaching a desired error level. The algorithm of coefficients estimation is shown in Algorithm 1. We also analyze the convergence of $\mu$ in Appendix C and find it insensitive to initialization. Runtime analysis of Algorithm 1 is provided in Appendix E, which demonstrates the advantage over the typical hyper-parameter search procedure for most other decoding methods (Meister et al. [2022]; Li et al. [2022]). ### 2.3.2 Conditional Sampling from EBM Sampling from the decoding distribution defined by EBM in Eq. (2) is non-trivial, given that it is globally normalized over the whole sequence space. We first present the conditional probability of sampling a continuation $x_{>t_0}$ from $p_{\theta,\mu}$ given a prefix $x_{\leq t_0}$: $$p_{\theta,\mu}(x_{>t_0}\|x_{\leq t_0}) = p_\theta(x_{>t_0}\|x_{\leq t_0}) \exp\left[-E_\mu(x_{\leq t_0}, x_{>t_0})\right]/Z(x_{\leq t_0}),$$ (4) where $Z(x_{\leq t_0}) = \mathbb{E}_{x'_{>t_0} \sim p_\theta(\cdot\|x_{\leq t_0})}[\exp(-E_\mu(x_{\leq t_0}, x'_{>t_0}))]$ is the marginalization over future tokens sampled from the conditional proposal given the prefix. The detailed derivation is provided in Appendix A.5.1. As direct sampling from this auto-regressive factorization is computationally prohibitive (Deng et al. [2020]), we instead turn to a particle-based approximation of $p_{\theta,\mu}$ using the sampling-importance-resampling (SIR) technique (DB [1988]; Smith & Gelfand [1992]). Specifically, we first leverage the given LM $p_\theta$ as a proposal to generate a set of $M$ plausible continuation candidates $\{\hat{x}^i_{>t_0}\}_{i=1}^M$ given the prefix $x_{\leq t_0}$ in parallel. Then the final generation result is resampled from the distribution defined by the importance weight which is proportional to $\exp(-E_\mu(x_{\leq t_0}, \hat{x}^i_{>t_0}))$ normalized over all candidates $\{\hat{x}^i_{>t_0}\}_{i=1}^M$. We present the SIR approximation procedure of the conditional probability $\hat{p}_{\theta,\mu}^M(\cdot\|x_{\leq t_0})$ in Appendix A.5.2. In the limit of $M \to \infty$, the empirical distribution $\hat{p}_{\theta,\mu}^M(\cdot\|x_{\leq t_0})$ induced by SIR recovers the exact conditional distribution $p_{\theta,\mu}(\cdot\|x_{\leq t_0})$ for arbitrary $x_{>t_0}$. Skare et al. [2003] proved that the point-wise relative error of the empirical distribution induced by SIR is $O(M^{-1})$ (see Theorem 2.1 in the original paper). In practice where $M$ is finite, we propose to sample from the temperature modulated proposal $\tilde{p}_\theta^\tau$ with lower temperature $\tau$ to increase the chance of obtaining high-quality candidates within a realistic computational budget. The algorithm of conditional sampling is shown in Algorithm 2. In fact, various existing sampling methods can be used for candidate sampling, we choose to use temperature sampling as it preserves the feasibility to compute perplexity (see Appendix A.3). We provide complexity and runtime analysis of Algorithm 2 and baseline decoding methods in Appendix F. Algorithm 1 $\mu_{\text{opt}}$ estimation with WIS Input: $p_\theta$, $F$, learning rate $\alpha$ Output: $\mu_{\text{opt}}$ 1: Initialize $\mu$ randomly 2: Sample trajectories $\{\hat{x}^i\}_{i=1}^N \sim p_\theta$ 3: repeat 4: \hspace{1em} $\hat{F} \leftarrow \frac{\sum_{i=1}^N \exp(-E_\mu(\hat{x}^i)) f(\hat{x}^i)}{\sum_{i=1}^N \exp(-E_\mu(\hat{x}^i))}$ 5: \hspace{1em} $\mu \leftarrow \mu - \alpha \nabla_\mu \sqrt{\frac{1}{K} \\|1 - \hat{F}/F\\|_2^2}$ 6: until convergence 7: $\mu_{\text{opt}} \leftarrow \mu$ Algorithm 2 Conditional Sampling with SIR Input: $p_\theta$, $E_\mu$, prefix $x_{\leq t_0}$, $M$, $\tau$ Output: continuation $x_{>t_0}$ 1: for $i \leftarrow 1$ to $M$ do \hspace{1em} ▷ In parallel 2: \hspace{2em} Sample $\hat{x}^i_{>t_0} \sim \tilde{p}_\theta^\tau(\cdot\|x_{\leq t_0})$ 3: \hspace{2em} Compute $w_i \leftarrow \exp(-E_\mu(x_{\leq t_0}, \hat{x}^i_{>t_0}))$ 4: end for 5: Sample $j \sim \text{Categorical}\left(\frac{w_1}{\sum_{i=1}^M w_i}, \ldots, \frac{w_M}{\sum_{i=1}^M w_i}\right)$ 6: Set $x_{>t_0} \leftarrow \hat{x}^j_{>t_0}$ 3 EXPERIMENT 3.1 DATASETS We evaluate our method on the Wikipedia and News domain for open-ended text generation. For the Wikipedia domain, the data comes from documents in the Wikitext-103 corpus (Merity et al., 2017). For the News domain, the data comes from news articles in Wikinews. We follow the data pre-processing procedure suggested by Li et al. (2022), and randomly select 512 samples as the development set for hyper-parameter tuning for all decoding methods. The data statistics of each domain and detailed data pre-processing steps are provided in Appendix J. 3.2 EVALUATION METRIC SETTINGS In this section, we introduce the set of evaluation metrics we consider in aligning with human texts, which correspond to $ f $ in Eq. (1). These metrics cover a wide range of aspects including repetition, coherence, diversity, and information content. Repetition. We evaluate repetition at both sequence level and token level. The sequence-level metric measures the portion of duplicate $ n $-grams in the generated texts (Welleck et al., 2020): \[ \text{SEQ-REP-N} = 100 \times \left(1 - \frac{\text{unique } n\text{-grams}(\hat{x})}{\text{total } n\text{-grams}(\hat{x})}\right) \] where $ \hat{x} $ is the generated text (SR-N in short). The token-level metric measures the average frequency of each generated token reoccurring in the previous $ l $ tokens (Fu et al., 2021; Ji & Huang, 2021): \[ \text{TOK-REP-L} = 100 \times \left(\frac{1}{\|\hat{x}\|} \sum_{t=1}^{\|\hat{x}\|} 1[\hat{x}_t \in \hat{x}_{t-l:t-1}]\right) \] (TR-L in short). We adopt SR-N with $ n = \{2, 3, 4\} $ and TR-L with $ l = \{8, 16, 32\} $, respectively. Coherence. We evaluate coherence following Su et al. (2022) by calculating the cosine similarity between the sentence embedding of the prefix $ x_{\leq t_0} $ and the generated continuation $ \hat{x}_{>t_0} $: \[ \text{COH} = 100 \times \cos(\text{emb}(x_{\leq t_0}), \text{emb}(\hat{x}_{>t_0})) \] where emb($\cdot$) is parametrized by the pre-trained sentence embedding model SimCSE (Gao et al., 2021) based on RoBERTa (Liu et al., 2019). Diversity. We evaluate diversity following Li et al. (2022) by aggregating the $ n $-gram repetition rate of $ n = \{2, 3, 4\} $: \[ \text{DIV} = 100 \times \prod_{n=2}^{4} \left(1 - \text{SEQ-REP-N}\right) \] DIV reflects the overall lexical diversity of the text at different levels of granularity. Information Content. We evaluate the average amount of information contained per word given the preceding contexts, by calculating the exponential of the entropy rate on the generated text $ \hat{x} $ using a language model: \[ e^{\text{ENT}} = \exp\left(-\frac{1}{\|\hat{x}\|} \sum_{t=1}^{T} \log p_{LM}(\hat{x}_t\|\hat{x}_{<t})\right) \] Shannon (1951); Braverman et al. (2020). A low $ e^{\text{ENT}} $ suggests that the information in the text is redundant, while a high $ e^{\text{ENT}} $ indicates high surprisal in the text according to the language model. In our experiment, we use a general-domain language model GPT-2 XL to calculate the log probability of the generated texts. Note that these metrics are also used in the automatic evaluation part of our experiment (\$3.5) to measure how well the generated texts align with human texts in different aspects. Thus, the criterion for these evaluation metrics is the closeness between the metric scores of generated texts and those of human references. In addition, we also report MAUVE score (Pillutla et al., 2021) (MAU in short) which measures the distributional similarity between the set of generated texts and that of references by calculating the area under the divergence frontier of the two empirical distributions. As this metric cannot provide an evaluation score that corresponds to a certain aspect for each text sample, we only adopt it to assess the final performance of different decoding methods. We use GPT-2 XL to extract features from texts which was restricted to a maximum length of 256. 3.3 BASELINES AND IMPLEMENTATION DETAILS We thoroughly compare Daemon with various sampling-based and search-based methods in particular. We consider three canonical sampling-based methods: Top-k sampling (Fan et al., 2018), Nucleus sampling (Holtzman et al., 2020) and Typical decoding (Meister et al., 2022). For search- --- http://www.wikinews.org https://en.wikipedia.org/wiki/Information_content \| Method \| Wikipedia \| News \| \|--------\|-----------\|------\| \| \| SR-4 \| TR-32 \| COH \| DIV \| e<sub>ENT</sub> \| MAU \| SR-4 \| TR-32 \| COH \| DIV \| e<sub>ENT</sub> \| MAU \| \| Reference \| 0.48 \| 21.3 \| 62.3 \| 92.5 \| 23.2 \| - \| 0.29 \| 18.7 \| 66.6 \| 94.1 \| 13.8 \| - \| \| Greedy \| 60.9 \| 65.5 \| 60.2 \| 8.03 \| 2.29 \| 59.7 \| 53.2 \| 58.2 \| 63.8 \| 13.2 \| 2.19 \| 65.2 \| \| Top-k \| 2.11 \| 23.4 \| 60.9 \| 87.8 \| 10.1 \| 77.8 \| 0.95 \| 20.3 \| 64.7 \| 91.7 \| 8.17 \| 96.3 \| \| Nucleus \| 1.19 \| 20.0 \| 57.3 \| 92.4 \| 17.3 \| 78.3 \| 0.80 \| 18.7 \| 60.8 \| 93.5 \| 11.0 \| 95.3 \| \| Typical \| 0.81 \| 17.4 \| 54.9 \| 94.5 \| 30.1 \| 78.7 \| 0.42 \| 16.9 \| 57.2 \| 95.3 \| 18.2 \| 95.0 \| \| CD \| 1.31 \| 28.2 \| 68.7 \| 85.9 \| 7.55 \| 77.8 \| 0.63 \| 23.2 \| 71.2 \| 90.5 \| 6.55 \| 95.1 \| \| CS \| 1.78 \| 23.0 \| 56.9 \| 90.6 \| 5.25 \| 83.3 \| 0.77 \| 19.2 \| 63.6 \| 94.1 \| 4.18 \| 95.7 \| \| DAEMON \| 0.42 \| 22.5 \| 62.5 \| 92.2 \| 22.8 \| 88.1 \| 0.18 \| 18.7 \| 66.3 \| 94.5 \| 13.7 \| 97.4 \| Table 1: Main results of automatic evaluation on the Wikipedia and News domain using GPT-2 XL and OPT-6.7B. For all metrics, the best scores are the closest to the human scores except for MAU, which is better when higher. The best score is in boldface and the second best is underlined. Based methods, besides vanilla Greedy decoding, we also consider two recent methods that maximize the contrastive objectives: Contrastive Decoding (CD) (Li et al., 2022) and Contrastive Search (CS) (Su et al., 2022). To demonstrate the effectiveness of our method across different language model families and scales, we consider GPT-2 XL (1.5B) (Radford et al., 2019) and OPT-6.7B (Zhang et al., 2022) as the base models for all decoding methods. For baselines, we follow the hyper-parameter settings in the original papers which are shown to work well in general. For DAEMON in the main results, we use the nine metrics (described in §3.2) in the constraints. During sampling, we set the size of candidate set from the proposal model $M = 25$ as it balances efficiency and performance. We set $\tau = 0.97$ for the Wikipedia domain and $\tau = 0.99$ for the News domain. We leave more implementation details of the baselines and DAEMON in Appendix J.2. ### 3.4 Human Evaluation We further conduct human evaluation to assess the quality of generated texts. We consider three widely used criteria for open-ended generation: Fluency, Coherence, and Informativeness (van der Lee et al., 2019). Specifically, Fluency is characterized by the grammatical correctness and naturalness of the text without repetition; Coherence is characterized by topic maintenance with the input prefix and well-structured discourse; Informativeness is characterized by the adequacy of elaborating engaging details in a coherent plot. We randomly sample 100 prefixes and conduct pair-wise comparisons between DAEMON and baselines. Three annotators on Amazon Mechanical Turk are hired to choose the better continuation (i.e., win, lose, or tie) from the ones generated by our method and the baselines in terms of the three criteria above. More detailed settings of human evaluation are provided in Appendix D. ### 3.5 Main Results Automatic Evaluation. We first present the main results obtained via automatic evaluation in Table 1, where we benchmark our method against strong baselines across different domains and model scales under the evaluation metrics described in §3.2. Due to the space limit, we only present representative metrics and leave the full results in Appendix I.3. DAEMON outperforms all baselines in aligning with human texts on four aspects including repetition, coherence, diversity, and information content, and it also achieves the highest MAUVE score across different domains (Wikipedia / News) and model scales (1.5B / 6.7B). The consistent performance improvement demonstrates the effectiveness and generalizability of DAEMON. Notably, DAEMON achieves the lowest sequence-level repetition (indicated by SR-4) across all settings and maintains human-level coherence and high MAUVE score at the meantime, compared with baselines that have a similar repetition level, e.g., \| Model \| Wikipedia \| News \| \|-----------\|-----------\|------\| \| \| ori \| imp \| ori \| imp \| \| GPT-2 XL \| 23.1 \| 22.0 \| 13.9 \| 13.1 \| \| OPT-6.7B \| 16.4 \| 16.2 \| 10.8 \| 10.2 \| Table 2: Perplexity evaluation results. “ori” is the original perplexity of the LM distribution. “imp” is the improved perplexity of the optimal decoding distribution. the Typical decoding method. The results indicate that DAEMON is the only method that effectively aligns multiple aspects with human texts without hard trade-off among those aspects. For sampling-based methods, Nucleus sampling generally preserves more diverse candidates than Top-k sampling and achieves better diversity at human-level at the cost of low coherence. Typical decoding over-emphasizes the long tail of the distribution, which produces texts with highly diverse lexicality (highest diversity) but severe topic shift (lowest coherence). On the other hand, search-based methods generally have substantially lower information score than sampling-based methods, which indicate the redundancy of information in the generated texts. Specifically, CD achieves the highest coherence score at the expense of having the worst diversity except for greedy decoding. By analyzing its generated texts, we found that CD tends to repeat tokens from the previous context (indicated by high $ \text{TR}_{-32} $), which could be a bias captured by SimCSE to compute representations with high similarity. Additionally, we found that the MAUVE score of CS is generally higher than CD when the evaluation length is set to 256, which contradicts the previous result (Li et al., 2022) where the length is truncated to 128. Our result is consistent with our observation that CD generates semantically more repetitive texts its longer generations. To demonstrate the universality of our method, we additionally conduct an experiment on a text summarization dataset in Appendix I. Perplexity Evaluation. We calculate the perplexity of DAEMON’s decoding distribution, which can be directly computed using a proposal model with temperature modulation. The derivation of perplexity calculation is presented in Appendix A.3. From the results in Table 2, we observe that the perplexity improvement over the original LM distribution is consistent across model sizes and data domains. Notably, the perplexity improvement is more obvious for the model with a smaller size (GPT-2 XL), which manifests the advantage of our approach in enhancing the capacity of the autoregressive LM in modeling language with a constrained computational budget. Human Evaluation. We conduct pairwise human evaluation and selectively compare our method against strong baselines including Contrastive Decoding (CD), Contrastive Search (CS), Nucleus sampling, and Typical decoding. As shown in Table 3, DAEMON is more preferred than all four baselines in fluency, coherence, and informativeness respectively on the domain of Wikipedia according to the human judgment. Specifically, DAEMON generates significantly more coherent texts compared to all the baselines. We provide a list of qualitative cases produced by different decoding methods in Appendix J.4 to help readers comprehend the difference in their generation quality. ### 3.6 Ablation Study Ablating Metrics in Constraints. We ablate the metrics in the constraints of DAEMON to investigate their individual contributions to the overall alignment performance. In Table 4, we present the results of ablating different metrics while keeping other settings verbatim. The ablation experiments are obtained using GPT-2 XL on the Wikipedia domain. We first observe that for most metrics, removing them lead to drastic deterioration in the corresponding metric scores, e.g., $ \text{SR}_{-4} $ (0.42 → 3.57), $ \text{COH} $ (62.5 → 57.6), $ \text{e}_{\text{ENT}} $ (22.8 → 19.9). We also observe clear inter-dependence between certain metrics pairs, as removing one leads to notable performance drop in another, e.g., $ \text{SR}_{-N} $ and $ \text{DIV} $, $ \text{e}_{\text{ENT}} $ and $ \text{TR}_{-L} $, which reflects the intrinsic correlation among the evaluated aspects. Number of Candidates for Resampling. We then study the impact of the number of candidates for resampling ($ M $ described in 2.3.2). In Figure 3, we present the results on five metrics and the relative decoding latency with a batch size of 1. We set the temperature of the proposal distribution to 1.0 to isolate the impact of $ M $. From the results, we observe that the alignment on all metrics generally improves with a larger $ M $, which indicates better SIR approximation to the optimal decoding. \| Metrics \| SR-4 \| TR-32 \| COH \| DIV \| e^{ENT} \| \|---------\|------\|-------\|-----\|-----\|--------\| \| Reference \| 0.48 \| 21.3 \| 62.3 \| 92.5 \| 23.2 \| \| DAEMON \| 0.42 \| 22.5 \| 62.5 \| 92.2 \| 22.8 \| \| w/o SR-N \| 3.57 \| 22.6 \| 63.0 \| 86.8 \| 22.1 \| \| w/o TR-L \| 0.19 \| 20.2 \| 62.0 \| 93.9 \| 25.1 \| \| w/o COH \| 0.37 \| 22.3 \| 57.6 \| 92.6 \| 23.8 \| \| w/o DIV \| 0.30 \| 22.1 \| 62.4 \| 93.0 \| 23.3 \| \| w/o e^{ENT} \| 0.55 \| 23.0 \| 63.1 \| 91.5 \| 19.9 \| Table 4: Ablation results of different metrics where rows in gray are results with corresponding metrics removed. “SR-N” and “TR-L” denote the set of metrics with $ n = 2, 3, 4 $ and $ l = 8, 16, 32 $ respectively. Figure 3: Ablation results of varying the number of candidates for resampling ($ M $). Results on the five metrics are compared with the reference and the latency is relative to Greedy decoding. Specifically, the convergence rate of different metric varies, e.g., DIV and $ e^{ENT} $ converge slower than TR-32 and COH with the increase of $ M $. Finally, increasing $ M $ inevitably incurs higher decoding latency, and thus we choose $ M = 25 $ with slightly lower temperature in the main result to maintain both efficiency and performance. We test the robustness of our method by investigating the performance of different metrics when we optimize a single metric in Appendix H. Temperature When Sampling from the Proposal Model. As suggested by (Caccia et al., 2020; Zhang et al., 2020), quality and diversity are two important aspects which can be traded off by sweeping hyper-parameters (e.g., temperature) to alter the sharpness of the distribution. For DAEMON, we tune the temperature of the proposal model ($ \tau $ described in §2.3.2) with other settings unchanged and plot the curve in the dimensions of coherence and diversity in Figure 2. We also plot the result of tuning the hyper-parameters of different baseline methods. We first observe that DAEMON dominates the compared baselines in terms of coherence for all interested diversity level. Second, DAEMON is able to achieve human-level performance on these two aspects by slightly tuning the temperature lower, which demonstrates the effectiveness and practicality of our approach. 4 Conclusion and Future Work In this study, we introduce Decoding as Direct Metrics Optimization (DAEMON), a decoding framework that explicitly aligns the generations with human texts under various aspects, e.g., coherence, repetition, and etc. The induced sampling distribution harnesses candidates generated by an auto-regressive LM, which are re-weighted according to a sequence-level energy function. We demonstrate both theoretical and empirical benefits of DAEMON, which outperforms strong decoding baselines in both human evaluation and automatic evaluation in terms of metrics alignment to human texts, perplexity improvement over the original LM, and superior quality-diversity trade-off. As for the future work of DAEMON, we consider exploring directions to generalize the framework, e.g., extending the equality constraints to more general constraint types, such as inequalities and structural equations. It is also necessary to consider more aspects along with evaluation metrics beyond text quality, e.g., human value. This can therefore complement with other training-time alignment methods, such as RLHF. And finally more efficient method to sample from the distribution defined by the EBM is also important to ensure the practicality of DAEMON. Overall, we firmly believe this work paves the way for advanced methods that guide the language model towards desired behavior by incorporating constraints that capture intended regularities. ACKNOWLEDGMENTS This work was supported by the National Science Foundation for Distinguished Young Scholars (with No. 62125604), the NSFC projects (with No. 62306160 and No. 61936010), the China National Postdoctoral Program for Innovative Talents (No. BX20230194), and the China Postdoctoral Science Foundation (No. 2023M731952). REFERENCES Kushal Arora, Layla El Asri, Hareesh Bahuleyan, and Jackie Chi Kit Cheung. Why exposure bias matters: An imitation learning perspective of error accumulation in language generation. In Smaranda Muresan, Preslav Nakov, and Aline Villavicencio (eds.), Findings of the Association for Computational Linguistics: ACL 2022, Dublin, Ireland, May 22-27, 2022, pp. 700–710. Association for Computational Linguistics, 2022. doi: 10.18653/v1/2022.findings-acl.58. 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EriR6Ec69a	Shouldn't a higher spectral norm contribute to higher dimensional dynamics? For a low spectral norm the network would quickly collapse onto a low-dimensional manifold. How do you explain then the RNNs have higher spectral norm and lower dimensionality in their dynamics?	LEVERAGING LOW-RANK AND SPARSE RECURRENT CONNECTIVITY FOR ROBUST CLOSED-LOOP CONTROL Neehal Tumma¹ Mathias Lechner² Noel Loo² Ramin Hasani² Daniela Rus² ¹Harvard University ²MIT CSAIL ABSTRACT Developing autonomous agents that can interact with changing environments is an open challenge in machine learning. Robustness is particularly important in these settings as agents are often fit offline on expert demonstrations but deployed online where they must generalize to the closed feedback loop within the environment. In this work, we explore the application of recurrent neural networks to tasks of this nature and understand how a parameterization of their recurrent connectivity influences robustness in closed-loop settings. Specifically, we represent the recurrent connectivity as a function of rank and sparsity and show both theoretically and empirically that modulating these two variables has desirable effects on network dynamics. The proposed low-rank, sparse connectivity induces an interpretable prior on the network that proves to be most amenable for a class of models known as closed-form continuous-time neural networks (CfCs). We find that CfCs with fewer parameters can outperform their full-rank, fully-connected counterparts in the online setting under distribution shift. This yields memory-efficient and robust agents while opening a new perspective on how we can modulate network dynamics through connectivity. 1 INTRODUCTION Building models that are robust under natural distribution shift has long been a goal in artificial intelligence (Taori et al., 2020). Existing techniques have sought to address this challenge through various approaches, including domain adaptation (Farahani et al., 2020), transfer learning (Zhuang et al., 2019) and data augmentation (Perez & Wang, 2017). However, these techniques come with drawbacks: domain adaptation and transfer learning can be computationally expensive, and data augmentation often suffers from robustness gains that are not uniform across corruption types (Ford et al., 2019). More generally, machine learning systems tend to perform poorly under distribution shift because approaches like these only serve to ameliorate a problem that is rooted in the model architecture itself. Natural learning systems serve to address this problem by interacting with their environment to understand the world. They make use of biologically-inspired frameworks that account for distribution shifts by modeling temporal structure (Hasani et al., 2020b). These models are particularly well-suited in a paradigm where they are fit on offline, passive datasets using imitation learning, but deployed in closed-loop, active testing settings (de Haan et al., 2019). This domain is commonly referred to as the open-loop to closed-loop causality gap (Lechner et al., 2022) in which generalization requires the agent to learn a coherent representation of its world (Lechner et al., 2020). One particularly effective framework known as closed-form continuous-time neural networks (CfCs) were shown to outperform many state-of-the-art recurrent models in closed-loop settings (Hasani et al., 2021). These models leverage a sparse wiring structure induced at initialization, yielding compact and interpretable networks (Chahine et al., 2023). However, the reason as to why sparse connectivity is useful in closed-loop systems is poorly understood. In this work, we will address this question by proposing and analyzing an interpretable parameterization of connectivity in CfCs. To do so, we turn to another class of models: low-rank recurrent neural networks (low-rank RNNs). A low-rank RNN is a network whose recurrent connectivity matrix is low-rank. Deriving inspiration from neural activity in the brain, these networks have provably low-dimensional patterns of activity which makes them successful in many simple neuroscience-based tasks (Mastrogiuseppe & Ostojic, 2018). In this paper, we leverage ideas from previous work on CfCs and low-rank RNNs in order to devise a novel parameterization of recurrent connectivity that improves robustness in closed-loop settings and offers interpretable measures for the network dynamics it incites. In doing so, we show the following: - Parameterizing recurrent connectivity as a function of rank and sparsity yields provable and interpretable network dynamics (Section 3.1) - Low-rank and sparse recurrent neural networks can outperform their full-rank, fully-connected counterparts under distribution shift in closed-loop environments (Section 4.1) - Pruning by inducing sparsity and pruning by enforcing a low-rank structure are distinct in the types of network dynamics each induces (Section 4.3) - CfCs are more amenable to a low-rank and sparse connectivity prior than other canonical recurrent architectures such as LSTMs (Section 4.4) 2 RELATED WORK In this section, we describe previous works that are closely related to the core findings of the paper. Robustness in closed-loop settings. A class of continuous-time recurrent models known as liquid neural networks (Hasani et al., 2020a) have been shown to achieve state-of-the-art performance under distribution shift by explicitly accounting for external interventions to environment conditions. Liquid neural networks are a prime example of natural learning systems that learn robust, and provably causal, representations (Vorbach et al., 2021). CfCs are a specific instance of liquid neural networks that can leverage sparse connectivity to learn robust representations in the closed-loop causality gap setting. Outside of modifying the model itself, other approaches to training robust models under an imitation learning framework include augmentation strategies, human interventions (Ross et al., 2010), goal-conditioning (Codevilla et al., 2017), reward conditioning (Srivastava et al., 2019), task-embedding (James et al., 2018) and meta-learning (Finn et al., 2017). These advances fail to consider the structure of the underlying policy; in contrast, liquid neural networks improve the robustness of the decision-making process in autonomous agents, leading to better generalization under the same training distribution (Vorbach et al., 2021). Relating connectivity and network dynamics. A long-standing goal in computational neuroscience is to understand the relationship between structure and function in the brain (Sporns, 2013). We can ask the analogous question in the context of artificial neural networks: what role does model connectivity at initialization play in learned network dynamics? Amongst recurrent neural networks, echo state networks (Goodfellow et al., 2016) present one example of a model that induces static sparsity in order to modulate the complexity of dynamics along the recurrent dimension. However, unlike in our case, echo state networks fix their recurrent connectivity during training. Other works have found that sparse networks can yield provably more robust models (Guo et al., 2018) that generalize across many tasks (Chen et al., 2022) outside the recurrent setting. With respect to modulating network rank, the advent of the low-rank RNN framework (Mastrogiuseppe & Ostojic, 2018) is one of the first attempts at explicitly modeling the low-dimensional dynamics of the brain. Some follow-ups on this work include examining how low-rank Figure 1: Open-loop systems receive ground-truth observations $x_i$. Closed-loop systems receive observations $x_i(a_{i-1})$ that are a function of the previous actions the agent takes. There is no external feedback to correct the agent in the closed-loop setting. connectivity arises in full-rank settings (Schuessler et al., 2020) and an analysis of how some tasks are more amenable to low-rank connectivity than others (Dubreuil, 2022). Our work borrows approaches and intuition from many of these prior studies, but is distinct with respect to the proposed connectivity, analysis of network dynamics and task setting. Pruning at initialization. Neural network pruning typically either removes connections in a costly iterative training-retraining paradigm (Frankle & Carbin, 2018) or modifies the objective function to promote sparsity (Goodfellow et al., 2016) which can lead to difficulties in optimization. Recently, pruning at initialization has emerged as an approach to resolve these issues. Pruning at initialization attempts to take a randomly initialized network and remove weights before training (Wang et al., 2021). Various approaches include using connectivity sensitivity (Lee et al., 2018), maintaining dynamical isometry (Lee et al., 2019) and training on supermasks (Ramanujan et al., 2019). Generally, these techniques outperform sparsity induced randomly at initialization. Another approach to pruning is using rank decomposition. One example is low-rank matrix factorization, which is good at reducing size but at the cost of performance (Sainath et al., 2013). The most effective low-rank compression techniques usually leverage a low-rank approximation on the full-rank matrix either during or after training (Xu et al., 2020). In contrast, in this work, we propose a parameterization of connectivity that randomly induces sparsity and leverages a low-rank decomposition of the recurrent weights at initialization – the most straightforward and least costly form of pruning. Alternative RNN parameterizations. Existing works proposed other recurrent architectures such as Lipschitz RNNs (Erichson et al., 2021), antisymmetric RNNs (Chang et al., 2019) and Cayley RNNs (Helfrich et al., 2018) which, like us, propose alternative parameterizations of recurrent connectivity. However, unlike our study, these works introduce changes to the underlying network architecture, whereas our work focuses on the applications of low-rank and sparse recurrent connectivity in more canonical recurrent networks. This is in line with previous works in the closed-loop causality gap setting (Chahine et al., 2023) which also restrict the scope of their models as such. ### 3 PARAMETERIZATION OF CONNECTIVITY Consider a standard RNN whose input is denoted by $ x_t \in \mathbb{R}^n $ and hidden state is given by $ h_t \in \mathbb{R}^h $ for time step $ t $, hidden size $ h $ and input size $ n $. Then the functional form of the RNN is given by \[ h_t = \tanh(W_{\text{rec}} h_{t-1} + W_{\text{inp}} x_t + b) \] such that $ W_{\text{rec}} \in \mathbb{R}^{h \times h} $ denotes the recurrent weights, $ W_{\text{inp}} \in \mathbb{R}^{n \times h} $ denotes the input weights and $ b $ denotes the bias. We now propose an alternative parameterization of connectivity for the recurrent weights. Consider a singular value decomposition of $ W_{\text{rec}} $ given by $ W_{\text{rec}} = U \Sigma V^T $. The canonical rank $ r $ approximation of $ W_{\text{rec}} $ is $ U_r \Sigma_r V_r^T $ such that $ \Sigma_r $ consists of the top-$ r $ singular values and $ U_r $ and $ V_r^T $ consist of the corresponding singular vectors. Using this low-rank approximation, we can construct the following parameterization of the recurrent weights for a given rank $ r $ and sparsity level $ s $: \[ W_{\text{rec}}(r, s) = W_1(r) W_2(r) \odot M(s) \] where $ W_1 = U_r (\Sigma_r)^{1/2} $, $ W_2 = (\Sigma_r)^{1/2} V_r^T $ and $ M \in \mathbb{F}_2 $ is a random binary mask such that each entry $ M_{ij} \sim 1 - \text{Bernoulli}(s) $. This low-rank and sparse parameterization of $ W_{\text{rec}} $ is a generalization of the parameterization studied in Herbert & Ostoic (2022) to recurrent connectivity of arbitrary rank and trainable weights $ W_1, W_2 $ as opposed to restricting ourselves to the setting of rank-1 matrices and fixed weights in which dynamical analysis is more tractable, but the network is constrained in expressivity. Note that the random mask $ M $ is fixed throughout training. ### 3.1 NETWORK DYNAMICS AT INITIALIZATION We propose this parameterization of recurrent connectivity as it allows us to modulate aspects of the network that instill an inductive bias, which proves to be beneficial for performance in the closed-loop setting, particularly under distribution shift. We care about three measures of $ W_{\text{rec}}(r, s) $ in particular: the spectral radius, the spectral norm and the rate of decay of the singular value spectrum. The spectral radius $\lambda_{\text{max}}$ of $W_{\text{rec}}(r, s)$ is widely accepted as a proxy for the rate at which the gradient evolves backwards in time (Pascanu et al., 2012). Namely, if $\lambda_{\text{max}} < 1$, an RNN has a vanishing gradient (refer to A.5.1 for details). In many time series applications, having a constant error flow is a desirable property, as arbitrary data sequences may have long-term relations. However, in the case of many closed-loop systems, learning long-term dependencies can be detrimental in the online setting due to the short-term causality inherent to the task (Lechner et al., 2020). To put this into context, in this work we consider agents learning to play games in various Arcade Learning Environments (ALEs). Agents that perform well in ALEs often use frame stacking (Horgan et al., 2018a) which enforces a strict short-horizon temporal prior by leveraging a look-back window of only a few input frames. Hence, networks like LSTMs or GRUs may capture spurious long-term dependencies and thus learn inadequate models. In contrast, the vanishing of gradients that tends to be more pronounced in RNNs counterintuitively enhances the performance of the agent, as it places a short-horizon prior on the temporal attention span of the network. We will motivate the importance of the singular values in the distribution shift setting. Consider an SVD on the recurrent weights given by $W_{\text{rec}}(r, s) = U \Sigma V^T$. Furthermore, consider a perturbation $e$ sampled uniformly at random from the set of norm-1 vectors. If we apply this perturbation to the hidden state $h_t$, then to measure robustness we want to quantify the deviation between $W_{\text{rec}}(r, s)h_t$ and $W_{\text{rec}}(r, s)(h_t + e)$. This is given by $W_{\text{rec}}(r, s)e = U \Sigma V^T e$. Note that both $U$ and $V^T$ are unitary matrices and thus do not affect the magnitude of $e$. This means that $\Sigma$ captures the nature of the transformation on $e$. The two relevant aspects of the transformation are its magnitude and direction. The spectral norm, the maximum singular value, measures the former (i.e. smaller spectral norm implies better robustness). To quantify the latter, we examine the rate at which the singular values decay. This provides a proxy for the effective number of directions $e$ is expanded in (i.e. faster decay implies better robustness). For details on this argument, refer to A.5.2. Now that we have motivated the spectral properties of $W_{\text{rec}}(r, s)$, we next aim to understand how they change as a function of rank and sparsity. We consider two random initialization schemes for $W_{\text{rec}}(r, s)$: Glorot uniform spectral initialization (GU-spec) and orthogonal spectral initialization (ortho-spec). To thoroughly motivate the proposed parameterization, we provide theoretical proof of the relationships between rank/sparsity and spectral radius/spectral norm for both initialization schemes where possible. For the cases we do not prove, we provide an empirical analysis instead. Theorem 1. Given recurrent weights with parametrization shown in Equation (2) and initialization scheme specified in parentheses, we prove the following: - The spectral radius of $W_{\text{rec}}(r, s)$ decreases as a function of $s$ (GU-spec) - The spectral radius of $W_{\text{rec}}(r, s)$ increases as a function of $r$ (ortho-spec) - The spectral norm of $W_{\text{rec}}(r, s)$ decreases as a function of $s$ (GU-spec) - The spectral norm of $W_{\text{rec}}(r, s)$ is constant as a function of $r$ (ortho-spec, GU-spec) For proofs of the above properties and an empirical analysis for the unproven cases (which follow the same trends as the proven cases), refer to A.6. And since our theoretical arguments do not readily extend to the full singular value spectrum, we provide an empirical analysis as a function of rank and sparsity for this as well (A.7) which shows that the rate of singular value spectrum decay increases as a function of rank and decreases as a function of sparsity. Figure 3: Online performance of recurrent networks under different ranks and sparsities in Seaquest environment. For offline performance, refer to A.9. a) In-distribution rewards in the online, closed-loop setting normalized by the rewards obtained by the expert in-distribution. b) Rewards averaged across 5 distribution shifts, normalized by the rewards obtained by the expert under distribution shift. 4 EXPERIMENTS Experimental setup. We study the offline-online generalization gap of various recurrent architectures parameterized by low-rank, sparse connectivity under an imitation learning framework. In particular, we measure the performance of our models in the Arcade Learning Environment (Bellemare et al., 2012) and MuJoCo (Todorov et al., 2012). For ALEs, we run experiments in the Seaquest and Alien environments and for MuloCo we explore the HalfCheetah environment. For each environment, we train a deep-Q network to generate expert trajectories that we then use to fit our recurrent networks in an offline setting. We evaluate the recurrent networks online, closed-loop when deployed into the environment, both in-distribution and under distribution shift. For more details on the experimental framework, refer to A.11. Models. We examine the following recurrent architectures: RNNs, LSTMs, GRUs, CNNs and CfCs. RNNs, LSTMs (Hochreiter & Schmidhuber, 1997) and GRUs (Cho et al., 2014) refer to their canonical implementations. CNNs serve to address whether supervision along the recurrent dimension is even necessary. CfCs (Hasani et al., 2021) are a modified version of an RNN that admit the following functional form: \[ h(t) = \sigma(F(h_{t-1}, x_t, \theta_F)) \odot G(h_{t-1}, x_t, \theta_G) + [1 - \sigma(F(h_{t-1}, x_t, \theta_f))] \odot H(h_{t-1}, x_t, \theta_H) \] Here, $ H $ and $ G $ are vanilla RNNs and $ F $ can be interpreted as an adaptive gating mechanism that interpolates between the state-space trajectories of $ H $ and $ G $ on a per-element basis in the hidden vector $ h(t) $ (refer to A.4 for details). Finally, recall in Section 3.1 that we proposed a low-rank, sparse parameterization of connectivity $ W_{rec}(r, s) $ for the recurrent weights in a vanilla RNN. We generalize this parameterization to the other recurrent architectures by simply parameterizing all recurrent weights in the network in the form of $ W_{rec}(r, s) $. For specifics, refer to A.11. 4.1 PERFORMANCE IN CLOSED-LOOP SETTINGS We first explore the impact of our proposed connectivity parameterization for various ranks $ r $ and sparsities $ s $ in the Seaquest environment in the online, in-distribution setting. We observe that the best performing models in-distribution tend to be low-sparsity, high-rank CfCs and LSTMs (Figure 3a). GRUs, on the other hand, tend to perform poorly for most $(r, s)$ which aligns with previous work which also shows that GRUs are not particularly well-suited for closed-loop settings (Chahine et al., 2023). Due to the poor performance of GRUs relative to LSTMs, we will restrict most of our discussion and analysis to CfCs, LSTMs and RNNs. The most notable finding from the in-distribution results is observed in the high-sparsity models; in particular, we find that LSTMs tend to perform much better than CfCs at high sparsities. Analogously, RNNs, which in general tend to perform worse than CfCs, show similarly worse performance at high sparsities relative to LSTMs. We offer intuition for these findings in Section 4.2 where we explore the relationship between sparsity and the recurrent memory horizon of the network. Under distribution shift, we find that the best performing models are low-rank, low-sparsity CfCs. More generally, across all recurrent architectures, we find that the low-rank models tend to perform on par with, and in many cases actually outperform, higher rank ones. This demonstrates that we can construct models with fewer parameters at initialization yet still learn more robust models. Interestingly, however, we find that the means through which we prune at initialization matters: in particular for high-sparsity models, we do not achieve the same robustness that we observe in low-rank ones. We will provide justification for the apparent disparity between pruning by increasing sparsity and pruning by decreasing rank in Section 4.3. For analogous results in the Alien and HalfCheetah environments, refer to A.9. ### 4.2 Modulating Recurrent Memory Here, we aim to gain intuition for the results we observed in the online, in-distribution setting and in particular offer an explanation as to why the only effective form of pruning in-distribution was inducing sparsity in LSTMs. Recall in Section 3.1 we motivated the parameterization $W_{\text{rec}}(r, s)$ with respect to three measures, one of which was spectral radius, a proxy for a model’s attention span across time. In particular, we showed at initialization that the spectral radius of $W_{\text{rec}}(r, s)$ decreases as a function of sparsity and increases as a function of rank. We first note that these trends persist after training (Figure 4a). In particular, in both CfCs and LSTMs, the low-rank and highly-sparse models tend to admit solutions with recurrent weights of low spectral radius. Comparing CfCs and LSTMs, we note that CfCs tend to admit solutions with significantly lower spectral radii. This is perhaps not surprising considering that LSTMs explicitly promote a more consistent error flow over time (Jozefowicz et al., 2015) via their forget gate, and are thus more likely to attend to distant observations. So, assuming that the spectral radius is a reasonable proxy for the recurrent memory-horizon of a model, it appears that LSTMs tend to have longer-term memory than CfCs. However, note that the quantity we care about in practice is $\left\\| \frac{\partial h_t}{\partial h_{k-1}} \right\\| = \\| J_k \\|$, as this is the true measure of gradient propagation backwards in time. Since $\\| J_k \\|$ depends not only on the recurrent weights but also on the hidden state (for details, refer to A.4), it is possible that the disparate functional forms of CfCs and LSTMs mitigates the efficacy of spectral radius as a proxy for attention across time (for more details on this intuition, refer to A.5.1). Let us define $G_t = J_k J_{k-1} \ldots J_{t-t+1}$. Assuming that time $k$ represents the end of the time series, $G_t$ represent the gradient backpropagated $t$ steps in time, which when analyzed as a function of $t$ provides a direct measure of how much importance a model assigns to inputs as a function of the time that has passed since it last observed them. In particular, we compute $\log \\| G_t \\|$ as a function of $t$ and find that gradients decay significantly faster across time in CfCs than LSTMs (Figure 4b). One, this demonstrates that spectral radius is an effective measure of recurrent memory across architectures in this task setting. Two, we observe that in their full-rank, fully-connected forms, LSTMs have a long recurrent memory-horizon, while CfCs do not. In the context of a closed-loop task, we know that it is beneficial to limit the network’s temporal attention span (Lechner et al., 2020). Since an LSTM’s recurrent gradient in its baseline form does not decrease particularly fast across time, inducing sparsity at initialization pushes the network into the vanishing gradient regime, reducing its attention across time. This result provides support as to why LSTMs are quite effective at navigating environments in-distribution in spite of high-sparsity in their recurrent weights (Figure 3a). In contrast, CfCs in their baseline form already lie in the vanishing gradient regime: intuitively, inducing sparsity is less effective when the baseline network inherently possesses an affinity to selectively attend to past observations. This aligns with an analogous trend observed in RNNs which are networks that, like CfCs, are amenable to learning a vanishing gradient (Figure 19). By analyzing the spectral radius of $W_{rec}(r,s)$ as well as the recurrent gradients $G_t$, we have characterized sparsity in its ability to modulate the network’s recurrent memory-horizon. However, note that we also observe that low-rank LSTMs tend to admit solutions with lower spectral radius as well (Figure 4b). Yet, low-rank LSTMs tend to perform worse in-distribution (Figure 3a) than their sparse counterparts. To understand why there is a disparity between reducing rank and increasing sparsity, we turn to an analysis of the singular values, which are best motivated in the distribution shift setting. ### 4.3 ROBUSTNESS UNDER DISTRIBUTION SHIFT In Section 4.1, we found that in the distribution shift setting, pruning by reducing rank improved performance (most notably in CfCs) whereas pruning by increasing sparsity did not. This is in stark juxtaposition to the in-distribution trends in which inducing sparsity in LSTMs was the only form of pruning that did not worsen performance (Section 4.2). Here, we attempt to interpret these results by understanding both why CfCs appear to be the most robust model type and why pruning by reducing rank is distinct from pruning by increasing sparsity. Recall in Section 3.1 we reduced the robustness of the hidden state under perturbation to two measures: the spectral norm and the decay of the singular value spectrum of $W_{rec}(r,s)$ (for intuition, refer to A.5.2). Regarding the former, across models we observe that CfCs have lower spectral norms than both RNNs and LSTMs (Figure 5b, Figure 23). While this offers intuition as to the robustness of CfCs, it is not sufficient as a standalone measure. This is because in practice, distribution shifts are applied to the input $x_t$, which in turn corrupts the hidden state $h_{t+1}$. Thus, while it remains important to analyze the spectral norm of the recurrent weights, it is also pertinent to analyze the spectral norm of the input weights. In doing so, we find that while there exist marginal differences across architectures, the input spectral norm does not vary to the extent we observe in the recurrent spectral norm (Figure 20b). This, along with the fact that CfCs learn significantly lower spectral norms in their recurrent weights, offers intuition as to the heightened robustness of CfCs under distribution shift. In addition, the (albeit loose) relationship between weights with lower spectral norms corresponding to networks with lower Lipschitz constants provides even further support as to why CfCs tend to express more robust functions (elaborated upon in A.5.2). Next, we address the observed disparity between sparse and low-rank recurrent connectivity. Across ranks and sparsities, the spectral norm decreases as a function of increasing sparsity and decreasing rank (Figure 5b), aligning with the trends induced at initialization. In Section 4.2, we similarly showed that spectral radius decreases as a function of increasing sparsity and decreasing rank in the trained networks. Thus, we cannot explain the apparent disparity between low-rank and sparse connectivity via these measures, and instead turn to the decay of the singular values. Again, aligning with the prior induced at initialization, we find that the decay of the singular values increases as a function of increasing sparsity but decreases as a function of decreasing rank (Figure 5a). And this is precisely where the two methods of pruning differ: since sparsity reduces the rate of spectral decay, the resulting transformation (induced by $ W_{\text{rec}} $) on a perturbed version of the hidden state vector expands in more directions (i.e., increasing the effect of the perturbation); lowering the rank does the opposite. We can make this notion more concrete by analyzing the dimensionality of the state-space trajectories of each model. Note that the trajectory induced by $ h_t $ can be decomposed into two parts: the recurrently-driven portion $ W_{\text{rec}} h_{t-1} $ and the input-driven portion $ W_{\text{inp}} x_t $. Under this formulation, we can consider the full state-space trajectory $ h_t $ to be driven by $ W_{\text{full}} = [W_{\text{rec}} \ W_{\text{inp}}] $. To measure the complexity of these state-space trajectories, we ran PCA on them to measure their effective dimensionalities. We find that with decreasing rank, the dimensionality of the recurrently-driven trajectories decreases, whereas with increasing sparsity it increases (Figure 6a, Figure 6b). Hence, our intuition is as follows: since the activity along the recurrent axis is constrained to lie in the subspace spanned by the vectors comprising $ W_{\text{rec}}(r, s) $, the recurrent dynamics of low-rank recurrent networks are lower-dimensional and hence simpler. We can imagine that in the in-distribution setting, it is not desirable to arbitrarily constrain the network’s capacity to learn recurrently. LSTMs also having a shorter recurrent memory-horizon, inducing sparsity into LSTMs does not worsen the in-distribution performance of the network like constraining rank does (Figure 3a). In contrast, recall that under distribution shift, we observed the opposite: inducing sparsity at initialization was detrimental whereas constraining the network to be low-rank improved robustness. Constraining the recurrently-driven portion of the state-space is one means of reducing the network’s variability in the presence of input perturbations. Supervision along the recurrent axis is pivotal when the agent is faced with environmental occlusions, as it needs to lean on some notion of the past to make a decision in the present. By making the recurrent state more robust, we are better able to generalize under distribution shift. We make one final note across the model axis to further justify why CfCs appear to be inherently more robust than LSTMs and RNNs. In particular, in both the input-driven and full state-space trajectories (each of which are little affected by changes in the recurrent connectivity), we find that for all pairs $(r, s)$, the dimensionality of LSTM trajectories is much higher than that of CfC trajectories, offering additional justification for the robustness observed in CfCs. But what is perhaps more surprising is that RNNs, despite their simpler functional form, also possess higher-dimensional state-space dynamics than CfCs (Figure 18). Since RNNs and CfCs differ only with respect to the time constant network $ F $, we explore this gating mechanism further in A.2. ![Figure 6](image-url) Figure 6: Effective dimensionalities of the recurrent, input and full state-space trajectories collected during online testing, in-distribution, as measured by the explained variance of the top 5 principal components. a) CfC state-space dynamics ($\pm 1$ SE). b) LSTM state-space dynamics ($\pm 1$ SE). Figure 7: We individually consider the Frobenius norm of the weights at initialization $W_0$ and the change in the weights after training $\Delta W$. Note that the models shown here have no sparsity. 4.4 Exploring the Task Dimension Gap We now have demonstrated two things: one is the efficacy of the proposed connectivity parameterization in units of its interpretability as a modulator of network dynamics. Second is the impact connectivity has on each of the networks we examined: in particular, by disentangling the effects of rank and sparsity, we showed why LSTMs are more amenable to sparse connectivity in-distribution whereas CICs tend to show the most promise with low-rank connectivity under distribution shift. Here, we further our intuition on the performance of CICs under distribution shift by understanding why they tend to be the most amenable architecture to low-rank recurrent connectivity. As we have noted, the efficacy of this parameterization rests upon the ability of the network to adhere to the prior throughout training. And recall that we found that the spectral norm of the recurrent weights in CICs remained much closer to 1 in the full-rank, fully-connected network than in either LSTMs or RNNs (Figure 5). To better understand this, we borrow from an analysis conducted by Schuessler et al. (2020) in which they decomposed the recurrent weights as follows: $W_{rec} = W_0 + \Delta W$ where $W_0$ denotes the weights at initialization and $\Delta W$ denotes the change in the weights after training. In their setting, the purpose of the decomposition was to demonstrate that in a set of simple tasks, the Frobenius norm of the weights at convergence $\|\|W_{rec}\|\|_F$ is dominated by the norm of the weights at initialization $\|\|W_0\|\|_F$. In spite of the recurrent connectivity they used being full-rank, they found that the changes in the weights learned during training were in fact low-rank as measured by $\|\|\Delta W\|\|_F$. In our analysis, we find the opposite: $\|\|W_0\|\|_F$ certainly does not dominate the norm of the final weights and is in fact lower than $\|\|\Delta W\|\|_F$ (Figure 7). This reinforces the notion of task dimension put forth by Schuessler et al. (2020) which describes the rank of the training-induced connectivity changes as a function of the task the network is trained on. In particular, in their work, they showed that the task dimension of the simple tasks they examined was low and hence a network with unconstrained, full-rank connectivity learned low-rank changes. In contrast, in our setting we consider a significantly more complex task domain which incites higher-rank changes in the recurrent connectivity. This brings forth the notion of a task dimension gap between the offline, open-loop and online, closed-loop settings: namely, the networks we examined are trained offline without being exposed to distribution shifts and hence learn higher-rank changes in their connectivity. In contrast, as we have shown, succeeding in the closed-loop setting under distribution shift means learning lower-rank dynamics. Thus, networks that are able to abide to our low-rank prior and avoid learning high-rank changes in connectivity are better at generalizing under distribution shift. This is precisely where CICs supersede LSTMs and to some extent RNNs as well. We find that despite each network starting at the same $\|\|W_0\|\|_F$, $\|\|\Delta W\|\|_F$ is lowest in CICs. 5 Conclusion In this work, we investigated the use of a low-rank, sparse parameterization of recurrent connectivity in various architectures as a means of improving model robustness in closed-loop environments. We showed that this type of connectivity was most amenable to CfCs and also showed promise in more canonical networks like LSTMs and RNNs. Furthermore, we demonstrated the interpretability of this prior by analyzing the network dynamics it induces as a function of both rank and sparsity. Our results represent an application in pruning recurrent networks at initialization to improve performance under distribution shift. ACKNOWLEDGMENTS Research was sponsored by the United States Air Force Research Laboratory and the Department of the Air Force Artificial Intelligence Accelerator and was accomplished under Cooperative Agreement Number FA8750-19-2-1000. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Department of the Air Force or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein. REPRODUCIBILITY STATEMENT To ensure the reproducibility of our work, we extensively detail the experimental setup in the appendix as well as provide information regarding the analyses we conducted. The bulk of these details can be found in appendix A.11 which describes how we constructed the dataset, how the models were trained (including details on hyperparameters), the evaluation metrics for the models and the intuition/implementation regarding the analyses that we performed on the trained models. A key portion of our results is driven by an initialization scheme we proposed that deviates from default initializers given in existing open-source implementations. We thoroughly describe how and why our proposed initialization differs in appendix A.10, so the reader can leverage it to reproduce our results. Regarding our theoretical work, we provide proofs for the claims we made in the main portion of the paper which can be found in appendix A.6. In that section, we clearly delineate the cases in which we were unable to prove certain claims and had to resort to an empirical analysis instead. 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79tJB1eTmb	Comparison with the gpt-3.5-turbo-based method (AutoCot) shows that the gap is 0.5 points. Comparing this with the data requirements that the proposed method imposes brings the approach's utility into question.	Meta-CoT: Generalizable Chain-of-Thought Prompting in Mixed-task Scenarios with Large Language Models Anonymous authors Paper under double-blind review Abstract Large language models (LLMs) have unveiled remarkable reasoning capabilities by exploiting chain-of-thought (CoT) prompting, which generates intermediate reasoning chains to serve as the rationale for deriving the answer. However, current CoT methods either simply employ general prompts such as Let’s think step by step, or heavily rely on handcrafted task-specific demonstrations to attain preferable performances, thereby engendering an inescapable gap between performance and generalization. To bridge this gap, we propose Meta-CoT, a generalizable CoT prompting method in mixed-task scenarios where the type of input questions is unknown. Meta-CoT firstly categorizes the scenario based on the input question and subsequently constructs diverse demonstrations from the corresponding data pool in an automatic pattern. Meta-CoT simultaneously enjoys remarkable performances on ten public benchmark reasoning tasks and superior generalization capabilities. Notably, Meta-CoT achieves the state-of-the-art result on SVAMP (93.7%) without any additional program-aided methods. Our further experiments on five out-of-distribution datasets verify the stability and generality of Meta-CoT. Code is available at Anonymous. 1 Introduction Large language models (LLMs) (Brown et al., 2020; Scao et al., 2022; Thoppilan et al., 2022; Chowdhery et al., 2022; Touvron et al., 2023; OpenAI, 2023) have exhibited commendable capabilities on complex reasoning by virtue of chain-of-thought (CoT) prompting (Wei et al., 2023). CoT prompting entails the generation of intermediate reasoning chains that serve as the rationale before deriving the answer. Current CoT prompting methods predominantly fall into two categories, which we dub as General Zero-Shot-CoT and Specific Few-Shot-CoT, respectively. The former leverages general prompts such as Let’s think step by step and appends them directly to the input question, aiming to summon up the step-by-step reasoning potential from LLMs (Kojima et al., 2023; Yang et al., 2023). The latter provides task-specific input-output pairs as in-context demonstrations and puts them before the input question, for the purpose of instructing LLMs to carry out multi-step reasoning with elaborately selected demonstrations (Wei et al., 2023; Zhang et al., 2023; Wan et al., 2023; Diao et al., 2023). Briefly, there are two major limitations in previous studies. On one hand, the General Zero-Shot-CoT pattern is endowed with favorable generalization ability as it does not need any task-related exemplars, but it often pales in terms of performance when compared with the few-shot pattern. On the other hand, the Specific Few-Shot-CoT pattern heavily leans on task-specific demonstrations to attain superior performances, yet fails to bear on decent generalization ability. Although recent works have made progress by either alleviating manual labor (Zhang et al., 2023) or promoting the quality of demonstrations (Arora et al., 2023; Wan et al., 2023; Diao et al., 2023), all of them rest on the task-associated perspective thus far. Nevertheless, in practical applications, LLMs tend to confront situations of mixed types of questions, where it cannot be clearly identified which task the question belongs to. On these occasions, it is neither reasonable to improvise several task-related examples by hand nor possible to manually search for which task it refers to, not to mention that the question encountered in actual cases is not... Figure 1: Comparison with existing paradigms of CoT prompting. General zero-shot-CoT and specific few-shot-CoT are from Kojima et al. (2023) and Wei et al. (2023), respectively. even from a pre-defined collection of tasks. Besides, naive use of general trigger prompts is likely to result in performance degradation as the lack of templated rationales often leads to spurious reasoning steps (Wan et al., 2023). Therefore, there exists an inescapable gap between performance and generalization, especially in realistic mixed-task scenarios. To mitigate this gap, a potential strategy is to explore the trade-off area between generality and performance while ensuring certain practical applicability. Motivated by the above ideas, we propose Meta-CoT: a generalizable CoT prompting method in mixed-task scenarios where the type of input questions is unknown. Meta-CoT comprises three phases: firstly, it gathers questions of various reasoning types from a collection of reasoning tasks and samples distinct questions as in-context learning (ICL) demonstrations. Those ICL demonstrations are used to categorize the scenario of the input question. Secondly, it automatically constructs diverse demonstrations from the corresponding data pool based on the classified scenario obtained in the first phase. Thirdly, it performs a final inference on the input question with the demonstrations elaborated in the second phase and delivers the feedback to the data pool. We evaluate our proposed Meta-CoT on ten benchmark reasoning tasks including: (i) arithmetic reasoning (MultiArith (Roy & Roth, 2015), GSM8K (Cobbe et al., 2021), AddSub (Hosseini et al., 2014), AQUA-RAT (Ling et al., 2017), SingleEq (Koncel-Kedziorski et al., 2015), SVAMP (Patel et al., 2021)); (ii) commonsense reasoning (CSQA (Talmor et al., 2019), StrategyQA (Geva et al., 2021)); (iii) symbolic reasoning (Last Letter Concatenation, Coin Flip) (Wei et al., 2023). In addition, we further validate the stability and generalization of Meta-CoT on five out-of-distribution datasets including ARC-challenge (Clark et al., 2018), ASDiv (Miao et al., 2020), CSQA2.0 (Talmor et al., 2021), Sports Understanding (Suzgun et al., 2022) and Creak (Onoe et al., 2021). Experimental results show that Meta-CoT simultaneously enjoys remarkable performances and superior generalization capabilities. Notably, Meta-CoT achieves the state-of-the-art result on SVAMP (93.7%) without any additional program-aided methods. Moreover, Meta-CoT achieves impressive performance on GSM8K (89.92%) even without in-context demonstrations from GSM8K itself. To sum up, our work has three major contributions as follows: (i) To the best of our knowledge, our work pioneers a novel setting of the mixed-task scenario for CoT prompting, which has significant practical application values. (ii) We propose a generalizable CoT prompting method in mixed-task scenarios, which not only bridges the gap between performance and generalization but also unearths their in-between mutual synergy by gaining performance improvements in sync with achieving generality. (iii) Our approach has shown impressive performance and superior generalization ability on a total of 15 in-distribution and out-of-distribution datasets. Notably, it achieves the state-of-the-art result on SVAMP (93.7%) without any additional program-aided methods. Table 1: Typical CoT techniques (ICL: in-context learning; FT: fine-tuning; KD: knowledge distillation). Segment 1: fine-tuning techniques; Segment 2: in-context learning techniques. To the best of our knowledge, our work is the first to apply CoT prompting to mixed-task scenarios with enjoyable generality and superior performance without additional manual labor. In our work, we focus on in-context learning techniques, eliminating the burden of fine-tuning LLMs. \| Model \| Training \| Generality \| w/o Manual Labor \| w/ Input-related Info. \| \|------------------------\|----------\|------------\|------------------\|------------------------\| \| Fine-tune-CoT \| KD \| X \| ✓ \| X \| \| LoRAHub \| FT \| ✓ \| ✓ \| X \| \| Zero-Shot-CoT \| ICL \| ✓ \| ✓ \| X \| \| Few-Shot-CoT \| ICL \| X \| X \| ✓ \| \| Self-Consistency-CoT \| ICL \| X \| X \| ✓ \| \| Least-to-Most Prompting\| ICL \| X \| X \| ✓ \| \| Auto-CoT \| ICL \| X \| ✓ \| ✓ \| \| Active Prompt \| ICL \| X \| X \| ✓ \| \| OPRO \| ICL \| X \| ✓ \| X \| \| Meta-CoT (our work) \| ICL \| ✓ \| ✓ \| ✓ \| 2 RELATED WORK Two lines of research are key to our work: CoT prompting and cross-task generalization. 2.1 CHAIN-OF-THOUGHT PROMPTING Recently, CoT prompting methods have pushed the multi-step reasoning abilities of LLMs to a remarkable aptitude by eliciting them to generate intermediate reasoning chains before deriving the final answer (Wei et al., 2023), of which some typical techniques are listed in Table 1. Currently, there are two flavors of research in CoT prompting: General Zero-Shot-CoT (Kojima et al., 2023) and Specific Few-Shot-CoT (Wei et al., 2023). The former merely appends a general prompt such as "Let's think step by step" to the input question, with the intuition that the step-by-step capabilities of LLMs can be conjured with simple natural language triggers. The latter leverages several task-specific input-output pairs as reasoning demonstrations and inserts them before the test question, in light of decent in-context learning capability of LLMs (Radford et al., 2019; Brown et al., 2020). General Zero-Shot-CoT. LLMs have proven to be competent zero-shot reasoners by Kojima et al. (2023), which has greatly broadened the generalizability of CoT techniques and liberated the need to prepare task-specific examples in advance. While benefiting from its task-agnostic property, it often fails to excel at performance in comparison with its few-shot rivals (Wei et al., 2023; Zhang et al., 2023). In order to further boost the performance, recent works have laid emphasis on the optimization of triggering prompts (Yang et al., 2023). In their work, LLMs are employed as optimizers, and new prompts are progressively generated based on the past optimization history. Despite the augmented performance, the optimization process for prompts reverts to a task-specific problem, and for unseen test questions in real-world scenarios, it may not be advisable to use LLMs to optimize prompts on the fly. Specific Few-Shot-CoT. Owing to the well-crafted in-context exemplars, Few-Shot-CoT achieves preferable performance, which consequently extends to a plethora of studies focusing on improvements upon it. According to the period of improvement, these studies are grouped into three categories: (i) pre-reasoning pattern; (ii) peri-reasoning pattern; and (iii) post-reasoning pattern. For the pre-reasoning pattern, current research attends to either alleviating manual labor when selecting demonstrations (Zhang et al., 2023; Wan et al., 2023), or promoting demonstration quality (Creswell et al., 2023; Madaan & Yazdanbakhsh, 2022; Arora et al., 2023; Diao et al., 2023). Auto-CoT (Zhang et al., 2023) exploited the benefits of diversity in demonstrations and automatically constructed the demonstrations without the need for additional manual labor. Active-Prompt (Diao et al., 2023) underscored the significance of uncertainty by intentionally selecting the most uncertain questions for annotation and utilizing them as demonstrations. For the peri-reasoning pattern, recent studies concentrate on fine-grained reasoning processes such as problem decomposition (Zhou et al., Figure 2: The ratio of wrong cases in task identification (a), ratio of wrong cases in category identification (b) and ratio of wrong cases falling into form identification (c). Press et al. (2022), Zhou et al. (2023) introduced least-to-most prompting, which reduced complex problems to sub-problems and then the sub-problems were solved sequentially. Self-ask (Press et al., 2022) specifically asked follow-up questions to the model and then answered them before responding to the initial question. For the post-reasoning pattern, related works principally enhanced the performance by verification (Weng et al., 2022; Lyu et al., 2023) or ensemble-like methods (Wang et al., 2023; Li et al., 2023; Wang et al., 2022; Yoran et al., 2023). Weng et al. (2022) computed an explainable answer verification score by taking turns masking the initial conditions and predicting their results. Wang et al. (2023) introduced a self-consistency decoding approach to sample multiple outputs of LLMs and then voted over the final answers. However, the aforementioned works, which mainly hinge on task-associated exemplars, fail to step outside the task-specific framework to pursue generalizability. In turn, there is an upper bound to the performance that a general Zero-Shot-CoT method can achieve, thus leading the current CoT prompting to a dilemma. Our work, in contrast, manages to find a way out of this dilemma by intuitively carrying out an upstream scenario identification task, making our proposed Meta-CoT applicable in realistic mixed-task scenarios. 2.2 Cross-task Generalization Cross-task generalization has been a long-standing research goal in natural language processing (NLP). The conventional pre-training and fine-tuning paradigm gains a foothold by pre-training on a large corpus of text to capture general knowledge and fine-tuning on specific tasks to acquire specific knowledge. Beyond this primitive paradigm, post pre-training and multi-task learning encourage further advancements in this research area. For instance, Yu et al. (2022) made progress in the science domain while Zhang & Zhao (2021) promoted the model’s performance on dialogue-related tasks by introducing two novel training objectives to incorporate the dialogue-like features. Furthermore, typical multi-task learning frameworks promote models to learn shared representations across tasks to achieve task generalization. For example, MT-DNN (Liu et al., 2019) leveraged a few task-aware output modules to tailor the shared representations to each task. Notably, Zhang et al. (2022) proposed a task prefix guided multi-task pre-training framework, under the motivation that there are potential relationships among tasks which can be helpful for task generalization. Our work, consequently, is inspired by the discovery that data from different tasks may have similarities, thus sensibly partitioning mixed questions is likely to detect the mutual synergy between generalization and performance. More recent works such as ExT5 (Aribandi et al., 2022), T0 (Sanh et al., 2022) and FLAN (Wei et al., 2022) strived to convert a variety of tasks into an identical text-to-text format, so that models can be trained on those tasks jointly. LoraHub (Huang et al., 2023) leveraged the composability of LoRA (Low-Rank Adaption of LLMs) modules to promote the task generalization ability of LLMs. Our work, however, manages to effectuate task generalization through timely and user-friendly in-context learning without any training. 3 CHALLENGES OF GENERALIZABLE CoT IN MIXED-TASK SCENARIOS Existing studies (Wei et al., 2023) commonly assume that the type of questions fed to the model is known and conduct each set of evaluations on the questions from the same dataset. However, a more realistic setting lies in mixed-task scenarios where the type of input questions is unknown and they come in an arbitrary manner. To address the mixed-task scenarios, we put forward a salient procedure, namely scenario identification to explore practical and efficient solutions in a plug-and-play fashion. Beforehand, we need to address the following two challenges: (i) How to effectively partition the mixed questions so that we can invoke the pre-defined solutions (e.g., scenario-wise ICL)? (ii) What information do LLMs need to know for efficient scenario identification? 3.1 PARTITIONING MIXED QUESTIONS In the first place, we investigate how to effectively partition the mixed questions. Following Kojima et al. (2023); Zhang et al. (2023), we adopt questions from ten reasoning tasks. Those questions cover three categories including arithmetic, commonsense and symbolic reasoning and involve three forms encompassing short-answer, multiple-choice, and yes-or-no questions. At the very beginning, we make a simple and naive attempt to test how well LLMs can identify various tasks. We randomly sample one question from each of the ten tasks. For each question, we retain the task name from which it originates so that we obtain ten question-task pairs, which we employ as in-context learning demonstrations for question type identification. As can be seen from Figure 2, the identification accuracy is only 42%. We then analyze the wrong examples and find that 92% and 64% of them belong to the same category and form as the correct task respectively. The results demonstrate that LLMs are not qualified for distinguishing task names, but possess a high probability of correctly discriminating their categories or forms. We speculate that the underlying reason can be two-fold: on one hand, task names themselves are too abstract for LLMs to well perceive their differences through in-context learning alone. On the other hand, there exist potential similarities and correlations among tasks themselves (Zhang et al., 2022), which enlightens us to disclose more rational partitioning strategies. Since the majority of cases that misidentify task names fall into the same category or form, we compare the identification accuracy with the following three variants of partitioning schemes: (i) Category-based scheme which separates mixed questions into diverse categories; (ii) Form-based scheme which segments data into different answer forms; (iii) <Category, Form>-based scheme which concurrently takes the two aspects into account. As is shown in the right parts of Figure 2, we find that for category- and form-based schemes, a particular group tends to dominate the wrong cases. For instance, 85% of wrong cases in category identification belong to the symbolic group. We discover that this is because the sampled symbolic group demonstrations do not cover symbolic yes-or-no question, thus hindering LLMs from accurately identifying this missing type. As such, partitioning mixed questions based on both its category and form is a sensible strategy, which adequately considers the two major natures of question data. The results in Figure 3 show that this strategy reaches high accuracy (99%). 3.2 IDENTIFYING SCENARIOS In this part, we analyze what information LLMs require for efficient scenario identification. We extract the questions (Q) from the original data files and obtain the corresponding rationales (CoT) and predicted answers (A) from the Zero-Shot-CoT log files from Kojima et al. (2023). Abiding by the <Category, Form>-based partitioning strategy discussed in Section 3.1, we consider four alternatives of input formats fed to LLMs for scenario identification: (i) [Q] which takes purely the question as input ; (ii) [Q, A] which concatenates the question and the corresponding predicted --- 1 More data information is shown in Appendix A.1 Input question: If John scored 100 on his first 3 tests and an 80 on his 4th, what was his average score across the 4 tests? Figure 4: Overview of Meta-CoT, which consists of three phases: (i) scenario identification: categorizes the scenario of the input question (left); (ii) demonstration selection: fetches the ICL demonstrations for the categorized scenario (middle); (iii) answer derivation: performs the answer inference by feeding the LLM with the prompt comprising the fetched ICL demonstrations and the input question (right). Results in Table 2 suggest that the question itself is sufficient for LLMs to perceive the scenario. Notably, the participation of CoT degrades the identification performance, which may reveal that LLMs only need to focus on the question itself and the rationales would distract LLMs, thus leading to identification errors. Therefore, the question-only pattern [Q] is a satisfactory input option for scenario identification with decent accuracy and generality. 4 Meta-CoT This section introduces Meta-CoT, which is illustrated in Figure 4. On a high level, Meta-CoT consists of three phases: (i) scenario identification: categorizes the scenario of the input question; (ii) demonstration selection: fetches the ICL demonstrations for the categorized scenario; (iii) answer derivation: performs the answer inference by feeding the LLM with the prompt comprising the fetched ICL demonstrations and the input question. We detail these phrases as follows. 4.1 Scenario Identification Given an input question $q_{in}$, the goal of the scenario identification phase is to categorize the scenario, e.g., the type of the question. To this end, we first prepare a few ICL demonstrations, each of which consists of a question $q_i$ and its scenario $s_i$. The ICL demonstrations will be concatenated with $q_{in}$ to prompt the LLM to infer the question scenario. At the very beginning, we leverage public off-the-shelf datasets and obtain $n$ data groups based on the <category, form> partitioning strategy to construct the ICL demonstrations. Now that we have $n$ data groups $[D_1, D_2, \ldots, D_n]$ as a mixed questions pool $MP$, we randomly sample one question from each data group and obtain a set of \| Input format \| Generality \| Accuracy \| \|-------------\|------------\|----------\| \| [Q] \| ✓ \| 99.00 \| \| [Q, A] \| ✗ \| 96.40 \| \| [Q, CoT] \| ✗ \| 90.30 \| \| [Q, CoT, A] \| ✗ \| 91.10 \| Table 2: Identification accuracy (%) with different input formats. questions $[q_1, q_2, \ldots, q_n]$, with $q_i \in D_i$. Let $s_i$ represent the scenario name for the data group $D_i$. The demonstration $d_i$ for data group $D_i$ is formed by: $d_i = [Q: q_i, \text{Scenario}: s_i]$. We run such a process for each data group to have $n$-shot demonstrations: $P_{icl} = [d_1, d_2, \ldots, d_n]$. Similarly, the prompted input for identification $P_{ide}$ can be formulated as $[Q: q_{in}, \text{Scenario}: ]$. Finally, we concatenate the demonstrations and the prompted input together: $[P_{icl}, P_{ide}]$ and feed it into LLMs to predict the scenario $s_{in}$ for $q_{in}$. ### 4.2 Demonstration Selection After categorizing the scenario $s_{in}$ for the input question $q_{in}$, we are able to construct scenario-wise demonstrations for in-context learning. Given the scenario $s_{in}$ for the input question obtained in Section 4.1, we fetch the corresponding scenario data group $D_{in} \in [D_1, D_2, \ldots, D_n]$. Therefore, we have the questions in $D_{in}$ under the same scenario with $q_{in}$. Then, we construct the few-shot demonstrations by sampling a few representative questions by $k$-means clustering and invoking Zero-Shot-CoT to obtain the reasoning chains following Auto-CoT (Zhang et al., 2023). Concretely, we leverage Sentence-BERT (Reimers & Gurevych, 2019) to obtain a vector representation for each candidate question in $D_{in}$. Afterward, we perform $k$-means clustering over the acquired contextualized representations. For each cluster $i$, we sort the questions in ascending order by distance from the cluster center. Then we iterate over the sorted question list and apply Zero-Shot-CoT to the current question, namely adding Let’s think step by step after the question, to get the rationale and predicted answer. Next, we follow prior works (Wei et al., 2023; Zhang et al., 2023) and conduct simple filtering operations on the question and rationale, which help obtain more effective demonstrations. Once the question-rationale pair is retained under the filtering operation, we stop functioning on other questions in cluster $i$. As a result, we manage to collect a total of $k$ representative and high-quality demonstrations for $D_{in}$: $[(q^1_{re}, r^1_{re}, a^1_{re}), (q^2_{re}, r^2_{re}, a^2_{re}), \ldots, (q^k_{re}, r^k_{re}, a^k_{re})]$, where $r^i_{re}$ and $a^i_{re}$ refer to the rationale and predicted answer of $q^i_{re}$ by invoking Zero-Shot-CoT. ### 4.3 Answer Derivation Now that we have $k$ typical demonstrations of the formerly classified scenario $s_{in}$, we execute a final inference to obtain the answer to $q_{in}$. Concretely, we construct each demonstration $d^i_{re}$ by: $d^i_{re} = [Q: q^i_{re}, A: r^i_{re}, a^i_{re}]$ where $q^i_{re}, r^i_{re}, a^i_{re}$ are from $D_{in}$. Then we prepare the templated input prompt for inference by $P_{inf} = [Q: q_{in}, A: \text{Prompt}]$, where Prompt refers to simple triggers such as Let’s think step by step. After that, the concatenated demonstrations $[d^1_{re}, d^2_{re}, \ldots, d^k_{re}]$ are inserted before the input prompt $P_{inf}$, which is eventually delivered to LLMs to derive the rationale $r_{in}$ and answer $a_{in}$ of input question $q_{in}$. Meanwhile, we obtain a new triple of the input question, rationale and answer $(q_{in}, r_{in}, a_{in})$, which is sent back to the identified data group $D_{in}$ to update the mixed questions pool $MP$. ## 5 Experiments This section will describe our experimental setup and present the main results. ### 5.1 Setup Tasks and Datasets. Our method is evaluated on 10 in-distribution benchmark datasets and 5 out-of-distribution datasets. The in-distribution datasets are from three categories of reasoning tasks: (i) arithmetic reasoning (MultiArith (Roy & Roth, 2015), GSM8K (Cobbe et al., 2021), AddSub (Hosseini et al., 2014), AQUA-RAT (Ling et al., 2017), SingleEq (Koncel-Kedziorski et al., 2015), SVAMP (Patel et al., 2021)); (ii) commonsense reasoning (CSQA (Talmor et al., 2019), StrategyQA (Geva et al., 2021)); (iii) symbolic reasoning (Last Letter Concatenation, Coin Flip) (Wei et al., 2023). The five out-of-distribution datasets include: ARC-challenge (Clark et al., 2018), ASDiv (Miao et al., 2020), CSQA2.0 (Talmor et al., 2021), Sports Understanding (Suzgun et al., 2022) and Creak (Onoe et al., 2021). --- 2More details are attached in Appendix B.1 Table 3: Accuracy (%) on ten in-distribution reasoning datasets. Segment 1: ICL methods without CoT; Segment 2: Task-specific CoT approaches; Segment 3: CoT techniques with generalization. † indicates the experiment is based on GPT-4, otherwise GPT-3.5-Turbo is employed by default. Results in bold and _underline_ are the best and second-best performances respectively. \| Method \| AQuA \| MultiArith \| AddSub \| GSM8K \| SingleEq \| SVAMP \| Letter \| Coin \| Strategy \| CSQA \| Avg. \| \|--------------\|------\|------------\|--------\|-------\|----------\|-------\|--------\|------\|----------\|------\|------\| \| Zero-Shot \| 29.1 \| 67.2 \| 84.5 \| 15.9 \| 83.1 \| 67.9 \| 4.8 \| 44.0 \| 65.3 \| 74.3 \| 53.6 \| \| Few-Shot \| 33.1 \| 87.5 \| 86.6 \| 22.8 \| 89.0 \| 79.1 \| 7.2 \| 64.4 \| 62.3 \| 81.0 \| 61.3 \| \| Few-Shot-CoT \| 54.3 \| 97.3 \| 89.1 \| 73.8 \| 92.9 \| 81.9 \| 73.2 \| 99.0 \| 63.7 \| 78.0 \| 80.3 \| \| Auto-CoT \| 49.6 \| 99.3 \| 89.6 \| 75.9 \| 92.3 \| 84.6 \| 81.2 \| 100.0 \| 64.6 \| 72.2 \| 80.9 \| \| Zero-Shot-CoT\| 51.6 \| 94.7 \| 84.2 \| 71.2 \| 91.1 \| 78.4 \| 85.8 \| 99.0 \| 62.6 \| 69.9 \| 78.8 \| \| General-CoT \| 46.9 \| 98.7 \| 87.9 \| 74.1 \| 92.9 \| 83.8 \| 75.2 \| 100.0 \| 63.4 \| 72.2 \| 79.5 \| \| Meta-CoT \| 54.7 \| 99.7 \| 90.9 \| 72.6 \| 93.5 \| 88.6 \| 77.2 \| 100.0 \| 64.5 \| 72.4 \| 81.4 \| \| Meta-CoT† \| 72.8 \| 99.0 \| 91.9 \| 89.9 \| 92.3 \| 93.7 \| 90.2 \| 100.0 \| 74.1 \| 86.4 \| 89.0 \| Implementation. We utilize the popular and publicly available GPT-3.5-Turbo and GPT-4 (OpenAI [2023]) from OpenAI API[^1]. Experimental results are based on GPT-3.5-Turbo by default unless otherwise specifically marked. The original mixed questions pool $MP$ is constructed based on the 10 in-distribution datasets. The number of data groups $n$ is 6 according to the partitioning scheme discussed in Section 3.1. Following Wei et al. (2023), the number of demonstrations $k$ is 8 except for <arithmetic, multiple-choice questions> and <symbolic, short-answer questions> (4), <commonsense, multiple-choice questions> (7) and <commonsense, yes-or-no questions> (6). Baselines. We compare Meta-CoT with 6 baselines, which can be divided into three groups: (i) ICL methods without CoT prompting including Zero-Shot (Kojima et al., 2023) and Few-Shot (Brown et al., 2020); (ii) task-specific CoT approaches involving Few-Shot-CoT (Wei et al., 2023) and Auto-CoT (Zhang et al., 2023); (iii) CoT techniques with generalization referring to Zero-Shot-CoT (Kojima et al., 2023) and General-CoT. General-CoT is a strong baseline that we specifically devise for generalization comparison. It randomly collects one demonstration from each partitioned question group in our mixed data pool ($MP$) and then leverages the gathered demonstrations as a generic inference prompt for all the input data.[^2] ### 5.2 Main Results Performance of Meta-CoT on 10 in-distribution datasets Table 3 presents the results on ten in-distribution reasoning tasks. Notably, Meta-CoT achieves a state-of-the-art result on SVAMP (93.7%) without any additional program-aided methods. Meta-CoT also attains impressive performance on GSM8K without in-context demonstrations from GSM8K itself. Furthermore, Meta-CoT towers above all the baseline methods from different angles. On one hand, compared with two typical task-specific CoT approaches, Meta-CoT not only surpasses them in performance but also enjoys the generalizable property, which means that the input question with an unknown type can be adapted to our method in an automatic and labor-free pattern. On the other hand, while the general CoT techniques both witness performance degradation (i.e., 80.9% → 78.8/79.5%), Meta-CoT stands out by continually boosting the performance (i.e., 80.9% → 81.4%), thus shedding light on the mutual synergy between performance and generalization of LLMs. Performance of Meta-CoT on five out-of-distribution datasets As our work aims to accomplish a generalizable CoT prompting method in mixed-task scenarios, we further conduct experiments on 5 out-of-distribution datasets to verify its generality. We observe from Table 4 that our approach is capable of achieving a decent performance while maintaining favorable stability. The results certify the applicability of Meta-CoT to realistic situations where the incoming data is not defined by a certain type. Besides, we surprisingly discover that comparable results are yielded with the demonstrations of <commonsense, yes-or-no questions> scenario. We analyze that it is probably due to the broad coverage of commonsense knowledge that assists in the generality of LLMs. [^1]: https://openai.com/blog/openai-api [^2]: More details are presented in Appendix B.2. Table 4: Accuracy (%) on five out-of-distribution datasets. SAQ: short-answer question; MCQ: multiple-choice question; Y/N: yes-or-no question. We report the mean (Avg.) and standard deviations (Std.). We calculate Std. based on different question groups. Segment 1: Methods that leverage demonstrations of a specified scenario; Segment 2: Our Meta-CoT method. Results in bold and underline are the best and second-best performances respectively. \| Method \| Creak \| Sports \| CSQA2.0 \| ASDiv \| ARC-c \| Avg. ± Std. \| \|-----------------\|-------\|--------\|---------\|-------\|-------\|-------------\| \| Symbolic, SAQ \| 10.8 \| 58.5 \| 22.4 \| 73.2 \| 66.6 \| 56.8±22.9 \| \| Symbolic, Y/N \| 28.3 \| 22.6 \| 33.3 \| 73.3 \| 60.9 \| 54.1±23.4 \| \| Arithmetic, SAQ \| 8.6 \| 43.6 \| 16.7 \| 77.2 \| 67.6 \| 55.9±28.9 \| \| Arithmetic, MCQ \| 18.8 \| 59.1 \| 28.5 \| 77.3 \| 70.0 \| 61.2±22.5 \| \| Commonsense, Y/N\| 85.7 \| 83.1 \| 65.2 \| 71.7 \| 76.6 \| 75.4 ± 3.3 \| \| Commonsense, MCQ\| 22.5 \| 25.5 \| 23.5 \| 74.0 \| 77.9 \| 58.6±30.2 \| \| Meta-CoT \| 85.1 \| 83.1 \| 62.3 \| 77.1 \| 77.6 \| 77.2±0.4 \| 6 ANALYSIS 6.1 METHODS OF CONSTRUCTING CoT DEMONSTRATIONS Since our work is situated in realistic mixed-task scenarios, accessing high-quality demonstrations in a labor-saving pattern is of crucial importance. Accordingly, we select two representative labor-free sampling methods for comparison: (i) Similarity-based which retrieves the most top-$k$ similar questions based on cosine similarity; (ii) Randomness-based which randomly samples $k$ demonstrations for each input question. Results in Table 5 show our proposed Meta-CoT performs best, illustrating the importance of diversity in demonstrations. Table 5: Accuracy (%) of different demonstration construction methods. \| Method \| AQuA \| Strategy \| ASDiv \| Creak \| CSQA2.0 \| ARC-c \| Avg. \| \|-----------------\|------\|----------\|-------\|-------\|---------\|-------\|------\| \| Meta-CoT \| 54.7 \| 64.5 \| 100.0 \| \| \| \| \| \| w/ similarity \| 49.6 \| 64.1 \| 99.2 \| \| \| \| \| \| w/ randomness \| 52.0 \| 61.2 \| 99.0 \| \| \| \| \| 6.2 EFFECT OF SCENARIO IDENTIFICATION In order to further explore the effect of scenario identification which plays a key role in generalization, we discard this identification phase and adopt an idealized strategy in which we assume that the model is given the gold scenario. Results in Table 6 reveal that only a trivial improvement is detected even with the correct scenario given (70.2% → 70.6%). This indicates that our method potentially arouses the self-determination ability of LLMs without the need for manual intervention. Table 6: Effect of scenario identification. We study the cases where the correct scenario for the input question is given and then compare them with our method, which adaptively predicts the scenario. \| Method \| AQuA \| Strategy \| ASDiv \| Creak \| CSQA2.0 \| ARC-c \| Avg. \| \|-----------------\|------\|----------\|-------\|-------\|---------\|-------\|------\| \| Meta-CoT \| 54.7 \| 64.5 \| 77.1 \| 85.1 \| 62.3 \| 77.6 \| 70.2 \| \| w/ correct scenario \| 52.8 \| 65.0 \| 77.2 \| 85.7 \| 65.2 \| 77.9 \| 70.6 \| 7 CONCLUSION In this work, we initially put forward a novel setting with significant application values, namely mixed-task scenarios where the type of input question is unknown. Upon this challenging setting, we propose Meta-CoT, a generalizable CoT prompting mechanism that first performs scenario identification based on the input data and then automatically constructs corresponding demonstrations for ICL. 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ISq7Hnln0t	According to last paragraph in section 3, during evaluation, point prompts are used by default for quantitative evaluations. Is the UAP created under point prompts still effective under other types of SAM prompts? We can never assume the user to stick at one single prompt type.	SEGMENT ANYTHING MEETS UNIVERSAL ADVERSARIAL PERTURBATION Anonymous authors Paper under double-blind review ABSTRACT As Segment Anything Model (SAM) becomes a popular foundation model in computer vision, its adversarial robustness has become a concern that cannot be ignored. This work investigates whether it is possible to attack SAM with image-agnostic Universal Adversarial Perturbation (UAP). In other words, we seek a single perturbation that can fool the SAM to predict invalid masks for most (if not all) images. We demonstrate conventional image-centric attack framework is effective for image-independent attacks but fails for universal adversarial attack. To this end, we propose a novel perturbation-centric framework that results in a UAP generation method based on self-supervised contrastive learning (CL), where the UAP is set to the anchor sample and the positive sample is augmented from the UAP. The representations of negative samples are obtained from the image encoder in advance and saved in a memory bank. The effectiveness of our proposed CL-based UAP generation method is validated by both quantitative and qualitative results. On top of the ablation study to understand various components in our proposed method, we shed light on the roles of positive and negative samples in making the generated UAP effective for attacking SAM. 1 INTRODUCTION With an increasingly important role in driving groundbreaking innovations in AI, deep learning has gradually transitioned from training models for specific tasks to a general-purpose foundation model [Bommasani et al., 2021]. For language foundation models like BERT [Devlin et al., 2018] and GPT [Radford et al., 2018, 2019], have made significant breakthroughs in the natural language processing (NLP) area and contributed to the development of various generative AI [Zhang et al., 2023a], including the text generation (ChatGPT [Zhang et al., 2023b]), text-to-image [Zhang et al., 2023c] and text-to-speech [Zhang et al., 2023d], text-to-3D [Li et al., 2023], etc. On top of the early successful attempts like masked encoder [Zhang et al., 2022a], Meta research team has recently proposed a vision foundation model called Segment Anything Model (SAM) [Kirillov et al., 2023], which mimics GPT to control the output with prompts. Such a prompt-guided approach alleviates the need for finetuning and thus has impressive zero-shot transfer performance. After the release of Segment Anything project, SAM has been widely used in various applications, such as image editing [Kevmo, 2023] and object tracking [Adamdad, 2023; Chen, 2023], etc. Therefore, it is critical to understand its robustness in various contexts. Early works [Qiao et al., 2023] have examined its generalization capabilities beyond natural images to medical images [Zhang et al., 2023e] and camouflaged images [Tang et al., 2023]. Follow-up works have further evaluated its robustness under style transfer, common corruptions, patch occlusion and adversarial perturbation. Attack-SAM is a pioneering work to study how to attack SAM with adversarial examples, but it mainly focuses on image-independent attacks [Zhang et al., 2023f]. In other words, the generated perturbation can only be used for attacking the model for a specific image, which requires generating a new perturbation when the image changes. By contrast, a universal adversarial attack seeks a single perturbation (termed UAP) that causes the adversarial effect to all images and leads to wrong label predictions for most images [Moosavi-Dezfooli et al., 2017a] in the context of image classification. With the image-agnostic property, the UAP can be generated beforehand and applied to any image for the attack purpose and thus is relatively more practical but also more challenging. Therefore, our work is devoted to studying whether it is possible to attack SAM with a UAP. Classical adversarial attack methods like DeepFool [Moosavi-Dezfooli et al., 2016] and PGD Madry et al., (2018) optimize the perturbation to make the output of the adversarial image different from that of the original clean image. The classical UAP algorithm introduced in Moosavi-Dezfooli et al., (2017a) is based on DeepFool and thus follows such an image-centric approach. This requires access to the original training data and thus FFF Mopuri et al., (2017a) studies PGD-based approaches for generating data-free UAP Mopuri et al., (2017a) with a relatively weaker attack performance. Prior works Qiao et al., (2023); Zhang et al., (2023f) show that such an image-centric approach is also effective for attacking SAM, but the investigation is limited to image-independent attacks. A major difference in generating UAP lies in changing the to-be-attacked training image in every iteration to avoid over-fitting to any specific image. We follow this practice to extend Attack-SAM from image-independent attacks to universal attacks, however, such a preliminary investigation leads to unsatisfactory performance. This is attributed to the change of optimization target from one image to another in the image-centric approach. To this end, this work proposes a new perturbation-centric attack method, by shifting the goal from directly attacking images to seeking augmentation-invariant property of UAP. Specifically, we optimize the UAP in the CL method where the UAP is chosen as the anchor sample. The positive sample is chosen by augmenting the anchor sample, while random natural images are chosen as the negative samples. For the proposed CL-based UAP generation method, we experiment with various forms of augmentations to generate a positive sample and find that augmenting the UAP by adding natural images yields the most effective UAP for universal adversarial attack. Beyond quantitative verification, we also show visualize the attack performance of the generated UAP under both point and box prompts. We have an intriguing observation that the predicted mask gets invalid under both types of prompts: getting smaller under point prompts and getting larger under box prompts. Moreover, we present a discussion to shed light on why our generated UAP is effective by analyzing different pairs of inputs for the encoded feature representations. It helps us understand the roles of positive samples and negative samples in our CL-based UAP method for crafting an effective UAP to attack SAM. 2 RELATED WORKS Segment Anything Model (SAM). SAM is a recent advancement in the field of computer vision that has garnered significant attention Ma and Wang (2023); Zhang et al., (2023e); Tang et al., (2023); Han et al., (2023); Shen et al., (2023); Kang et al., (2022). Unlike traditional deep learning recognition models focusing solely on label prediction, SAM performs mask prediction tasks using prompts. This innovative approach allows SAM to generate object masks for a wide range of objects, showcasing its remarkable zero-shot transition performance. Researchers have explored the reliability of SAM by investigating its susceptibility to adversarial attacks and manipulating label predictions. Furthermore, SAM has been extensively utilized in various applications, including medical imaging Ma and Wang (2023); Zhang et al., (2023e), and camouflaged object detection Tang et al., (2023). It has also been combined with other models and techniques to enhance its utility, such as combining with Grounding DINO for text-based object detection and segmentation IDEA-Research (2023) and integrating with BLIP or CLIP for label prediction Chen et al., (2023); Park (2023); Li et al., (2022); Radford et al., (2021). SAM has found applications in image editing Rombach et al., (2022), inpainting Yu et al., (2023), and object tracking in videos Yang et al., (2023); Zxyang (2023). More recently, MobileSAM Zhang et al., (2023g), which is significantly smaller and faster than the original SAM, realizes lightweight SAM on mobile devices by decoupled knowledge distillation. With the advent of MobileSAM, it is expected more and more SAM-related applications will emerge, especially in the computation-constrained edge devices. This yields a need to understand how SAM works, for which Zhang et al., (2023h) performs a pioneering study and shows that SAM is biased towards texture rather than shape. Moreover, multiple works Qiao et al., (2023); Zhang et al., (2023i) have shown that SAM is vulnerable to the attack of adversarial examples. Our work also investigates the adversarial robustness of SAM, but differentiates by focusing on universal adversarial attack. Universal Adversarial Attack. Universal adversarial perturbation (UAP) has been first introduced in Moosavi-Dezfooli et al., (2017a) to fool the deep classification model by making wrong label predictions for most images. Unlike the vanilla universal attack by the projected algorithm to generate the perturbations, the SV-UAP Khrulkov and Oseledets (2018) adopts singular vectors to craft UAPs, where the method is data-efficient with only 64 images used to iteratively craft the perturbations. Inspired by the Generative Adversarial Networks (GAN), NAG Mopuri et al. and GAP [Perolat et al., 2018] focus on obtaining the distribution of UAPs. To compute the UAPs, these approaches use a subset of the training dataset, however, the attacker might be limited in accessing the training data. Therefore, multiple works explore data-free to generate UAPs. FFF [Mopuri et al., 2017b] is pioneering to propose a data-independent approach to generate the UAPs, adopting fooling the features learned at multiple layers. GD-UAP [Mopuri et al., 2018b] can generate universal perturbations and transfer to multiple vision tasks. Class-discriminative UAP has been investigated in [Zhang et al., 2020a; Benz et al., 2020] to fool the model for a subset of classes while minimizing the adversarial effect on other classes of images. They opt to train the UAP with Adam Optimizer [Kingma and Ba, 2015] instead of adopting sign-based PGD algorithms [Goodfellow et al., 2015; Madry et al., 2018], and such a practice has also been adopted in [Zhang et al., 2020b; 2021]. In contrast to prior works adopting image-centric DeepFool or PGD to optimize the UAP, our work proposes a perturbation-centric framework with a new UAP generation method based on contrastive learning. Self-supervised Contrastive Learning (CL). With the goal of learning augmentation-invariant representation, for which CL is a milestone development of unsupervised learning [Schroff et al., 2015; Wang and Gupta, 2015; Sohn, 2016; Misra et al., 2016; Federici et al., 2020]. CL consists of positive pair and negative pairs. Unlike the negative pairs, the positive pair are obtained from the same image but differ in augmentation to ensure they have similar semantic information. Earlu works on CL have adopted margin-based contrastive losses [Hadsell et al., 2006; Wang and Gupta, 2015; Hermans et al., 2017], and NCE-like loss [Wu et al., 2018; Oord et al., 2018] has later emerged to become the de facto standard loss in CL. For example, classical CL methods like SimCLR [Chen et al., 2020a] and MoCo families [He et al., 2020; Chen et al., 2020b] adopt the InfoNCE loss which combines mutual information and NCE. Specifically, it maximizes the mutual information between the representation of different views in the same scene. 3 BACKGROUND AND PROBLEM FORMULATION 3.1 PROMPT-GUIDED IMAGE SEGMENTATION Segment Anything Model (SAM) consists of three components: an image encoder, a prompt encoder, and a lightweight mask decoder. The image encoder adopts the MAE [He et al., 2022] pre-trained Vision Transformer (ViT), which generates the image representation in the latent space. The prompt encoder utilizes positional embeddings to represent the prompt (like points and boxes). The decoder takes the outputs of image and prompt encoders as the inputs and predicts a valid mask to segment the object of interest. In contrast to classical semantic segmentation performing pixel-wise label prediction, the SAM generates a label-free mask. With $x$ and $p$ denoting the image and prompt, respectively, we formalize the mask prediction of SAM as follows: $$y = \text{SAM}(x, p; \theta),$$ where $\theta$ represents the parameter of SAM. Given a image $x \in \mathbb{R}^{H \times W \times C}$, the shape of $y$ is $\mathbb{R}^{H \times W}$. We set the $x_{ij}$ as the pixel values at the image $x$ with the coordinates $i$ and $j$. $x_{ij}$ belongs to the masked area if the pixel value $y_{ij}$ is larger than the threshold of zero. 3.2 UNIVERSAL ADVERSARIAL ATTACK ON SAM Here, we formalize the task of universal adversarial attack on SAM. Let $\mu$ denote the distribution of images in $\mathbb{R}^{H \times W \times C}$. In the image recognition tasks, the adversary goal is to fool the model to predict wrong labels. Universal adversarial attack, under the assumption that the predicted labels of clean images are the correct ones, seeks a single perturbation vector $v \in \mathbb{R}^{H \times W \times C}$ termed UAP to cause label changes for most images [Moosavi-Dezfooli et al., 2017a]. In other words, it aims to maximize the adversarial effect of the UAP in terms of the fooling rate, the ratio of images whose predicted label changes after adding the UAP [Moosavi-Dezfooli et al., 2017a]. In the context of SAM, the predicted outputs are masks instead of labels and thus the attack goal is to cause mask changes. We follow Attack-SAM to adopt the widely used Intersection over Union (IoU) in image segmentation to evaluate such mask changes. The mIoU calculates the mean IoU for $N$ pairs of clean mask $Mask_{clean}$ and adversarial mask $Mask_{adv}$ shown in Equation 2. \[ mIoU = \frac{1}{N} \sum_{n=1}^{N} IoU(Mask_{clean}^{(n)}, Mask_{adv}^{(n)}), \] (2) where all the adversarial masks $ Mask_{adv} $ are generated for all $ N $ images by a single UAP. The goal of universal adversarial attack on SAM is to seek such a single perturbation $ v $ to decrease the mIoU defined in Eq. 2 as much as possible. The UAP $ v $ is bounded by a $ l_p $ norm, which is set to $ l_\infty $ norm conventions in prior works on SAM [Moosavi-Dezfooli et al., (2017a,b)]. Implementation details. Considering the image-agnostic property, $ N $ in Eq. 2 needs to be larger than 1 and is set to 100 in this work. For the prompts, we randomly choose point prompts unless specified otherwise. Specifically, we randomly select 100 test images from the SA-1B dataset [Kirillov et al., (2023)] for evaluating the generated UAP. Note that the test images cannot be used for generating the UAP. Following the existing works on the universal adversarial attacks in computer vision, we use 10/255 as the maximum limit for the perturbation. In other words, the allowed maximum change on each pixel can be no bigger than 10/255. ## 4 Method ### 4.1 Existing Image-Centric Attack Framework For the task of adversarial attack, the goal is to make the deep model predict invalid output after adding a small perturbation on the input image. Therefore, numerous attack methods, including classical DeepFool [Moosavi-Dezfooli et al., (2016)] and PGD [Madry et al., (2018)], optimize such an adversarial perturbation to make the output of adversarial image different from that of its clean image. Such an image-centric approach consists of two steps. First, it predicts the output of clean image $ y_{clean} $ and saves it as the ground-truth$^1$. Second, the perturbation in the adversarial image is optimized to make $ y_{adv} $ different from the ground-truth $ y_{clean} $. Universal adversarial attack requires the perturbation to be effective at random unseen images. Therefore, the to-be-attacked training image needs to be changed in every iteration of the optimization process to avoid over-fitting on any single training image. Such an image-centric approach has been adopted in [Zhang et al., (2023)] to demonstrate successful image-independent attacks, and we have adapted it to image-agnostic, universal adversarial attacks. The results in Table 1 show that the generated UAP performs much better than random uniform noise sampled between $-10)/255 and 10/255. Nonetheless, the value of mIoU (59.50%) is still quite high, demonstrating that the UAP is not sufficiently effective for causing mask changes. We also experiment with not changing the to-be-attacked image, which fixes the same optimization goal and results in a successful image-dependent attack with a mIoU of 0.0%. This suggests that a successful attack in SAM requires a consistent optimization target (like attacking a single image). However, such success is limited to image-dependent attacks due to overfitting and cannot be generalized to unseen test images. \| Input \| Image-dependent \| Image-agnostic \| \|------------------------\|-----------------\|---------------\| \| Uniform noise \| 86.97 \| 86.97 \| \| Adversarial attack \| 0.0 \| 59.50 \| ### 4.2 Proposed Perturbation-Centric Attack Framework The above image-centric method is suitable for image-independent attack on SAM but fails for universal attack. The image-centric method is in essence a supervised approach where \( y_{clean} $ plays the role of ground-truth and the added perturbation is optimized to make $ y_{adv} $ far from $ y_{clean} $. Such a supervised approach inevitably causes a dramatic change to the optimization goal when the training --- 1the ground-truth output might be given at first in some cases, where this step can be skipped. image is changed at every iteration. In other words, the failure of image-centric approach for universal attack is conjectured to be the inconsistent optimization goal caused by the change of training image at every iteration. Therefore, we shift the perspective from image to perturbation, which results in our proposed perturbation-centric method. Specifically, in contrast to the predicted masks of the clean and adversarial images, we focus on the independent features of the UAP, which is motivated by perceiving the UAP as an independent input considering its image-agnostic property. How to optimize the UAP in such a perturbation-centric approach, however, is a non-trivial task. It cannot be straightforwardly optimized in a supervised manner as in the image-centric method. To this end, we turn to a widely used self-supervised approach known as Contrastive Learning (CL). The difference between image-centric and perturbation-centric framework is summarized in Figure 1. ![Figure 1](image) Figure 1: Difference between image-centric (left) and perturbation-centric (right) attack frameworks. CL-based UAP Generation Method. Outperforming its supervised counterpart, self-supervised learning has become a dominant approach for pre-training an backbone encoder, where CL is a widely adopted method. In the classical CL, there are three types of samples: anchor sample, positive sample, and negative sample. The anchor sample is the sample of interest, while the positive sample is augmented from the anchor sample. Other random images are chosen as the negative samples, and we adopt the same practice in our CL-based UAP generation method. What makes it different from the classical CL method lies in the choice of anchor sample. Specifically, the UAP ($v$) is chosen as the anchor sample because it is the input of interest in this context. For the positive sample, we obtain it by augmenting the anchor sample UAP, which will be discussed in detail. The NCE-like loss (often termed InfoNCE loss) has been independently introduced in multiple works and constitutes as the de-facto standard loss for CL. Following [He et al., 2020], we denote the encoded features of the anchor sample, positive sample, and negative sample with $q$, $k_+$, and $k_-$, respectively. Note that the encoded features are often L2 normalized to remove scale ambiguity, based on which the InfoNCE loss adopted in the CL-based UAP generation method is shown as follows: $$L_{infonce} = -\log \frac{\exp(q \cdot k_+ / \tau)}{\exp(q \cdot k_+ / \tau) + \sum_{i=1}^{K} \exp(q \cdot k_i^- / \tau)},$$ (3) where $\tau$ represents the temperature controlling the hardness-aware property and thus has an implicit influence on the size of negative samples [Wang and Liu, 2021; Zhang et al., 2022b]. A large negative sample size is required to better sample the high-dimensional visual space [He et al., 2020]. We follow prior works to save the encoded features of negative samples in a list termed as memory bank [Wu et al., 2018] or dictionary [He et al., 2020]. Since the to-be-attacked SAM encoder does not change during the optimization of UAP, the list does not need to be updated as in classical CL method [Wu et al., 2018; He et al., 2020]. In other words, the $k^-$ in Eq[3] can be generated once and then saved for sampling during the optimization of UAP. In the classical CL method, augmentation is applied to ensure augmentation-invariant property for the encoder learning meaningful representations. In our CL-based UAP method, augmentation is also essential for making the generated UAP cause augmentation-invariant feature response on the encoder. This yields two intertwined questions: (1) how should we choose such augmentation for making the UAP effective? (2) why does such augmentation-invariant property make the UAP effective? The following section performs an empirical study to shed insight on these two intertwined questions. 5 EXPERIMENTAL RESULTS AND ANALYSIS 5.1 TOWARDS FINDING EFFECTIVE AUGMENTATION Preliminary investigation. In the classical CL method, there are mainly two types of augmentations [Chen et al., 2020a]. The first type involves spatial transformation like crop/resize and cutout. The second type involves no spatial transformation but causes appearance change by adding low-frequency content (like color shift) or high-frequency content (like noise). We experiment with both types of augmentation and the results are shown in Table 2. We observe that the mIoU values with augmentation crop/resize and cutout consistently remain high, at 85.11% and 75.48%, respectively. It suggests that the spatial transformation is not an effective augmentation type in our UAP generation method. For the second type of adding content, adding uniform noise is also not effective with a mIoU value of 81.14%. By contrast, the augmentation of color shift yields a mIoU of 61.64%, which is comparable to that of the image-centric method (see 59.5% in Table 1). Table 2: Comparison of different augmentations. The Crop size is 200×200 out of 1024×1024, cutout size is 200×200. The uniform noise and color shift are ranged from 0 to 255. Adding natural images achieves significantly better performance than other augmentations. \| Augmentation type \| mIoU (↓) \| \|-----------------------\|----------\| \| Crop/Resize \| 85.11 \| \| Cutout \| 75.48 \| \| Uniform noise \| 81.14 \| \| Color shift \| 61.64 \| \| Adding natural images \| 15.01 \| From color shift to natural images. Our preliminary investigation suggests that color shift is the most effective augmentation among those we investigate. We believe that this might be connected to how the generated UAP is applied to attack the model in practice. Since UAP is directly added to the images without spatial transformation, which explains why spatial transformation is less effective. Moreover, natural images have the property of being locally smooth and thus mainly contain low-frequency content, which justifies why the color shift is relatively more effective than adding noise. Motivated by the above interpretations, we conjecture that replacing the color shift images with random natural images for additive augmentation is beneficial for higher attack performance, which is supported by the results in Table 2. Here, for simplicity, the weight of the augmented natural images is set to 1. However, it can be set to values different from 1 (see the ablation study results in Figure 4). 5.2 QUALITATIVE RESULTS It is worth highlighting that our generated UAP has one hidden merit it can generalize to all prompts because the UAP is optimized only on the SAM encoder. In other words, it is truly universal in the sense of being both image-agnostic and prompt-agnostic. In the above, we only report the quantitative results under random point prompts. Here, for qualitative results, we visualize the attack performance under both point prompts and box prompts, with results shown in Figure 2 and Figure 3, respectively. We find that the single UAP causes the model to produce invalid masks for both types of prompts but with an intriguing distinction. Specifically, under the point prompts, the predicted mask region gets smaller with a boundary close to the chosen point prompt. Under the box prompt, however, the predicted mask gets larger than the original mask. We have no clear explanation for this intriguing phenomenon. A possible explanation is that the UAP tends to cause the predicted output to have similar values, i.e. causing confusion between the original masked regions and unmasked regions. For the point prompt, the unmasked region tends to be much larger than that of the masked region and thus the predicted mask gets smaller after UAP. By contrast, the box prompts tends to predict a mask inside the box, and thus tends to make the predicted mask boundary get larger and vague. Note that we can still observe the glass mask in the third row of Figure 3 but the mask boundary gets blurred. Figure 2: Qualitative results under point prompts. Column (a) and (b) shows the clean and adversarial images with the point prompt marked in a green star, with their predicted masks shown in column (c) and (d), respectively. The UAP makes the mask invalid by removing it (or making it smaller). Figure 3: Qualitative results under box prompts. Column (a) and (b) refers to the clean and adversarial images with the box prompt marked with green lines, with their predicted masks shown in column (c) and (d), respectively. The UAP makes the mask invalid by making it larger and blurry. 5.3 Ablation Study Weight of Augmented Images. Here, we first conduct an ablation study on the weight of the augmented images. The results are shown in Figure 4. We observe that the mIoU value decreases first increases and then decreases when the weight value is increased from 0.2 to 2 with an interval of 0.1. The strongest attack performance with the mIoU value of 14.21 appears when the weight is set to 1.2. Overall the mIoU value stays low for a relatively wide range of augmentation weight, suggesting our proposed method is moderately sensitive to the choice of augmentation weight. ![Figure 4: The mIoU (%) results for different weights of the augmented images.](image) Size of Negative Sample. For negative samples in contrastive learning, unlike the positive samples that aim to attract the anchor, our objective is to create a repelling effect on the anchor. This enables the anchor to more effectively focus on independent features by being drawn towards the positive samples. To accomplish this, it is essential to incorporate a diverse set of negative sample representations, thus avoiding repetitive generation. Therefore, we implement the memory bank mechanism, as do in prior work. We use various sample numbers (1, 2, 5, 10, 20, 50, 100) as our memory bank. As shown in Table 3, we observe a significant increase in universal attack performance as the number of samples increases. This indicates that augmenting diverse negative sample representations through the memory bank is beneficial for UAP training. To further augment diverse negative sample representations, \| N \| 1 \| 2 \| 5 \| 10 \| 20 \| 50 \| 100 \| \|-----\|------\|------\|------\|------\|------\|------\|------\| \| mIoU (↓) \| 38.91 \| 30.71 \| 24.83 \| 19.88 \| 17.63 \| 15.92 \| 15.01 \| Temperature. Temperature is widely known to have a large influence on the performance of CL method [Wang and Liu (2021); Zhang et al. (2022b)]. The influence of temperature in our CL-based UAP method is shown in Table 4. By default, the temperature is set to 0.1 in this work. We observe that the temperature significantly decreases when the temperature is set to a large value. The reason is that a smaller temperature causes more weight on spent on those hard negative samples [Wang and Liu (2021); Zhang et al. (2022b)]. As revealed in [Zhang et al. (2022b)], a small temperature is equivalent to choosing a small negative sample size. Therefore, it is well expected that the attack performance decreases when the temperature is set to a sufficiently small value because a relatively large negative sample size is required for CL. Unlike classical CL, a relatively large temperature does not cause a performance drop. Table 4: The mIoU (%) results on different InfoNCE temperatures. \| Temperature \| 0.005 \| 0.01 \| 0.05 \| 0.1 \| 0.5 \| 1 \| \|-------------\|-------\|------\|------\|-----\|-----\|-----\| \| mIoU (↓) \| 64.61 \| 60.58\| 22.78\| 15.01\| 13.28\| 13.48\| 5.4 DISCUSSION To shed more light on why the generated UAP is effective in attacking unseen images, we analyze the cosine similarity of different pairs of inputs for the encoded feature representations, and the results are shown in Table 5. The positive sample pairs have a much higher cosine similarity than that of the negative sample pairs, which aligns with our training objective in Eq[3]. The cosine similarity between pairs of adversarial images and its clean images is higher than that of the negative sample pairs, which is expected because the adversarial image consists of a random natural image and the UAP. The fact that the cosine similarity between positive sample pairs is very high (0.87) suggests that the UAP has independent features and it can be robust against the augmentation of image addition, which aligns with the finding in Zhang et al. (2020b). This partly explains why the cosine similarity between pairs of clean images and adversarial images is relatively low (0.40), causing a successful universal attack. In other words, how the generated UAP attacks the model does not intend to identify the vulnerable spots in the clean images to fool the model as suggested in Moosavi-Dezfooli et al. (2017a,b), but instead form its own augmentation-invariant features. For the role of negative samples in Eq[3], we find that it is can be at least partially attributed to the existence of common feature representations regardless of the image inputs for the image encoder, which is supported by a Cosine similarity value of 0.55 higher than zero for pairs of random images. With a list of negative samples in Eq[3], the UAP is expected to be optimized to offset such common features, thus causing adversarial effects. This interpretation is partially supported by the comparison between 0.40 and 0.55 in Table 5. Overall, the success of Eq[3] for generating an effective UAP can be interpreted as follows: the role of the positive sample is to make the UAP have independent features that are robust against the disturbance of natural image, while the role of negative images facilitates the UAP to find more effective directions to cause adversarial effects by partially canceling out the common feature representations in the image encoder. We leave further detailed analysis to future works. Table 5: Cosine similarity analysis with different pairs of inputs. \| Input pairs \| Cosine similarity \| \|-------------------------------------------------\|-------------------\| \| Positive sample pairs (UAP and augmented UAP) \| 0.87 \| \| Negative sample pairs (UAP and random image) \| 0.34 \| \| Pairs of adversarial image and its clean image \| 0.40 \| \| Pairs of two random images \| 0.55 \| 6 CONCLUSION Our work is the first to study how to perform adversarial attack SAM with a single UAP. We demonstrate that existing image-centric attack framework is effective for image-dependent attacks but fails to achieve satisfactory performance for universal adversarial attacks. We propose a perturbation-centric attack framework resulting in a new generation method based on contrastive learning, where the UAP is set to the anchor sample. We experiment with various forms of augmentations and find that augmenting the UAP by adding a natural image yields the most effective UAP among all augmentations we have explored. The effectiveness of our proposed method has been verified with both qualitative and quantitative results. Moreover, we have presented and analyzed different pairs of inputs for the encoded feature representations, which shed light on the roles of positive samples and negative samples in our CL-based UAP method for crafting an effective UAP to attack SAM. 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v8eWha27jw	In Section 3.3 “distribution-aware unbiased quantization”, this work proposes two optimization problems to find the optimal quantization values to reduce NMSE. In the first optimization problem on page 4, the notations $S(z, x)$ and $R(x)$ are a bit confusing. Are $S$ and $R$ two functions to be optimized?	ABSTRACT Distributed Mean Estimation (DME), in which $ n $ clients communicate vectors to a parameter server that estimates their average, is a fundamental building block in communication-efficient federated learning. In this paper, we improve on previous DME techniques that achieve the optimal $ O(1/n) $ Normalized Mean Squared Error (NMSE) guarantee by asymptotically improving the complexity for either encoding or decoding (or both). To achieve this, we formalize the problem in a novel way that allows us to use off-the-shelf mathematical solvers to design the quantization. 1 INTRODUCTION Federated learning [McMahan et al., 2017; Kairouz et al., 2019], is a technique to train models across multiple clients without having to share their data. During each training round, the participating clients send their model updates (hereafter referred to as gradients) to a parameter server that calculates their mean and updates the model for the next round. Collecting the gradients from the participating clients is often communication-intensive, which implies that the network becomes a bottleneck. A promising approach for alleviating this bottleneck and thus accelerating federated learning applications is compression. We identify the Distributed Mean Estimation (DME) problem as a fundamental building block that is used for that purpose either to directly communicate the gradients [Suresh et al., 2017; Konečný & Richtárik, 2018; Vargaftik et al., 2021, 2022; Davies et al., 2021] or as part of more complex acceleration mechanisms [Richtárik et al., 2021, 2022; Gorbunov et al., 2021; Szlendak et al., 2022; Condat et al., 2022b; Basu et al., 2019; Condat et al., 2022a; Condat & Richtárik, 2022; Horváth et al., 2023; Tyurin & Richtárik, 2023; He et al., 2023]. DME is defined as follows. Consider $ n $ clients with $ d $-dimensional vectors (e.g., gradients) to report; each client sends an approximation of its vector to a parameter server (hereafter referred to as ‘server’) which estimates the vectors’ mean. We briefly survey the most relevant and recent related works for DME. Common to these techniques is that they preprocess the input vectors into a different representation that allows for better compression, generally through quantization of the coordinates. For example, in Suresh et al. (2017), each client, in $ O(d \cdot \log d) $ time, uses a Randomized Hadamard Transform (RHT) to preprocess its vector and then applies stochastic quantization. The transformed vector has a smaller coordinate range (in expectation), which reduces the quantization error. The server then aggregates the transformed vectors before applying the inverse transform to estimate the mean, for a total of $ O(n \cdot d + d \cdot \log d) $ time. Such a method has a Normalized Mean Squared Error (NMSE) that is bounded by $ O(\log d/n) $ using $ O(1) $ bits per coordinate. Hereafter, we refer to this method as ‘Hadamard’. This work also suggests an alternative method that uses entropy encoding to achieve an NMSE of $ O(1/n) $, which is optimal. However, entropy encoding is a compute-intensive process that does not efficiently translate to GPU execution, resulting in a slow decode time. A different approach to DME computes the Kashin’s representation [Kashin, 1977; Lyubarskii & Vershynin, 2010] of a client’s vector $ x $ before applying quantization [Caldas et al., 2018; Safaryan et al., 2020]. Intuitively, this replaces the $ d $-dimensional input vector by $ O(d) $ coefficients, each bounded by $ O(\\|x\\|_2/\sqrt{d}) $. Applying quantization to the coefficients instead of the original vectors allows the server to estimate the mean using $ O(1) $ bits per coordinate with an $ O(1/n) $ NMSE. However, computing the coefficients requires applying multiple RHTs, asymptotically slowing down its encoding time from Hadamard’s $ O(d \cdot \log d) $ to $ O(d \cdot \log d \cdot \log(n \cdot d)) $. The works of Vargaftik et al. (2021, 2022) transform the input vectors in the same manner as Suresh et al. (2017), but with two differences: (1) clients must use independent transforms; (2) clients use deterministic (biased) quantization, derived using existing information-theoretic tools like the Lloyd-Max quantizer, on their transformed vectors. Interestingly, the server still achieves an unbiased estimate of each client’s input vector after multiplying the estimated vector by a real-valued ‘scale’. \| Algorithm \| Enc. complexity \| Dec. complexity \| NMSE \| \|-----------------\|-----------------\|-----------------\|----------\| \| QSGD (Alistarh et al., 2017) \| $O(d)$ \| $O(n \cdot d)$ \| $O(d/n)$ \| \| Hadamard (Suresh et al., 2017) \| $O(d \cdot \log d)$ \| $O(n \cdot d + d \cdot \log d)$ \| $O(\log d/n)$ \| \| Kashin (Caldas et al., 2018; Safaryan et al., 2020) \| $O(d \cdot \log d \cdot \log(n \cdot d))$ \| $O(n \cdot d + d \cdot \log d)$ \| $O(1/n)$ \| \| EDEN-RHT (Vargaltik et al., 2022) \| $O(d \cdot \log d)$ \| $O(n \cdot d \cdot \log d)$ \| $O(1)$ \| \| EDEN-URR (Vargaltik et al., 2022) \| $O(d^3)$ \| $O(n \cdot d^3)$ \| $O(1/n)$ \| \| QUIC-FL (New) \| $O(d \cdot \log d)$ \| $O(n \cdot d + d \cdot \log d)$ \| $O(1/n)$ \| Table 1: DME worst-case guarantees (without variable-length encoding; see App. B) for $b = O(1)$. Figure 1: Normalized Mean Squared Error vs. processing time. (that is sent by the client) and applying the inverse transform. Using uniform random rotations, which RHT approximates, such a process achieves $O(1/n)$ NMSE and is empirically more accurate than Kashin’s representation. With RHT, their encoding complexity is $O(d \cdot \log d)$, matching that of Suresh et al. (2017). However, since the clients transform their vectors independently of each other (and thus the server must invert their transforms individually, i.e., perform $n$ inverse transforms), the decode time is asymptotically increased to $O(n \cdot d \cdot \log d)$ compared to Hadamard’s $O(n \cdot d + d \cdot \log d)$. Further, with RHT the algorithm is biased, and thus its worse-case NMSE does not decrease in $1/n$; empirically, it works well for gradient distributions, but we show in Appendix A there are adversarial cases. While the above methods suggest aggregating the gradients directly using DME, recent works leverage it as a building block. For example, in EF21 Richtárık et al. (2021), each client sends the compressed difference between its local gradient and local state, and the server estimates the mean to update the global state. Similarly, DIANA Mishchenko et al. (2019) uses DME to estimate the average gradient difference. Thus, better DME techniques can improve their performance (see Appendix J.2). We defer further discussion of frameworks that use DME as a building block to Appendix B. In this work, we present Quick Unbiased Compression for Federated Learning (QUIC-FL), a DME method with $O(d \cdot \log d)$ encode and $O(n \cdot d + d \cdot \log d)$ decode times, and the optimal $O(1/n)$ NMSE. As summarized in Table 1, QUIC-FL asymptotically improves over the best encoding and/or decoding times of techniques with this NMSE guarantee. In QUIC-FL, each client applies RHT and quantizes its transformed vector using an unbiased method we develop to minimize the quantization error. Critically, all clients use the same transform, thus allowing the server to aggregate the results before applying a single inverse transform. QUIC-FL’s quantization features two new techniques; first, we present Bounded Support Quantization (BSQ), where clients send a small fraction of their largest (transformed) coordinates exactly, thus minimizing the difference between the largest quantized coordinate and the smallest one and thereby the quantization error. Second, we design a near-optimal distribution-aware unbiased quantization. To the best of our knowledge, such a method is not known in the information-theory literature and may be of independent interest. We implement QUIC-FL in PyTorch Paszke et al. (2019) and TensorFlow Abadi et al. (2015) and evaluate it on different FL tasks (Section 4). We show that QUIC-FL can compress vectors with over 33 million coordinates within 44 milliseconds and is markedly more accurate than existing $O(n \cdot d)$ and $O(n \cdot d + d \cdot \log d)$ decode time approaches such as QSGD Alistarh et al. (2017), Hadamard Suresh et al. (2017), and Kashin Caldas et al. (2018); Safaryan et al. (2020). Compared with DRIVE Vargaltik et al. (2021) and EDEN Vargaltik et al. (2022), QUIC-FL has a competitive NMSE while asymptotically improving the estimation time, as shown in Figure 1. Recent academic and industry sources (e.g., McMahan et al., 2022; Bonawitz et al., 2019) discuss FL deployments with thousands to tens of thousands of clients per round; thus, this speedup can lead to large savings in time and/or resources. The figure illustrates the encode and decode times vs. NMSE for $b = 4$ bits per coordinate, $d = 2^{20}$ dimensions, and $n = 256$ clients. Our code will be released upon publication. 2 PRELIMINARIES Notation. Capital letters denote random variables (e.g., $I_c$) or functions (e.g., $T(\cdot)$); overlines denote vectors (e.g., $\overline{x}_c$); calligraphic letters stand for sets (e.g., $\mathcal{X}_b$) with the exception of $\mathcal{N}$ and $\mathcal{U}$ that denote the normal and uniform distributions; and hats denote estimators (e.g., $\hat{x}_{avg}$). Problems and Metrics. Given a nonzero vector $\overline{x} \in \mathbb{R}^d$, a vector compression protocol consists of a client that sends a message to a server that uses it to estimate $\overline{x} \in \mathbb{R}^d$. The vector Normalized Mean Squared Error ($vNMSE$) of the protocol is defined as $$\frac{\mathbb{E}\left[\\|\hat{x} - x\\|^2_2\right]}{\\|x\\|^2_2}.$$ The above generalizes to Distributed Mean Estimation (DME), where each of $n$ clients has a nonzero vector $\overline{x}_c \in \mathbb{R}^d$, where $c \in \{0, \ldots, n-1\}$, that they compress and communicate to a server. We are interested in minimizing the Normalized Mean Squared Error ($NMSE$), defined as $$\frac{\mathbb{E}\left[\\|\hat{x}_{avg} - \frac{1}{n} \sum_{c=0}^{n-1} \overline{x}_c\\|^2_2\right]}{\frac{1}{n} \sum_{c=0}^{n-1} \\|x_c\\|^2_2},$$ where $\hat{x}_{avg}$ is our estimate of the average $\frac{1}{n} \cdot \sum_{c=0}^{n-1} \overline{x}_c$. For unbiased algorithms and independent estimates, we that $NMSE = vNMSE/n$. Randomness. We use global (common to all clients and the server) and client-specific shared randomness (one client and server). Client-only randomness is called private. 3 THE QUIC-FL ALGORITHM We first describe our design goals in Section 3.1. Then, in Sections 3.2 and 3.3, we successively present two new tools we have developed to achieve our goals, namely, bounded support quantization and distribution-aware unbiased quantization. In Section 3.4, we present QUIC-FL’s pseudocode and discuss its properties and guarantees. Finally, in Section 3.5, we overview additional optimizations. 3.1 DESIGN GOALS We aim to develop a DME technique that requires less computational overhead while achieving the same accuracy at the same compression level as the best previous techniques. As shown by recent works Suresh et al. (2017); Lyubarskii & Vershynin (2010); Caldas et al. (2018); Safaryan et al. (2020); Vargaftik et al. (2021, 2022), a preprocessing stage that transforms each client’s vector to a vector with a different distribution (such as applying a uniform random rotation or RHT) can lead to smaller quantization errors and asymptotically lower $NMSE$. However, in existing DME techniques that achieve the asymptotically optimal $NMSE$ of $O(1/n)$, such preprocessing incurs a high computational overhead on either the clients (i.e., Lyubarskii & Vershynin (2010); Caldas et al. (2018); Safaryan et al. (2020)) or the server (i.e., Lyubarskii & Vershynin (2010); Caldas et al. (2018); Safaryan et al. (2020); Vargaftik et al. (2021, 2022)). The question is then how to preserve the appealing $NMSE$ of $O(1/n)$ but reduce the computational burden? In QUIC-FL, similarly to previous DME techniques, we use a preprocessing stage where each client applies a uniform random rotation on its input vector. After the rotation, the coordinates’ distribution approaches independent normal random variables for high dimensions Vargaftik et al. (2021). We use our knowledge of the resulting distribution to devise a fast and near-optimal unbiased quantization scheme that both preserves the appealing $O(1/n)$ $NMSE$ guarantee and is asymptotically faster than existing DME techniques with similar $NMSE$ guarantees. A particularly important aspect of our scheme is that we can avoid decompressing each client’s compressed vector at the server by having all clients use the same rotation (determined by shared randomness), so that the server can directly sum the compressed results and perform a single inverse rotation. 3.2 BOUNDED SUPPORT QUANTIZATION Our first contribution is the introduction of bounded support quantization (BSQ). For a parameter $p \in (0, 1]$, we pick a threshold $t_p$ such that up to $d \cdot p$ values can fall outside $[-t_p, t_p]$. BSQ separates the vector into two parts: the small values in the range $[-t_p, t_p]$, and the remaining (large) values. The large values are sent exactly (matching the precision of the input), whereas the small values are stochastically quantized and sent using a small number of bits each. This approach decreases the error of the quantized values by bounding their support at the cost of sending a small number of values exactly. 1 In Section 3.5 move to the computationally efficient RHT instead, while preserving Table 1’s guarantees. For the exactly sent values, we also need to send their indices. There are different ways to do so. For example, it is possible to encode these indices using $ \log \left( \frac{d}{d \cdot p} \right) \approx d \cdot p \cdot \log(1/p) $ bits at the cost of higher complexity. When the $ d \cdot p $ indices are uniformly distributed (which will be essentially our case later), then delta coding methods can be applied (see, e.g., Section 2.3 of Vaidya et al. [2022]). Alternatively, we can send these indices without any additional encoding using $ d \cdot p \cdot \lceil \log d \rceil $ bits (i.e., $ \lceil \log d \rceil $ bits per transmitted index) or transmit a bit-vector with an indicator for each value whether it is exact or quantized. Empirically, sending the indices using $ \lceil \log d \rceil $ bits each without encoding is most useful, as $ p \cdot \log d \ll 1 $ in our settings, resulting in fast processing time and small bandwidth overhead. In Appendix C, we prove that BSQ admits a worst-case NMSE of $ \frac{1}{n \cdot p \cdot (2^b - 1)^2} $ when using $ b $ bits per quantized value. In particular, when $ p $ and $ b $ are constants, we get an NMSE of $ O(1/n) $ with encoding and decoding times of $ O(d) $ and $ O(n \cdot d) $, respectively. However, the linear dependence on $ p $ means that the hidden constant in the $ O(1/n) $ NMSE is often impractical. For example, if $ p = 2^{-5} $ and $ b = 1 $, we need three bits per value on average: two for sending the exact values and their indices (assuming values are single precision floats and indices are 32-bit integers) and another for stochastically quantizing the remaining values using 1-bit stochastic quantization. In turn, we get an NMSE bound of $ \frac{1}{n \cdot 2^{-5} \cdot (2^1 - 1)^2} = 32/n $. In the following, we show that combining BSQ with our chosen random rotation preprocessing allows us to get an $ O(1/n) $ NMSE with a much lower constant for small values of $ p $. For example, a basic version of QUIC-FL with $ p = 2^{-9} $ and $ b = 1 $ can reach an NMSE of $ 8.58/n $, a $ 3.72 \times $ improvement despite using $ 2.66 \times $ less bandwidth (i.e., 1.125 bits per value instead of 3). ### 3.3 Distribution-Aware Unbiased Quantization The first step towards our goal involves randomly rotating and scaling an input vector and then using BSQ to send values (rotated and scaled coordinates) outside the range $[-t_p, t_p]$ exactly. The values in the range $[-t_p, t_p]$ are sent using stochastic quantization, which ensures unbiasedness for any choice of quantization-values that cover that range. Now we seek quantization-values that minimize the estimation variance and thereby the NMSE. We take advantage of the fact that, after randomly rotating a vector $ x \in \mathbb{R}^d $ and scaling it by $ \sqrt{d}/\\|x\\|_2 $, the rotated and scaled coordinates approach the distribution of independent normal random variables $ N(0, 1) $ as $ d $ increases [Vargaltik et al. 2021, 2022]. We thus choose to optimize the quantization-values for the normal distribution and later show that it yields a near-optimal quantization for the actual rotated coordinates (see Appendix D for further discussion). That is, since we know both the distribution of the coordinates after the random rotation and scaling and we know the range of the values we are stochastically quantizing, we can design an unbiased quantization scheme that is optimized for this specific distribution rather than using, e.g., the standard approach of uniformly sized intervals. Formally, for $ b $ bits per quantized value and a BSQ parameter $ p $, we find the set of quantization-values $ Q_{b,p} $ that minimizes the estimation variance of the random variable $ Z \mid Z \in [-t_p, t_p] $ where $ Z \sim N(0, 1) $, after stochastically quantizing it to a value in $ Q_{b,p} $ (i.e., the quantization is unbiased). Then, we show how to use this precomputed set of quantization-values $ Q_{b,p} $ on any preprocessed vector. Consider parameters $ p $ and $ b $ and let $ X_b = \{0, \ldots, 2^b - 1\} $. Then, for a message $ x \in X_b $, we denote by $ S(z, x) $ the probability that the sender quantizes a value $ z \in [-t_p, t_p] $ to $ R(x) $, the value that the receiver associates with $ x $. With these notations at hand, we solve the following optimization problem to find the set $ Q_{b,p} $ that minimizes the estimation variance (we are omitting the constant factor $ 1/\sqrt{2\pi} $ in the normal distribution’s pdf from the minimization as it does not affect the solution): \[ \begin{align} \text{minimize} & \quad \int_{-t_p}^{t_p} \sum_{x \in X_b} S(z, x) \cdot (z - R(x))^2 \cdot e^{-z^2/2} dz \\ \text{subject to} & \quad \sum_{x \in X_b} S(z, x) \cdot R(x) = z \quad \forall z \in [-t_p, t_p] \\ & \quad \sum_{x \in X_b} S(z, x) = 1 \quad \forall z \in [-t_p, t_p], \quad S(z, x) \geq 0 \quad \forall z \in [-t_p, t_p], \quad x \in X_b \end{align} \] Observe that $ Q_{b,p} = \{R(x) \mid x \in X_b\} $ is the set of quantization-values that we are seeking. We note that the problem is known to be non-convex for any $ b \geq 2 $ [Faghri et al. 2020, Appendix B]. While there exist solutions to this problem excluding the unbiasedness constraint (e.g., the Lloyd-Max Scalar Quantizer [Lloyd (1982); Max (1969)]), we are unaware of existing methods for solving the above problem analytically. Instead, we propose a discrete relaxation, allowing us to approach the problem with a solver. To that end, we discretize the problem by approximating the truncated normal distribution using a finite set of $ m $ quantiles. Denote $ I_m = \{0, \ldots, m-1\} $ and let $ Z \sim N(0, 1) $. Then, $ A_{p,m} = \{A_{p,m}(i) \| i \in I_m\} $, where the quantile $ A_{p,m}(i) $ satisfies \[ \Pr[Z \leq A_{p,m}(i) \| Z \in [-t_p, t_p]] = \frac{i}{m-1}. \] We find it convenient to denote $ S'(i, x) = S(A_{p,m}(i), x) $. Accordingly, the discretized unbiased quantization problem is defined as (we omit the $ 1/m $ constant as it does not affect the solution): \[ \begin{align} \text{minimize} & \quad \sum_{i \in I_m, x \in X_b} S'(i, x) \cdot (A_{p,m}(i) - R(x))^2 \\ \text{(Unbiasedness)} & \quad \sum_{x \in X_b} S'(i, x) \cdot R(x) = A_{p,m}(i) \quad \forall i \in I_m \\ \text{(Probability)} & \quad \sum_{x \in X_b} S'(i, x) = 1 \quad \forall i \in I_m, \quad S'(i, x) \geq 0 \quad \forall i \in I_m, \; x \in X_b \end{align} \] The solution to this optimization problem yields the set of quantization-values $ Q_{b,p} = \{R(x) \| x \in X_b\} $ we are seeking. A value $ z \in [-t_p, t_p] $ (not just the quantiles) is then stochastically quantized to one of the two nearest values in $ Q_{b,p} $. Such quantization is optimal for a fixed set of quantization-values, so we do not need $ S $ at this point. Unlike in vanilla BSQ (Section 3.2), in QUIC-FL, as implied by the optimization problem, the number of values that fall outside the range $[-t_p, t_p]$ may slightly deviate from $ d \cdot p $ (and our guarantees are unaffected by this). This is because we precompute the optimal quantization-values set $ Q_{b,p} $ for a given $ b $ and $ p $ and set $ t_p $ according to the $ N(0, 1) $ distribution. In turn, this allows the clients to use $ Q_{b,p} $ when encoding rather than compute $ t_p $ and then $ Q_{b,p} $ for each preprocessed vector separately. This results in a near-optimal quantization for the actual rotated and scaled coordinates, in the sense that: (1) for large $ d $ values, the distribution of the rotated and scaled coordinates converges to that of independent normal random variables; (2) for large $ m $ values, the discrete problem converges to the continuous one. ### 3.4 Putting it All Together The pseudo-code of QUIC-FL appears in Algorithm 1. As mentioned, we use the uniform random rotation as a preprocessing stage done by the clients. Crucially, similarly to [Suresh et al. (2017)], and unlike in [Vargatuk et al. (2021); (2022)], all clients use the same rotation, which is a key ingredient in achieving fast decoding complexity. To compute this rotation (and its inverse by the server), the parties rely on global shared randomness as mentioned in Section 2. In practice, having shared randomness only requires the round’s participants and the server to agree on a pseudo-random number generator seed, which is standard practice. Clients. Each client $ c $ uses global shared randomness to compute its rotated vector $ T(\bar{x}_c) $. Importantly, all clients use the same rotation. As discussed, for large dimensions, the distribution of each entry in the rotated vector converges to $ N(0, \\| \bar{x}_c \\|_2^2 / d) $. Thus, $ c $ normalizes it by $ \sqrt{d} / \\| \bar{x}_c \\|_2 $, so the values of $ \bar{Z}_c $ are approximately distributed as $ N(0, 1) $ (line 1). (Note that we do not assume the values are actually normally distributed; this is not required for our algorithm or our analysis.) Next, the client divides the preprocessed vector into large and small values (lines 2-4). The small values (i.e., whose absolute value is smaller than $ t_p $) are stochastically quantized (i.e., in an unbiased manner) to values in the precomputed set $ Q_{b,p} $. We implement $ Q_{b,p} $ as an array where $ Q_{b,p}[x] $ stands for the $ x $’th quantization-value; this allows us to transmit just the quantization-value indices over the network (line 5). Finally, each client sends to the server the vector’s norm $ \\| \bar{x}_c \\|_2 $, the indices $ \bar{X}_c $ of the quantization-values of $ \bar{V}_c $ (i.e., the small values), and the exact large values with their indices in $ \bar{Z}_c $ (line 6). Server. For each client $ c $, the server uses $ \bar{X}_c $ to look up the quantization-values $ \hat{\bar{V}}_c $ of the small coordinates (line 8) and constructs the estimated scaled rotated vector $ \hat{\bar{Z}}_c $ using $ \hat{\bar{V}}_c $ and the accurate --- 2We use the Gekko [Beal et al. (2018)] software package that provides a Python wrapper to the APMonitor [Hedengren et al. (2014)] environment, running the solvers IPOPT [IPO] and APOPT [APO]. Algorithm 1 QUIC-FL Input: Bit budget $b$, BSQ parameter $p$, and their threshold $t_p$ and precomputed quantization-values $Q_{b,p}$. Client $c$: 1: $\mathcal{Z}_c \leftarrow \frac{\sqrt{d}}{\\|\mathbf{x}_c\\|_2} \cdot T(\mathbf{x}_c)$ 2: $\mathcal{U}_c \leftarrow \{\mathcal{Z}_c[i] \mid \|\mathcal{Z}_c[i]\| > t_p\}$ 3: $\mathcal{T}_c \leftarrow \{i \mid \|\mathcal{Z}_c[i]\| > t_p\}$ 4: $\mathcal{V}_c \leftarrow \{\mathcal{Z}_c[i] \mid \|\mathcal{Z}_c[i]\| \leq t_p\}$ 5: $\mathcal{X}_c \leftarrow$ Stochastically quantize $\mathcal{V}_c$ using $Q_{b,p}$ 6: Send $(\\|\mathbf{x}_c\\|_2, \mathcal{U}_c, \mathcal{T}_c, \mathcal{X}_c)$ to server Server: 7: For all $c$: 8: $\hat{\mathcal{V}}_c \leftarrow \{Q_{b,p}[x] \text{ for } x \text{ in } \mathcal{X}_c\}$ 9: $\hat{\mathcal{Z}}_c \leftarrow$ Merge $\hat{\mathcal{V}}_c$ and $(\mathcal{U}_c, \mathcal{T}_c)$ 10: $\hat{\mathcal{Z}}_{avg} \leftarrow \frac{1}{n} \cdot \sum_{c=0}^{n-1} \frac{\\|\mathbf{x}_c\\|_2}{\sqrt{d}} \cdot \hat{\mathcal{Z}}_c$ 11: $\hat{\mathbf{x}}_{avg} \leftarrow T^{-1}(\hat{\mathcal{Z}}_{avg})$ Information about the large coordinates $\mathcal{U}_c$ and their indices $\mathcal{T}_c$ (line 9). Then, the server computes the estimate $\hat{\mathcal{Z}}_{avg}$ of the average rotated and scaled vector by averaging the reconstructed clients’ scaled and rotated vectors and multiplying the results by the inverse scaling factor $\frac{\\|\mathbf{x}_c\\|_2}{\sqrt{d}}$ (line 10). Finally, the server performs a single inverse rotation using the global shared randomness to obtain the estimate of the mean vector $\hat{\mathbf{x}}_{avg}$ (line 11). In Appendix E, we formally establish the following error guarantee for QUIC-FL (i.e., Algorithm 1). Theorem 3.1. Let $Z \sim N(0,1)$ and let $\hat{Z}$ be its estimation by our distribution-aware unbiased quantization scheme. Then, for any number of clients $n$ and any set of $d$-dimensional input vectors $\{\mathbf{x}_c \in \mathbb{R}^d \mid c \in \{0,\ldots,n-1\}\}$, we have that QUIC-FL’s NMSE respects $$NMSE = \frac{1}{n} \cdot \mathbb{E}\left[(Z - \hat{Z})^2\right] + O\left(\frac{1}{n} \cdot \sqrt{\log d/d}\right).$$ The theorem accounts for the cost of quantizing the actual rotated and scaled coordinates (which are not independent and follow a shifted-beta distribution) instead of independent and truncated normal variables. The difference manifests in the $O(1/n \cdot \sqrt{\log d/d}) = O(1/n)$ term; this quickly decays with the dimension and number of clients. As the theorem suggests, $NMSE \approx \frac{1}{n} \cdot \mathbb{E}[(Z - \hat{Z})^2]$ for QUIC-FL in settings of interest. Moreover, $$\mathbb{E}\left[(Z - \hat{Z})^2 \mid Z \in [-t_p,t_p]\right] \cdot \Pr[Z \in [-t_p,t_p]]$$ $$+ \mathbb{E}\left[(Z - \hat{Z})^2 \mid Z \notin [-t_p,t_p]\right] \cdot \Pr[Z \notin [-t_p,t_p]],$$ where the first summand is exactly the quantization error of our distribution-aware unbiased BSQ, and the second summand is 0 as such values are sent exactly. This means that for any $b$ and $p$, we can exactly compute $\mathbb{E}[(Z - \hat{Z})^2]$ given the solver’s output (i.e., the precomputed quantization values). For example, it is $\approx 8.58$ for $b = 1$ and $p = 2^{-9}$. Another important corollary of Theorem 3.1 is that the convergence speed with QUIC-FL matches the vanilla SGD since its estimates are unbiased and with an $O(1/n)$ NMSE (e.g., see Remark 5 in Karimireddy et al. (2019)). 3.5 Optimizations We introduce two optimizations for QUIC-FL: we further reduce NMSE with client-specific shared randomness and then accelerate the processing time via the randomized Hadamard transform. QUIC-FL with client-specific shared randomness. Past works (e.g., Ben Basat et al. (2021); Chen et al. (2020); Roberts (1962b)) on optimizing the quantization-bandwidth tradeoff show the benefit of using shared randomness to reduce the quantization error. Here, we show how to leverage this (client-specific) shared randomness to design near-optimal quantization of the rotated and scaled vector. To that end, in Appendix F, we first extend our optimization problem to allow client-specific shared randomness and then derive the related discretized problem. Importantly, we also discretize the client-specific shared randomness where each client, for each rotated and quantized coordinate, uses a shared random $\ell$-bit value $H \sim U[\mathcal{H}_\ell]$ where $\mathcal{H}_\ell = \{0,\ldots,2^\ell - 1\}$. The resulting optimization problem is given as follows (additions are highlighted in red): \[ \begin{align} \text{minimize} & \quad \sum_{h \in H_t, i \in I_m, x \in X_b} S'(h, i, x) \cdot (A_{p,m}(i) - R(h, x))^2 \\ \text{subject to} & \\ (\text{Unbiasedness}) & \quad \frac{1}{2^t} \cdot \sum_{h \in H_t, x \in X_b} S'(h, i, x) \cdot R(h, x) = A_{p,m}(i) \quad \forall i \in I_m \\ (\text{Probability}) & \quad \sum_{x \in X_b} S'(h, i, x) = 1 \quad \forall h \in H_t, i \in I_m, \quad S'(h, i, x) \geq 0 \quad \forall h \in H_t, i \in I_m, x \in X_b \end{align} \] Here $S'(h, i, x) = S(h, A_{p,m}(i), x)$ represents the probability that the sender sends the message $x \in X_b$ given the shared randomness value $h$ for the input value $A_{p,m}(i)$. Similarly, $R(h, x)$ is the value the receiver associates with the message $x$ when the shared randomness is $h$. We explain how to use $R(h, x)$ to determine the appropriate message for the sender on a general input $z$, along with further details, in Appendix F. We note that Theorem 3.1 trivially applies to QUIC-FL with client-specific shared randomness as this only lowers the quantization’s expected squared error, i.e., $\mathbb{E}[(Z - \hat{Z})^2]$, and thus the resulting NMSE. Here, we provide an example based on the solver’s solution for the case of using a single shared random bit (i.e., $H \sim \mathcal{U}([H_1])$), a single-bit message ($b = 1$), and $p = 2^{-9}$ ($t_p \approx 3.097$); We can then use the following algorithm, where $X$ is the sent message and $\alpha = 0.7975, \beta = 5.397$ are constants: \[ X = \begin{cases} 1 & \text{if } H = 0 \text{ and } Z \geq 0 \\ 0 & \text{if } H = 1 \text{ and } Z < 0 \\ \text{Bernoulli}\left(\frac{2Z}{\alpha + \beta}\right) & \text{If } H = 1 \text{ and } Z \geq 0 \\ 1 - \text{Bernoulli}\left(\frac{-2Z}{\alpha + \beta}\right) & \text{If } H = 0 \text{ and } Z < 0 \end{cases} \] \[ \hat{Z} = \begin{cases} -\beta & \text{if } H = X = 0 \\ -\alpha & \text{if } H = 1 \text{ and } X = 0 \\ \alpha & \text{if } H = 0 \text{ and } X = 1 \\ \beta & \text{if } H = X = 1 \end{cases} \] For example, consider $Z = 1$, and recall that $H = 0$ w.p. $1/2$ and $H = 1$ otherwise. Then: - If $H = 0$, we have $X = 1$ and thus $\hat{Z} = \alpha$. - If $H = 1$, then $X = 1$ w.p. $\frac{2}{\alpha + \beta}$ and we get $\hat{Z} = \beta$. Otherwise (if $X = 0$), we get $\hat{Z} = -\alpha$. Indeed, we have that the estimate is unbiased since: \[ \mathbb{E}[\hat{Z} \| Z = 1] = \frac{1}{2} \cdot \alpha + \frac{1}{2} \cdot \left(\frac{2}{\alpha + \beta} \cdot \beta + \frac{\alpha + \beta - 2}{\alpha + \beta} \cdot (-\alpha)\right) = 1. \] We next calculate the expected squared error (by symmetry, we integrate over positive $z$): \[ \mathbb{E}\left[(Z - \hat{Z})^2\right] = \sqrt{\frac{2}{\pi}} \left(\int_0^{t_p} \frac{1}{2} \cdot \left((z - \alpha)^2 + \frac{2z}{\alpha + \beta} \cdot (z - \beta)^2 + \frac{\alpha + \beta - 2z}{\alpha + \beta} \cdot (z + \alpha)^2\right) \cdot e^{-z^2/2} dz\right) \approx 3.29. \] Observe that it is significantly lower than the 8.58 quantization error obtained without shared randomness. As we illustrate (Figure 2), the error further decreases when using more shared random bits. Accelerating QUIC-FL with RHT. Similarly to previous algorithms that use random rotations as a preprocessing state (e.g., Suresh et al. (2017); Vargaftik et al. (2021, 2022)) we propose to use the Randomized Hadamard Transform (RHT) (Ailon & Chazelle (2009)) instead of uniform random rotations. Although RHT does not induce a uniform distribution on the sphere, it is considerably more efficient to compute, and, under mild assumptions, the resulting distribution is close to that of a uniform random rotation (Vargaftik et al. (2021)). Nevertheless, we are interested in establishing how using RHT instead of a uniform random rotation affects the formal guarantees of QUIC-FL. As shown in Appendix G, QUIC-FL with RHT remains unbiased and has the same asymptotic guarantee as with random rotations, albeit with a larger constant (constant factor increases in the fraction of exactly sent values and NMSE). See also Appendix D for further discussion and references. We note that these guarantees are still stronger than those of DRIVE (Vargaftik et al. (2021) and EDEN (Vargaftik et al. (2022)), which only prove RHT bounds for vectors whose coordinates are sampled i.i.d. from a distribution with finite moments, and are not applicable to adversarial vectors. For example, when $p = 2^{-9}$ and we use $\ell = 4$ shared random bits per quantized coordinate, our analysis shows that the NMSE for $b = 1, 2, 3, 4$ is bounded by $4.831/n, 0.692/n, 0.131/n, 0.0272/n$, accordingly, and that the expected number of coordinates outside $[-t_p, t_p]$ is bounded by $3.2 \cdot p \cdot d \approx 0.006 \cdot d$. We note that this result does not have the $O(1/n \cdot \sqrt{\log d/d})$ Figure 2: The NMSE of QUIC-FL (with $ n = 256 $ clients) as a function of the bit budget $ b $, fraction $ p $, and shared random bits $ \ell $. In the leftmost figure, $ p = 2^{-9} $, while the other two use $ b = 4 $. Figure 3: Comparison to alternatives with $ n $ clients that have the same LogNormal(0, 1) input vector. The default values are $ n = 256 $ clients, $ b = 4 $ bit budget, and $ d = 2^{20} $ dimensions. additive NMSE term. The reason is that we directly analyze the error for the Hadamard-rotated coordinates (whereas Theorem [3,1] relies on analyzing the error in quantizing normal variables and factoring in the difference in distributions). In particular, we get that for $ p = 2^{-9}, b \in \{1, 2, 3\} $, running QUIC-FL with Hadamard and $ (b + 1 + 2.2 \cdot p) \approx b + 1.0043 $ bits per coordinate has lower NMSE than $ b $-bits QUIC-FL with uniform random rotation. That is, one can compensate for the increased error caused by using RHT by adding one bit per coordinate. In practice, as shown in the evaluation, the actual performance is (as one might expect) actually close to the theoretical results for uniform random rotations; improving the bounds is left as future work. Finally, Table 1 summarizes the theoretical guarantees of QUIC-FL in comparison to state-of-the-art DME techniques. The encoding complexity of QUIC-FL is dominated by RHT and is done in $ O(d \cdot \log d) $ time. The decoding of QUIC-FL only requires the addition of all estimated rotated clients’ vectors and a single inverse RHT transform resulting in $ O(n \cdot d + d \cdot \log d) $ time. As mentioned, the NMSE with RHT remains $ O(1/n) $. Observe that QUIC-FL has an asymptotic speed improvement either at the clients or the server among the techniques that achieve $ O(1/n) $ NMSE. A lower bound on the continuous problem. QUIC-FL obtains a solution for the above problem via the discretization of the distribution and shared randomness. To obtain a lower bound on the vNMSE of the continuous problem, we can use the Lloyd-Max quantizer, which finds the optimal biased quantization for a given distribution. In particular, we get that the optimal (non-discrete) vNMSE is at least 0.35, 0.11, 0.031, 0.0082 for $ b = 1, 2, 3, 4 $, accordingly. Compared to unbiased QUIC-FL’s vNMSE of 1.52, 0.223, 0.044, 0.0098. Note that as $ b $ grows, QUIC-FL’s vNMSE quickly approaches the Lloyd-Max lower bound for biased quantization. 4 EVALUATION In this section, we evaluate the fully-fledged version of QUIC-FL that leverages RHT and client-specific shared randomness, as given in Appendix E and Algorithm 3. Parameter selection. We experiment with how the different parameters (number of quantiles $ m $, the fraction of coordinates sent exactly $ p $, the number of shared random bits $ \ell $, etc.) affect the performance of our algorithm. As shown in Figure 2, introducing shared randomness significantly decreases the NMSE compared with Algorithm 1 (i.e., $ \ell = 0 $). We note that these results are essentially independent of the input data (because of the RHT). Additionally, the benefit from adding each additional shared random bit diminishes, and the gain beyond $ \ell = 4 $ is negligible, especially for large $ b $. Accordingly, we hereafter use $ \ell = 6 $ for $ b = 1 $, $ \ell = 5 $ for $ b = 2 $, and $ \ell = 4 $ for $ b \in \{3, 4\} $. With respect to $ p $, we determined $ 1/512 $ as a good balance between the NMSE and bandwidth overhead for accurately sent values and their indices. Comparison to state-of-the-art DME techniques. Next, we compare the performance of QUIC-FL to the baseline algorithms in terms of NMSE, encoding speed, and decoding speed, using an NVIDIA 3080 RTX GPU machine with 32GB RAM and i7-10700K CPU @ 3.80GHz. Specifically, we compare with inputs where each coordinate is independently $\text{LogNormal}(0, 1)$ \citep{chmiel2020hadamard,suresh2017kashin}. Kashin’s representation \citep{caldas2018fedavg,safaryan2020fedavg}, QSGD \citep{alistarh2017qsgd}, and EDEN \citep{vargaftik2022eden}. We evaluate two variants of Kashin’s representation: (1) The TensorFlow (TF) implementation \citep{tensorflow} that, by default, limits the decomposition to three iterations, and (2) the theoretical algorithm that requires $O(\log(n \cdot d))$ iterations. For this experiment, the coordinates are $A$s shown in Figure 3. QUIC-FL has significantly faster decoding than EDEN (as previously conveyed in Figure 1), the only alternative with competitive NMSE. QUIC-FL is also significantly more accurate than all other approaches. We observe that the default TF configuration of Kashin’s representation suffers from a bias, and therefore its NMSE is not $O(1/n)$. In contrast, the theoretical algorithm is unbiased but has an asymptotically slower encoding time. We observed similar trends for different $n$, $b$, and $d$ values. We consider the algorithms’ bandwidth over all coordinates (i.e., with $b + \frac{64}{572}$ bits for QUIC-FL, namely a float and a 32-bit index for each accurately sent entry). We evaluate the algorithms on additional input distributions and report similar results in Appendix H. Overall, the empirical measurements fall in line with the bounds in Table 1. ![Figure 4: FedAvg over the Shakespeare next-word prediction task at various bit budgets (rows). We report training accuracy per round with a rolling mean of 200 rounds.](image) Federated Learning Experiments. We evaluate QUIC-FL over the Shakespeare next-word prediction task \citep{mcmahan2017communication} using an LSTM recurrent model. It was first suggested in \citep{mcmahan2017communication} to naturally simulate a realistic heterogeneous federated learning setting. We run FedAvg \citep{mcmahan2017communication} with the Adam server optimizer \citep{kingma2015adam} and sample $n = 10$ clients per round. We use the setup from the federated learning benchmark of \citep{reddi2021adaptive}, restated for convenience in Appendix I. Figure 4 shows how QUIC-FL is competitive with the asymptotically slower EDEN and markedly more accurate than other alternatives. Due to space limits, experiments for image classification (Appendix J.1), a framework that uses DME as a building block (Appendix J.2), and power iteration (Appendix J.3), appear in the appendix. ## 5 RELATED WORKS In Section 1, we gave an extensive overview of most related works, namely, other DME methods. In Appendix B, we give a broader overview of other compression and acceleration techniques, including frameworks that use DME as a building block; bounded support quantization alternatives; distribution-aware quantization; Entropy encoding techniques; methods that use client-side memory; error feedback solutions; opportunities in aggregating things other than gradients (such as gradient differences); in-network aggregation; sparsification approaches; shared randomness applications; non-uniform quantization; improvements by leveraging gradient correlations; and privacy concerns. ## 6 LIMITATIONS We view the main limitation of QUIC-FL as its inability to leverage structure in the gradient (e.g., correlations across coordinates). While some structure (e.g., sparsity) is extractable (e.g., by encoding just the non-zero coordinates and separately encoding the coordinate positions that are zero), other types of structure may be ruined by applying RHT. 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5T46w5X3Go	The major concern I have is that the paper can separate the common features and task-specific features explicitly. In other words, in Option A and Option B, we can directly know that $w$ is a common feature, while $q$ is not. In practice, we do not have any idea about which part of the feature should be useful and we should try to figure out the common features and task-specific features rather than assuming we know that.	THEORETICAL ANALYSIS ON THE GENERALIZATION POWER OF OVERFITTED TRANSFER LEARNING Anonymous authors Paper under double-blind review ABSTRACT Transfer learning is a useful technique for achieving improved performance and reducing training costs by leveraging the knowledge gained from source tasks and applying it to target tasks. Assessing the effectiveness of transfer learning relies on understanding the similarity between the ground truth of the source and target tasks. In real-world applications, tasks often exhibit partial similarity, where certain aspects are similar while others are different or irrelevant. To investigate the impact of partial similarity on transfer learning performance, we focus on a linear regression model with two distinct sets of features: a common part shared across tasks and a task-specific part. Our study explores various types of transfer learning, encompassing two options for parameter transfer. By establishing a theoretical characterization on the error of the learned model, we compare these transfer learning options, particularly examining how generalization performance changes with the number of features/parameters in both underparameterized and overparameterized regimes. Furthermore, we provide practical guidelines for determining the number of features in the common and task-specific parts for improved generalization performance. For example, when the total number of features in the source task’s learning model is fixed, we show that it is more advantageous to allocate a greater number of redundant features to the task-specific part rather than the common part. Moreover, in specific scenarios, particularly those characterized by high noise levels and small true parameters, sacrificing certain true features in the common part in favor of employing more redundant features in the task-specific part can yield notable benefits. 1 INTRODUCTION Transfer learning is a powerful technique that enhances the learning performance of a target task by leveraging knowledge from a related source task (Pan & Yang, 2010). There are two main categories of transfer learning: parameter transfer and sample transfer. In parameter transfer, the learned parameters from the source task are directly copied to the target task’s learning model. In sample transfer, training samples from the source task are integrated into the target task’s dataset and contribute to its training process. Comparing these two methods, sample transfer can provide additional valuable information and allow for preprocessing of the transferred samples to better align them with the target task, while parameter transfer offers significant savings in training costs and thus is very helpful for models with a large number of parameters such as deep neural networks (DNNs). Despite the proven effectiveness of transfer learning with DNNs in various real-world applications, a comprehensive theoretical understanding of its performance remains under-explored. DNNs are typically overparameterized, allowing them to fit all training samples while maintaining relatively good generalization performance. This behavior challenges our understanding of the classical bias-variance trade-off. Recent studies have explored the phenomenon of “double-descent” or “benign overfitting” in certain linear regression setups, where the test error descends again in the overparameterized region, shedding light on this mystery. However, most of the existing literature focuses on single-task learning. The existence of a similar phenomenon in transfer learning, even in the simple linear regression setting, remains insufficiently explored. The additional transfer process in transfer learning makes the analysis of the generalization performance in the underparameterized and overparameterized regimes considerably more complex. Furthermore, quantifying task similarity necessitates the development of appropriate analytical methods to establish a connection with the generalization performance of transfer learning. The contribution of this paper is as follows. In this paper, we investigate the generalization performance of transfer learning in linear regression models under both the underparameterized and overparameterized regimes. Compared to the existing literature that considers a general noisy linear relation between the true parameters of the source and target tasks, we delve into the separation between common and task-specific features in greater detail. Specifically, we partition the feature space into a common part and a task-specific part. This setup enables us to analyze how the number of parameters in different parts influences the generalization performance of the target task. By characterizing the generalization performance, we offer insightful findings on transfer learning. For instance, when the total number of features in the source task’s learning model is fixed, our analysis reveals the advantage of allocating more redundant features to the task-specific part rather than the common part. Additionally, in specific scenarios characterized by high noise levels and small true parameters, sacrificing certain true features in the common part in favor of employing more redundant features in the task-specific part can yield notable benefits. 1.1 Related Work “Benign overfitting” and “double-descent” have been discovered and studied for overfitted solutions in single-task linear regression. Some works have explored double-descent with minimum $\ell_2$-norm overfitted solutions (Belkin et al., 2018; 2019; Bartlett et al., 2020; Hastie et al., 2019; Muthukumar et al., 2019) or minimum $\ell_1$-norm overfitted solutions (Mitra, 2019; Ju et al., 2020), while employing simple features such as Gaussian or Fourier features. In recent years, other studies have investigated overfitted generalization performance by utilizing features that approximate shallow neural networks. For example, researchers have explored random feature (RF) models (Mei & Montanari, 2019), two-layer neural tangent kernel (NTK) models (Arora et al., 2019; Satpathi & Srikant, 2021; Ju et al., 2021), and three-layer NTK models (Ju et al., 2022). Note that all of these studies have focused solely on a single task. There are only a limited number of studies on the theoretical analysis of transfer learning. Lampinen & Ganguli (2019) investigate the generalization dynamics in transfer learning by multilayer linear networks using a student-teacher scenario where the teacher network generates data for the student network, which is different from our setup where the data of the source task and the target task are independently generated by their own ground truth. Dhifallah & Lu (2021) focus on the problem of when transfer learning is beneficial using the model of the single-layer perceptron. Gerace et al. (2022) study a binary classification problem by transfer learning of the first layer in a two-layer neural network. However, both Dhifallah & Lu (2021) and Gerace et al. (2022) include an explicit regularization term in their models, which prevents overfitting. There are also some recent studies of transfer learning on linear models (Bastani, 2021; Li et al., 2022; Tian & Feng, 2022; Li et al., 2023; Tripuraneni et al., 2020; Zhang et al., 2022; Lin & Reimherr, 2022). For example, Bastani (2021) and Li et al. (2022) investigate estimation and prediction in high-dimensional linear models. Tian & Feng (2022) and Li et al. (2023) further extend the setup to high-dimensional generalized linear models. Tripuraneni et al. (2020) consider the case where source and target tasks share a common and low-dimensional linear representation. Lin & Reimherr (2022) study transfer learning in a functional linear regression where the similarity between source and target tasks is measured using the Reproducing Kernel Hilbert Spaces norm. Zhang et al. (2022) provide minimax bounds on the generalization performance but do not overfit the training data. In particular, none of these studies have considered the task similarity structure of interest in this paper, nor investigated the generalization performance in both overparameterized and underparameterized regimes. The most related work to ours is Dar & Baraniuk (2022). Specifically, Dar & Baraniuk (2022) studies the double descent phenomenon in transfer learning, which is also our focus in this paper. However, Dar & Baraniuk (2022) does not consider an explicit separation of the feature space by the common part and the task-specific part like we do in this paper. As we will show, such a separation in the system model enables us to analyze the double descent phenomenon under different options for transfer learning, including two options for parameter transfer and two options for data transfer. In contrast, Dar & Baraniuk (2022) only studies one option of parameter transfer. Therefore, our analysis is quite different from that of Dar & Baraniuk (2022). 2 SYSTEM MODEL 2.1 LINEAR GROUND TRUTH INVOLVING MULTIPLE TASKS In a classical single-task linear regression, ground truth parameters are treated as one vector, and all corresponding features (each feature is a scalar) are also treated as one vector. However, when involving multiple tasks, due to the partial similarity among different tasks, using only one vector to represent the ground truth parameters and features is no longer enough. A finer linear model should consider the common part and the task-specific part separately. Here we consider one training (source) task and one test (target) task, respectively referred to as the first and second task from now on. We consider a linear model for each task; i.e., for the $i$-th task with $i \in \{1 \text{ (source)}, 2 \text{ (target)}\}$, samples are generated by $$y(i) = x^T w(i) + z^T q(i) + \epsilon(i),$$ (1) where $\hat{x} \in \mathbb{R}^s$ denotes the value of the features that correspond to the similar/common parameters $w(i) \in \mathbb{R}^s$, $\hat{z}(i) \in \mathbb{R}^{s(i)}$ denotes the value of the features that correspond to the task-specific parts $q(i) \in \mathbb{R}^{s(i)}$, and $\epsilon(i) \in \mathbb{R}$ denotes the noise. Here, $s$ denotes the number of common features and $s(i)$ denotes the number of $i$-th task-specific features. Let $\hat{S}(i)$ denote the set of features corresponding to $\hat{z}(i)$ and $\hat{S}_{co}$, the set of features corresponding to $\hat{x}$ (so their cardinality $\|\hat{S}(i)\| = s_i$ and $\|\hat{S}_{co}\| = s$). Representative motivating example: In real-world applications, many tasks actually have such a partial similarity structure. For example, for image recognition tasks, some low-level features are common (e.g., skin texture of animals, surface of a machine) among different tasks even if the objectives of those tasks are completely different (e.g., classifying cat and airplane, or classifying dog and automobile). These low-level features are usually captured by convolutional layers in DNNs, while the remaining parts of the DNNs (e.g., full-connected layers) are used to extract task-specific features. Even for a simple linear regression model, a theoretical explanation of the effect of common features and task-specific features on the generalization performance of transfer learning may provide useful insights on designing more suitable real-world transfer learning model structures (e.g., how many neurons to use in convolutional layers of DNNs to extract common low-level features to transfer). 2.2 FEATURE SELECTION FOR LEARNING From the learner’s point of view, the true feature sets $\hat{S}_{co}$ and $\hat{S}(i)$ are usually unknown for many real-world applications. In the overparameterized regime, redundant parameters (along with redundant features) are used/selected more than necessary, which is characterized by the following assumption. Choosing redundant features also means that the learner does not need to be very precise in distinguishing the common and task-specific features, since the learner can include “suspicious” features in the common feature set. Definition 1. $\hat{S}_{co} \subseteq S_{co}$ and $\hat{S}(i) \subseteq S(i)$ for all $i \in \{1, 2\}$, where $S_{co}$ denotes the set of selected features for the common part, and $S(i)$ denotes the set of selected task-specific features. Define $p := \|S_{co}\|$ and $p(i) := \|S(i)\|$. Let $\tilde{w} \in \mathbb{R}^p$ denote the parameters to learn the common part and $q(i) \in \mathbb{R}^{p(i)}$ the parameters to learn the $i$-th task’s specific part. With Definition 1, we construct $w(i) \in \mathbb{R}^p$ (corresponding to $S_{co}$) from $\tilde{w}(i)$ (corresponding to $\hat{S}_{co}$) by filling zeros in the positions of the redundant features (corresponding to $S_{co} \setminus \hat{S}_{co}$). We similarly construct $q(i) \in \mathbb{R}^{p(i)}$ from $\tilde{q}(i)$. Thus, Eq. (1) can be alternatively expressed as $$y(i) = x^T w(i) + z^T q(i) + \epsilon(i),$$ (2) where $x \in \mathbb{R}^p$ are the features of $\hat{S}_{co}$ and $z(i) \in \mathbb{R}^{p(i)}$ are the features of $S(i)$. Notice that the ground truth (i.e., input and output) does not change with $p$ or $p(i)$ (since it only changes how many additional zeros are added). For analytical tractability, we adopt Gaussian features and noise, which is formally stated by the following assumption. Assumption 1. All features follow i.i.d.$^1$ standard Gaussian $N(0, 1)$. The noise also follows the Gaussian distribution. Specifically, $\epsilon_{(1)} \sim N\left(0, \sigma^2_{(1)}\right)$ and $\epsilon_{(2)} \sim N\left(0, \sigma^2_{(2)}\right)$. Remark 1. If there exist some missing features$^2$ in $S_{co}$ and $S_{(i)}$ (i.e., Definition 1 is not satisfied), then the effect of these missing features is the same as the noise since we adopt i.i.d. Gaussian features. Thus, our methods and results still hold by redefining $\sigma^2_{(1)}$ and $\sigma^2_{(2)}$ as the total power of the noise and the missing features, i.e., $\sigma^2_{(i)} \leftarrow \sigma^2_{(i)} + \\|\hat{w}^{missing}_{(i)}\\|^2 + \\|\hat{q}^{missing}_{(i)}\\|^2$ where $\hat{w}^{missing}_{(i)}$ and $\hat{q}^{missing}_{(i)}$ denote the sub-vectors for the missing features of $\hat{w}_{(i)}$ and $\hat{q}_{(i)}$, respectively. 2.3 Training samples and training losses Let $n_{(i)}$ denote the number of training samples for task $i \in \{1, 2\}$. We stack these $n_{(i)}$ samples as matrices/vectors $X_{(i)} \in \mathbb{R}^{p_{(i)} \times n_{(i)}}$, $Z_{(i)} \in \mathbb{R}^{p_{(i)} \times n_{(i)}}$, $y_{(i)} \in \mathbb{R}^{n_{(i)}}$, where the $j$-th column of $X_{(i)}$, the $j$-th column of $Z_{(i)}$, and the $j$-th element of $y_{(i)}$ correspond to $(x, z_{(i)}, y_{(i)})$ in Eq. (2) of the $j$-th training sample. Now Eq. (2) can be written into a matrix equation for training samples: \[ y_{(i)} = X_{(i)}^T w_{(i)} + Z_{(i)}^T q_{(i)} + \epsilon_{(i)}, \] where $\epsilon_{(i)} \in \mathbb{R}^{n_{(i)}}$ is the stacked vector that consists of the noise in the output of each training sample (i.e., $\epsilon_{(i)}$ in Eq. (2)). We use mean squared error (MSE) as the training loss for the $i$-th task with the learner’s parameters $\tilde{w}, \tilde{q}$ as: \[ L_{(i)}^{\text{train}}(\tilde{w}, \tilde{q}) := \frac{1}{n_{(i)}} \\|y_{(i)} - X_{(i)}^T \tilde{w} - Z_{(i)}^T \tilde{q}\\|^2. \] 2.4 Options of parameter transfer The process of transfer learning by transferring parameters consists of three steps: step 1) train for the source task using samples $(X_{(1)}, Z_{(1)}; y_{(1)})$; step 2) select the parameters for the common features $S_{co}$ from the learned result of the source task and then send them to the target task model; and step 3) determine/train the parameters for the target task using its own samples $(X_{(2)}, Z_{(2)}; y_{(2)})$ based on the transferred parameters in step 2. Step 1 is similar to a classical single-task linear regression. The training process will converge to a solution $\tilde{w}_{(1)}, \tilde{q}_{(1)}$ that minimizes this training loss, i.e., $(\tilde{w}_{(1)}, \tilde{q}_{(1)}) := \arg \min_{\tilde{w}, \tilde{q}} L_{(1)}^{\text{train}}(\tilde{w}, \tilde{q})$. When $p + p_{(1)} > n_{(1)}$ (overparameterized), there exist multiple solutions that can make the training loss zero (with probability 1). In this situation, we will choose the one with the smallest $\ell_2$-norm $(\tilde{w}_{(1)}, \tilde{q}_{(1)})$ which is defined as the solution of the following optimization problem: \[ \min_{\tilde{w}, \tilde{q}} \\|\tilde{w}\\|^2 + \\|\tilde{q}\\|^2 \quad \text{subject to} \quad X_{(1)}^T \tilde{w} + Z_{(1)}^T \tilde{q} = y_{(1)}. \] We are interested in this minimum $\ell_2$-norm solution among all overfitted solutions because it corresponds to the convergence point of stochastic gradient descent (SGD) or gradient descent (GD) training with zero initial point (see proof in Lemma 5). Steps 2 and 3 jointly determine the learned result for the target task $\tilde{w}_{(2)}$ and $\tilde{q}_{(2)}$. In this paper, we analyze two possible options differentiated by the usage of the transferred common part $\tilde{w}_{(1)}$. Option A (Transfer and Fix): We directly copy the learned result, i.e., $\tilde{w}_{(2)} := \tilde{w}_{(1)}$. For the training of the target task, only the task-specific parameters are trained. In other words, $\tilde{q}_{(2)} := \arg \min_{\tilde{q}} L_{(2)}^{\text{train}}(\tilde{w}, \tilde{q})$ when underparameterized. When $p_{(i)} > n_{(2)}$ (overparameterized), there exist multiple solutions that can make the training loss zero. We then define $\tilde{q}_{(2)}$ as the minimum $\ell_2$-norm overfitted solution, i.e., $\tilde{q}_{(2)}$ is defined as the solution of the following optimization problem: \[ \min_{\tilde{q}} \\|\tilde{q}\\|^2 \quad \text{subject to} \quad X_{(2)}^T \tilde{w}_{(1)} + Z_{(2)}^T \tilde{q} = y_{(2)}. \] Option B (Transfer and Train): We only use the learned common part as an initial training point of $\tilde{w}_{(2)}$. In this option, both $\tilde{w}_{(2)}$ and $\tilde{q}_{(2)}$ are determined by the training of the source task. Specifically, $(\tilde{w}_{(2)}, \tilde{q}_{(2)}) := \arg \min_{\tilde{w}, \tilde{q}} L_{(2)}^{\text{train}}(\tilde{w}, \tilde{q})$ when underparameterized. When $p + p_{(2)} > n_{(2)}$, --- $^1$In Appendix F, we numerically check our results and insights in the situation of non-i.i.d. settings. $^2$A missing feature means that a true feature is not included in the data. there are multiple solutions that can make $ \mathcal{L}_{(2)}(\tilde{w}, \tilde{q}) = 0 $. We then define $(\tilde{w}_{(2)}, \tilde{q}_{(2)})$ as the convergence point of SGD/GD starting from $(\tilde{w} = \tilde{w}_{(1)}, \tilde{q} = 0)$. Indeed, $(\tilde{w}_{(2)}, \tilde{q}_{(2)})$ corresponds to the smallest $ \ell_2 $-norm of the difference between the result and the initial point (see proof in Lemma 5): \[ \min_{\tilde{w}, \tilde{q}} \quad \\| \tilde{w} - \tilde{w}_{(1)} \\|^2 + \\| \tilde{q} \\|^2 \quad \text{subject to} \quad X_{(2)}^T \tilde{w} + Z_{(2)}^T \tilde{q} = y_{(2)}. \] ### 2.5 Performance Evaluation We define the model error for the target task as \[ \mathcal{L} := \\| \tilde{w}_{(2)} - w_{(2)} \\|^2 + \\| \tilde{q}_{(2)} - q_{(2)} \\|^2. \] (4) It can be proven that the model error $ \mathcal{L} $ is the expected test loss on noiseless test samples. To make our results in the following sections concise, we define \[ \mathcal{L}_{co} := \mathbb{E}_{x_{(1)}, z_{(1)}, e_{(1)}} \\| w_{(2)} - \tilde{w}_{(1)} \\|^2 \quad \text{(transferring error)}, \] (5) \[ \mathcal{L}_{co}^{\text{noiseless}} := \mathcal{L}_{co} \|_{\sigma_{(1)} = 0} \quad \text{(transferring error when } \sigma_{(1)} = 0), \] (6) \[ \delta := \\| w_{(2)} - w_{(1)} \\| \quad \text{(similarity on common features)}, \] (7) \[ r := 1 - \frac{n_{(1)}}{p + p_{(1)}} \quad \text{(overparameterized ratio in step 1)}. \] Intuitively, $ \mathcal{L}_{co} $ describes how well the common part learned from the source task estimates the target task’s common part, $ \delta $ reflects the similarity between the common parts of the source task and the target task, and $ r $ can be regarded as the overparameterization ratio in step 1 introduced in Section 2.4. ### 3 Main Results for Parameter Transfer For the scheme of transferring parameters (Section 2.4), we will establish three theorems corresponding to the performance of the transferring error$^3$, the model error of Option A, and the model error of Option B, respectively. Theorem 1 (transferring error). The transferring error (defined in Eq. (5)) is given by \[ \mathcal{L}_{co} = \begin{cases} \mathcal{L}_{co}^{\text{noiseless}} + b_{\text{noise}}, & \text{for } p + p_{(1)} > n_{(1)} + 1, \\ \delta^2 + \frac{p \sigma_{(1)}^2}{n_{(1)} - (p + p_{(1)}) - 1}, & \text{for } n_{(1)} > p + p_{(1)} + 1, \end{cases} \] (8) where $ 0 \leq \mathcal{L}_{co}^{\text{noiseless}} \leq \min_{i=1,2,3} \bar{b}_i^2 $, and \[ \bar{b}_1 := \delta + \sqrt{r (\\| w_{(1)} \\|^2 + \\| q_{(1)} \\|^2)}, \] (10) \[ \bar{b}_2 := \\| w_{(2)} \\| + \sqrt{1 - r} \\| w_{(1)} \\| + \sqrt{\min\{r, 1 - r\}} \\| q_{(1)} \\|, \] (11) \[ \bar{b}_3 := \sqrt{r} \\| w_{(1)} \\| + \delta + \sqrt{\min\{r, 1 - r\}} \\| q_{(1)} \\|, \] (12) \[ b_{\text{noise}} := \frac{p}{p + p_{(1)}} \cdot \frac{n_{(1)} \sigma_{(1)}^2}{p + p_{(1)} - n_{(1)} - 1}. \] (13) Theorem 2 (Option A). For Option A, we must have \[ \mathbb{E}[\mathcal{L}] = \begin{cases} \mathcal{L}_{co} + \frac{n_{(2)}}{p_{(2)} - n_{(2)} - 1} \left( \mathcal{L}_{co} + \sigma_{(2)}^2 \right) + \left( 1 - \frac{n_{(2)}}{p_{(2)}} \right) \\| q_{(2)} \\|^2, & \text{for } p_{(2)} > n_{(2)} + 1, \\ \mathcal{L}_{co} + \frac{p_{(2)}}{n_{(2)} - p_{(2)} - 1} \left( \mathcal{L}_{co} + \sigma_{(2)}^2 \right), & \text{for } n_{(2)} > p_{(2)} + 1. \end{cases} \] (14) (15) $^3$The error caused by the transferred parameters. The precise definition is given in Eq. (5). Figure 1: Generalization performance of transfer learning under different setups, where $ s = s_1 = s_2 = 5 $, $ n(1) = 100 $, $ n(2) = 50 $, $ w(1) = w(2) $. Each point is the average of 100 random runs. Other settings for each subfigure: (a) $ \\|q(2)\\| = \\|w(1)\\| = 1 $; (b) $ \\|w(1)\\| = \\|q(1)\\| = 1 $; (c) $ \\|\sigma(1)\\| = 1 $, $ \\|\sigma(2)\\| = 0.2 $, $ \\|q(1)\\| = \\|q(2)\\| = \\|w(1)\\| = 0.1 $. Theorem 3 (Option B). For Option B, we must have \[ E[L] = \begin{cases} \left(1 - \frac{n(2)}{p + p(2)}\right) \left(\mathcal{L}_{co} + \\|q(2)\\|^2\right) + \frac{n(2)\sigma^2(2)}{p + p(2) - n(2) - 1}, & \text{for } p + p(2) > n(2) + 1, \\ \frac{(p + p(2))\sigma^2(2)}{n(2) - (p + p(2)) - 1}, & \text{for } n(2) > p + p(2) + 1. \end{cases} \] The proofs of Theorems 1 to 3 are given in Appendices B to D, respectively. Theorems 1 to 3 provide some interesting insights, which we now discuss in Sections 3.1 to 3.3. ### 3.1 Common Insights for Options A and B (1) Benign overfitting $^4$ w.r.t. $ p(1) $ needs large $ \sigma(1) $. For the overparameterized regime result in Eq. (8) of Theorem 1, when $ \sigma(1) $ is large, the term $ b_{noise} $ (defined in Eq. (13)) dominates $ \mathcal{L}_{co} $ and is monotone decreasing w.r.t. $ p(1) $. When $ p(1) \to \infty $, we have $ b_{noise} \to 0 $. In contrast, for the underparameterized regime result in Eq. (9), Term O1 (noise effect) is always larger than $ \frac{p\sigma^2(1)}{n(1)} $, which can be worse than that of the overparameterized regime when $ p(1) $ is sufficiently large. By Theorems 2 and 3, we know that $ L $ decreases when $ \mathcal{L}_{co} $ decreases. Therefore, in the situation of large $ \sigma(1) $, increasing $ p(1) $ in the overparameterized regime (of step 1) w.r.t. $ p(1) $ can reduce the generalization error, which implies the existence of benign overfitting. We also numerically verify the impact of $ \sigma(1) $ on the benign overfitting in Fig. 1(a), where we plot the empirical average of $ L $ w.r.t. $ p(1) $. The two curves of $ \sigma(1) = 3 $ with markers “×” descend in the $^4$i.e., the test error of the overparameterized regime is lower than that of the underparameterized regime. overparameterized regime ($p_{(1)} > 80$) and can be lower than their values in the underparameterized regime. In contrast, the two curves of $\sigma_{(1)} = 0.1$ with markers “+” increase in most parts of the overparameterized regime and are higher than the underparameterized regime. Such a contrast indicates the benign overfitting w.r.t. $p_{(1)}$ needs large $\sigma_{(1)}$. (2) Benign overfitting w.r.t. $p_{(2)}$ needs large $\sigma_{(2)}$. For Eq. (15) (underparameterized regime of Option A), $\mathbb{E}[L]$ is always larger than $L_{co}(1 + \frac{1}{n_{(2)}})$. In contrast, for Eq. (14) (overparameterized regime of Option A), when $\sigma_{(2)}$ is much larger than $\\|q_{(2)}\\|^2$, then Term A2 is negligible and Term A1 dominates. In this situation, $\mathbb{E}[L]$ is monotone decreasing w.r.t. $p_{(2)}$ and will approach $L_{co}$ when $p_{(2)} \to \infty$. In other words, benign overfitting exists. Similarly, by Theorem 3, benign overfitting exists when $\sigma_{(2)}^2$ is much larger than $L_{co} + \\|q_{(2)}\\|^2$. In Fig. 1(b), the two curves with markers “∇” denote the model error of Option A and Option B when $\sigma_{(2)}$ is large ($\sigma_{(2)} = 2$). They have a descending trend in the entire overparameterized regime. In contrast, the two curves with markers “+”, which denote the model error for the situation of small $\sigma_{(2)}$ ($\sigma_{(2)} = 0.2$), only decrease w.r.t. $p_{(2)}$ at the beginning of the overparameterized regime, while increasing thereafter. (3) A descent floor$^5$ w.r.t. $p_{(2)}$ sometimes exists. For Eq. (14) of Option A, Term A1 is monotone decreasing w.r.t. $p_{(2)}$, while Term A2 is monotone increasing w.r.t. $p_{(2)}$. When $p_{(2)}$ is a little larger than $n_{(2)}$, the denominator $p_{(2)} - n_{(2)} - 1$ in Term A1 is close to zero, and thus Term A1 dominates and causes $\mathbb{E}[L]$ to be decreasing w.r.t. $p_{(2)}$. When $p_{(2)}$ gradually increases to infinity, $\mathbb{E}[L]$ will approach $L_{co} + \\|q_{(2)}\\|^2$. By calculating $\partial \mathbb{E}[L]/\partial p_{(2)}$, we can tell that if $L_{co} + \sigma_{(2)}^2 < \\|q_{(2)}\\|^2$, in the overparameterized regime, $\mathbb{E}[L]$ will first decrease and then increase, which implies a descent floor (by Lemma 9 in Appendix A.1). Similarly, by calculating $\partial \mathbb{E}[L]/\partial p_{(2)}$ for Eq. (16) of Option B, if $\sigma_{(2)}^2 < L_{co} + \\|q_{(2)}\\|^2$, in the overparameterized regime, $\mathbb{E}[L]$ will have a descent floor w.r.t. $p_{(2)}$ (by Lemma 10 in Appendix A.1). An interesting observation related to the descent floor is that the condition of the existence of the descent floor is different for Option A and Option B, where Option A needs small $L_{co}$ but Option B needs large $L_{co}$. In Fig. 1(b), we see that both curves with markers “+” have a descent floor in the overparameterized regime. In contrast, for the two curves with markers “×” where $\sigma_{(1)}$ is large, only Option B has a descent floor while Option A does not. Since large $\sigma_{(1)}$ implies large $L_{co}$, such a difference confirms that the descent floor of Option A needs small $L_{co}$ while the one of Option B needs large $L_{co}$. (4) The effect of $q_{(1)}$ is negligible when heavily or slightly overparameterized in step 1. The effect of $q_{(1)}$ on $L$ is through $L_{noiseless}$. By Eqs. (8) and (10) to (12), the coefficient of $\\|q_{(2)}\\|$ is $\min\{r, 1 - r\}$. When heavily overparameterized in step 1, we have $p + p_{(1)} \gg n_{(1)}$ and thus $r \approx 0$. When slightly overparameterized in step 1, we have $p + p_{(1)} \approx n_{(1)}$ and thus $r \approx 1$. In both situations, we have the coefficient $\min\{r, 1 - r\} \approx 0$, which implies that the effect of $q_{(1)}$ is negligible when heavily or slightly overparameterized in step 1. In Fig. 1(a), we compare two curves with markers “△” (for large $q_{(1)}$ that $\\|q_{(1)}\\| = 5$) against two curves with markers “+” (for small $q_{(1)}$ that $\\|q_{(1)}\\| = 1$). We observe for both Option A and Option B that the curves with markers “△” overlap the curves with markers “+” at the beginning and the latter part of the overparameterized regime. This phenomenon validates the implication (4) which is inferred from the factor $\min\{r, 1 - r\}$ in Eqs. (11) and (12). 3.2 Insights for Option A (A1) Benign overfitting w.r.t. $p_{(2)}$ is easier to observe with small knowledge transfer. In the underparameterized regime, by Eq. (15), $\mathbb{E}[L]$ is at least $L_{co} + \frac{L_{co} + \sigma_{(2)}^2}{n_{(2)}}$. In contrast, for the overparameterized regime, when $L_{co}$ is large, Term A1 of Eq. (14) dominates $\mathbb{E}[L]$. When $p_{(2)}$ increases to $\infty$, Term A1 will decrease to $L_{co}$. Notice that large $L_{co}$ implies small knowledge transfer from the source task to the target task. Thus, benign overfitting w.r.t. $p_{(2)}$ appears when knowledge transfer is small. $^5$i.e., the descent of the test error stops at a certain point (which is like a floor) In Fig. 1(b), we let the ground-truth parameters be very small compared with the noise level, so the error $ L $ in Fig. 1 is mainly from noise. The blue curve with markers “×” has larger $ \sigma_{(1)} $ (with $ \sigma_{(1)} = 3 $) compared with the blue curve with markers “∇” (with $ \sigma_{(1)} = 0.1 $), and consequently, larger $ L_{co} $ and smaller knowledge transfer. We observe from Fig. 1(b) that the blue curve with markers “×” descends w.r.t. $ p_{(2)} $ in the entire overparameterized regime, while the blue curve with markers “∇” descends at the beginning of the overparameterized regime and ascends in the remainder of the overparameterized regime. Such a phenomenon validates the insight (A1). (A2) Larger $ p $ is not always good to reduce the noise effect when overparameterized. By Theorems 1 and 2, we know that the direct effect of $ p $ on noise in the overparameterized regime is only through the term $ b_{noise} $ in $ L_{co} $. By checking the sign of $ \frac{\partial b_{noise}}{\partial p} $, we can prove that $ b_{noise} $ increases w.r.t. $ p $ when $ p^2 < p_{(1)}(p_{(1)} - n_{(1)} - 1) $, and decreases when $ p^2 > p_{(1)}(p_{(1)} - n_{(1)} - 1) $ (see calculation details in Lemma 11 in Appendix A.1). In Fig. 1(c), the blue curve with markers “∇” depicts how the model error $ L $ of Option A changes w.r.t. $ p $ in the overparameterized regime ($ p + p_{(1)} > n_{(1)} $). This curve first increases and then decreases, which validates the insight (A2). 3.3 Insights for Option B (B1) Benign overfitting w.r.t. $ p_{(2)} $ is easier to observe with large knowledge transfer and small target task-specific parameters. In Eq. (16), small $ L_{co} + \\| q_{(2)} \\|^2 $ implies that Term B2 dominates the value of $ E[L] $. As we explained previously in (2) of Section 3.1, benign overfitting exists in this situation. Meanwhile, small $ L_{co} $ and $ \\| q_{(2)} \\| $ imply large knowledge transfer and small target task-specific parameters, respectively. In Fig. 1(b), the orange curve with markers “◇” denotes the model error $ L $ of Option B w.r.t. $ p_{(2)} $ when $ \sigma_{(1)} $ and $ q_{(2)} $ are small, i.e., large knowledge transfer and small target task-specific parameters. Compared with the orange curve with markers “×”, this curve descends in the entire overparameterized regime and can achieve a lower value than that of the underparameterized regime. This phenomenon validates the insight (B1). (B2) Multiple descents of noise effect when increasing $ p $ in the overparameterized regime. Different from Option A where $ p $ only affects the consequence of the noise in the source task (since no $ p $ appears in Eq. (14) except $ L_{co} $), for Eq. (16) of Option B, we see that $ p $ not only affects $ L_{co} $ but also Term B2, which implies that $ p $ relates to the noise effect in both the source task and the target task. Specifically, the trend of $ E[L] $ w.r.t. $ p $ is determined by $ (1 - \frac{n_{(2)}}{p+p_{(2)}})b_{noise} $ and Term B2 in Eq. (16). In (A2) of Section 3.2, we show that $ b_{noise} $ sometimes first increases and then decreases. The factor $ 1 - \frac{n_{(2)}}{p+p_{(2)}} $ is monotone increasing w.r.t. $ p $. Term B2 in Eq. (16) is monotone decreasing w.r.t. $ p $. Thus, the overall noise effect may have multiple descents w.r.t. $ p $. In Fig. 1(c), the orange curve with markers “∇” provides an example of how the model error $ L $ of Option B behaves in the overparameterized regime. We see that this curve has multiple descents, which validates the insight (B2). We also run additional simulations in Appendix F with a neural network, and we can observe the descent w.r.t. the number of parameters of the transferred part. 4 Further Discussion 4.1 Which option performs better in the overparameterized regime? (C1) First, by comparing the coefficients of $ L_{co} $ in Eq. (14) and Eq. (16), we know that the effect of the error in step one deteriorates in the model error $ L $ of Option A (since the coefficient of $ L_{co} $ in Eq. (14) is larger than 1), whereas this is mitigated in the model error of Option B (since the coefficient of $ L_{co} $ in Eq. (16) is smaller than 1). (C2) Second, by comparing the coefficients of $ \\| q_{(2)} \\|^2 $ and $ \sigma_{(2)}^2 $ in Eqs. (14) and (16) under the same $ p $ and $ p_{(2)} $, we know that Option B is worse to learn $ q_{(2)} $, but is better to reduce the noise effect of $ \sigma_{(2)} $ than Option A (since $ 1 - \frac{n_{(2)}}{P_{(2)}} < 1 - \frac{n_{(2)}}{p+p_{(2)}} $). and $ \frac{n_{(2)}}{p_{(2)} - n_{(2)} - 1} > \frac{n_{(2)}}{p + p_{(2)} - n_{(2)} - 1} $. (C3) Third, by letting $ p_{(2)} \to \infty $ in Eqs. (14) and (16), the model error $ L $ of both Option A and Option B approaches the same value $ L_{co} + \\| q_{(2)} \\|^2 $. Intuitive Comparison of Options A and B: An intuitive explanation of the reason for these differences is that Option B does train the common part learned by the source task but Option A does not. Thus, Option B should do better to learn the common part. At the same time, since Option B uses more parameters ($ p + p_{(2)} $) than Option A ($ p $) to learn the target task’s samples, the noise effect is spread among more parameters in Option B than in Option A, and thus Option B can mitigate the noise better than Option A. However, those additional $ p $ parameters interfere with the learning of $ q_{(2)} $ since those $ p $ parameters correspond to the features of the common part $ S_{co} $, not the target task-specific features $ S_{(2)} $, which implies that Option B is worse in learning $ q_{(2)} $ than Option A. In Fig. 1(b), when overparameterized (i.e., $ p_{(2)} > 50 $ for Option A, and $ p_{(2)} > 30 $ for Option B), Option A is slightly better than Option B around $ p_{(2)} = 70 $ under the situation "$ \sigma_{(1)} = 0.1, \sigma_{(2)} = 0.2, \\| q_{(2)} \\| = 1 $" (i.e., the two curves with markers "+"). Notice that this situation has the smallest $ \sigma_{(1)}, \sigma_{(2)} $ and the largest $ \\| q_{(2)} \\| $. Thus, insights (C1),(C2) are verified. Besides, in Fig. 1(b), in every situation, the curves of Option A and Option B overlap when $ p_{(2)} $ is very large, which validates insight (C3). ### 4.2 The common part or the task-specific part? When the total number of parameters is fixed, it is better to use more parameters on the task-specific parts. Specifically, we have the following proposition. Proposition 4. When $ p + p_{(1)} = C $ is fixed, $ L_{co} $ is monotone increasing with respect to $ p $. Therefore, in order to minimize $ L_{co} $ when Definition 1 is assured, the best choice is $ p = s, p_{(1)} = C - s $. Sometimes it is even better to sacrifice certain true features in the common part in favor of employing more redundant features in the task-specific part. We still consider the case of fixed $ p + p_{(1)} = C $. In certain situations (especially when the noise level is large and some true parameters are very small), it is better to make $ p $ even smaller than $ s $, i.e., it is better to violate Definition 1 deliberately (in contrast to Remark 1 where Definition 1 is violated unconsciously). We now construct an example of this situation. Let $ \\| q_{(1)} \\|^2 = 0, \\| w_{(2)} \\| + \\| w_{(1)} \\| = 1 $ (so $ b_2^2 \leq 1 $ by Eq. (11)). Suppose there are only 2 true common features (i.e., $ s = 2 $) and $ C > n_{(1)} + 1 $. If we do not violate Definition 1, then by Proposition 4, the best choice is to let $ p = 2 $. By Theorem 1 we know that $ L_{co} $ is at least $ Q_1 := \frac{2}{C} \cdot \frac{n_{(1)}(\sigma_{(1)}^2 + 0.1^2)}{C - n_{(1)} - 1} $ (since $ L_{noiseless} \geq 0 $). In contrast, if we violate Definition 1 deliberately by sacrificing one true common feature with parameter value 0.1 for the source task and value 0 for the target task, then the only effect is enlarging the source task’s noise level by $ \sigma_{(1)}^2 \leftarrow \sigma_{(1)}^2 + 0.1^2 $. Thus, by Theorem 1, we know that $ L_{co} $ is at most $ Q_2 := 1 + \frac{1}{C} \cdot \frac{n_{(1)}(\sigma_{(1)}^2 + 0.1^2)}{C - n_{(1)} - 1} $ (since $ b_2^2 \leq 1 $). We can easily find a large enough $ \sigma_{(1)}^2 $ to make $ Q_1 > Q_2 $, which leads to our conclusion. ### 5 Conclusion Our study on transfer learning in linear regression models provides valuable insights into the generalization performance of the target task. We propose a comprehensive framework that considers task similarity in terms of both parameter distance and feature sets. Our analysis characterizes the double descent of transfer learning for two different options of parameter transfer. Further investigation reveals that allocating more redundant features to the task-specific part, rather than the common part, can enhance performance when the total number of features is fixed. Moreover, sometimes sacrificing true features in the common part in favor of employing more redundant features in the task-specific part can yield notable benefits, especially in scenarios with high noise levels and small numbers of true parameters. These findings contribute to a better understanding of transfer learning and offer practical guidance for designing effective transfer learning approaches. There are some interesting directions for future work. First, we can use our current framework of partial similarity to analyze the performance of sample transfer. Second, going beyond the linear models of Gaussian features, we can use models that are closer to actual DNNs (such as neural tangent kernel models) to study the generalization performance of overfitted transfer learning. 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Fq8tKtjACC	In Table 2, it looks like for all prior models as well as phi-1-base (the model without finetuning), there is a significant gap between the new score and the HumanEval one. However, in both finetunened phi-1 models this gaps is removed. Is it not possible that this means that while the finetuning data may be unrelated to the new evaluation, it contains considerable leakage with HumanEval?	TEXTBOOKS ARE ALL YOU NEED Anonymous authors Paper under double-blind review ABSTRACT We introduce phi-1, a new large language model for code, with significantly smaller size than competing models: phi-1 is a Transformer-based model with 1.3B parameters, trained for 4 days on 8 A100s, using a selection of “textbook quality” data from the web (6B tokens) and synthetically generated textbooks and exercises with GPT-3.5 (1B tokens). Despite this small scale, phi-1 attains pass@1 accuracy 50.6% on HumanEval and 55.5% on MBPP. It also displays surprising emergent properties compared to phi-1-base, our model before our fine-tuning stage on a coding exercises dataset, and phi-1-small, a model with 350M parameters trained with the same pipeline that still achieves 45% on HumanEval. 1 INTRODUCTION The art of training large artificial neural networks has made extraordinary progress in the last decade, especially after the discovery of the Transformer architecture [Vaswani et al., 2017], yet the science behind this success remains limited. Amidst a vast and confusing array of results, a semblance of order emerged around the same time as Transformers were introduced, namely that performance improves somewhat predictably as one scales up either the amount of compute or the size of the network [Hestness et al., 2017], a phenomenon which is now referred to as scaling laws [Kaplan et al., 2020]. The subsequent exploration of scale in deep learning was guided by these scaling laws [Brown et al., 2020], and discoveries of variants of these laws led to rapid jump in performances [Hoffmann et al., 2022]. In this work, following the footsteps of Eldan and Li [Eldan & Li, 2023], we explore the improvement that can be obtained along a different axis: the quality of the data. It has long been known that higher quality data leads to better results, e.g., data cleaning is an important part of modern dataset creation [Raffel et al., 2020], and it can yield other side benefits such as somewhat smaller datasets [Longpre et al., 2023; Yu et al., 2023] or allowing for more passes on the data [Muenmghoff et al., 2023]. The recent work of Eldan and Li on TinyStories (a high quality dataset synthetically generated to teach English to neural networks) showed that in fact the effect of high quality data extends well past this: improving data quality can dramatically change the shape of the scaling laws, potentially allowing to match the performance of large-scale models with much leaner training/models. In this work we go beyond the initial foray of Eldan and Li to show that high quality data can even improve the SOTA of large language models (LLMs), while dramatically reducing the dataset size and training compute. Importantly, smaller models requiring less training can significantly reduce the environmental cost of LLMs [Bender et al., 2021]. We focus our attention on LLMs trained for code, and specifically writing simple Python functions from their docstrings as in [Chen et al., 2021]. The evaluation benchmark proposed in the latter work, HumanEval, has been widely adopted for comparing LLMs’ performance on code. We demonstrate the power of high quality data in breaking existing scaling laws by training a 1.3B-parameter model, which we call phi-1, for roughly 8 passes over 7B tokens (slightly over 50B total tokens seen) followed by finetuning on less than 200M tokens. Roughly speaking we pretrain on “textbook quality” data, both synthetically generated (with GPT-3.5) and filtered from web sources, and we finetune on “textbook-exercise-like” data. Despite being several orders of magnitude smaller than competing models, both in terms of dataset and model size (see Table 1), we attain 50.6% pass@1 accuracy on HumanEval and 55.5% pass@1 accuracy on MBPP (Mostly Basic Python Programs), which are one of the best self-reported numbers using only one LLM generation. In Section 2, we give some details of our training process, and we discuss evidence for the importance of our data selection process in achieving this result. Moreover, despite being trained on much fewer tokens compared to existing models, phi-1 still displays emergent properties. In Section 3 we discuss these \| Date \| Model \| Model size (Parameters) \| Dataset size (Tokens) \| HumanEval (Pass@1) \| MBPP (Pass@1) \| \|----------\|------------------------\|-------------------------\|-----------------------\|--------------------\|---------------\| \| 2021 Jul \| Codex-300M [Chen et al., 2021] \| 300M \| 100B \| 13.2% \| - \| \| 2021 Jul \| Codex-12B [Chen et al., 2021] \| 12B \| 100B \| 28.8% \| - \| \| 2022 Mar \| CodeGen-Mono-350M [Nijkamp et al., 2023b] \| 350M \| 577B \| 12.8% \| - \| \| 2022 Mar \| CodeGen-Mono-16.1B [Nijkamp et al., 2023b] \| 16.1B \| 577B \| 29.3% \| 35.3% \| \| 2022 Apr \| PalLM-Coder [Chowdhery et al., 2022] \| 540B \| 780B \| 35.9% \| 47.0% \| \| 2022 Sep \| CodeGex [Zheng et al., 2023] \| 13B \| 850B \| 22.9% \| 24.4% \| \| 2022 Nov \| GPT-3.5 [OpenAI, 2023] \| 175B \| N.A. \| 47% \| - \| \| 2022 Dec \| SantaCoder [Aliai et al., 2023] \| 1.1B \| 236B \| 14.0% \| 35.0% \| \| 2023 Mar \| GPT-4 [OpenAI, 2023] \| N.A. \| N.A. \| 67% \| - \| \| 2023 Apr \| Repli [Repli, 2023] \| 2.7B \| 525B \| 21.9% \| - \| \| 2023 Apr \| Repli+mine [Repli, 2023] \| 2.7B \| 525B \| 30.5% \| - \| \| 2023 May \| CodeGen-1B [Nijkamp et al., 2023a] \| 1B \| N.A. \| 10% \| - \| \| 2023 May \| CodeGen-2.7B [Nijkamp et al., 2023a] \| 7B \| N.A. \| 19.1% \| - \| \| 2023 May \| StarCoder [Li et al., 2023] \| 15.5B \| IT \| 33.6% \| 52.7% \| \| 2023 May \| StarCoder-Prompted [Li et al., 2023] \| 15.5B \| IT \| 40.8% \| 49.5% \| \| 2023 May \| PalLM-2 [Anil et al., 2023] \| N.A. \| N.A. \| 37.6% \| 50.0% \| \| 2023 May \| CodeT5 [Wang et al., 2023] \| 2B \| 52B \| 24.2% \| - \| \| 2023 May \| InstructCodeT5+ [Wang et al., 2023] \| 16B \| 52B \| 35.0% \| - \| \| 2023 Jun \| WizardCoder [Luó et al., 2023] \| 16B \| IT \| 57.3% \| 51.8% \| \| 2023 Jun \| phi-1 \| 1.3B \| 7B \| 50.6% \| 55.5% \| Table 1: We use self-reported scores whenever available. Despite being trained at vastly smaller scale, phi-1 outperforms several competing models on HumanEval and MBPP. Emergent properties, and in particular we confirm the hypothesis that the number of parameters plays a key role in emergence (see e.g., [Wei et al., 2022]), by comparing the outputs of phi-1 with those of phi-1-small, a model trained with the same pipeline but with only 350M parameters. The methodology used in this section is reminiscent of the Sparks of AGI paper [Bubeck et al., 2023] for beyond-benchmark evaluation. Finally in Section 4 we discuss alternative benchmarks to evaluate the model and in Section 5 we study possible contamination of our training data with respect to HumanEval. We release the model for usage and evaluation by the broader community, but omit some details of the synthetic data generation, for proprietary reasons.\footnote{In recent past, other highly influential papers like [Brown et al., 2020] and [Lewkowycz et al., 2022] have also similarly withheld dataset details for competitive advantage.} More related works. Our work is part of the recent program of using LLMs for program synthesis, see [Chen et al., 2021; Nijkamp et al., 2022] for more references on this. Our approach is also part of the emerging trend of using existing LLMs to synthesize data for the training of new generations of LLMs, [Wang et al., 2022; Taori et al., 2023; Mukherjee et al., 2023; Lin et al., 2023; Jung et al., 2023]. There is an ongoing debate about whether such “recursive training” might lead to narrower scope for the resulting LLM [Shumailov et al., 2023; Gudiband et al., 2023], see [Mukherjee et al., 2023] for a counterviewpoint. Note that in this paper we focus on a narrow task, similarly to [Jung et al., 2023], where it is plausible to improve upon the teacher LLM (as is argued in the latter paper). 2 TRAINING DETAILS AND THE IMPORTANCE OF HIGH-QUALITY DATA As alluded to in the title of the paper, the central ingredient our model relies on textbook-quality training data. We devote this section primarily to our data curation ideas.\footnote{Our model architecture and training methods are largely conventional and discussed in the Appendix D.} Previous work used standard sources of text and code data for code generation, such as The Stack [Kocetkov et al., 2022] and other web-based datasets (e.g., StackOverflow). While these form large and diverse corpus covering broad range of topics and use cases, we argue that these sources are not optimal for teaching the model how to reason and plan algorithmically. Based on manual inspection we observe that many of these snippets are not very instructive for learning the basics of coding: - Many samples are not self-contained, meaning that they depend on other modules or files that are external to the snippet, making them hard to understand without additional context. - Typical examples do not involve any meaningful computation, but rather consist of trivial or boilerplate code, such as defining constants, parameters, or configuring GUI elements. - Samples that do contain algorithmic logic are often buried inside complex or poorly documented functions, making them difficult to follow or learn from. - The examples are skewed towards certain topics or use cases, resulting in an unbalanced distribution of coding concepts and skills across the dataset. Figure 1: Pass@1 accuracy (%) on HumanEval. The grouping of bar plots correspond to the usual scaling dimensions of either increasing the compute time (more passes on the data, here from 26B tokens seen to 76B) or increasing the number of parameters of the model (here from 350M to 1.3B). Each column within a group corresponds to different training datasets: (A) The first (orange) column represents the performance of models trained on the standard datasets of deduplicated Python files from The Stack and StackOverflow; (B) The second (light green) column represents the performance of models trained with our new dataset composition CodeTextbook; (C) Finally, the third (dark green) column corresponds to the respective second column models finetuned on our new CodeExercises dataset. For the 1.3B models, phi-1 and phi-1-base are checkpoints after training on 51B tokens and The Stack+ model was trained for 76B tokens. We highlight that even without any finetuning, our phi-1-base model trained on CodeTextbook dataset achieves 29% HumanEval performance with a mere 1.3B parameter model. The previous smallest model that achieves close to 30% performance on HumanEval was Replit-Finetuned at 2.7B parameters, which was trained with 100 times more training tokens than us [Replit(2023)]. On top of this, finetuning on our CodeExercises dataset to obtain phi-1 not only gives us our top performance of 51% on HumanEval, but also unlocks unexpected coding capabilities (see Section 3). One can only imagine how frustrating and inefficient it would be for a human learner to try to acquire coding skills from these datasets, as they would have to deal with a lot of noise, ambiguity, and incompleteness in the data. We hypothesize that these issues also affect the performance of language models, as they reduce the quality and quantity of the signal that maps natural language to code. We conjecture that language models would benefit from a training set that has the same qualities as a good “textbook”: it should be clear, self-contained, instructive, and balanced. In this work, we address this challenge directly and show that by intentionally selecting and generating high-quality data, we can achieve state-of-the-art results on code-generation tasks with a much smaller model and less compute than existing approaches. Our training relies on three main datasets: - A filtered code-language dataset, which is a subset of The Stack and StackOverflow, obtained by using a language model-based classifier (consisting of about 6B tokens). - A synthetic textbook dataset of <1B tokens of GPT-3.5 generated Python textbooks. - A small synthetic exercises dataset of ~180M tokens of Python exercises and solutions. We describe those datasets in more detail in the next subsections. Taken together, the above datasets contain less than 7B tokens. We refer to the combination of filtered code-language and synthetic textbook datasets as “CodeTextbook” and use it in the pretraining phase to obtain our base model phi-1-base—this model already achieves a competitive HumanEval performance of 29%. Then we use the 180M token synthetic exercises dataset, referred to as “CodeExercises”, to finetune our phi-1-base model to obtain phi-1. Despite the small size of the “CodeExercises” dataset, finetuning with this dataset is crucial not only for large improvements in generating simple Python function as shown in Figure 1, but more broadly to unlock many interesting emergent capabilities in our phi-1 model that are not observed in phi-1-base (see Section 3). 2.1 Filtering of Existing Code Datasets Using a Transformer-Based Classifier We begin with publicly available Python code datasets: we use the Python subset of the deduplicated version of The Stack and the StackOverflow, which together contain over 35 million files/samples, totalling over 35B tokens. We annotate the quality of a small subset of these files (about 100k samples) using GPT-4: given a code snippet, the model is prompted to “determine its educational value for a student whose goal is to learn basic coding concepts”. We then use this annotated dataset to train a random forest classifier that predicts the quality of a file/sample using its output embedding from a pretrained codegen model as features. We note that unlike GPT-3.5, which we use extensively to generate synthetic content (discussed below), we use GPT-4 minimally only for annotations on the quality of a small subset of The Stack and StackOverflow samples. We thus view our usage of GPT-4 as merely a way to avoid tedious human-annotation efforts [Dubois et al., 2023]. Our filtering boosts model performance significantly even without the synthetic datasets discussed below: for 350M parameter models trained on unfiltered Stack (deduplicated python) and StackOverflow, the HumanEval performance saturates at 12.19% even after training for 96k steps (200B tokens), while training on the filtered subset achieves 17.68% on HumanEval after 36k steps. We further improve this to 20.12% (reported in Figure 1) by training on a combination of the filtered dataset and the synthetic textbooks dataset discussed below. ### 2.2 Creation of Synthetic Textbook-Quality Datasets One of the main challenges in creating a high-quality dataset for code generation is ensuring that the examples are diverse and non-repetitive. By diversity, we mean that the examples should cover a wide range of coding concepts, skills, and scenarios, and that they should vary in their level of difficulty, complexity, and style. Diversity is important for several reasons: it exposes the language model to different ways of expressing and solving problems in code, it reduces the risk of overfitting or memorizing specific patterns or solutions, and it increases the generalization and robustness of the model to unseen or novel tasks. However, achieving diversity is not trivial, especially when using synthetic data generated by another language model. Simply prompting the model to produce a coding textbook or a set of exercises, even with some variation in the instructions or the parameters, will likely result in a very homogeneous and redundant dataset, where the same concepts and solutions are repeated over and over with minor changes. This is because language models tend to follow the most probable or common paths given their training data and their priors, and they lack the creativity or the incentive to explore alternative or novel ways of generating code. Therefore, one needs to find the right “trick” that will induce the language model to be more creative and diverse in its output, while still maintaining the quality and the coherence of the examples. Inspired by Eldan & Li [2023], where a diverse set of short stories were created by including a random subset of words chosen from some fixed vocabulary in the prompt and requiring that they would be somehow combined in the generated text, we look for ways to inject randomness into the prompt in a way that gives rise to the generation of a diverse dataset. THE SYNTHETIC TEXTBOOK DATASET This dataset consists of less than 1B tokens of GPT-3.5 generated Python textbooks, synthesized to provide a high-quality source of natural language heavy text interleaved with relevant code snippets. We further targeted the content of these textbooks to cover topics that promote reasoning and basic algorithmic skills. Here, diversity is obtained by providing constraints on topics and target audience of the generated textbook. The following is an example text from the synthetic textbook: ``` To begin, let us define singular and nonsingular matrices. A matrix is said to be singular if its determinant is zero. On the other hand, a matrix is said to be nonsingular if its determinant is not zero. Now, let's explore these concepts through examples. Example 1: Consider the matrix $ A = \text{np.array}([[1, 2], [2, 4]]) $. We can check if this matrix is singular or nonsingular using the determinant function. We can define a Python function, `is_singular(A)`, which returns true if the determinant of $ A $ is zero, and false otherwise. ```python import numpy as np def is_singular(A): det = np.linalg.det(A) if det == 0: return True else: return False A = np.array([[1, 2], [2, 4]]) print(is_singular(A)) # True ``` THE CODEEXERCISES DATASET This is a small synthetic exercises dataset consisting of less than 180M tokens of Python exercises and solutions. Each exercise is a docstring of a function that needs to be completed. The goal of this dataset is to align the model to perform function completion tasks based on natural language instructions. This dataset was also generated by GPT-3.5, where the main means of eliciting diversity is by constraining the function names. For this dataset in particular, we conduct explicit decontamination and alternative evaluations in the following sections to ensure that problems similar to those from HumanEval benchmark are not seen during finetuning. Example exercise: ```python def valid_guessing_letters(word: str, guesses: List[str]) -> List[str]: """ Returns a list of valid guessing letters, which are letters that have not been guessed yet and are present in the word. Parameters: word (str): The word to guess. guesses (List[str]): A list of letters that have already been guessed. Returns: List[str]: A list of valid guessing letters. """ valid_letters = [] for letter in word: if letter not in guesses and letter not in valid_letters: valid_letters.append(letter) return valid_letters ``` 3 SPIKES OF MODEL CAPABILITY AFTER FINETUNING ON CODEEXERCISES Figure 1 showed that the largest improvement in HumanEval resulted from finetuning on the small CodeExercises dataset (<200M tokens). CodeExercises consist exclusively of short Python tasks using only basic Python libraries. In this section, we demonstrate that, quite remarkably the model after finetuning also exhibits a substantial improvement in executing tasks that are not featured in the finetuning dataset. This includes managing intricate algorithmic tasks and using external libraries. This suggests that our finetuning process might have helped the model in reorganizing and consolidating the knowledge acquired during pretraining, even if such knowledge is not explicitly present in our CodeExercises dataset. In this section we will focus on qualitatively comparing and contrasting the capabilities of our finetuned model phi-1 and its pretrained base model phi-1-base. 3.1 Finetuning Improves the Model’s Understanding Using a simple Python function that we created ourselves, we observe in Figure 2 that the model shows a much higher level of understanding and compliance with instructions after finetuning. In particular, phi-1-base struggles with the logical relationships in the prompts, while phi-1 can interpret the question and generate the answer correctly. In this example, even our 350M phi-1-small model shows some level of understanding of the problem even though the final solution is wrong. Prompt: "* Alice, Bob and Charles are playing games. Alice first pick a random number from 333 to 678, Bob then pick a number starting from Alice’s number to 888, Charles then pick a number starting from 123 to Bob’s number. Alice gets one point if Alice’s number minus Charles’s number is divisible by the floor of the square root of Bob’s number, otherwise Bob gets one point. Simulate Alice’s and Bob’s points in 30 iterations."* \| phi-1 \| phi-1-base \| phi-1-small \| \|-------\|------------\|-------------\| \| points = [0, 0] \| self.points = 0 \| indices = {} \| \| for i in range(30): \| self.alice = 333 \| import random \| \| alice_num = random. \| self.bob = 678 \| alice_points = 0 \| \| randint(333, 678) \| self.charles = 888 \| bob_points = 0 \| \| bob_num = random. \| self.bob_points = 0 \| for i in range(30): \| \| randint(alice_num, 888) \| self.alice_points = 0 \| alice_num = random.randint \| \| charles_num = random. \| self.iterations = 30 \| (333, 678) \| \| randint(123, bob_num) \| self.alice_turn = True \| bob_num = random.randint \| \| if (alice_num - \| self.bob_turn = True \| (123, bob_num) \| \| charles_num) % int( \| self.bob_turn_points = 0 \| if (alice_num - bob_num) % \| \| bob_num ** 0.5) == 0: \| self.alice_turn_points = 0 \| math.floor(math.sqrt(bob_num)) == 0: \| \| points[0] += 1 \| _history = [] \| alice_points += 1 \| \| else: \| self.bob_turn_points \| if (i % 2) == 0: \| \| points[1] += 1 \| _history = [] \| bob_points += 1 \| \| return f"Alice: {points[0]}, Bob: {points[1]}" \| ... \| print(f"Alice gets {alice_points} points, while Bob gets {bob_points} points.") \| Figure 2: Model performance with a multi-step algorithmic prompt, comparing the effects of finetuning and scale. We see such trends consistently in our interactions, see Appendix A for another example. 3.2 Finetuning Improves the Model’s Ability to Use External Libraries We demonstrate here that finetuning on CodeExercises unexpectedly improves the model’s ability to use external libraries such as Pygame, Tkinter, and pytorch, eventhough our exercises do not contain these libraries. This suggests that our finetuning not only improves the tasks we targeted, but also makes unrelated tasks easier to distill from pretraining. As an example, Figure 3 shows a PyGame example that asks the model to generate code to move a ball, where we see that phi-1 shows phenomenal improvement over phi-1-base model. See Appendix A for additional examples. 4 Evaluation on Unconventional Problems with LLM Grading A potential concern with the surprisingly good performance of phi-1 on HumanEval (see Table 1 and Figure 1) is that there might be memorization stemming from contamination of the synthetic CodeExercises dataset. We study this potential contamination directly in Section 5, while this section addresses the concern with a new evaluation that is designed to be unconventional enough to be unlikely to appear in our training data. To minimize bias and leakage, the new evaluation problems were created by a dedicated team that did not access the CodeExercises dataset or the final model. They created 50 new problems in the format as HumanEval with instructions to design problems that are unlikely to appear in real-world code bases or as coding exercises. Here is an example: ```python def sort_concat_square_deduplicate(list1, list2, my_threshold): """This functions takes two lists of integers, sorts each of them in ascending order, concatenates them, squares the entries at even indices, filters out entries smaller than my_threshold and then removes duplicates. The resulting list is returned.""" ``` One of the challenges of evaluating language models on coding tasks is that the output of the model is often binary: either the code passes all the unit tests or it fails. However, this does not capture the nuances of the model’s performance, as it might have produced a code that is almost correct but has a minor error, or a code that is completely wrong but coincidentally passes some tests. Arguably, a more informative way of assessing the model’s coding skills is to compare its output with the correct solution and grade it based on how well it matches the expected logic. This is similar to how humans are evaluated on coding interviews, where the interviewer does not only run the code but also examines the reasoning and the quality of the solution. To evaluate candidate solutions, we therefore adopt the approach of using GPT-4 to grade the solution (such as in Eldan & Li (2023)). This approach has two distinct advantages: (1) by using GPT-4 as a grader, we can leverage its knowledge and generative abilities to obtain a more fine-grained and meaningful signal of the student model’s coding capabilities, and (2) it obviates the need for tests. Our prompt instructs the LLM to evaluate a student’s solution first in a short verbal evaluation followed by grades from 0 to 10. See Table 2 for our results with phi-1 and competing models. The grades on our new unconventional problems give the same ranking as HumanEval (see Table 1). phi-1 again achieves a score significantly higher than StarCoder, as it did on HumanEval. Given that the new problems have had no chance to contaminate the training data and, furthermore, were designed to be outside the training distribution, these results greatly increase our confidence in the validity of phi-1’s performance. --- 3 Developing rigorous sets of tests can be a significant undertaking, as demonstrated by Liu et al. (2023). \| Model \| Size \| Train tokens \| Score \| HumanEval \| \|------------------------\|--------\|--------------\|-------\|-----------\| \| CodeGen-Mono-350M \| 350M \| 577B \| 19% \| 13% \| \| CodeGen-Mono-16.1B \| 16.1B \| 577B \| 38% \| 29% \| \| Replit \| 2.7B \| 525B \| 37% \| 22% \| \| StarCoder \| 15.5B \| 1T \| 51% \| 34% \| \| phi-1-base \| 1.3B \| 7B \| 37% \| 29% \| \| phi-1-small \| 350M \| 7B \| 45% \| 45% \| \| phi-1 \| 1.3B \| 7B \| 52% \| 51% \| Table 2: LLM graded Understanding scores on 50 new unconventional coding problems. 5 DATA PRUNING FOR UNBIASED PERFORMANCE EVALUATION In Figure 1, we see that training on CodeExercises leads to a substantial boost in the performance of the model on the HumanEval benchmark. To investigate this boost, we propose to prune the CodeExercises dataset by removing files that are “similar” to those in HumanEval. This process can be viewed as a “strong form” of data decontamination. We then retrain our model on such pruned data, and still observe strong performance on HumanEval. In particular, even after aggressively pruning more than 40% of the CodeExercises dataset (this even prunes files that are only vaguely similar to HumanEval, see Appendix C), the retrained phi-1 still outperforms StarCoder. We believe that such data pruning experiment is a fair way to evaluate performance, and is more insightful than standard “contamination” studies in the literature that are usually based on measures of overlap between training and test data (e.g., Section 4.8 of Austin et al. (2021)). For sake of completeness we start this section by conducting a standard contamination experiment, which shows that CodeExercises is not contaminated by HumanEval in this standard sense. 5.1 N-GRAM OVERLAP N-gram measures the similarity of text segments based on the shared n-word sequences. We calculate the n-gram overlap between the docstrings of each humaneval question and each exercise in the CodeExercises dataset that was generated. We found 4 humaneval questions with 13-gram overlap with at least one of the entries in our dataset. After further investigating, we found out that all the 4 overlap cases in the 13-gram are all false positives (see examples shown in Appendix C). 5.2 EMBEDDING AND SYNTAX-BASED SIMILARITY ANALYSIS As we just saw, the n-grams are not refined enough to find similar code snippets between HumanEval and CodeExercises. Instead we use a combination of embedding and syntax-based distances. For the embedding distance we compute the L2 distance between the embedding of the code snippets where the embedding is derived from a pre-trained CodeGen-Mono 350M model (Nijkamp et al., 2023b). We observe that the embedding distance is successful in capturing code pairs where the overall code semantics are similar, which can be inferred via the Python Docstring, function/class names, as well as the code structure. For the syntax-based distance we calculate the (string) edit distance between the abstract syntax trees (ASTs) of two given code snippets. The AST distance successfully identifies overlapping sections between code pairs while being agnostic to non-syntax text such as variable/function naming, comments, and Python Docstrings. See Appendix C for examples of code pairs that are captured at various $\tau$ and embedding distances. For our pruning experiments on CodeExercises, we fix a threshold for the embedding distance, and we test several match rate $\tau$ for the AST distance. We vary $\tau$ between 0.95 and 0.8, which corresponds to 4% to 40% of problems in CodeExercises, respectively. Table 3 summarizes the performance of our retrained phi-1 on pruned datasets (with $\tau = 0.95, 0.9, 0.85$ and 0.8) versus the original phi-1 trained on full CodeExercises and the 15.5B-parameter StarCoder-prompted. We divide the HumanEval problems into two subsets (“similar” and “non-similar”) based on whether or not they have at least one close match (for this given $\tau$) inside the original CodeExercises dataset. We then report the accuracy of the models on each subset of HumanEval separately. As one can see, even after heavily pruning our dataset, phi-1 still outperforms StarCoder-Prompted by a large margin, which validates that our performance boost is not due to dataset “contamination”, even when the latter term is understood loosely. \| $\tau$ \| Problem Count \| phi-1 \| phi-1 retrained on pruned data \| StarCoder-Prompted [Li et al., 2023] \| \|-------\|---------------\|-------\|-------------------------------\|----------------------------------\| \| \| similar \| 71 \| 81.7% \| 74.6% \| 57.7% \| \| 0.95 \| non-similar \| 93 \| 26.9% \| 32.3% \| 29.0% \| \| \| total \| 164 \| 50.6% \| 50.6% \| 41.5% \| \| \| similar \| 93 \| 63.4% \| 51.6% \| 48.4% \| \| 0.9 \| non-similar \| 71 \| 33.8% \| 36.6% \| 32.4% \| \| \| total \| 164 \| 50.6% \| 45.1% \| 41.5% \| \| \| similar \| 106 \| 62.3% \| 52.8% \| 47.2% \| \| 0.85 \| non-similar \| 58 \| 29.3% \| 34.5% \| 31.0% \| \| \| total \| 164 \| 50.6% \| 46.3% \| 41.5% \| \| \| similar \| 116 \| 59.5% \| 52.6% \| 45.7% \| \| 0.8 \| non-similar \| 48 \| 29.2% \| 27.1% \| 31.2% \| \| \| total \| 164 \| 50.6% \| 45.1% \| 41.5% \| Table 3: Percentage of similar versus non-similar HumanEval problems correctly solved by different models. Similarity is determined based on whether or not the corresponding HumanEval problem has any close matches inside the CodeExercises dataset (for a given $\tau$). The problem count denotes the number of HumanEval problems within each subset. Here, $\tau$ is the threshold on AST-based match rate between codes for similarity check. 6 CONCLUSION Just as a comprehensive, well-crafted textbook can provide a student with the necessary knowledge to master a new subject, our work demonstrates the remarkable impact of high-quality data in honing a language model’s proficiency in code-generation tasks. By crafting “textbook quality” data we were able to train a model that surpasses almost all open-source models on coding benchmarks such as HumanEval and MBPP despite being 10x smaller in model size and 100x smaller in dataset size. We hypothesize that such high quality data dramatically improves the learning efficiency of language models for code as they provide clear, self-contained, instructive, and balanced examples. There remains a number of limitations of our model compared to larger models for code. Firstly, phi-1 is specialized in Python coding, which restricts its versatility compared to multi-language models. Secondly, phi-1 lacks the domain-specific knowledge of larger models such as programming with specific APIs or using less common packages. Lastly, due to the structured nature of the datasets and the lack of diversity in terms of language and style, phi-1 is less robust to stylistic variations or errors in the prompt (for instance, its performance substantially degrades with grammatical mistakes in the prompt). We expand on these limitations and other failure modes of phi-1 in Appendix B. None of these limitations seem fundamental, and with more work our approach could be used to tackle each one of them, although it is unclear what scaling might be necessary to overcome them (both for the model size and the dataset size). We also believe that significant gains could be achieved by using GPT-4 to generate the synthetic data instead of GPT-3.5, as we noticed that GPT-3.5 data has a high error rate. It is interesting that phi-1 is able to achieve such high coding proficiency despite those errors (a similar phenomenon was observed in Allen-Zhu & Li (2023) where a language model can be trained on data with 100% error rate and still generate correct answers at test time). More generally, our work provides evidence that developing good methodology for creating high-quality datasets is a central direction of research for advancing natural language processing and related fields (see also Jung et al. (2023) for further evidence). However, creating high-quality datasets is not a trivial task, and it poses several challenges that need to be addressed. One challenge is to ensure that the dataset covers all the relevant content and concepts that one wants the model to learn, and that it does so in a balanced and representative way. Another challenge is to ensure that the dataset is truly diverse and non-repetitive, so that the model does not simply overfit to the data or memorize specific patterns or solutions. This requires finding ways to inject randomness and creativity into the data generation process, while still maintaining the quality and the coherence of the examples. Moreover, even after creating such datasets, we lack a good methodology to measure and evaluate the amount of diversity and redundancy in the data. For example, if we have a dataset with coding exercises, it is hard to determine how many different variations of each exercise exist, and how they are distributed across the dataset. 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PFdjJiZjPj	I do not understand what the following statement means: “the test cases generated by LLMs can show a descent pass rate, and this pass rate is even higher than the code pass rate on HumanEval+, which holds for both large and small LLMs.”	THE PROGRAM TESTING ABILITY OF LARGE LANGUAGE MODELS FOR CODE Anonymous authors Paper under double-blind review ABSTRACT Recent development of large language models (LLMs) for code like CodeX and CodeT5+ demonstrates tremendous promise in achieving code intelligence. Their ability of synthesizing code that completes a program for performing a pre-defined task has been intensively tested and verified on benchmark datasets including HumanEval and MBPP. Yet, evaluation of these LLMs from more perspectives (than just program synthesis) is also anticipated, considering their broad scope of applications in software engineering. In this paper, we explore the ability of LLMs for testing programs/code. By performing thorough analyses of recent LLMs for code in program testing, we show a series of intriguing properties of these models and demonstrate how program testing ability of LLMs can be improved. Following recent work which utilizes generated test cases to enhance program synthesis, we further leverage our findings in improving the quality of the synthesized programs and show +11.77% and +4.22% higher code pass rates on HumanEval+ comparing with the GPT-3.5-turbo baseline and the recent state-of-the-art, respectively. 1 INTRODUCTION The community has witnessed a surge in the development of large language models (LLMs), which have achieved incredible ability in understanding and generating not only texts but also code. LLMs for code (CodeX [Chen et al., 2021], StarCoder [Li et al., 2023b], CodeT5+ [Wang et al., 2023b], etc) have been widely adopted to a variety of applications to achieve code intelligence. However, current evaluation of these LLMs mostly focuses on program completion/synthesis, despite the models can also be utilized in other applications. As the field continues to advance, evaluation of these models from more perspectives is anticipated, which could facilitate deeper understanding of the LLMs. The ability of automatically generating proper test suites is of great desire to software engineering, yet challenging. Being learning-based or not, current test generation efforts (e.g., fuzzing) primarily focus on creating diverse test inputs to identify faults in the code as much as possible via maximizing their coverage, e.g., line coverage and branch coverage [Fioraldi et al., 2020; Tufano et al., 2022; Dinella et al., 2022; Lemieux et al., 2023; Xia et al., 2023]. Although such test inputs try to verify the (non-)existence of crashes and hangs of the tested code, they lack the ability of determining whether the code adheres to the aim of the function which is represented by input-output relationships. Automatic test case generation for this aim not only requires an high coverage of the code being tested but also necessitates a correct understanding of the “true” desired input-output relationships in the tested code, leaving it a challenging open problem. Being capable of synthesizing correct code implementations given docstrings, LLMs for code seem capable of understanding the desired input-output relationship of a function described in natural language. This capability inspires applying these LLMs to generating test cases automatically [Chen et al., 2021]. However, the ability of these models for program testing has not been systematically evaluated. In this paper, we systematically compare the ability of recent LLMs for code in testing from two perspectives focusing on both the correctness and diversity of the test cases, considering that 1) program testing is of great interest in software engineering and software security as mentioned and 2) automatically generated test cases can further be adopted to improve the program synthesis performance [Chen et al., 2023]. Our analyses focus on algorithmic coding, based on the popular 164 problems from HumanEval+ [Liu et al., 2023a] and 427 sanitized problems from MBPP [Austin et al., 2021]. It is worth noting that the model may encounter various scenarios when generating test cases. It may generate test cases when provided with only natural language descriptions of the desire of the program, or it could generate test cases when given an “optimal” oracle implementation. In more complex situations, it may even need to test its own imperfect generated code or the code generated by other models. We consider 4 test-case generation settings (i.e., “self-generated” which uses each LLM to test code synthesized by the LLM itself, “all-generated” which lets all LLMs to test the same code synthesized by a group of four LLMs, “oracle” which tests an oracle implementation, and the “placeholder” in Figure1) and test a collection of 11 competitive LLMs for code. We conducted a variety of experiments, from which intriguing takeaway messages are delivered. As previously mentioned, several very recent papers (Shi et al., 2022; Li et al., 2023a; Chen et al., 2023) have shown that appropriate usage of generated test cases can improve the quality of program synthesis. Yet, the quality of generated test cases largely impacts the performance of such methods. Due to the lack of systematic evaluation of the testing ability of LLMs for code, it is unclear how to craft test cases that could be potentially more helpful to program synthesis. The studies in this paper also shed light on this. We will show that, substantially improved program synthesis performance can be gained by utilizing takeaway messages in our studies. Specifically, we can achieve +11.77% higher code pass rate on HumanEval+, in comparison with the GPT-3.5-turbo baseline. Compared with a very recent state-of-the-art called CodeT, our solution gains +4.22% higher code pass rate. 2 EVALUATION METRICS To make the evaluation more reliable and comprehensive, it is crucial to first design some suitable metrics, like BLEU (Papineni et al., 2002), ROUGE (Lin, 2004), and the pass rate (Chen et al., 2021) for evaluating machine translation, text summarization, and program synthesis, respectively. In this section, we specify two main evaluation metrics to evaluate the program testing ability of LLMs, from the perspective of correctness and diversity. Pass rate In software engineering, we expect test cases to represent some desired “ground-truth” functionality of the tested program/code. In practice, such “ground-truth” functionality can be described in the header comments of a function (i.e., docstrings of the function) and tested using the oracle implementation, as in HumanEval (Chen et al., 2021) and MBPP (Austin et al., 2021). The oracle program/code should be able to pass the test, if a generated test case is correct. Therefore, we leverage the pass rate as a measure to evaluate the correctness of the generated test cases. For a fair comparison, we instruct each model to generate three test cases in the prompt, and, when a model generates more than three test cases, we select the first three for evaluation. Assuming that there are in total $M$ programming problems in an experimental dataset and, for each problem, we have $N$ program/code implementations to be generated test cases for. Each model has only one chance to generate these test cases for each program/code. Then, we calculate the pass rate as: $$P = \frac{1}{MN} \sum_{i=1}^{M} \sum_{j=1}^{N} \frac{p_{ij}}{n_{ij}},$$ where $n_{ij}$ is the number of test cases in $Q_{ij}$ which includes no more than three test cases generated for the $j$-th program/code implementation of the $i$-th problem by the evaluated LLM at once, i.e., $Q_{ij} = \{(x_{ijk}, y_{ijk})\}_k$, and $p_{ij}$ is the number of test cases (in $Q_{ij}$) that do not fail the oracle. The pass rate defined in Eq. (1) measures correctness of the generated test cases. However, as can be seen in Figure1, the model can generate duplicate test cases that are less helpful, even though they are correct. To avoid such an evaluation bias, we further advocate deduplication in the set of test cases that are considered as correct, which leads to computation of a deduplicated pass rate defined as $P' = \frac{1}{MN} \sum \sum p'_{ij}/n'_{ij}$, where we use ‘ to denote the numbers of unique test cases. Coverage rate In addition to the above pass rates, we further consider coverage rate as a more fine-grained metric for evaluating the diversity of the generated test cases. According to its definition, converge rate computes the degree to which the code is executed, given a test case. Since, for each program/code, we keep no more than three test cases at once, we calculate how much percentage of the control structure is covered given these test cases. Similar to Eq. (1), we evaluate the performance of testing all programs/code over all $M \times N$ times of generation, i.e., we calculate $$C = \frac{1}{MN} \sum_{i=1}^{M} \sum_{j=1}^{N} c_{ij},$$ where $c_{ij}$ is the per-test-case branch coverage rate. We apply the `pytest` library to evaluate the branch coverage for all the three test cases for each code and average the results for all programs/code and all problems. Apparently, $C \leq 1$, and a higher $C$ shows better testing ability of an LLM, since we expect all parts of the programs/code to be executed to find our all potential bugs. 3 LARGE LANGUAGE MODELS FOR CODE In this section, we outline the evaluated models. We adopt some “small” models whose numbers of parameters are around 1B (to be more specific, from 770M to 1.3B in our choices) and some larger models that achieve state-of-the-art performance in the task of program synthesis. For the small models, we use InCoder (1.3B) (Fried et al., 2023), CodeGen2 (1B) (Nijkamp et al., 2023a), CodeT5+ (770M) (Wang et al., 2023b), and SantaCoder (1.1B) (Allal et al., 2023). InCoder is a unified generative model that can perform program/code synthesis as well as code editing, and it combines the strengths of causal language modeling and masked language modeling. The CodeGen2 model was trained on a deduplicated subset of the Stack v1.1 dataset (Kocetkov et al., 2023), and its training is formatted with a mixture of objectives for causal language modeling and span corruption. CodeT5+ is an encoder-decoder model trained on several pre-training tasks including span denoising and two variants of causal language modeling. SantaCoder was trained on the Python, Java, and JavaScript code in the Stack dataset. The pass rate (Chen et al., 2021) of programs generated by these models is compared in Table 1. When evaluating the (program) pass rate, we let the model generate 200 code implementations for each problem, and we set the temperature to 0.2, 0.6, and 0.8 for calculating pass@1, pass@10, and pass@100, respectively. As for larger models that achieve state-of-the-art program synthesis performance, we use CodeGen2 (16B) (Nijkamp et al., 2023a), CodeGen-Multi (16B) (Nijkamp et al., 2023b), CodeGen-Mono (16B) (Nijkamp et al., 2023b), StarCoder (15B) (Li et al., 2023b), WizardCoder (15B) (Luo et al., 2023), CodeGeeX2 (6B) (Zheng et al., 2023), and GPT-3.5-turbo. CodeGen-Multi and CodeGen-Mono are two large models from the first version of CodeGen. CodeGen-Multi was first trained on the pile dataset (Gao et al., 2020) and then trained on a subset of the publicly available BigQuery dataset which contains code written in C, C++, Go, Java, JavaScript, and Python. Based on the 16B CodeGen-Multi model, CodeGen-Mono (16B) was obtained by further tuning on a set of Python code collected from GitHub. Given a base model that was pre-trained on 1 trillion tokens from the Stack dataset, the 15B StarCoder model was obtained by training it on 35B tokens of Python code. WizardCoder further empowers StarCoder with instruction tuning, following a similar instruction evolution strategy as in WizardLM (Xu et al., 2023). CodeGeeX2, the second generation of a multilingual generative model for code, is implemented based on the ChatGLM2 architecture and trained on more code data. GPT-3.5-turbo is a very capable commercial LLM developed by OpenAI and we accessed it in August, 2023. For these large LLMs, we tested pass@1 of all models except GPT-3.5-turbo (whose result can be directly collected from Liu et al. (2023a)’s paper). By sorting their pass@1 from high to low, they are ranked as: GPT-3.5-turbo (61.7%), WizardCoder (46.23%, 15B), CodeGeeX2 (29.97%, 6B), StarCoder (27.9%, 15B), CodeGen-Mono (26.15%, 16B), CodeGen2 (19.33%, 16B), CodeGen-Multi (15.35%, 16B). The ranks on the MBPP dataset are similar. 4 CODE TO BE TESTED For evaluating the testing ability of LLMs, we need an oracle to express the ground-truth functionality of the tested code. Fortunately, current datasets for evaluating program synthesis performance often provide such oracles (see HumanEval (Chen et al., 2021) and MBPP (Austin et al., 2021)). In our experiments, we utilize an amended version of HumanEval called HumanEval+ (Liu et al., 2023a), together with MBPP (the sanitized version). These datasets are established to evaluate basic Python programming performance of LLMs, and they contain 164 and 427 problems, respectively. 4.1 IMPERFECT CODE IMPLEMENTATIONS In order to simulate real-world scenarios where the tested code is often buggy, we first adopt synthesized programs/code as the programs/code to be tested, considering that the synthesis of even --- 1https://pytest.org state-of-the-art LLMs is still imperfect. We evaluate the performance of each LLM in testing code that was generated by itself (which is denoted as “Self-generated”) and code in a set consisting of program completion results of several different LLMs (which is denoted by “All-generated”). That said, the compared LLMs take different code implementations when generating test cases for each programming problem in the self-generated setting. Whereas, in the all-generated setting, the same program/code implementations are given to different LLMs for generating test cases for comparison. In practice, we apply InCoder (1.3B), CodeGen2 (1B), CodeT5+ (770M), and SantaCoder (1.1B) to construct the all-generated program/code set, while, in the self-generated setting, each LLM first synthesizes code and complete a program to fulfill the requirement of each programming problem, and the LLM then generates test cases with the synthesized programs/code in its prompts. The temperature for all LLMs is uniformly set to 0.2 for synthesizing the programs/code in both settings. We obtain 100 program/code completions for each problem and we prompt each LLM to generate 3 test cases for every program/code implementation in the self-generated setting, and we sampled 100 implementations from the synthesis results of InCoder (1.3B), CodeGen2 (1B), CodeT5+ (770M), and SantaCoder (1.1B) to form the all-generated code set, i.e., we have $N = 100$ for these settings. We follow the same way of generating code as introduced in the papers of these LLMs. For model without instruction tuning, like InCoder and CodeT5+, we synthesize programs/code using the default prompt given by each programming problem in the test dataset, while, for models that have adopted instruction tuning, e.g., WizardCoder, we use the recommended prompt in their papers. ### 4.2 Optimal Code Implementations (Oracle) As a reference, we also report the performance of generating accurate and diverse test cases when the written code is perfectly correct, which is achieved by adopting the oracle as the programs/code to be tested (and such a setting is denoted by “Oracle”). Since Liu et al. (2023a) have reported that some oracle code in the HumanEval dataset can be incorrect, we adopt the amended oracle set in HumanEval+ in this setting. We further used the revised oracle code implementations instead of the original ones in evaluating the pass rate (i.e., $P'$) of the generated test cases. Considering that the public datasets often only provide one oracle implementation for each problem, and to keep the uncertainty of evaluation results consistent, we copy the oracle implementation by $100\times$ and we \| Model \| Size \| Pass@1 \| Pass@10 \| Pass@100 \| \|-------------\|--------\|--------\|---------\|----------\| \| InCoder \| 1.3B \| 6.95% \| 14.06% \| 23.76% \| \| CodeGen2 \| 1B \| 9.19% \| 17.50% \| 25.90% \| \| CodeT5+ \| 770M \| 12.95% \| 28.02% \| 37.56% \| \| SantaCoder \| 1.1B \| 15.21% \| 29.42% \| 43.80% \| Table 1: Program synthesis performance of the small LLMs (whose number of parameters is around 1 billion) evaluated on HumanEval+/MBPP (sanitized). prompt to generate 3 test cases for each of these copies. It can be regarded as letting $N = 100$, just like in the previous settings in Section 4.1. ### 4.3 No Implementation (Placeholder) In certain scenarios, we require test cases before the function/program has been fully implemented, hence we also evaluate in a setting where the main body of a tested function/program is merely a placeholder, as depicted in Figure 1(b). This scenario often occurs when the main code has not yet been implemented for a function/program or the test engineer does not want to introduce implementation bias to the LLM when generating test cases for a function/program. We denote such a setting as “Placeholder” in this paper. We also let $N = 100$, as in the oracle setting. ## 5 Test Case Generation In this section, we introduce how test cases can be generated, when the implementation of a function/program is given as described in Section 4. In this paper, a desired test case is a pair of input and its expected output for the function/program defined in the context. As an example, Figure 1 demonstrates some test cases for the programming problem of checking whether the two words satisfy a specific rotation pattern. To generate test cases, we use the LLMs introduced in Section 3. We wrote extra prompts to instruct the LLMs to generate three test cases for each given code which include docstrings that describe the purpose of this function, as depicted in Figure 1. Our instruction commands the LLMs (1) to “check the correctness of this function with three test” and (2) to start writing test code with an “assert” statement and the tested function, which specifies the format of the test cases as input-output pairs that can be parsed. For instance, given the example in Figure 1, the extra prompt should be “# Check the correctness of this function with three test cases \n assert cycpattern_check”. We then concatenate the extra prompt with the code and feed the concatenation into each LLM, for extracting test cases from the model output. The LLM will try to complete the given input by generating one or more “assert” statement(s), and we split the generation results into sub-strings, with “assert” as the separator. Each sub-string is then considered as a test statement, and we only take the first three statements if there exist more than three statements, as has been introduced in Section 2. Such a split can be considered as an effective post-processing operation which largely improves the quality of the generated test code, considering that some non-sense code pieces may be generated in the output of the LLMs. When using HumanEval+ and MBPP, we try removing test cases in the docstrings of the function, if there exist any, just to get rid of the broad hints from the docstrings (Chen et al., 2023). The temperature for generating test cases is kept as 0.2. Once obtained, the generated test cases are then compiled, and evaluated for their correctness and diversity to report the pass rate $P'$ and the coverage rate $C$. When calculating, for each problem and every set of completions generated, we create a temporary folder. ## 6 Main Results for Test Case Generation The experiment results of small and large LLMs on HumanEval+ can be found in Table 2 and Table 3 respectively. Table 4 shows the results on MBPP. There are several takeaways from these tables. - First, the test cases generated by LLMs can show a descent pass rate, and this pass rate is even higher than the code pass rate on HumanEval+, which holds for both large and small \| Model \| Size \| Oracle \| Self-generated \| All-generated \| Placeholder \| \|------------\|--------\|--------------\|----------------\|---------------\|-------------\| \| InCoder \| 1.3B \| 21.31% (61.43%) \| 23.37% (59.36%) \| 22.72% (61.10%) \| 25.19% (62.75%) \| \| CodeGen2 \| 1B \| 31.63% (71.55%) \| 30.62% (69.38%) \| 30.93% (69.70%) \| 30.69% (69.00%) \| \| CodeT5+ \| 770M \| 35.43% (71.45%) \| 32.34% (70.45%) \| 31.49% (69.75%) \| 32.67% (70.67%) \| \| SantaCoder \| 1.1B \| 30.97% (71.46%) \| 30.43% (70.81%) \| 30.13% (70.55%) \| 30.78% (71.24%) \| Table 2: The pass rates (and coverage rate) of the test cases generated on HumanEval+ in different settings for LLMs with around 1 billion parameters. \| Model \| Size \| Oracle \| Self-generated \| All-generated \| Placeholder \| \|----------------\|------\|------------\|----------------\|---------------\|-------------\| \| CodeGen-Multi \| 16B \| 43.88% (67.91%) \| 41.85% (69.30%) \| 40.38% (66.97%) \| 39.74% (68.28%) \| \| CodeGen2 \| 16B \| 46.34% (73.07%) \| 45.44% (73.17%) \| 42.00% (72.45%) \| 42.69% (72.86%) \| \| CodeGen-Mono \| 16B \| 49.03% (74.82%) \| 45.73% (73.74%) \| 43.91% (73.66%) \| 44.92% (73.63%) \| \| StarCoder \| 15B \| 55.07% (76.02%) \| 52.52% (72.45%) \| 48.20% (72.30%) \| 50.58% (74.52%) \| \| CodeGeeX2 \| 6B \| 57.03% (74.42%) \| 53.16% (73.55%) \| 49.28% (70.32%) \| 51.78% (73.08%) \| \| WizardCoder \| 15B \| 53.89% (77.87%) \| 55.47% (76.07%) \| 48.02% (75.27%) \| 49.89% (75.12%) \| \| GPT-3.5-turbo \| - \| 71.03% (77.85%) \| 72.45% (77.24%) \| 59.24% (74.99%) \| 66.28% (74.03%) \| Table 3: The pass rates (and coverage rate) of the test cases generated on HumanEval+ in different settings for LLMs whose parameters are obviously more than 1 billion. Figure 2: The correlation between code past rate and test pass rate in the “Oracle” setting. Figure 3: How the correctness of the test cases changes with their order when being generated. LLMs. Such a result is consistent with intuitions from previous work which rejects code that cannot pass the generated tests to improve the quality of program synthesis. • Second, the correctness of the generated test cases is positively correlated with the LLM’s ability of generating code (see Figure 2, where each red cross represents the performance of a model), which means an LLM showing the state-of-the-art program synthesis performance is possibly also the state-of-the-art LLM for program testing. As shown in Tables 2 and 3, GPT-3.5-turbo, which synthesizes programs/code with the highest correctness, provides test cases with the highest pass rate (71.03%) on HumanEval+. For an LLM, the more accurate it is capable of synthesizing programs/code on a dataset, the more powerful testing ability will probably be exhibited on the same dataset. There also exist a few exceptions, e.g., SantaCoder (1.1B) outperforms CodeT5+ (770M) and CodeGen2 (1B) in generating code, but it shows inferior performance in program testing on HumanEval+. By carefully examining the test cases yielded by SantaCoder on HumanEval+, we found that it tends to generate more complex and longer test cases than CodeT5+ for several problems on HumanEval+, which are often more desirable in program testing. This is also why the SantaCoder test cases show higher coverage rates in Table 2. To be concrete, in Problem 131 in HumanEval+, where the program is required to return the product of all digits with an odd position in a positive integer $ n $ (which is the input), the test input provided by CodeT5+ tends to be small for this problem, e.g., $ n = 2 $, while the SantaCoder test cases tend to have more digits (e.g., $ n = 12358 $), which is helpful in digging out hidden bugs. Yet, generating longer and more complex test cases is more challenging, and the correctness can be lower. • Third, as can be seen in Tables 3 and 4, generating test cases using large LLMs with their self-generated code (in the prompts) often leads to a higher level of correctness, compared with the placeholder results. This observation is in fact unsurprising, considering that generating code first and test case afterwards resembles the chain-of-thought prompting (Wei et al., 2022) (if adopting the placeholder is regarded as a plain prompting), which is beneficial to reasoning. Moreover, the self-generated performance of an LLM sometimes even outperforms its testing performance with an oracle, and we ascribe this to: 1) randomness in the style of the oracles which are few in number and/or 2) less distribution shift between self-generated code in prompt and the training code, for some powerful LLMs. \| Model \| Size \| Oracle \| Self-generated \| All-generated \| Placeholder \| \|---------------\|--------\|--------------\|----------------\|---------------\|-------------\| \| InCoder \| 1.3B \| 21.56% (46.81%) \| 17.98% (46.11%) \| 19.53% (46.45%) \| 22.58% (46.72%) \| \| CodeGen2 \| 1B \| 25.61% (54.26%) \| 21.85% (53.09%) \| 23.15% (50.43%) \| 22.81% (52.11%) \| \| CodeT+ \| 770M \| 29.02% (56.86%) \| 24.44% (52.31%) \| 24.84% (53.20%) \| 25.59% (55.81%) \| \| SantaCoder \| 1.1B \| 32.37% (55.68%) \| 26.40% (52.38%) \| 26.20% (52.83%) \| 26.53% (53.86%) \| \| CodeGen-Multi \| 16B \| 41.32% (60.63%) \| 35.96% (59.03%) \| 34.17% (58.09%) \| 34.84% (58.92%) \| \| CodeGen2 \| 16B \| 45.30% (62.15%) \| 38.67% (60.16%) \| 36.77% (58.59%) \| 37.27% (59.16%) \| \| CodeGen-Mono \| 16B \| 50.24% (64.39%) \| 43.94% (62.94%) \| 39.55% (61.99%) \| 42.41% (62.31%) \| \| StarCoder \| 15B \| 54.84% (65.10%) \| 46.77% (63.60%) \| 42.80% (61.95%) \| 45.35% (62.66%) \| \| CodeGeeX2 \| 6B \| 52.45% (64.64%) \| 44.52% (63.72%) \| 41.72% (60.48%) \| 43.86% (63.51%) \| \| WizardCoder \| 15B \| 57.85% (66.68%) \| 46.56% (64.86%) \| 41.62% (60.72%) \| 47.45% (64.54%) \| \| GPT-3.5-turbo \| - \| 74.30% (66.19%) \| 66.14% (65.30%) \| 49.56% (62.95%) \| 63.34% (64.72%) \| Table 4: The pass rates (and coverage rate) of the test cases generated on MBPP. • Fourth, with only a few exception, test cases obtained using the oracle code exhibit slightly higher code coverage, while the coverage rate achieved in the other settings (i.e., the self-generated, all-generated, and the placeholder settings) is often slightly lower. The above four takeaway messages can all be inferred from Tables 2, 3, and 4. In addition to all these results, we conduct more experiments to achieve the following takeaway messages. • Fifth, by analyzing the relationship between the quality of code in prompts and the correctness of test, we found that correct code implementation in the prompt often leads to higher quality of test code generation than the case when some incorrect code is given. We conducted an experiments where we first select programming problems in HumanEval+, where the code pass rate of an LLM is neither 0% or 100%. Then we separate self-generated programs/code of the model into two groups, with one group only contains programs/code that are considered as correct and the other only contains incorrect programs/code. In Table 5, we compare the performance of using these two sorts of code in the prompt, for generating test cases using the same LLM. Apparently, the quality of test cases obtained with correct programs/code is obviously higher. We further evaluate the overall testing performance of LLMs with only correct self-generated programs/code, if there exists any, in their prompts. Unlike in Table 5, where we do not take problems that can be 100% or 0% solved, we take all given problems in this evaluation, except, for every problem, we eliminate all incorrect self-generated programs/code if there exist at least one correct implementation synthesized by the evaluated LLM. By doing so, we can observe substantially improved program testing ability on HumanEval+ (i.e., 74.95% for GPT-3.5-turbo, 56.87% for WizardCoder, 54.33% for CodeGeeX2, and 53.24% for StarCoder), comparing with the original self-generated results in Table 5. The same on MBPP. • Sixth, by conducting an additional experiment, we further compare the quality of test cases collected from different positions in the generation results. For every set of the three generated test cases, we analyze the relationship between their correctness and the order when they are generated. The results are illustrated in Figure 3. As can be seen in the figure, the first generated test case often shows the best correctness and the latterly generated ones are more incorrect. This may be due to the fact that the model tends to first generate content with a high level of confidence (which is also more likely to be correct). 7 Improving Program Synthesis Using the Generated Test Cases High quality test cases are not only desired in program analyses, but also helpful to program synthesis. Previous methods have successfully used generated test cases to improve the performance of LLMs in synthesizing programs/code. For instance, [Li et al. (2023a)] designed a special prompt which involves the test cases as an preliminary, if they are available, for generating programs/code. One step further, [Chen et al. (2023)] proposed CodeT, which leverages the LLM to obtain test cases first and tests all synthesized programs/code with these test cases by performing a dual execution agreement, and it picks the code in the largest consensus set (i.e., the consensus set with the most code implementations and test cases) as output to obtain state-of-the-arts program synthesis performance. We encourage interested reader to read the original paper. \| Model \| Size \| w/ correct code \| w/ incorrect code \| #Problem \| \|---------------\|------\|-----------------\|-------------------\|----------\| \| InCoder \| 1.3B \| 28.55% \| 27.39% \| 27 \| \| CodeGen2 \| 1B \| 27.25% \| 25.74% \| 11 \| \| CodeT5+ \| 770M \| 40.19% \| 36.78% \| 27 \| \| SantaCoder \| 1.1B \| 37.45% \| 34.08% \| 24 \| \| CodeGen-Multi \| 16B \| 55.49% \| 50.06% \| 32 \| \| CodeGen2 \| 16B \| 43.56% \| 39.31% \| 29 \| \| CodeGen-Mono \| 16B \| 45.18% \| 42.86% \| 56 \| \| StarCoder \| 15B \| 58.16% \| 57.08% \| 68 \| \| CodeGeeX2 \| 6B \| 52.84% \| 48.63% \| 51 \| \| WizardCoder \| 15B \| 48.02% \| 45.12% \| 54 \| \| GPT-3.5-turbo \| - \| 75.39% \| 68.52% \| 126 \| Table 5: With the correct (self-generated) code, the LLMs show stronger ability of generating correct test cases on HumanEval+ (evaluated only on those problems that can neither be 0% solved nor 100% solved), than in the case where incorrect self-generated code is given in the prompts. Since most LLMs cannot generate any correct code for many hard problems while they often generate incorrect code even for easy problems, the number of tested problems in this experiment increases with the power of the tested LLM, as shown in the rightmost column. In the previous section, we have obtained results about many intriguing properties of the program testing performance of LLMs for code. In this section, we would like to drive the readers to think whether it is possible to utilize these results to improve the program synthesis performance, considering that the test cases (hand-crafted and given or automatically generated in particular) are widely and successfully used in program synthesis. We shall demonstrate that, by utilizing takeaway messages in Section 6, the program synthesis performance of previous methods can be improved significantly. Taking CodeT as an example of the previous state-of-the-art, the method uses a placeholder to generate test cases and treats all the test cases as equally correct as a prior. However, as discussed in our third takeaway message, using self-generated code helps to achieve more powerful ability in generating correct test cases. Moreover, if multiple test cases are provided in a single run of generation given an LLM, the correctness of the test cases decreases with their generation order, as shown in our fifth point. Hence, to obtain superior program synthesis performance, we introduce two simple modifications to it: 1) we employ the “self-generated” setting instead of the “placeholder” setting for generating test cases, which means we utilized synthesize programs in prompts when generating test cases for each program, 2) we assign different weights to the generated test cases based on their order in each generation result, which means we used the rank of each generated test case to re-weight its contribution to the consensus set it belongs to. We test the effectiveness of using 1) the prompt which involves self-generated (SG) code as the test cases generated in this setting show higher correctness than the baseline placeholder setting and 2) the rank-based re-weighted (RW) test cases, in improving program synthesis performance on HumanEval+. Following Chen et al. [2023], we used a temperature of 0.8 to generate code and self-generated test cases. After obtaining the consensus set, we re-weight test case by $p^{i-1}$ with $i$ being its order in the model output, and we let $p = 0.8$. That is, instead of directly using their counting numbers, we use the sum of $p^{i-1}$ and the final score of a consensus set is then the sum of a) $\sum p^{i-1}$ and b) the number of code implementations in the consensus set, and code implementations in the consensus set with the highest score are considered as the best solutions. Table 6 shows the results. We compare CodeT with CodeT+SG, CodeT+RW, and CodeT+SG+RW. For CodeT, we follow their official implementation and generate $100 \times 5$ test cases for each problem. For fair comparison, we ensure that our solutions with SR and/or RW generate the same numbers of program implementations and test cases as CodeT does. Hence, for each problem in HumanEval+, we synthesize a program together with its 5 test cases for 100 times when SR and/or RW are incorporated, i.e., we have $i \in \{1, 2, 3, 4, 5\}$. It can be seen from the table that both SG and WR improves the program synthesis performance considerably on most LLMs, except for Incoder, CodeGen2-1B, CodeT5+, and SantaCoder for which the test cases generated in the placeholder setting show similar or even higher correctness than in the self-generated setting and SG fails with them. For some LLMs, SG is more powerful, while, on the other models including SantaCoder and StarCoder, RW is more powerful. By combining SG and RW, the program synthesis performance of most powerful LLMs in Table 6 improves, comparing to only using one of the two. On GPT-3.5-turbo and WizardCoder, which are the best two models in synthesizing programs on HumanEval+, we achieve +4.22% and +3.04% performance gains for CodeT, respectively, with SG & RW. \| Model \| Size \| Baseline \| CodeT \| + SG \| + RW \| + SG & RW \| \|----------------\|-------\|----------\|-------\|------\|------\|-----------\| \| InCoder \| 1.3B \| 6.99% \| 9.85% \| 9.45%\| 10.26%\| 9.98% \| \| CodeGen2 \| 1B \| 9.19% \| 15.15%\| 14.89%\| 15.67%\| 15.35% \| \| CodeT5+ \| 770M \| 12.95% \| 16.57%\| 16.28%\| 17.19%\| 16.98% \| \| SantaCoder \| 1.1B \| 15.21% \| 18.43%\| 18.17%\| 18.75%\| 18.63% \| \| CodeGen-Multi \| 16B \| 15.35% \| 24.50%\| 25.71%\| 25.72%\| 26.95% \| \| CodeGen2 \| 16B \| 19.33% \| 27.56%\| 28.51%\| 28.43%\| 29.63% \| \| CodeGen-Mono \| 16B \| 26.15% \| 35.63%\| 36.69%\| 36.63%\| 37.95% \| \| StarCoder \| 15B \| 27.90% \| 40.46%\| 41.21%\| 42.12%\| 43.15% \| \| CodeGeeX2 \| 6B \| 29.97% \| 44.16%\| 45.23%\| 44.92%\| 46.32% \| \| WizardCoder \| 15B \| 46.23% \| 58.41%\| 60.13%\| 59.60%\| 61.45% \| \| GPT-3.5-turbo \| - \| 61.70% \| 69.25%\| 72.45%\| 70.75%\| 73.47% \| Table 6: Program synthesis performance (Pass@1) of LLMs can be significantly improved by using our takeaway messages in Section 6. The experiment is on HumanEval+. 8 RELATED WORK Test case generation via program analysis. Generating reasonable test cases for analyzing programs is a long-standing problem in the software engineering community. Various program analysis techniques, e.g., fuzzing, have been developed for achieving this goal. AFL++ (Fioraldi et al., 2020) is the most popular tool which incorporates many techniques in this category. A major weakness of these techniques is understandability of the generated test cases. Test case generation via deep learning. The invention of transformer and self-supervised pre-training have brought a breakthrough to programming language processing and program testing (Fioraldi et al., 2020; Tufano et al., 2022; Dinella et al., 2022). After being trained in a self-supervised manner on a large and diverse code corpus, LLMs have demonstrated remarkable abilities in understanding and synthesizing programs. We have also witnessed the adaptation of pre-trained LLMs (e.g., ChatGPT) to fuzzing (Xia et al., 2023) very recently. Similarly, Lemieux et al. (2023) utilized Codex to provide example test cases for under-covered functions, which prevents the coverage improvements stall. Nevertheless, there still lack and require in-depth analyses and intensive comparisons of different LLMs in program testing, considering that powerful LLMs emerge continuously. For instance, the recent WizardCoder (Luo et al., 2023) exhibits an obvious program synthesis superiority over other contemporary open-source LLMs. In our study, we focus on the analyses and comparison of the LLMs in writing test code and generating test cases. Evaluation of Large Language Model. Recently, large language models (LLMs) has incited substantial interest in both academia and industry. In order to evaluate the capabilities of large language models, a variety of effort have been devoted from the perspectives of natural/programming language processing accuracy, robustness, ethics, biases, and trustworthiness, etc. For instance, PromptBench (Zhu et al., 2023) demonstrates that current LLMs are sensitive to adversarial prompts, and careful prompt engineering is necessary for achieving descent performance with them. Another example, DecodingTrust (Wang et al., 2023a), offers a multifaceted exploration of trustworthiness of the GPT models, especially GPT-3.5 and GPT-4. The evaluation expands beyond the typical trustworthiness concerns to include several new critical aspects. Agentbench (Liu et al., 2023b) evaluates LLM as agents on challenging tasks in interactive environments. Their experimental results show that, while top commercial LLMs present a strong ability of acting as agents in complex environments, there is a significant disparity in performance between them and their open-source competitors. 9 CONCLUSION In this paper, we have performed thorough analyses of recent LLMs (mostly LLMs for code) in testing programs/code. Through comprehensive experiments with 11 LLMs on programming benchmark datasets including HumanEval+ and MBPP (the sanitized version), we have uncovered a range of intriguing characteristics of these LLMs for program/code testing. We have illustrated how the program testing capabilities of these LLMs can be enhanced in comparing intensive empirical results in four different settings. Based on our findings, we are also capable of improving the performance of state-of-the-art LLMs in synthesizing programs/code with test cases of higher quality. As a preliminary research work, we believe our paper can provide new research insights and spark new ideas in program/code synthesis, test-case generation, and LLM understanding, and we look forward to future exploration in this direction in future work. REFERENCES Loubna Ben Allal, Raymond Li, Denis Kocetkov, Chenghao Mou, Christopher Akiki, Carlos Munoz Ferrandis, Niklas Muennighoff, Mayank Mishra, Alex Gu, Manan Dey, et al. Santacoder: don’t reach for the stars! arXiv preprint arXiv:2301.03988, 2023. Jacob Austin, Augustus Odena, Maxwell Nye, Maarten Bosma, Henryk Michalewski, David Dohan, Ellen Jiang, Carrie Cai, Michael Terry, Quoc Le, et al. Program synthesis with large language models. arXiv preprint arXiv:2108.07732, 2021. Bei Chen, Fengji Zhang, Anh Nguyen, Daoguang Zan, Ziqi Lin, Jian-Guang Lou, and Weizhu Chen. Codet: Code generation with generated tests. In The Eleventh International Conference on Learning Representations, 2023. 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x2rZGCbRRd	Another thought concerns use of the proposed DAG. Post-treatment variables are presumably omnipresent, and it can be difficult to know how to proceed when they are around (motivating this paper). For applicability in practice, I do not know when or whether investigators would be willing to assume the proposed graph in Figure 1c. Knowing more about the limitations of the proposed approach would help (e.g., discussion of limitations [the paper does not seem to currently have a limitations section]).	EXTRACTING POST-TREATMENT COVARIATES FOR HETEROGENEOUS TREATMENT EFFECT ESTIMATION Anonymous authors Paper under double-blind review ABSTRACT The exploration of causal relationships between treatments and outcomes, and the estimating causal effects from observational data, have garnered considerable interest in the scientific community recently. However, traditional causal inference methods implicitly assume that all covariates are measured prior to treatment assignment, while in many real-world scenarios, some covariates are affected by the treatment and collected post-treatment. In this paper, we demonstrate how ignoring or mishandling post-treatment covariates can lead to biased estimates of treatment effects, referred to as the "post-treatment bias" problem. We discuss the possible cases in which post-treatment bias may appear and the negative impact it can have on causal effect estimation. To address the challenge, we propose a novel variable decomposition approach to account for post-treatment covariates and eliminate post-treatment bias, based on a newly proposed causal graph for post-treatment causal inference analyses. Extensive experiments on synthetic, semi-synthetic, and real-world data demonstrate the superiority of our proposed method over state-of-the-art models for heterogeneous treatment effect estimation. 1 INTRODUCTION The estimation of treatment effects plays a pivotal role in decision-making across several influential domains, such as epidemiology (Dechartres et al., 2013), economics (Lin & Ye, 2007), and social sciences (Heckman, 1991). It enables the identification of causal relationships between treatment, such as smoking, and outcome of interest, for instance, heart disease. In recent years, a plethora of methods (Liu et al., 2020; Rosenbaum, 1987; Rosenbaum & Rubin, 1983; Wu et al., 2022; Frangakis & Rubin, 2002; Hullsiek & Louis, 2002; Athey & Imbens, 2016; Chipman et al., 2010; Wager & Athey, 2018; Atan et al., 2018; Hassanpour & Greiner, 2019; Johansson et al., 2016; Shalit et al., 2017; Yao et al., 2018) have emerged with a primary focus on addressing the estimation of causal effects using observational data. Nevertheless, these methods primarily center on mitigating the confounding bias introduced by confounders within the observational data through statistical or spatial mapping techniques. Other formidable challenges have not received adequate attention and resolution, just as we are about to discuss in this work, where we will explore and address the estimation bias stemming from post-treatment variables (Holland, 1986; Pearl, 2015). Despite the considerable success of the current methods in estimating treatment effects, they implicitly assume that all covariates are measured before the treatment or intervention is imposed and their values and distributions are not affected by the intervention, known as pre-treatment variables (Yao et al., 2021), and they mainly focus on eliminating the confounding bias. However, in many real-world scenarios (e.g., medical health), a significant proportion of covariates will be affected by the intervention, which are referred to as post-treatment variables (Holland, 1986; Pearl, 2015). For instance, in a study investigating the effect of smoking on the incidence of heart disease, post-treatment variables could be the occurrence of side effects (e.g., headache) or a certain medical measurement (e.g., blood pressure). Practitioners have increasingly directed their attention toward the role of post-treatment variables in causal inference. For instance, in (Bareinboim & Pearl, 2012; Bareinboim & Tian, 2015; Correa et al., 2018; Bareinboim et al., 2022), researchers utilized post-treatment variables to recover unbiased causal effects from selection bias. (Zhang et al., 2020) utilized post-treatment variables to remove confounding bias in image classification task. In this work, we focus on another problem: ignoring or mishandling post-treatment variables can lead to post-treatment bias (Montgomery et al., 2018; Coppock, 2019). Figure 1: (a)-(b) illustrate two cases of post-treatment bias. (c) shows the proposed causal graph with observed covariates $X$, unmeasured variables $U$, treatment $T$, and outcome $Y$. $C$, $Z_m$, and $Z_c$ denote confounders, mediation, and collider post-treatment variables, respectively. For example, as presented in Figure 1, the treatment variable $T$ indicates whether a person smokes, the post-treatment variable $Z$ represents whether a person’s blood pressure or pulse rate is normal or not, the outcome variable $Y$ represents the incidence of heart disease, $C$ is the confounder (e.g., Age), and $U$ is the risk factor that affects $Z$ and $Y$. For the sake of clarity, we treat $U$ as an unobserved variable (e.g., Health Status) here. We will elaborate on other scenarios in subsequent sections of this paper. In Figure 1(a), the causal effect of $T$ on $Y$ not only includes the direct effect from $T$ to $Y$, but also involves a mediating effect caused by the post-treatment variable $Z$. Noting that in this work we focus on the total treatment effect, which is more practical in real-world scenarios. If we fail to identify and separate $Z$ from other covariates, such as confounders $C$, when we adjust for confounders, the treatment effect through the mediation pathway can be lost, leading to biased treatment effect estimation. In Figure 1(b), the post-treatment variable $Z$ is a collider affected by both treatment $T$ and risk factor $U$. If we condition on blood pressure $Z$ equaling normal, the treated group, i.e., the smoking population, may consist of more people with better health status than the control group. This creates an unblocked path between $T$ and $U$ in the causal graph, introducing another post-treatment bias due to the imbalance of health status between different groups resulting from conditioning on $Z$. Although recent studies $[Kuroki, 2000; Pearl, 2015; VanderWeele, 2009]$ have discussed the harm caused by ignoring post-treatment variables in causal inference, they either address only experimental studies $[Coppock, 2019; Homola et al., 2020; King, 2010; Montgomery et al., 2018]$ or consider post-treatment bias due to mediators alone $[Li et al., 2022]$, ignoring other situations that may lead to post-treatment bias. In this study, we tackle the challenge of post-treatment bias mitigation by employing representation learning techniques to derive post-treatment variables from observed covariates. We delve into the examination of two distinct scenarios capable of inducing post-treatment bias and introduce a comprehensive framework namely PoNet. Within this framework, we focus on inferring representations of confounding factors and post-treatment variables directly from the observed covariates. Subsequently, we put forth an inference policy designed to facilitate the estimation of heterogeneous treatment effects, aiming to achieve estimates of interest with addressing post-treatment bias. ## Preliminaries ### Post-treatment Bias We begin with the concepts of post-treatment variables that can result in post-treatment bias. As shown in Figure 1, Mediation Post-treatment Variable, denoted by $Z_m$, refers to the variables that is affected by the treatment $T$ and influences the outcome $Y$; Collider Post-treatment Variable, denoted by $Z_c$, refers to the variables that is affected by both treatment $T$ and risk factor $U$ but has no direct effect on outcome. Incorrect handling of the two aforementioned variables can lead to post-treatment bias. To illustrate this bias, we use the linear structural causal model as an example, demonstrating the consequences of ignoring or mishandling each of the two post-treatment variables. For the case of mediation post-treatment variable in Figure 1(a), assuming the causal model is formulated as $Y = \tau T + \beta C + \eta Z = (\tau + \eta \gamma)T + \beta C$, the total treatment effect of $T$ on $Y$ is $\tau + \eta \gamma$, then the estimated average treatment effect from observational data can be formulated as: \[ \Delta_a = E(Y\|T = 1) - E(Y\|T = 0) = \tau + \beta(E(C\|T = 1) - E(C\|T = 0)) + \eta(E(Z\|T = 1) - E(Z\|T = 0)), \] it is well known that eliminating confounding bias is an essential step in causal inference, and the common practice is to adjust for the confounder. However, if the mediation post-treatment variable $ Z $ is not extracted and separated from confounders, the post-treatment variable $ Z $ will be incorrectly adjusted (i.e., $ E(Z\|T = 1) - E(Z\|T = 0) $ tends to be 0) when we adjust for the confounders, then the estimated average treatment effect of $ T $ on $ Y $ would be biased that $ \Delta_a = \tau \neq \tau + \eta \gamma $, which is the mediation post-treatment bias. For the collider post-treatment variables in Figure 1(b), assuming the causal model is $ Y = \tau T + \beta U $ where $ U $ is the unmeasured variable that affects the post-treatment variable $ Z $ and outcome $ Y $. In this model, noting that $ T $ and $ U $ are independent and the causal effect of $ T $ on $ Y $ is $ \tau $. Similarly, the estimated average treatment effect from observational data in this model can be formulated as: \[ \Delta_b = E(Y\|T = 1) - E(Y\|T = 0) = \tau + \beta(E(U\|T = 1) - E(U\|T = 0)), \] in this model, if we condition on the post-treatment variable $ Z $, a backdoor path will be open between $ T $ and $ U $, which means $ T $ and $ U $ are no longer independent, then $ E(U\|T = 1) - E(U\|T = 0) $ in the last equation is not equal to 0 if there is an imbalance in the distribution of $ U $ between the control and treated group. Therefore, the estimated treatment effect of $ T $ on $ Y $ is biased that $ \Delta_b = \tau + \beta c \neq \tau $ where $ c $ represents the discrepancy in distributions of $ U $ between two groups. The detailed derivation of equation (1) and (2) can be found in Appendix. ### 2.2 Causal Mechanism for Post-treatment Modeling Causal Effect Identification. We propose a new causal graph in Figure 1(c) to account for post-treatment bias. Let $ X, T $ and $ Y $ denote the observed covariates, treatment and outcome, respectively. $ Z_c, Z_m, C $ and $ U $ represent the collider, mediation post-treatment variables, confounders and unmeasured risk factor. Here, we provide a formal theorem about the identification of heterogeneous treatment effects: Theorem 1. (Identifiability of Heterogeneous Treatment Effect) If we can recover, $ p(Z_m\|T, X) $ and $ p(C\|X) $ from the observational data, then we can recover and identify the intervention distribution for estimating heterogeneous treatment effect of $ T $ on $ Y $, which can be expressed by: \[ p(Y_u\|do(T), X) = \int_{C,Z_m} p(Y_u\|T, Z_m, C)p(C\|X)p(Z_m\|T, X), \] where $ Y_u $ represents the observed outcome with underlying risk factor $ U = u $. It is noteworthy that $ U $ and $ T $ are marginally independent. Therefore, neglecting to account for the presence of $ U $ in our estimates does not introduce bias in our assumption. This theorem indicates that the probability distribution of an outcome under an intervention $ T $ is determined by the distribution of confounders and mediation post-treatment variables, rather than by collider post-treatment variables. This aligns directly with our analysis in Section 2.1. Proof of the theorem can be found in the Appendix. Minimally Sufficient Guarantee: Building upon Theorem 1, which establishes that the treatment effect can be identified through the recovery of confounders ($ C $) and mediation post-treatment variables ($ Z_m $), we propose a further theorem that asserts that these variables are minimally sufficient (Silvey, 2017), for the optimal parameters $ \theta $ for estimating unbiased treatment effects. Theorem 2. The joint set of inferred factors for $ C, Z_m $ is minimally sufficient for the optimal parameters $ \theta $ which estimation of unbiased treatment effects needs. This theorem implies that the inferred factors for $ C $ and $ Z_m $ encapsulate all the necessary information that is required for optimally estimating the parameters $ \theta $ for the recovery of the treatment effects. A detailed proof of this theorem is provided in the Appendix. $ C, Z_m $ and even $ Y $ could be the risk factor. In our earlier analysis, we elucidated the bias-creation process resulting from unmeasured risk factors. It is only natural to ponder the following questions: (1) What happens if $ C $ exerts a causal influence on $ Z_c $? (2) What if $ Z_m $ causally affects $ Z_c $? (3) What if outcome $ Y $ affects $ Z_c $? For case (1), conditioning on $ Z_c $ introduces a new pathway between $ T $ and $ Y $ through confounder $ C $. The resulting bias in this case is analogous to confounding bias due to coincidence resulting from the distribution imbalance of confounder $C$. This bias can be mitigated by adjusting for the confounders. For case (2), a similar situation arises. Conditioning on $Z_c$ introduces an additional unblocked pathway between $T$ and $Y$ through $Z_m$, and the distribution imbalance on $Z_m$ becomes the source of bias. However, unlike confounder $C$, we cannot adjust or balance $Z_m$ in the same way, as doing so would falsely erase the mediating treatment effect from $T$ to $Y$. For case (3) as mentioned in (Hernan & Robins, 2020), conditioning on collider $Z_c$ will create a new causal path between $T$ and $Y$, which will disrupt the true causal effect of $T$ on $Y$. In summary, the critical point lies in isolating $Z_c$ from the observed covariates and excluding it during inference. It is worth noting that the model and inference policy we propose in the following sections are capable of addressing both of these scenarios as long as we can recover $Z_c$ from observed covariates. Therefore, we omit these two causal pathways in the proposed causal graph, given the analysis provided above. 3 METHODOLOGY 3.1 REPRESENTATION LEARNING FOR THE THREE UNDERLYING FACTORS Learning of post-treatment variables. We employ neural networks to infer the representations of post-treatment variables, $Z_m$ and $Z_c$, from observed covariates. Given the distinct nature of post-treatment variables under different treatment assignments, we construct two separate neural network channels to infer these representations. To be more precise, we seek to learn two representation functions, $f_{me}(x, t)$ and $f_{co}(x, t)$, with respect to the treatment assignment, mapping the observed covariates $X \in \mathbb{R}^d$ to an $m$-dimensional latent space. Each treatment assignment is accommodated by parametrizing these representation learning functions through the stacking of multiple fully connected layers, resulting in representations $z_m$ and $z_c$ for the mediation and collider post-treatment variables, respectively. Learning of confounders. Analogously, we establish a mapping function $f_c(x)$, also with $X \in \mathbb{R}^d \rightarrow \mathbb{R}^m$, to derive representations of confounders from the observed covariates. This function is parametrized using multiple fully connected layers, and the resultant confounder representation is denoted as $c$. Balancing confounders by optimal transport theory. To control the confounding bias, we need to balance the distribution of the inferred representation of confounders between treated and control groups. Optimal transport theory (Villani et al., 2009; Torres et al., 2021) is a mathematical framework that allows us to measure the distance between two probability distributions. Here, we adopt the Wasserstein distance (Villani & Villani, 2009) and minimize it between the treated and control group in terms of representations of confounders. We denote the distance as $L_{wass}$ and feed it into the loss function for optimization. More details can be found in the Appendix. 3.2 RECONSTRUCTION MODULE The causal graph shows that the post-treatment variable $Z_c$ is affected only by treatment and observed covariates, and has no direct impact on outcome $Y$. However, the supervised information of the training model only comes from factual outcomes in most cases, thus the lack of supervised information on $Z_c$ in training data makes it challenging to learn its representations. To model confounders and post-treatment variables more effectively, we propose a neural network-based reconstruction module. This module incorporates learned representations of confounders, collider and mediation post-treatment variables to generate an output that closely resembles the original covariates. The reconstruction module can be formulated as: $$\hat{x} = \Psi(z_m, z_c, c),$$ where $\hat{x}$ is the reconstructed covariates, $\Psi$ is a decoder function which is parameterized by multiple fully connected layers. 3.3 MUTUAL INFORMATION REGULARIZER BY KERNEL DENSITY ESTIMATION Separating confounders and post-treatment variables is essential for unbiased treatment effect estimation. When confounders’ representation includes information from mediation post-treatment variables $Z_m$, controlling for confounders may introduce mediation post-treatment bias. If $Z_m$ contains confounder information, addressing confounding bias might not be fully effective. Moreover, if $Z_m$ contains collider post-treatment information, conditioning on $Z_m$ can lead to collider post-treatment bias. Precise differentiation between confounders and post-treatment variables is thus critical for reliable treatment effect estimation. To achieve the goal of separating confounders and post-treatment variables, we design a Mutual Information Minimization Regularizer (MIMR) based on the following corollary yielded from the causal graph in Figure 1(c): Corollary 1. Given covariates $X$ and $T$, the confounders $C$, mediation post-treatment variables $Z_m$ and collider post-treatment variables $Z_c$ are independent to each other, i.e., $C \perp Z_m \perp Z_c \|\| X, T$. Specifically, We propose to utilize kernel density estimation (Terrell & Scott [1992]), a non-parametric method, to fit the distributions of the representations of these variables and measure their independence. Here we take the kernel density estimation of the representations of $C$ and $Z_m$ as an example. Let $\{c^0, c^1, ..., c^N\}$ be the representation samples of confounders $C$ drawn from the marginal distribution $D_C(\cdot)$, $\{z^0_m, z^1_m, ..., z^N_m\}$ be the representation samples of mediation post-treatment variables $Z_m$ drawn from the marginal distribution $D_{Z_m}(z_m)$, the kernel density estimates of the marginal distribution $D_C(\cdot)$, $D_{Z_m}(\cdot)$ and the joint distribution $D_{CZ_m}(\cdot)$ are given by: $$\hat{D}_C(c) = \frac{1}{N} \sum_{i=1}^{N} K_h(c - c^i),$$ $$\hat{D}_{Z_m}(z_m) = \frac{1}{N} \sum_{i=1}^{N} K_h(z_m - z^i_m),$$ $$\hat{D}_{CZ_m}(c, z_m) = \frac{1}{N} \sum_{i=1}^{N} K_h((c - c^i)\|\| (z_m - z^i_m)),$$ where $K(\cdot)$ is the kernel function, $h$ is the bandwidth parameter that controls the smoothness of the estimate, $\|\|$ denotes the concatenation, and $K_h(\cdot)$ is called the scaled kernel. The kernel function can be any non-negative function that integrates to 1. In this work, we adopt the Gaussian kernel as the kernel function. Then the mutual information between $Z_m$ and $C$ can be estimated by: $$\hat{I}(Z_m, C) = \sum_c \sum_{z_m} \hat{D}_{CZ_m}(c, z_m) \log \frac{\hat{D}_{CZ_m}(c, z_m)}{\hat{D}_C(c)\hat{D}_{Z_m}(z_m)},$$ similarly, we can obtain the estimated mutual information $\hat{I}(Z_c, C)$ and $\hat{I}(Z_m, Z_c)$. ### 3.4 Objective Function and Inference Policy Loss for predicting potential outcomes. With the inferred representations of confounders, mediation post-treatment variables $c$, $z_m$ and the treatment assignment $t \in \{0, 1\}$, we can develop a prediction function $\hat{y}_t^i = f_y(c^i, z^i_m, t_i)$ parameterized by stacking fully connected layers, then minimize the mean square error (MSE) $\mathcal{L}_y = \frac{1}{N} \sum_{i=1}^{N} (\hat{y}_t^i - y_t)^2$. Loss for covariate reconstruction. A loss function is typically defined to measure the discrepancy between the reconstructed covariate $\hat{x}$ and the true covariate $x$. One common loss function is the mean squared error (MSE), which is given by $\mathcal{L}_{re} = \frac{1}{N} \sum_{i=1}^{N} \|\|x_i - \hat{x}_i\|\|_2^2$. Other loss functions, such as the binary cross-entropy (De Boer et al. [2005]) or Kullback-Leibler divergence (Joyce [2011]), can also be used depending on the nature of the input data and the modeling objective. Loss for mutual information minimization regularizer. Here we combine the three terms of estimated mutual information by kernel density estimation to guarantee the independence of confounders, mediation post-treatment variables and collider post-treatment variables from each other: $$\mathcal{L}_{MIMR} = \hat{I}(Z_m, C) + \hat{I}(Z_c, C) + \hat{I}(Z_m, Z_c).$$ Overall loss function. The overall objective function of PoNet is defined by: $$\mathcal{L} = \mathcal{L}_y + \alpha \mathcal{L}_{MIMR} + \beta \mathcal{L}_{re} + \gamma \mathcal{L}_{wass} + \eta \|\|\Theta\|\|_2^2,$$ where $\alpha, \beta, \gamma, \eta$ are hyper-parameters to control the trade-off of the corresponding terms with the other terms, $\|\|\Theta\|\|_2^2$ is imposed on the learning weights $\Theta$ of the model to avoid the over-fitting. Inference Policy. Based on the previous analysis, to avoid the post-treatment bias in treatment effect estimation, the inference policy should be subject to the following two rules: First, do not condition on collider post-treatment variables $Z_c$; Second, condition on the mediation post-treatment variables $Z_m$ but do not adjust them when conducting the inference. 4 EXPERIMENTS 4.1 EXPERIMENT SETTING Baselines. We compare our proposed model with several baselines, which can fall into three categories: (1) Linear regression based models, including OLS1 (Shalit et al., 2017): An S-learner using linear regression, treating the treatment variable as just another covariate, OLS2 (Shalit et al., 2017): A T-learner that trains separate linear regression models for treated and control individuals; (2) Tree based models, including BART (Chipman et al., 2010): a nonparametric Bayesian regression approach based on multiple tree models, Causal Forest (Wager & Athey, 2018): a extension of random forest model for estimating treatment effects in causal perspective; (3) Neural network-based models, including Counterfactual Regression (CFR) (Shalit et al., 2017): A deep learning-based estimator that balances the distribution of confounders’ representations, TARNet (Johansson et al., 2016): A variant of CFR that removes the built-in representation balancing component, GAN-ITE (Yoon et al., 2018): Uses Generative Adversarial Nets to capture uncertainty in counterfactual distributions and estimate the treatment effect, CEVAE (Louizos et al., 2017): A deep latent variable model that leverages VAE (Kingma & Welling, 2013) and proxy learning to estimate the causal effect, TEDVAE (Zhang et al., 2021): A variational inference approach that infers latent factors from observed variables and disentangles them for treatment effect estimation. Evaluation Metrics. In this work, we adopt two widely used metrics for evaluating the performance of causal estimators. First, we adopt Rooted Precision in Estimation of Heterogeneous Effect ($\sqrt{\epsilon_{PEHE}}$) to measure the accuracy of conditional average treatment effect (CATE): \[ \sqrt{\epsilon_{PEHE}} = \sqrt{\frac{1}{N} \sum_{i=1}^{N} (\tau_i - \hat{\tau}_i)^2} \] where $\tau_i = y_i^{t_i=1} - y_i^{t_i=0}$ and $\hat{\tau}_i = \hat{y}_i^{t_i=1} - \hat{y}_i^{t_i=0}$ are the ground truth CATE and the estimated CATE, respectively. Second, we also adopt the mean square error (MSE) for measuring the accuracy of predicting outcomes: \[ \epsilon_{MSE} = \frac{1}{N} \sum_{i} (\hat{y}_i - y_i)^2 \] in some experiments. Please refer to the Appendix for a more detailed description of the experiment setting. 4.2 SYNTHETIC DATA First, we evaluate the proposed model on synthetic data. Here we only introduce the outline of the synthetic dataset due to the page limit, more details about the data generation process can be found in the Appendix. Roughly speaking, we generate the confounders (denoted by $x_C$) with the dimension $d_C$ from a multivariate Gaussian distribution, then generate the treatment $T$ from the Bernoulli distribution based on the generated confounders. After getting the treatment assignment, we generate the mediation and collider post-treatment variables (denoted by $x_{Z_m}$ and $x_{Z_c}$) based on the generated treatment with the dimension $d_{Z_m}$ and $d_{Z_c}$, respectively. Then we combine the three generated factors $\{x_C, x_{Z_m}, x_{Z_c}\}$ as the covariates $x$. Capability of identifying each underlying factor. Here we want to verify if the proposed model PoNet can identify the three underlying factors $\{x_C, x_{Z_m}, x_{Z_c}\}$ from the observed covariates $x$. In the proposed model, we develop three networks $f_c(\cdot)$, $f_{me}(\cdot)$ and $f_{co}(\cdot)$ for learning the representations of factors of confounders, mediation and collider post-treatment variables respectively. Taking the representation learning network $f_{me}(\cdot)$ for learning $Z_m$ as an example, the first layer’s dimension of the learned weights of network $f_{me}$ is $(d_C + d_{Z_m} + d_{Z_c}) \times K$ where $K$ is the dimension of the hidden layer. We can partition the learned weight matrix into two slices: (1) $S_{Z_m}$ with dimension $d_{Z_m} \times K$, that connects the variables belonging to $x_{Z_m}$ to the representation network $f_{me}$, (2) $S_{other}$ with dimension $(d_C + d_{Z_c}) \times K$, that connects other variables not belonging to $x_{Z_m}$ to the representation network $f_{me}$. For the network $f_{me}(\cdot)$ which can identify the mediation post-treatment variables, it is expected that the network can filter out the information of confounders $C$ and collider post-treatment variables $Z_c$, and retain the information of mediation post-treatment. --- 1The anonymous link of the source code of the proposed model PoNet is: [https://anonymous.4open.science/r/Ponet-37F2/](https://anonymous.4open.science/r/Ponet-37F2/) variables $Z_m$. In other words, if the network $f_{me}$ can accurately identify $x_{Z_m}$, the neuron links connected to $x_{Z_m}$ are more active than those connected to $x_C$ and $x_{Z_c}$, which can be reflected in the values of the learned weights, i.e., the average absolute values of $S_{Z_m}$ is higher than that of $S_{other}$. As shown in Figure 2, we plot the radar charts to visualize the capability of PoNet in identifying the three underlying factors. Each vertex on the circles represents the dimension of $\{x_C, x_{Z_m}, x_{Z_c}\}$, each vertex of the polygon measures the learned weights’ the average absolute value for each dimension setting. We can see that for each underlying factor, the average absolute value of $S_\ast$ ($\ast = Z_m, Z_c$ or $C$) is higher than that in $S_{other}$, which is consistent with what we expect. Therefore, it empirically shows that the proposed model is capable of identifying different underlying factors. ### 4.3 Semi-synthetic Data We then evaluate the proposed model PoNet using the semi-synthetic dataset PeerRead (Kang et al., 2018). The PeerRead dataset comprises peer reviews of computer science papers, with each entry representing an author. The features of each entry are bag-of-word representations extracted from the titles and abstracts of their papers. In this dataset, each author is categorized based on whether their papers contain specific keywords, and the outcome variable is the number of citations their papers receive. To simulate the necessary variables, we generate confounders $C$ and treatment assignments and introduce artificial mediation post-treatment variables and collider post-treatment variables $Z_m$ and $Z_c$, respectively, based on the generated treatments. For more information on the detailed data generation process, please refer to the Appendix. Treatment effect estimation: Here we consider the different dimension of the post-treatment variables as $d = 50, 100, 200$ and evaluate the performance of PoNet in comparison to other baselines for treatment effect estimation. The results of the experiment are presented in Table 1. Notably, PoNet outperforms the state-of-the-art methods in treatment effect estimation, as it effectively addresses the issue of post-treatment bias that is often neglected by other approaches. Verifying the effectiveness of inference policy. Based on the previous analysis, the inference policy is to condition on the mediation post-treatment variables but not on the collider post-treatment variables. To validate the effectiveness of this policy, we introduce two variants of the inference policy in our model: (1) PoNet with $Z_c$, which conditions on the collider post-treatment variables $Z_c$ while also conditioning on the mediation post-treatment variables $Z_m$; (2) PoNet w/o $Z_m$, which neither conditions on the mediation post-treatment variables $Z_m$ nor on the collider post-treatment variables $Z_c$. We compare the performance of these two policy variants with that of the original inference policy, and the experimental results are illustrated in Figure 3. Please note that the standard deviation line has been scaled down to enhance the clarity of the results. The findings indicate that the two variants of the inference policy do not perform as well as the original policy. The measurement in terms of $\sqrt{\epsilon_{PEHE}}$ provides evidence of the existence of post-treatment bias, and it \| Methods \| $d = 50$ \| $d = 100$ \| $d = 200$ \| \|-------------\|----------------\|----------------\|----------------\| \| OLS1 \| 2.241 ± 0.481 \| 3.052 ± 1.013 \| 3.193 ± 1.944 \| \| OLS2 \| 2.002 ± 0.396 \| 2.742 ± 0.751 \| 3.023 ± 1.354 \| \| BART \| 2.258 ± 0.498 \| 2.915 ± 0.998 \| 3.385 ± 1.958 \| \| Causal Forest\| 2.088 ± 0.440 \| 2.609 ± 0.846 \| 2.973 ± 1.743 \| \| CEVAE \| 2.303 ± 0.196 \| 3.037 ± 0.340 \| 3.188 ± 0.731 \| \| GANITE \| 2.414 ± 0.290 \| 2.756 ± 0.422 \| 2.529 ± 1.100 \| \| TEDVAE \| 2.150 ± 0.737 \| 2.669 ± 0.809 \| 2.602 ± 1.678 \| \| TARNet \| 2.420 ± 0.288 \| 2.582 ± 0.720 \| 2.722 ± 1.072 \| \| CFR \| 2.437 ± 0.287 \| 2.611 ± 0.733 \| 2.720 ± 1.098 \| \| PoNet \| 1.393 ± 0.178 \| 1.869 ± 0.502 \| 2.053 ± 0.829 \| Table 1: $\sqrt{\epsilon_{PEHE}}$ performance comparison on PeerRead, lower is better, $d$ denotes the dimension of post-treatment variables. Figure 3: Performance comparison between original inference policy and that w/o $Z_m$ or with $Z_c$. Further demonstrates that the proposed model with the original inference policy effectively captures the post-treatment variables and mitigates the post-treatment bias. Additionally, the measurement in terms of $\epsilon_{MSE}$ reveals that the decomposition of post-treatment variables contributes to the prediction of outcomes. It is intuitive that incorporating $Z_c$ into the outcome prediction introduces noise, while excluding $Z_m$ from the input results in the loss of valuable information, both of which can compromise the accuracy of the predictive model. Ablation Study. We further investigate the impact of different components of the proposed model PoNet on the treatment effect estimation. Specifically, we conduct the ablation study by deriving the following variants of the proposed model PoNet: (1) PoNet w/o Confounder Balancing, denoted by PoNet w/o CB; (2) PoNet w/o Reconstruction Module, denoted by PoNet w/o RM; (3) PoNet w/o Mutual Information Regularizer, denoted by PoNet w/o MI. We compare the performance of the three variants with the original model PoNet, the comparison results between the three variants and the original model are presented in Table 2. We can see that the original PoNet outperforms the other three variants, due to the following reasons: (1) PoNet w/o CB fails to adequately adjust for confounders, leading to confounding bias; (2) PoNet w/o RM is unable to effectively model the underlying factors, particularly the collider post-treatment variables that do not contribute to the outcome. Consequently, this can introduce potential post-treatment bias; (3) PoNet w/o MI is incapable of accurately separating the three underlying factors from each other, resulting in the potential generation of post-treatment and confounding bias. ### Table 2: Ablation Study on PeerRead in terms of $\sqrt{\epsilon_{PEHE}}$ \| Variants \| $d = 50$ \| $d = 100$ \| $d = 200$ \| \|-------------------\|--------------\|--------------\|--------------\| \| PoNet w/o CB \| 1.468 ± 0.186\| 1.919 ± 0.464\| 2.127 ± 0.887\| \| PoNet w/o RM \| 1.521 ± 0.186\| 1.965 ± 0.454\| 2.157 ± 0.910\| \| PoNet w/o MI \| 1.445 ± 0.202\| 1.939 ± 0.514\| 2.185 ± 0.900\| \| PoNet \| 1.393 ± 0.178 \| 1.869 ± 0.502 \| 2.053 ± 0.829 \| #### 4.4 Real-world Data MIMIC-III (Johnson et al., 2016) is a publicly available dataset of de-identified health-related data for over 40,000 patients who were admitted to the intensive care units of the Beth Israel Deaconess Medical Center between 2001 and 2012. The dataset includes data on patient demographics, vital signs, laboratory test results, medications, diagnoses, procedures, and imaging reports. Here we follow (Melnychuk et al., 2022) and use 25 vital signs and 3 static features as the covariates, whether using vasopressor as the treatment, the blood pressure as the outcome. Given that treatment can have an influence on numerous vital signs, it is essential to take into account the effects of post-treatment variables when estimating causal effects in real-world scenarios, particularly within healthcare data. In order to showcase the experimental results, we randomly sampled data from two distinct time steps, denoted as $t_1$ and $t_2$, with each time step consisting of 6133 samples. Given that true counterfactuals are no longer accessible in real-world data, we evaluate the performance of predicting factual outcomes. The results are presented in Table 3. Our proposed model surpasses all state-of-the-art baselines, providing evidence of the superiority of our approach. In addition to excelling in causal effect estimation, our model also demonstrates strong performance in outcome prediction tasks. This can be attributed to our precise segmentation of covariates into distinct factors, thereby eliminating irrelevant variables from consideration. TEDVAE stands out among the baselines, showcasing commendable performance. This can be attributed to its variable decomposition approach, modeling irrelevant variables subsequently discarding them. However, an important limitation is its disregard for modeling post-treatment variables, which reduces its accuracy in prediction. More importantly, we also want to verify if the proposed model PoNet can distinguish the three different underlying factors (e.g., confounders $C$, mediation post-treatment variables $Z_m$ and collider post-treatment variables $Z_c$). We employ t-SNE to reduce the dimensionality of the representations... Table 3: Performance of factual outcome prediction on real-world data MIMIC-III, lower is better. \| $\epsilon_{MSE}$ \| $t_1$ In-Sample \| $t_1$ Out-of-Sample \| $t_2$ In-Sample \| $t_2$ Out-of-Sample \| \|------------------\|-----------------\|--------------------\|-----------------\|--------------------\| \| OLS1 \| 0.351 ± 0.004 \| 0.378 ± 0.005 \| 0.410 ± 0.004 \| 0.431 ± 0.006 \| \| OLS2 \| 0.330 ± 0.004 \| 0.343 ± 0.005 \| 0.393 ± 0.005 \| 0.394 ± 0.005 \| \| BART \| 0.383 ± 0.007 \| 0.335 ± 0.023 \| 0.370 ± 0.006 \| 0.368 ± 0.019 \| \| Causal Forest\| 0.361 ± 0.012 \| 0.374 ± 0.039 \| 0.404 ± 0.013 \| 0.435 ± 0.043 \| \| CEVAE \| 0.315 ± 0.003 \| 0.328 ± 0.004 \| 0.357 ± 0.003 \| 0.359 ± 0.005 \| \| GANITE \| 0.335 ± 0.004 \| 0.327 ± 0.003 \| 0.363 ± 0.004 \| 0.361 ± 0.005 \| \| TEDVAE \| 0.284 ± 0.003 \| 0.293 ± 0.004 \| 0.304 ± 0.004 \| 0.331 ± 0.003 \| \| Tarnet \| 0.302 ± 0.004 \| 0.318 ± 0.003 \| 0.340 ± 0.004 \| 0.360 ± 0.005 \| \| CFR \| 0.301 ± 0.003 \| 0.308 ± 0.004 \| 0.339 ± 0.003 \| 0.361 ± 0.004 \| \| PoNet \| 0.281 ± 0.004 \| 0.283 ± 0.004 \| 0.279 ± 0.003 \| 0.320 ± 0.004 \| of the three underlying factors computed by the PoNet model. The representations are reduced to 2 dimensions and plotted using kernel density estimate to visualize the distribution of the three factors in the low-dimensional space as shown in Figure 4. The result clearly indicates that the inferred representations of the three factors from the proposed model exhibit significantly different distributions. This observation provides strong evidence for PoNet’s ability to effectively distinguish between the three underlying factors, even from the real-world cases. 5 RELATED WORKS Previous causal inference works mainly adjust confounders to control the confounding bias. Reweighting methods (Ciu et al., 2020; Rosenbaum, 1987; Rosenbaum & Rubin, 1983; Wu et al., 2022) alter the instance weighting, such as by Inverse Propensity Weighting (IPW) (Glynn & Quinn, 2010), to create a more balanced comparison group. Stratification methods (Frangakis & Rubin, 2002; Hullsiek & Louis, 2002) divide the population into subgroups with similar covariate distributions to infer causal effects within each subgroup. Tree and forest-based methods like BART (Hill, 2011), Causal Forest (Wager & Athey, 2018), and Recursive partitioning (Athey & Imbens, 2016) estimate treatment effects for different subgroups of the population by building decision trees or random forests. Representation-based learning methods like CFR (Shalit et al., 2017), SITE (Yao et al., 2018), TARNet (Johansson et al., 2016), GANITE (Yoon et al., 2018), CEVAE (Louizos et al., 2017), TEDVAE (Zhang et al., 2021) map observed covariates to latent space, reducing the distribution discrepancy between treated and control groups, and have shown to be superior in estimating causal effects. The above methods assume that all variables are pre-treatment. However, post-treatment variables can introduce bias, as discussed in Montgomery et al. (2018). Various studies (Coppock, 2019; Homola et al., 2020; King, 2010) outline ways to avoid post-treatment bias, but they mainly focus on experimental studies. A recent causal model (Li et al., 2022) addresses the inference of mediation post-treatment variables but does not consider collider post-treatment variables, thus potential collider post-treatment bias could be introduced. 6 CONCLUSION In this study, we examine the sources and mechanisms of post-treatment bias and introduce a novel deep learning-based approach for decomposing variables and inferring post-treatment variables from observed covariates, utilizing a newly proposed causal graph specifically designed for post-treatment analysis. We also develop various components to infer the representations of confounders and post-treatment variables, thereby eliminating both confounding bias and post-treatment bias. Through extensive experiments on synthetic, semi-synthetic, and real-world datasets, we demonstrate the superior performance of our model compared to other state-of-the-art models in estimating heterogeneous treatment effects. REFERENCES Onur Atan, James Jordon, and Mihaela Van der Schaar. Deep-treat: Learning optimal personalized treatments from observational data using neural networks. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 32, 2018. Susan Athey and Guido Imbens. Recursive partitioning for heterogeneous causal effects. Proceedings of the National Academy of Sciences, 113(27):7353–7360, 2016. Susan Athey and Guido W Imbens. Machine learning methods for estimating heterogeneous causal effects. stat, 1050(5):1–26, 2015. Elias Bareinboim and Judea Pearl. Controlling selection bias in causal inference. In Artificial Intelligence and Statistics, pp. 100–108. PMLR, 2012. Elias Bareinboim and Jin Tian. Recovering causal effects from selection bias. 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EHKS0oXuku	In the experiments, are you evaluating deterministic predictions using the mean parameters of the variational distribution, or are you using BMA through sampling from the variational distribution? In the context of BNNs, performance evaluation with BMA is necessary.	JENSEN-SHANNON DIVERGENCE BASED NOVEL LOSS FUNCTIONS FOR BAYESIAN NEURAL NETWORKS Anonymous authors Paper under double-blind review ABSTRACT We aim to overcome the limitations of Kullback-Leibler (KL) divergence-based variational inference (VI) used in Bayesian Neural Networks (BNNs), which stem from the lack of boundedness of KL-divergence. These limitations include unstable optimization, poor approximation, and difficulties in approximating light-tailed posteriors, which are well documented in the literature. To overcome these limitations, we propose two novel loss functions for BNNs based on Jensen-Shannon (JS) divergences, which are more general, and one of them is bounded. We employ a constrained optimization framework to formulate these loss functions due to the intractability of the JS divergence-based VI. Further, we show that the two loss functions presented here generalize the conventional KL divergence-based loss function for BNNs. In addition to establishing stability in optimization, we perform rigorous theoretical analysis, and empirical experiments to evaluate the performance of the proposed loss functions. The empirical experiments are performed on the Cifar-10 data set with various levels of added noise and a highly biased histopathology data set. Our analysis and experiments suggest that the proposed losses perform better than the KL divergence-based loss and significantly better than their deterministic counterpart. Similar improvements by the present approach are also observed on the Cifar-100 data set. We also perform experiments on six other regression datasets and compare the performance with the existing VI approaches for BNNs. 1 INTRODUCTION Despite the widespread success of deep neural networks (DNNs) and convolutional neural networks (CNNs) in numerous applications (Samarasinghe [2016], Li et al. [2021]), they suffer from overfitting when the data set is small, noisy, or biased (Buda et al. [2018], Thiagarajan et al. [2021]). Further, due to deterministic parameters, CNNs cannot provide a robust measure of uncertainty. Without a measure of uncertainty in the predictions, erroneous predictions by these models may lead to catastrophic failures in applications that necessitate high accuracy such as autonomous driving and medical diagnosis. Several methods were developed to provide prediction intervals as a measure of uncertainty in neural networks (Kabir et al. [2018]). Amongst these, Bayesian methods have gained eminence due to their rigorous mathematical foundation for uncertainty quantification through their stochastic parameters (Jospin et al. [2022], Kabir et al. [2018]). A Bayesian neural network (BNN) has stochastic parameters whose posterior distribution is learned through the Bayes rule (Tishby et al. [1989], Denker & LeCun [1990], Goan & Fookes [2020], Gal [2016]). Since the posterior distribution of parameters is intractable, the two most commonly used techniques to approximate them are the Variational Inference (VI) (Hinton & Van Camp [1993], Barber & Bishop [1998], Graves [2011], Hernández-Lobato & Adams [2015]) and the Markov Chain Monte Carlo Methods (MCMC) (Neal [2012], Welling & Teh [2011]). MCMC methods comprise a set of algorithms to sample from arbitrary and intractable probability distributions. Inference of posterior using MCMC algorithms can be very accurate but they are computationally demanding (Robert et al. [2018]). An additional limitation of MCMC algorithms is that they do not scale well with the model size. The VI is a technique to approximate an intractable posterior distribution by a tractable distribution called the variational distribution. The variational distribution is learned by minimizing an objective function derived from its dissimilarity with respect to the true posterior (Blundell et al., 2015). VI methods are efficient and they scale well for larger networks and have gained significant popularity. Most of the VI techniques in the literature use the KL divergence as a measure of the aforementioned dissimilarity. However, the KL divergence is unbounded which may lead to failure during training as reported in Hensman et al. (2014); Dieng et al. (2017); Deasy et al. (2020). In addition, KL divergence is asymmetric and thus it does not qualify as a metric. Therefore, it is imperative to explore alternative divergences for VI that can alleviate these limitations. In regards to exploring alternate divergences, Renyi’s α-divergences have been introduced for VI in Li & Turner (2016). They proposed a family of variational methods that unified various existing approaches. A χ-divergence-based VI has been proposed in Dieng et al. (2017) that provides an upper bound of the model evidence. Additionally, their results have shown better estimates for the variance of the posterior. Along these lines, an f-Divergence based VI has been proposed in Wan et al. (2020) to use VI for all f-divergences that unified the Reyni divergence (Li & Turner, 2016) and χ divergence (Dieng et al., 2017) based VIs. While these recent works (Li & Turner, 2016; Dieng et al., 2017; Wan et al., 2020) mainly focused on obtaining a generalized/unified VI framework, the present work specifically attempts to alleviate the limitations (unboundedness and asymmetry) of the KL divergence-based VI through the Jensen-Shanon (JS) divergence. As a result, two novel loss functions are proposed, which outperform the KL loss in applications that require regularization. A modification to the skew-geometric Jensen-Shanon (JS) divergence has been proposed in Deasy et al. (2020) to introduce a new loss function for Variational Auto Encoders (VAEs), which has shown a better reconstruction and generation as compared to existing VAEs. 1.1 Key contributions In the present work, we propose two novel loss functions for BNNs, which are based on: 1) the skew-geometric JS divergence (denoted as JS-G) and 2) a novel modification to the generalized JS divergence (denoted as JS-A). The primary contribution of this work is that it resolves the unstable optimization issue by leveraging the boundedness of the novel JS-A divergence. We show that these JS divergence-based loss functions are generalizations of the state-of-the-art KL divergence-based ELBO loss function. In addition to addressing the stability of the optimization, through rigorous analysis we explain why these loss functions should perform better. In addition, we derive the conditions under which the proposed skew-geometric JS divergence-based loss function regularises better than that of the KL divergence. Further, we show that the loss functions presented in this work perform better for image classification problems where the data set has noise or is biased towards a particular class. In our work, we provide both closed-form and MC-based algorithms for implementing the two JS divergences. The MC implementation can include priors of any family. The present work is different from the existing work on JS divergence-based VI (Deasy et al., 2020) for the following reasons: (i) The JS-G divergence proposed in the previous work is unbounded like KL which is resolved by the JS-A divergence proposed in this work. (ii) Deasy et al. (2020) introduced the JS-G divergence-based loss for variational autoencoders (VAEs). In the present work, the distributions of parameters of BNNs are learned, which are numerous, as opposed to a small number of latent factors typically found in VAEs. (iii) The previous work is restricted to Gaussian priors due to the closed-form implementation, which this work overcomes through MC implementation. 2 Mathematical Background 2.1 Background: KL and JS divergences The KL divergence between two random variables $ P $ and $ Q $ on a probability space $ \Omega $ is defined as \[ KL[p \|\| q] = \int_{\Omega} p(x) \log \left[ \frac{p(x)}{q(x)} \right] dx, \] where $ p(x) $ and $ q(x) $ are the probability distributions of $ P $ and $ Q $ respectively. The KL divergence is widely used in literature to represent the dissimilarity between two probability distributions for applications such as VI. However, it has limitations such as the asymmetric property, i.e. $ KL[p \|\| q] \neq KL[q \|\| p] $, and unboundedness, i.e. the divergence is infinite when $ q(x) = 0 $ and $ p(x) \neq 0 $. These limitations may lead to difficulty in approximating light-tailed posteriors as reported in Hensman et al. (2014). To overcome these limitations a symmetric JS divergence can be employed which is defined as $ \text{JS}[p \|\| q] = \frac{1}{2} \text{KL}[p \|\| (p + q)/2] + \frac{1}{2} \text{KL}[q \|\| (p + q)/2] $. It can be further generalized as, \[ \text{JS}^{A_\alpha}[p \|\| q] = (1 - \alpha) \text{KL}(p \|\| A_\alpha) + \alpha \text{KL}(q \|\| A_\alpha) \] where, $ A_\alpha $ is the weighted arithmetic mean of $ p $ and $ q $ defined as $ A_\alpha = (1 - \alpha)p + \alpha q $. Although this JS divergence is symmetric and bounded, unlike the KL divergence its analytical expression cannot be obtained even when $ p $ and $ q $ are Gaussians. To overcome this difficulty a generalization of the JS divergence using the geometric mean was proposed in Nielsen (2019). By using the weighted geometric mean $ G_\alpha(x, y) = x^{1-\alpha}y^\alpha $, where $ \alpha \in [0, 1] $, for two real variables $ x $ and $ y $, they proposed the following family of skew geometric divergence \[ \text{JS}^{G_\alpha}[p \|\| q] = (1 - \alpha) \text{KL}(p \|\| G_\alpha(p, q)) + \alpha \text{KL}(q \|\| G_\alpha(p, q)) \] The parameter $ \alpha $ called a skew parameter, controls the divergence skew between $ p $ and $ q $. However, the skew-geometric divergence in Eq. 2 fails to capture the divergence between $ p $ and $ q $ and becomes zero for $ \alpha = 0 $ and $ \alpha = 1 $. To resolve this issue, Deasy et al. (2020) used the reverse form of geometric mean $ G'_\alpha(x, y) = x^\alpha y^{1-\alpha} $, with $ \alpha \in [0, 1] $ for JS divergence to use in variational autoencoders. Henceforth, only this reverse form is used for the geometric mean. The JS divergence with this reverse of the geometric mean is given by \[ \text{JS-G}[p \|\| q] = (1 - \alpha) \text{KL}(p \|\| G'_\alpha(p, q)) + \alpha \text{KL}(q \|\| G'_\alpha(p, q)) \] This yields KL divergences in the limiting values of the skew parameter. Note that for $ \alpha \in [0, 1] $, \[ \text{JS-G}(p\|\|q)_\alpha = \text{JS-G}(p\|\|q)_{1-\alpha} \] which is not symmetric. However, for $ \alpha = 0.5 $, the JS-G is symmetric with $ \text{JS-G}(p\|\|q)_{\alpha=0.5} = \text{JS-G}(p\|\|q)_{\alpha=0.5} $. The geometric JS divergences, $ \text{JS}^{G_\alpha} $ and $ \text{JS-G} $ given in Eq. 2 and Eq. 3 respectively, have analytical expressions when $ p $ and $ q $ are Gaussians. However, they are unbounded like the KL divergence. Whereas, the generalized JS divergence $ \text{JS}^{A_\alpha} $ in Eq. 1 is both bounded and symmetric. ### 2.2 Background: Variational Inference Given a set of training data $ D = \{x_i, y_i\}_{i=1}^N $ and test input, $ x \in \mathbb{R}^P $, we learn a data-driven model (e.g., a BNN) to predict the probability $ P(y\|x, D) $ of output $ y \in Y $, where $ Y $ is the output space. The posterior probability distribution ($ P(w\|D) $) of the parameters $ w $ of BNN, can be obtained using the Bayes’ rule: \[ P(w\|D) = \frac{P(D\|w)P(w)}{P(D)} \] Where $ P(D\|w) $ and $ P(w) $ are the likelihood term and the prior distribution respectively. The term $ P(D) $, called the evidence, involves marginalization over the distribution of weights: \[ P(D) = \int_{\Omega_w} P(D\|w)P(w)dw \] Using the posterior distribution of weights, the predictive distribution of the output can be obtained by marginalizing the weights as $ P(y\|x, D) = \int_{\Omega_w} P(y\|x, w)P(w\|D)dw $. The term $ P(D) $ in the Bayes’ rule is intractable due to marginalization over $ w $, which in turn makes $ P(w\|D) $ intractable. To alleviate this difficulty, the posterior is approximated using variational inference. In variational inference, the unknown intractable posterior $ P(w\|D) $ is approximated by a known simpler distribution $ q(w\|\theta) $ called the variational posterior having parameters $ \theta $. The set of parameters $ \theta $ for the model weights are learned by minimizing the divergence (e.g., KL divergence) between $ P(w\|D) $ and $ q(w\|\theta) $ as shown in Blundell et al. (2015). \[ \theta^* = \arg \min_\theta \text{KL}[q(w\|\theta) \|\| P(w\|D)] = \arg \min_\theta \int q(w\|\theta) \left[ \log \frac{q(w\|\theta)}{P(w)P(D\|w)} + \log P(D) \right] dw \] Note that the term $ \log P(D) $ in Eq. 4 is independent of $ \theta $ and thus can be eliminated. The resulting loss function $ F(D, \theta) $, which is to be minimised to learn the optimal parameters $ \theta^* $ is expressed as: \[ F_{KL}(D, \theta) = \text{KL}[q(w\|\theta) \|\| P(w)] - \mathbb{E}_{q(w\|\theta)}[\log P(D\|w)] \] This loss is known as the variational free energy or the evidence lower bound (ELBO) (Graves, 2011; Blundell et al., 2015). 3 METHODS In this section, we provide a modification to the generalized JS divergence, formulations of JS divergence-based loss functions for BNNs, and insights into the advantages of the proposed loss. 3.1 PROPOSED MODIFICATION TO THE GENERALIZED JS DIVERGENCE The generalised JS divergence given in Eq. [1] fails to capture the divergence between $ p $ and $ q $ in the limiting cases of $ \alpha $ since, \[ [\text{JS}^{\alpha}(p \|\| q)]_{\alpha=0} = 0; \quad [\text{JS}^{\alpha}(p \|\| q)]_{\alpha=1} = 0 \] (6) To overcome this limitation we propose to modify the weighted arithmetic mean as $ A'_\alpha = \alpha p + (1 - \alpha)q, \alpha \in [0, 1] $ which modifies the generalized JS divergence as, \[ \text{JS-A}(p \|\| q) = (1 - \alpha)\text{KL}(p \|\| A'_\alpha) + \alpha\text{KL}(q \|\| A'_\alpha) \] (7) Hence, this yields KL divergences in the limiting cases of $ \alpha $ as, \[ [\text{JS-A}(p \|\| q)]_{\alpha=0} = \text{KL}(p \|\| q); \quad [\text{JS-A}(p \|\| q)]_{\alpha=1} = \text{KL}(q \|\| p) \] (8) Eq. [7] ensures that $ \text{JS}(P \|\| q) = 0 $ if and only if $ P = q, \forall \alpha \in [0, 1] $. This is necessary since the divergence is a metric to represent statistical dissimilarity. Theorem 1: Boundedness of the modified generalized JS divergence For any two distributions $ P_1(t) $ and $ P_2(t), t \in \Omega $, the value of the JS-A divergence is bounded such that, \[ \text{JS-A}(P_1(t) \|\| P_2(t)) \leq -(1 - \alpha)\log \alpha - \alpha \log(1 - \alpha), \quad \text{for } \alpha \in (0, 1) \] (9) The proof of Theorem 1 is presented in App. B. Due to this boundedness property of the JS-A divergence, the ensuing loss functions overcome the instability in optimization that is encountered in the KL divergence-based loss. We provide a comparison of symmetry (at $ \alpha = 0.5 $) and boundedness for divergences used in this work in Table 1 and in App. A and B. \| Divergence \| Bounded \| Symmetric \| Analytical expression \| \|------------\|---------\|-----------\|----------------------\| \| KL \| ✗ \| ✗ \| ✓ \| \| JS-A \| ✓ \| ✓ \| ✗ \| \| JS-G \| ✗ \| ✓ \| ✓ \| Table 1: Properties of various divergences 3.2 INTRACTABILITY OF THE JS DIVERGENCE-BASED LOSS FUNCTIONS FORMULATED THROUGH THE VARIATIONAL INFERENCE APPROACH In this subsection, we demonstrate that the JS divergence-based variational inference is intractable. If the JS-G divergence is used instead of the KL divergence in the VI setting (see Eq. [4]), the optimization problem becomes, \[ \theta^* = \arg \min_\theta \text{JS-G}[q(w\|\theta) \|\| P(w\|\mathcal{D})] \] (10) The loss function can then be written as, \[ F_{\text{JSG}}(\mathcal{D}, \theta) = \text{JS-G}[q(w\|\theta) \|\| P(w\|\mathcal{D})] = (1 - \alpha)\text{KL}(q \|\| G'_\alpha(q, P)) + \alpha\text{KL}(P \|\| G'_\alpha(q, P)) \] (11) Where, $ G'_\alpha(q, P) = q(w\|\theta)^{\alpha}P(w\|\mathcal{D})^{(1-\alpha)} $. Rewriting the first and the second term in Eq. [11] as, \[ T_1 = (1 - \alpha)^2 \int q(w\|\theta) \log \left[ \frac{q(w\|\theta)}{P(w\|\mathcal{D})} \right] dw; \quad T_2 = \alpha^2 \int P(w\|\mathcal{D}) \log \left[ \frac{P(w\|\mathcal{D})}{q(w\|\theta)} \right] dw \] (12) A detailed derivation of terms $ T_1 $ and $ T_2 $ is given in App. C. Term $ T_1 $ is equivalent to the loss function in Eq. [5] multiplied by a constant $ (1 - \alpha)^2 $. The term $ P(w\|\mathcal{D}) $ in $ T_2 $ is intractable as explained in section 2.2. Therefore the JS-G divergence-based loss function given in Eq. [11] cannot be used to find the optimum parameter $ \theta^* $ which contrasts the KL divergence-based loss function in Eq. [5]. Similarly, the JS-A divergence-based loss function obtained through VI is also intractable. We address this issue of intractability in the following subsection. 3.3 Proposed JS divergence-based loss functions formulated through a constrained optimization approach To overcome the intractability of the variational inference, we propose to use a constrained optimization framework, following Higgins et al. (2017); Deasy et al. (2020), to derive JS divergence-based loss functions for BNNs. We also show that such a loss function is a generalization of the loss function obtained through the variational inference. Given a set of training data $ \mathcal{D} $, we are interested in learning the probability distribution $ q(w\|\theta) $ of network parameters such that, the likelihood of observing the data given the parameters is maximized. Thus, the optimization problem can be written as \[ \max_{\theta} \mathbb{E}_{q(w\|\theta)} [\log P(\mathcal{D}\|w)] \] (13) Where $ \theta $ is a set of parameters of the probability distribution $ q(w\|\theta) $. This optimization is constrained to make $ q(w\|\theta) $ similar to a prior $ P(w) $. This leads to a constrained optimization problem as given below: \[ \max_{\theta} \mathbb{E}_{q(w\|\theta)} [\log P(\mathcal{D}\|w)] \quad \text{subject to } D(q(w\|\theta) \|\| P(w)) < \epsilon \] (14) where $ \epsilon $ is a real number that determines the strength of the applied constraint and $ D $ is a divergence measure. Following the KKT approach, the Lagrangian function corresponding to the constrained optimization problem can be written as \[ L = \mathbb{E}_{q(w\|\theta)} [\log P(\mathcal{D}\|w)] - \lambda (D(q(w\|\theta) \|\| P(w)) - \epsilon) \] (15) Since $ \epsilon $ is a constant it can be removed from the optimization. Also changing the sign of the above equations leads to the following loss function that needs to be minimized, \[ \tilde{\mathcal{F}}_D = \lambda D(q(w\|\theta) \|\| P(w)) - \mathbb{E}_{q(w\|\theta)} [\log P(\mathcal{D}\|w)] \] (16) This loss function reproduces the ELBO loss (Blundell et al., 2015) when KL divergence is used and $ \lambda $ is taken as 1. In the following, we obtain loss functions for two JS divergences, namely, the geometric JS divergence, and the modified generalised JS divergence. 3.3.1 Geometric JS divergence Using the modified skew-geometric JS divergence (JS-G) as the measure of divergence in Eq. 16 leads to the following loss function: \[ \tilde{\mathcal{F}}_{JSG} = \lambda \text{JS-G}(q(w\|\theta) \|\| P(w)) - \mathbb{E}_{q(w\|\theta)} [\log P(\mathcal{D}\|w)] \] (17a) \[ = \lambda (1-\alpha) \text{KL}(q \|\| G'_\alpha(q,P_w)) + \lambda \alpha \text{KL}(P_w \|\| G'_\alpha(q,P_w)) - \mathbb{E}_{q(w\|\theta)} [\log P(\mathcal{D}\|w)] \] (17b) Note, \[ \text{KL}(q \|\| G'_\alpha(q,P_w)) = \int q(w\|\theta) \log \frac{q(w\|\theta)}{q(w\|\theta)^{\alpha} P(w)^{1-\alpha}} dw = (1-\alpha) \int q(w\|\theta) \log \frac{q(w\|\theta)}{P(w)} dw \] \[ \text{KL}(P_w \|\| G'_\alpha(q,P_w)) = \int P(w) \log \frac{P(w)}{q(w\|\theta)^{\alpha} P(w)^{1-\alpha}} dw = \alpha \int P(w) \log \frac{P(w)}{q(w\|\theta)} dw \] Hence, the loss function can be written as, \[ \tilde{\mathcal{F}}_{JSG} = \lambda (1-\alpha)^2 \mathbb{E}_{q(w\|\theta)} \left[ \log \frac{q(w\|\theta)}{P(w)} \right] + \lambda \alpha^2 \mathbb{E}_{P(w)} \left[ \log \frac{P(w)}{q(w\|\theta)} \right] - \mathbb{E}_{q(w\|\theta)} [\log P(\mathcal{D}\|w)] \] (18) In Eq. 18, the first term is the mode seeking reverse KL divergence $ \text{KL}(q(w\|\theta)\|\|P(w)) $ and the second term is the mean seeking forward KL divergence $ \text{KL}(P(w)\|\|q(w\|\theta)) $. Therefore, the proposed loss function offers a weighted sum of the forward and reverse KL divergences in contrast to only the reverse KL divergence in ELBO. Whereas the likelihood part remains identical. The relative weighting between the forward and the reverse KL divergences can be controlled by the parameter $ \alpha $. The proposed loss function would ensure better regularisation by imposing stricter penalization if the posterior is away from the prior distribution which will be demonstrated in detail in Sec. 3.4.1. The parameters $ \lambda $ and $ \alpha $ can be used to control the amount of regularisation. --- 1 The constrained optimization approach-based loss functions are marked by an overhead tilde. 2 $ \lambda $ is taken as 1 for $ \tilde{\mathcal{F}}_{JSG} $ in this work unless otherwise stated. 3.3.2 Modified Generalised JS Divergence Using the modified Generalised JS divergence (JS-A) as the measure of divergence in Eq. 16 leads to the following loss function: \[ \tilde{F}_{JSA} = \lambda \text{JS-A}(q(w\|\theta) \|\| P(w)) - \mathbb{E}_{q(w\|\theta)}[\log P(D\|w)] = \lambda(1-\alpha)\text{KL}(q \|\| A'_\alpha(q,P_w)) + \lambda\alpha\text{KL}(P_w \|\| A'_\alpha(q,P_w)) - \mathbb{E}_{q(w\|\theta)}[\log P(D\|w)] \] (19) Where, $A'_\alpha(q,P_w) = \alpha q + (1-\alpha)P_w$. The above equation, Eq. 19 can be expanded as, \[ \tilde{F}_{JSA} = \lambda(1-\alpha)\mathbb{E}_{q(w\|\theta)}\left[\log \frac{q(w\|\theta)}{A'_\alpha(q,P_w)}\right] + \lambda\alpha\mathbb{E}_{P(w)}\left[\log \frac{P(w)}{A'_\alpha(q,P_w)}\right] - \mathbb{E}_{q(w\|\theta)}[\log P(D\|w)] \] (20) Note that the proposed loss functions in Eq. 18 and Eq. 20 yield the ELBO loss for $\alpha = 0$ and $\lambda = 1$. The minimization algorithms for the loss functions Eq. 18 and Eq. 20 are given in the App. D. 3.4 Insights into the Proposed JS Divergence-Based Loss Functions To better understand the proposed JS divergence-based loss functions, we use a contrived example to compare them against the conventional KL divergence-based loss function. In the following, we explore the regularization ability of the proposed loss functions. Further insights on Monte Carlo estimates are given in App. E. 3.4.1 Regularisation Performance of JS Divergences Let two Gaussian distributions $q = \mathcal{N}(\mu_q, \sigma^2_q)$ and $P = \mathcal{N}(\mu_p, \sigma^2_p)$ represent the posterior and the prior distribution of a parameter in a BNN. The KL, JS-A, and JS-G divergences are evaluated by varying the mean and variance of the distribution $q$. This emulates the learning of the network parameter during training. Fig. 1 shows that as the posterior distribution ($q$) moves away from the prior distribution ($P$), the JS divergences increase more rapidly than the KL divergence. In the case of the JS-A divergence in Fig. 1b and 1d, this is achieved by a higher value of $\lambda$. This implies that a greater penalization is offered by JS divergences than the KL divergence as the posterior deviates away from the prior. Thus, by assuming small values for the means of prior distributions we can regularize better by the proposed JS divergences. In practice, zero mean Gaussian priors are widely accepted for BNNs. For such priors, higher penalization of the loss function implies pushing the parameters’ mean closer to zero while learning the complexity of the data. In doing this, we can achieve better regularization. This regularization process requires finding optimal values of $\alpha$ and $\lambda$ through hyperparameter optimization. In the following subsection, we theoretically analyze the regularization performance of the JS-G divergence. ![Figure 1](image-url) Figure 1: Comparison of the KL and the JS divergences of distributions $P$ and $q$. (a) and (b) $\sigma^2_q, \mu_p, \sigma^2_p$ are fixed and $\mu_q$ is varied. (c) and (d) $\mu_q, \mu_p, \sigma^2_p$ are fixed and $\sigma^2_q$ is varied. The fixed values of the parameters are $\mu_q = 0.1, \sigma^2_q = 0.01, \mu_p = 0, \sigma^2_p = 0.1$ 3.4.2 Condition for Better Regularisation of $\tilde{F}_{JSG}$ The above example shows that the JS-G divergence is greater than the KL for the given Gaussian distributions. To generalize it further, we propose the following theorems that hold for any two arbitrary distributions. Theorem 2. For any two arbitrary distributions $ P $ and $ q $ such that $ P \neq q $, $ \tilde{F}_{JSG} > F_{KL} $ if and only if \[ \alpha > \frac{2 \text{KL}(q\|\|P)}{\text{KL}(q\|\|P) + \text{KL}(P\|\|q)} \in (0, \infty) \] Proof: Assuming, $ \tilde{F}_{JSG} - F_{KL} > 0 $ and from Eq. 5 and Eq. 18 we have, \[ (1 - \alpha)^2 \text{KL}(q\|\|P) + \alpha^2 \text{KL}(P\|\|q) - \text{KL}(q\|\|P) > 0 \] \[ (\alpha^2 - 2\alpha) \text{KL}(q\|\|P) + \alpha^2 \text{KL}(P\|\|q) > 0 \] This leads to, \[ \alpha > \frac{2 \text{KL}(q\|\|P)}{\text{KL}(q\|\|P) + \text{KL}(P\|\|q)} \] This proves that if $ \tilde{F}_{JSG} > F_{KL} $ then $ \alpha > \frac{2 \text{KL}(q\|\|P)}{\text{KL}(q\|\|P) + \text{KL}(P\|\|q)} $. The converse can be proved similarly. A detailed proof is shown in App. F. Theorem 3. If $ P = N(\mu_p, \sigma_p^2) $ and $ q = N(\mu_q, \sigma_q^2) $ are Gaussian distributions and $ P \neq q $, then \[ \frac{2 \text{KL}(q\|\|P)}{\text{KL}(q\|\|P) + \text{KL}(P\|\|q)} < 1 \] if and only if $ \sigma_p^2 > \sigma_q^2 $. Proof: Assuming $ \frac{2 \text{KL}(q\|\|P)}{\text{KL}(q\|\|P) + \text{KL}(P\|\|q)} < 1 $, we get \[ \text{KL}(P\|\|q) > \text{KL}(q\|\|P) \] Since $ P = N(\mu_p, \sigma_p^2) $ and $ q = N(\mu_q, \sigma_q^2) $, Eq. 21 can be written as, \[ \ln \frac{\sigma_q^2}{\sigma_p^2} + \frac{\sigma_p^2 + (\mu_q - \mu_p)^2}{\sigma_q^2} - 1 > \ln \frac{\sigma_p^2}{\sigma_q^2} + \frac{\sigma_q^2 + (\mu_p - \mu_q)^2}{\sigma_p^2} - 1 \] Denoting $ \gamma = \frac{\sigma_p^2}{\sigma_q^2} $, we get, \[ \gamma - \frac{1}{\gamma} + \ln \frac{1}{\gamma} - \ln \gamma + \frac{(\mu_q - \mu_p)^2}{\sigma_q^2} - \frac{(\mu_p - \mu_q)^2}{\gamma \sigma_q^2} > 0 \] or, \[ \ln \left[ \frac{1}{\gamma^2} \exp \left( \gamma - \frac{1}{\gamma} \right) \right] + \frac{(\mu_q - \mu_p)^2}{\sigma_q^2} \left( 1 - \frac{1}{\gamma} \right) > 0 \] (22) This condition Eq. 22 is satisfied only when $ \gamma > 1 $, which implies $ \sigma_p^2 > \sigma_q^2 $. Thus if \[ \frac{2 \text{KL}(q\|\|P)}{\text{KL}(q\|\|P) + \text{KL}(P\|\|q)} < 1 \] then $ \sigma_p^2 > \sigma_q^2 $. This result is also observed in Fig. 1c. The converse can be proved similarly as shown in App. G. Corollary: From Theorem 2 and 3: $ \tilde{F}_{JSG} > F_{KL} $ if $ \sigma_p^2 > \sigma_q^2 $ and $ \forall \alpha \in (0, 1] $ such that $ \alpha > \frac{2 \text{KL}(q\|\|P)}{\text{KL}(q\|\|P) + \text{KL}(P\|\|q)} $. Where, $ P $ and $ q $ are Gaussians and $ P \neq q $. 4 EXPERIMENTS In order to demonstrate the advantages of the proposed losses in comparison to the KL loss, we performed experiments. We have implemented the divergence part of the JS-G loss and the JS-A loss via a closed-form expression and a Monte-Carlo method respectively, in these experiments. 4.1 DATA SETS The following experiments were performed on two data sets: the Cifar-10 data set (Krizhevsky et al., 2009) and a histopathology data set (Janowczyk & Madabhushi, 2016; Cruz-Roa et al., 2014; Paul Mooney, 2017). To demonstrate the effectiveness of regularisation, varying levels of Gaussian noise were added to the normalized Cifar-10 data set for training, validation, and testing. We also used a histopathology data set which is highly biased towards one class. Further details on these data sets and the pre-processing steps used here are provided in App. H. 4.2 Hyperparameter Optimisation and Network Architecture Hyperparameters for all the networks considered here are chosen through hyperparameter optimization. A Tree-structured Parzen Estimator (TPE) algorithm (Bergstra et al., 2011) is used which is a sequential model-based optimization approach. A python library Hyperopt (Bergstra et al., 2013) is used to implement this optimization algorithm over a given search space. An optimization is performed to maximize the validation accuracy for different hyperparameter settings of the network. The results of the hyperparameter optimization are given in App. I. The architecture of all the networks used in this work follows the ResNet-18 V1 model (He et al., 2016) without the batch normalization layers. The network parameters are initialized with the weights of ResNet-18 trained on the Imagenet data set (Krizhevsky et al., 2012). 5 Results and Discussions This section presents the classification results and the performance comparison between the KL loss and the proposed JS losses. Performance evaluations on the Cifar-100 dataset along with the comparison between the proposed losses and deterministic networks, $\lambda$KL loss, and unaltered versions of JS divergences are provided in App. J. Computational costs of the losses are compared in App. K. 5.1 Training and Validation Three Bayesian CNNs were trained by minimizing the KL loss and the proposed JS losses. Training of the networks is done until the loss converges or the validation accuracy starts to decrease. Training of the Cifar-10 data set is performed with varying levels of noise intensity. Accuracy of training and validation sets for noise $N(\mu = 0, \sigma = 0.9)$ is presented for both KL loss and the proposed JS losses in Fig. 2a. For the histopathology data set, a learning rate scheduler is used during training in which the learning rate is multiplied by a factor of 0.1 in the 4th, 8th, 12th, and 20th epochs. Fig. 2b shows the accuracy of training and validation of the histopathology set for the KL loss and the proposed JS losses. It is evident that the KL loss learns the training data too well and fails to generalize for the unseen validation set on both data sets. Whereas, the proposed JS losses regularise better and provide more accurate results for the validation set. 5.2 Testing Results obtained on the test sets of the Cifar-10 data set and the histopathology data set are presented in this section. The test results correspond to the epoch in which the validation accuracy was maximum. Five runs were performed with different mutually exclusive training and validation tests to compare the results of the KL loss and the proposed JS losses. The accuracy of the noisy Cifar-10 test data set at varying noise levels is presented in Fig. 3a and Fig. 3b. It is evident that the accuracy of both the proposed JS losses is better than KL for all the noise level cases. Further, the difference in accuracy between KL loss and the JS losses shows an increasing trend with increasing noise levels. This demonstrates the regularising capability of the proposed JS losses. The results of the five runs Figure 3: Accuracy on (a) and (b) the Cifar-10 test data at different noise levels (c) histopathology test data. Each box chart displays the median as the center line, the lower and upper quartiles as the box edges, and the minimum and maximum values as whiskers. (d) ROC curves and (e)-(g) Confusion matrices for different losses for the histopathology data set. of the KL loss and the proposed JS losses on the biased histopathology data set are compared in Fig. 3c. It is evident that both the proposed JS losses perform better than the KL loss in all five runs with different training and validation sets. Since this data set is biased toward the negative class, the improvement in performance shown by the proposed JS losses is attributed to better regularisation and generalization capabilities of the loss functions. The receiver operating characteristic (ROC) curve is plotted in Fig. 3d, for the classification of the histopathology data set. The proposed JS losses perform better than the KL loss in terms of the area under the curve (AUC). The confusion matrices in Fig. 3e-3g show that in addition to improving the accuracy of predictions, the proposed JS-G and the JS-A losses reduce the number of false negative predictions by 11.7% and 12.8% respectively, as compared to the KL loss. Given that the data set is biased towards the negative class, this is a significant achievement. 6 LIMITATIONS The proposed loss functions have two additional hyperparameters that need to be optimized to realize their full potential, which increases computational expenses. Whenever such expenses can not be afforded, the parameters can be set to the fixed values $\alpha = 0$ and $\lambda = 1$ to recover the KL loss. 7 CONCLUSIONS We summarize the main findings of this work in the following. Firstly, the bounded JS-A divergence introduced in this work resolves the issue of unstable optimization associated with KL divergence-based loss functions. Secondly, we introduced two novel loss functions for Bayesian neural networks utilizing JS divergences through a rigorous theoretical formulation. The proposed loss functions encompass the KL divergence-based loss and extend it to a wider class of symmetric and bounded divergences. Thirdly, better regularization performance by the proposed loss functions compared to the state-of-the-art is established analytically and numerically. Fourthly, empirical experiments on standard data sets having bias or with various degrees of added noise, demonstrate performance enhancement by the proposed loss functions in comparison to the existing methods. REFERENCES David Barber and Christopher M Bishop. Ensemble learning in bayesian neural networks. Nato ASI Series F Computer and Systems Sciences, 168:215–238, 1998. James Bergstra, Rémi Bardenet, Yoshua Bengio, and Balázs Kégl. Algorithms for hyper-parameter optimization. In 25th annual conference on neural information processing systems (NIPS 2011), volume 24. Neural Information Processing Systems Foundation, 2011. James Bergstra, Daniel Yamins, and David Cox. Making a science of model search: Hyperparameter optimization in hundreds of dimensions for vision architectures. In International conference on machine learning, pp. 115–123. Proceedings of Machine Learning Research, 2013. Charles Blundell, Julien Cornebise, Koray Kavukcuoglu, and Daan Wierstra. 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4SrzKsJocx	And the measure RC_0 is introduced because the ideal uncorrelation is not achievable if the sample is few. But RC_0 is computed based on multiple random trials. That is to say, the evaluation metric is not deterministic.	SIMULTANEOUS DIMENSIONALITY REDUCTION: A DATA EFFICIENT APPROACH FOR MULTIMODAL REPRESENTATIONS LEARNING Anonymous authors Paper under double-blind review ABSTRACT Current experiments frequently produce high-dimensional, multimodal datasets—such as those combining neural activity and animal behavior or gene expression and phenotypic profiling—with the goal of extracting useful correlations between the modalities. Often, the first step in analyzing such datasets is dimensionality reduction. We explore two primary classes of approaches to dimensionality reduction (DR): Independent Dimensionality Reduction (IDR) and Simultaneous Dimensionality Reduction (SDR). In IDR methods, of which Principal Components Analysis is a paradigmatic example, each modality is compressed independently, striving to retain as much variation within each modality as possible. In contrast, in SDR, one simultaneously compresses the modalities to maximize the covariation between the reduced descriptions while paying less attention to how much individual variation is preserved. Paradigmatic examples include Partial Least Squares and Canonical Correlations Analysis. Even though these DR methods are a staple of statistics, their relative accuracy and data set size requirements are poorly understood. We use a generative linear model to synthesize multimodal data with known variance and covariance structures to examine these questions. We assess the accuracy of the reconstruction of the covariance structures as a function of the number of samples, signal-to-noise ratio, and the number of varying and covarying signals in the data. Using numerical experiments, we demonstrate that linear SDR methods consistently outperform linear IDR methods and yield higher-quality, more succinct reduced-dimensional representations with smaller datasets. Remarkably, regularized CCA can identify low-dimensional weak covarying structures even when the number of samples is much smaller than the dimensionality of the data, which is a regime challenging for all dimensionality reduction methods. Our work corroborates and explains previous observations in the literature that SDR can be more effective in detecting covariation patterns in data. These findings strengthen the intuition that SDR should be preferred to IDR in real-world data analysis when detecting covariation is more important than preserving variation. 1 INTRODUCTION Many modern experiments across various fields generate massive multimodal data sets. For instance, in neuroscience, it is common to record the activity of a large number of neurons while simultaneously recording the resulting animal behavior [Stringer et al., 2019; Steinmetz et al., 2021; Urai et al., 2022; Krakauer et al., 2017]. Other examples include measuring gene expressions of thousands of cells and their corresponding phenotypic profiles, or integrating gene expression data from different experimental platforms, such as RNA-Seq and microarray data [Clark et al., 2013; Zheng et al., 2017; Svensson et al., 2018; Huntley et al., 2015; Lorenzi et al., 2018]. In economics, important variables such as inflation are often measured using combinations of macroeconomic indicators as well as indicators belonging to different economic sectors [Gosselin & Tkacz, 2001; Baillie et al., 2002; Freyaldenhoven, 2022; Rudd, 2020]. In all of these examples, an important goal is to estimate statistical correlations among the different modalities. Analyses usually begin with dimensionality reduction (DR) into a smaller and more interpretable representation of the data. We distinguish two types of DR: independent (IDR) and simultaneous (SDR) (Martini & Nemenman, 2023). In the former, each modality is reduced independently, while aiming to preserve its variation, which we call self signal. In the latter, the modalities are compressed simultaneously, while maximizing the covariation (or the shared signal) between the reduced descriptions and paying less attention to preserving the individual variation. It is not clear if IDR techniques, such as the Principal Components Analysis (PCA) (Hotelling, 1933), are well-suited for extracting shared signals since they may overlook features of the data that happen to be of low variance, but of high covariance (Colwell et al., 2014; Borga et al., 1997). In particular, poorly sampled weak shared signals, common in high-dimensional datasets, can exacerbate this issue. SDR techniques, such as Partial Least Squares (PLS) (Wold et al., 2001) and Canonical Correlations Analysis (CCA) (Hotelling, 1936), are sometimes mentioned as more accurate in detecting weak shared signal (Chin & Newsted, 1999; Hair et al., 2011; Pacharawongsakda & Theeramunkong, 2016). However, the relative accuracy and data set size requirements for detecting the shared signals in the presence of self signals and noise remain poorly understood for both classes of methods. In this study, we aim to assess the strengths and limitations of linear IDR, represented by PCA, and linear SDR, exemplified by PLS and CCA, in detecting weak shared signals. For this, we use a generative linear model that captures key features of relevant examples, including noise, the self signal, and the shared signal components. Using this model, we analyze the performance of the methods in different conditions. Our goal is to assess how well these techniques can (i) extract the relevant shared signal and (ii) identify the dimensionality of the shared and the self signals from noisy, undersampled data. We investigate how the signal-to-noise ratios, the dimensionality of the reduced variables, and the method of computing correlations combine with the sample size to determine the quality of the DR. We propose best practices for achieving high-quality reduced representations with small sample sizes using these linear methods. 2 MODEL 2.1 RELATIONS TO PREVIOUS WORK The extraction of signals from large-dimensional data sets is a challenging task when the number of observations is comparable to or smaller than the dimensionality of the data. The undersampling problem introduces spurious correlations that may appear as signals, but are, in fact, just statistical fluctuations. This poses a challenge for DR techniques, as they may retain unnecessary dimensions or identify noise dimensions as true signals. Here, we focus exclusively on linear DR methods. For these, the Marchenko-Pastur (MP) distribution of eigenvalues of the covariance matrix of pure noise derived using the Random Matrix Theory (RMT) methods (Marchenko & Pastur, 1967) has been used to introduce a cutoff between noise and true signal in real datasets. However, recent work (Fleig & Nemenman, 2022) has shown that, when observations are a linear combination of uncorrelated noise and latent low-dimensional self signals, then the self signals alter the distribution of eigenvalues of the sampling noise, questioning the validity of this naive approach. Moving beyond a single modality, Bouchaud et al. (2007) calculated the singular value spectrum of cross-correlations between two nominally uncorrelated random signals. However, it remains unknown whether the linear mixing of self signals and shared signals affects the spectra of noise, and how all of these components combine to limit the ability to detect shared signals between two modalities from data sets of realistic sizes. Filling in this gap using numerical simulations is the main goal of this paper, and analytical treatment of this problem will be left for the future. The linear model and linear DR approaches studied here do not capture the full complexity of real-world data sets and state-of-the-art algorithms. However, if sampling issues and self signals limit the ability of linear DR methods to extract shared signals, it would be surprising for nonlinear methods to succeed in similar scaling regimes on real data. Thus extending the previous work to explicitly study the effects of linear mixtures of self signals, shared signals, and noise on limitations of DR methods is likely to endow us with intuition that is useful in more complex scenarios routinely encountered in different domains of science. Examples of scenarios with shared and self signals include inference of dynamics of a system through a latent space (Creutzig et al., 2009; Chen et al., 2022), where shared signals correspond to latent factors that are relevant for predicting the future of the system from its past, while self signals correspond to nonpredictive variation (Bialek et al., 2001). In economics, shared and self signals correspond to diverse macroeconomic indicators that are grouped into correlated distinct categories in structural factor models (Forni & Gambetti, 2010; Gosselin & Tkacz, 2001; Rudd, 2020; Baillie et al., 2002). In neuroscience, shared signals can correspond to the latent space, by which neural activity affects behavior, while self signals encode neural activity that does not manifest in behavior and behavior that is not controlled by the part of the brain being recorded from (Sponberg et al., 2015; Stringer et al., 2019; Natraj et al., 2022; Sani et al., 2021; Pang et al., 2016; Urai et al., 2022; Krakauer et al., 2017). Interestingly, in the context of the neural control of behavior, it was noticed that SDR reconstructs the shared neuro-behavioral latent space more efficiently and using a smaller number of samples than IDR (Sani et al., 2021). Similar observations have been made in more general statistical contexts (Chin & Newsted, 1999; Hair et al., 2011; Pacharawongsakda & Theeramunkong, 2016; Vogelstein et al., 2021), though the agreement is not uniform (Goodhue et al., 2006; 2012; 2013). Because of this, most practical recommendations for detecting shared signals are heuristic (Hair Jr et al., 2021), with widely acknowledged, but poorly understood limitations and possible resolutions (Kock & Hadaya, 2018). Our goal is to ground such rules in numerical simulations and scaling arguments. ### 2.2 Linear Model with Self and Shared Signals We consider a linear model with noise, $ m_{\text{self},X} $ and $ m_{\text{self},Y} $ self signals that are relevant to each modality independently, as well as $ m_{\text{shared}} $ shared signals that capture the interrelationships between modalities. It results in $ T $ observations of two high-dimensional standardized observables, $ X $ and $ Y $: \[ \begin{align} \tilde{X} \in \mathbb{R}^{N_X} &= R_X + U_X V_X + PQ_X, \\ \tilde{Y} \in \mathbb{R}^{N_Y} &= R_Y + U_Y V_Y + PQ_Y, \end{align} \] where $ R_X $ and $ R_Y $ are independent white noise components with variances $ \sigma^2_{R_X} $ and $ \sigma^2_{R_Y} $. $ U_X $ and $ U_Y $ are self-signal components residing in lower-dimensional subspaces $ \mathbb{R}^{m_{\text{self},X}} $ and $ \mathbb{R}^{m_{\text{self},Y}} $ with variances $ \sigma^2_{U_X} $ and $ \sigma^2_{U_Y} $. $ P $ is a shared-signal component in a shared lower-dimensional subspace $ \mathbb{R}^{m_{\text{shared}}} $ with variance $ \sigma^2_P $. These components are projected into their respective high-dimensional spaces $ \mathbb{R}^{N_X} $ and $ \mathbb{R}^{N_Y} $ using fixed quenched projection matrices $ V_X, V_Y, Q_X, $ and $ Q_Y $ with specified variances $ \sigma^2_{V_X}, \sigma^2_{V_Y}, \sigma^2_{Q_X}, $ and $ \sigma^2_{Q_Y} $, all respectively. Entries in these matrices are drawn from a Gaussian distribution with a zero mean and the corresponding variances. Further, division by $ \sigma_X $ and $ \sigma_Y $ standardizes each column of the data matrices by their empirical standard deviations. The total variance in the matrix $ \tilde{X} $ can be calculated as the sum of the variances of its individual components: \[ \sigma^2_{\tilde{X}} = \sigma^2_{R_X} + m_{\text{self},X} \times \sigma^2_{U_X} \sigma^2_{V_X} + m_{\text{shared}} \times \sigma^2_P \sigma^2_{Q_X}. \] A similar calculation can be done for the total variance in $ \tilde{Y} $. We define self and shared signal-to-noise ratios $ \gamma_{\text{self},X/Y}, \gamma_{\text{shared},X/Y} $ as the relative strength of signals compared to background noise per component in each modality. These definitions allow us to examine how easily self or shared signals in each dimension can be distinguished from the noise. \[ \gamma_{\text{self},X/Y} = \frac{\sigma^2_{U_{X/Y}} \sigma^2_{V_{X/Y}}}{\sigma^2_{R_{X/Y}}}, \quad \gamma_{\text{shared},X/Y} = \frac{\sigma^2_P \sigma^2_{Q_{X/Y}}}{\sigma^2_{R_{X/Y}}} \] Our main goal is to evaluate the ability of linear SDR and IDR methods to reconstruct the shared signal $ P $, while overlooking the effects of the self signals $ U_{X/Y} $ on the statistics of the shared ones. --- 1This model is an extension of the model introduced by Fleig & Nememani (2022), and its probabilistic form has been studied by Murphy (2022). In its turn, the latter is an extension of work by Klami et al. (2012), and Bach & Jordan (2005). However, within this model, we focus on the intensive limit, common in RMT (Potters & Bouchaud, 2020), where the number of observations scales as the number of observed variables. This scenario is common in many real-world applications. 3 METHODS We apply DR techniques to $X$ and $Y$ to obtain their reduced dimensional forms $Z_X$ and $Z_Y$, respectively. $Z_X, Z_Y$ are of sizes that can range from $T \times 1$ to $T \times N_X$ and $T \times N_Y$, respectively. As an IDR method, we use PCA (Hotelling [1933]). As SDR methods, we apply PLS (Wold et al. [2001]) and CCA (Hotelling [1936]; Vinod [1976]; Arup Nielsen et al. [1998]), including both normal and regularized versions of the latter. Each of these methods focuses on specific parts of the overall covariance matrix $$C_{X,Y} = \begin{bmatrix} C_{XX} & C_{XY} \\ C_{YX} & C_{YY} \end{bmatrix} = \begin{bmatrix} \frac{1}{T} X^T X & \frac{1}{T} X^T Y \\ \frac{1}{T} Y^T X & \frac{1}{T} Y^T Y \end{bmatrix}. \quad (4)$$ PCA aims to identify the most significant features that explain the majority of the variance in $C_{XX}$ and $C_{YY}$, independently. PLS, on the other hand, focuses on singular values and vectors that explain the covariance component $C_{XY}$. Along the same lines, CCA aims to find linear combinations of $X$ and $Y$ that are responsible for the correlation ($C_{XY}/\sqrt{C_{XX}C_{YY}}$) between $X$ and $Y$ (Borga et al. [1997]). See Appendix A.1 for a detailed description of these methods. For every numerical experiment, we generate training and test data sets $(X_{\text{train}}, Y_{\text{train}})$ and $(X_{\text{test}}, Y_{\text{test}})$ according to Eqs. (1-3). We apply PCA, PLS, CCA, and regularized CCA (rCCA) to the training to obtain the singular directions $W_{X_{\text{train}}}$ and $W_{Y_{\text{train}}}$ for each method (see Appendix A.1). We then obtain the projections of the test data on these singular directions $$Z_X = X_{\text{test}} W_{X_{\text{train}}}, \quad Z_Y = Y_{\text{test}} W_{Y_{\text{train}}}. \quad (5)$$ Finally, we evaluate the reconstructed correlations metric $\mathcal{RC}'$, which measures how well these singular directions recover the shared signals in the data, corrected by the expected positive bias due to the sampling noise, see Appendix A.2 for details. $\mathcal{RC}' = 0$ corresponds to no overlap between the true and the recovered shared directions, and $\mathcal{RC}' = 1$ corresponds to perfect recovery. 4 RESULTS We perform numerical experiments to explore the undersampled regime, $T \ll N_X, N_Y$. We use $T = \{100, 300, 1000, 3000\}$ samples, $N_X = N_Y = 1000$. We explore the case of one shared signal only, $m_{\text{shared}} = 1$ and we mask this shared signal by a varying number of self signals and noise. We vary the number of retained dimensions, ($\|Z_X\|, \|Z_Y\|$), and explore how many of them are needed to recover the shared signal in the noise and the self signal background with different SNR. For brevity, we explore two cases: (1) One self-signal in $X$ and $Y$ in addition to the shared signal ($m_{\text{self}} = 1$); (2) many self-signals in $X$ and $Y$. For both cases, we calculate the quality of reconstruction as the function of the shared and the self SNR, $\gamma_{\text{shared}}$ and $\gamma_{\text{self}}$. In all figures, we show $\mathcal{RC}'$ for severely undersampled (first row, $T = 300$) and relatively well sampled (second row, $T = 3000$) regimes. We also show the value of $\mathcal{RC}_0$, the bias that we removed from our reconstruction quality metric, for completeness, see Appendix A.2 for details. Experiments at different parameter values can be found in Appendix A.4. Figure 1 shows that, in Case 1, when one dimension is retained in DR of $X$ and $Y$, PCA populates the compressed variable with the largest variance signals and hence struggles to retain the shared signal when $\gamma_{\text{self}} > \gamma_{\text{shared}}$, regardless of the number of samples. However, both PLS and rCCA excel in achieving nearly perfect reconstructions. When $T \ll N_X$, straightforward CCA cannot be applied (see A.1.3, A.1.4), but it too achieves a perfect reconstruction when $T > N_X$. In Fig. 2, we allow two dimensions in the reduced variables. For PCA, we expect this to be sufficient to preserve both the self and the shared signals. Indeed, PCA now works for all $\gamma$s and $T$, although with a slightly reduced accuracy for large shared signals compared to Fig. 1. PLS and rCCA continue to deliver highly accurate reconstructions. So does the CCA for $T > N_X$. Spurious correlations, as measured by $\mathcal{RC}_0$, grow slightly with the increasing dimensionality of $Z_X, Z_Y$ compared to Fig. 1. This is expected since more projections must now be inferred from the same amount of data. --- 2 We fix $\sigma^2_{R_{X/Y}}, \sigma^2_{V_{X/Y}}, \sigma^2_{Q_{X/Y}}$ and allow $\sigma^2_{P}$ to vary when we choose $\gamma_{\text{self}}, X/Y, \gamma_{\text{shared}}, X/Y$. We first generate the fixed projection matrices $V_{X/Y}, Q_{X/Y}$, and we vary $R_{X/Y}, U_{X/Y}, P$ for each trial. Figure 1: Performance of PCA, PLS, CCA, rCCA, and noise in recovery of the shared signal for $\|Z_X\| = \|Z_Y\| = 1 = m_{\text{self}}$. The rows are undersampled and relatively well-sampled scenarios respectively. PCA struggles to detect shared signals when they are weaker than the self signals, even with more samples. PLS and rCCA demonstrate nearly perfect reconstruction. CCA displays no reconstruction in the undersampled regime $T \ll N_X$, and it is nearly perfect for large $T$. Figure 2: Same as Fig. 1 but for $\|Z_X\| = \|Z_Y\| = 2 = m_{\text{self}} + m_{\text{shared}}$. Now there are enough compressed variables for PCA to detect the shared signal. Other methods perform similarly to Fig. 1 albeit the noise is larger. We now turn to $m_{\text{self}} \gg m_{\text{shared}}$. We use $m_{\text{shared}} = 1$, $m_{\text{self}} = 30$ for concreteness. We expect that the performance of SDR methods will degrade weakly, as they are designed to be less sensitive to the masking effects of the self signals. In contrast, we expect IDR to be more easily confused by the many strong self-signals, degrading the performance. Indeed, Fig. 3 shows that PCA now faces challenges in detecting shared signals, even when the self signals are weaker than in Fig. 1. Increasing $T$ improves its performance only slightly. Somewhat surprisingly, PLS performance also degrades, with improvements at $T \gg N_X$. CCA again displays no reconstruction when $T \ll N_X$, switching to near perfect reconstruction at large $T$. Crucially, rCCA again shines, maintaining its strong performance, consistently demonstrating nearly perfect reconstruction. Since one retained dimension is not sufficient for PCA to represent the shared signal when $\gamma_{\text{shared}} < \gamma_{\text{self}}$, we increase the dimensionality of reduced variables $\|Z_X\| = \|Z_Y\| = m_{\text{self}} \gg m_{\text{shared}}$, cf. Fig. 4. PCA now detects shared signals even when they are weaker than the self-signals, $\gamma_{\text{shared}} < \gamma_{\text{self}}$, but at a cost of the reconstruction accuracy plateauing significantly below 1. In other words, when self and shared signals are comparable, they mix, allowing for partial reconstruction. However, even at $T \gg N_X$, PCA cannot break into the phase diagram’s lower right corner. Other methods perform similarly, reconstructing shared signals over the same or wider ranges of sampling and the SNR ratios than in Fig. 3. For all of them, the improvement comes at the cost of the decreased asymptotic performance. The most distinct feature of this regime is the dramatic effect of noise, where 30-dimensional compressed variables can accumulate enough sampling fluctuations to recover correlations that are supposedly nearly twice as high as the data actually has. Figure 3: Reconstruction results for $m_{\text{self}} = 30$, $m_{\text{shared}} = 1$, and $\|Z_X\| = \|Z_Y\| = 1$. PCA struggles to detect any shared signals when they are even comparable to the self ones. PLS performance also degrades. CCA displays its usual impotence at small $T$. Finally, rCCA demonstrates nearly perfect reconstruction for all parameter values. Figure 4: DR performance for $\|Z_X\| = \|Z_Y\| = m_{\text{self}} > m_{\text{shared}}$. PCA now detects shared signals even when they are weaker than the self signals. However, the quality of reconstruction is significantly lower than in Fig. 2. PLS detects signals in a larger part of the phase space, but also with a significant reduction in quality, which improves with sampling. CCA has its usual problem for $T \ll N_X$, and, like PLS, it has a significantly lower reconstruction quality than in the regime in Fig. 3. rCCA is able to detect the signal in the whole phase space, but again with worse quality. Finally, spurious correlations are high, though they decrease with better sampling. Figure 5 explores a regime when the dimensionality of the compressed variables is enough to store both the self and the shared interactions at the same time, $\|Z_X\| = \|Z_Y\| = m_{\text{self}} + m_{\text{shared}} = 31$. With just one more dimension than Fig. 4, PCA abruptly transitions to being able to recover shared signals for all SNRs, albeit still saturating at a far from perfect performance at large $T$. PLS, CCA, rCCA, and noise show behavior remain similar to Fig. 4. Our analysis suggests that there are three relevant factors that determine the ability of DR to reconstruct shared signals. The first is the strength of the shared and the self signals compared to each other and to noise. For brevity, in the following analysis, we fix $\gamma_{\text{self}}$ and define the ratio $\tilde{\gamma} = \gamma_{\text{shared}}/\gamma_{\text{self}}$ to represent this effect. The second factor affecting the performance is the ratio between the number of shared and self signals, denoted by $\tilde{m} = m_{\text{shared}}/m_{\text{self}}$. The third factor is the number of samples per dimension of the reduced variable, denoted by $\tilde{q} = T/\|Z\|$. In Fig. 6, we illustrate how these parameters influence the performance of DR, $RC'$. Each subplot varies $\tilde{q}$, while holding $T$ constant and changing $\|Z_X\|$. We compare the results of PCA (representing IDR) and rCCA (representing SDR). Each curve is averaged over 10 trials, with error bars indicating 1 standard deviation around the mean, using algorithmic parameters as described in Appendix A.3. We see that the relative strength of signals, as represented by $\tilde{\gamma}$, plays a significant role in determining which method performs better. If the shared signals are larger (bottom) both approaches work. Figure 5: PCA, PLS, CCA, rCCA, and noise results when 31 dimensions are kept after reduction ($\|Z_X\| = \|Z_Y\| = m_{self} + m_{shared}$). PCA now can detect more shared signals when they are weaker than the self signals (A1), however, with a significantly lower quality compared to figure 2, but suddenly explores the whole phase space, still with lower accuracy than Case 1. PLS, CCA, rCCA, and noise show similar behavior to figure 4. However, for weak shared signals (top), SDR is generally more effective. Further, the ratio between the number of shared and self signals, $\tilde{m}$, also plays an important role. When $\tilde{m}$ is large (left), IDR is more likely to detect the shared signal before the self signals, and it approaches the performance of SDR. However, when $\tilde{m}$ is small, IDR is more likely to capture the self signals before moving on to the shared signals, degrading performance (right). Finally, not surprisingly, the number of samples per dimension of the compressed variables, $\tilde{q}$, is also critical to the success. If $\tilde{q}$ is small, the signal is drowned in the sampling noise, and adding more retained dimensions hurts the DR process. This expresses itself as a peak for SDR performance around $\|Z_X\| = m_{shared}$. For IDR, the peak is around $\|Z_X\| = m_{self} + m_{shared}$, thus requiring more data to achieve performance similar to SDR. We observe that the performance of rCCA (SDR) is almost independent of changing $\tilde{m}$ or $\tilde{\gamma}$, indicating that it focuses on shared dimensions even if the latter is masked by self signals. The algorithm crucially depends on $\tilde{q}$, where adding more dimensions (decreasing $\tilde{q}$) than needed hurts the reduction. This is because, for a fixed number of samples, the reconstruction of each dimension then gets worse. In contrast, for PCA (IDR), the performance depends on all three relevant parameters, $\tilde{q}$, $\tilde{m}$, and $\tilde{\gamma}$. At some parameter combinations, the performance of IDR in reconstructing shared signals approaches SDR. However, in all cases, SDR never performs worse than IDR on this task. Further application of the identical methodology to a nonlinear Noisy MNIST dataset is presented in Appendix A.5. This analysis suggests that our conclusions hold beyond the relatively simple Gaussian mixture model synthetic data. 5 DISCUSSION We used a generative linear model which captures multiple desired features of multimodal data with shared and non-shared signals. The model focused only on data with two measured modalities. However, while not a part of this study, the model can be readily extended to accommodate more than two modalities (e.g., $X_i = R_i + U_i V_i + P Q_i$ for $i = 1, \ldots, n$, where $n$ represents the number of modalities). Then, methods such as Tensor CCA, which can handle more than two modalities (Luo et al., 2015), can be used to get insight into DR on such data. We analyzed different DR methods on data from this model in different parameter regimes. Linear SDR methods were clearly superior to their IDR counterparts for detecting shared signals. We observed similar results on a nonlinear dataset as well. We thus make a strong practical suggestion that, whenever the goal is to reconstruct a low dimensional representation of covariation between two components of the data, IDR methods (PCA) should always be avoided in favor of SDR. Of the examined SDR approaches, rCCA is a clear winner in all parameter regimes and should always be preferred. These findings explain the results of, for example, Sani et al. (2021) and others that SDR can recover joint neuro-behavioral latent spaces with fewer latent dimensions and using fewer samples than IDR methods. Further, our observation that SDR is always superior to IDR in the context of our model corroborates the theoretical findings of Martini & Nemenman (2023), who proved a similar result in the context of discrete data and a different SDR algorithm, namely the Symmetric Figure 6: Performance of PCA (IDR) and rCCA (SDR) for different values of the relevant parameters of the model: the number of samples per dimension of the compressed variable ($\tilde{q}$), the strength of shared signals relative to the self ones ($\tilde{\gamma}$), and the ratio of the number of shared to self signal components ($\tilde{m}$), while fixing the number of samples ($T = 1000$) and the number of shared dimensions ($m_{\text{shared}} = 10$). Note that increasing $1/\tilde{q}$ (left to right) corresponds to increasing the dimension of the latent space $\|Z_X\|$ at a fixed number of samples $T$. Information Bottleneck (Friedman et al., 2013; Vogelstein et al., 2021) made similar conclusions using conditional covariance matrices for the reduction in the context of classification. More recent work of Anonymous (2023) showed similar results using deep variational methods. Collectively, these diverse investigations, linear and nonlinear, theoretical, computational, and empirical, provide strong evidence that generic (not just linear) SDR methods are likely to be more efficient in extracting covariation than their IDR analogs. Our study also answers an open question in the literature surrounding the effectiveness of SDR techniques. Specifically, there has been debate about whether PLS, an SDR method, is effective at low sampling (Chin & Newsted, 1999; Hair et al., 2011; Goodhue et al., 2006, 2012). Our results show that SDR is not necessarily effective in the undersampled regime. It works well when the number of samples per retained dimension is high (even if the number of samples per observed dimension is low), but only when the dimensionality of the reduced description is matched to the actual dimensionality of the shared signals. Finally, our results can be used as a diagnostic test to determine the number of shared versus self signals in data. As demonstrated in Fig. 6, total correlations between $Z_X$ and $Z_Y$ obtained by applying PCA and rCCA increase monotonically as the dimensionality of $Z_S$ increases, until this dimensionality becomes larger than the signal dimensionality. For PCA, the signal dimensionality is equal to the sum of the number of the shared and the self signals, $m_{\text{shared}} + m_{\text{self}}$. For rCCA, it is only the number of the shared signal. Thus increasing the dimensionality of the compressed variables and tracking the performance of rCCA and PCA until they diverge can be used to identify the number of self signals in the data, provided that the data, indeed, has a low-dimensional latent structure. This approach can be a valuable tool in various applications, where the characterization of shared and self signals in complex systems can provide insights into their structure and function. In summary, we highlight a general principle that, when searching for a shared signal between different modalities of data, SDR methods are preferable to IDR methods. Additionally, the differences in performance between the two classes of methods can tell us a lot about the underlying structure of the data. Finally, for a limited number of samples, naive approaches, such as increasing the number of compressed dimensions indefinitely to overcome the masking of shared signals by self signals are infeasible. Thus, the use of SDR methods becomes even more essential in such cases. 6 LIMITATIONS, AND FUTURE WORK While this work has provided useful insight, the assumptions made here may not fully capture the complexity of real-world data. Specifically, our data is generated by a linear model with random Gaussian features. It is unlikely that real data have this exact structure. Therefore, there is a need for further exploration of the advantages and limitations of linear DR methods on data that have a low-dimensional, but nonlinear shared structure. This can be done using more complex nonlinear generative models, such as nonlinearly transforming the data generated by Eq. (12), or random feature two-layered neural network models (Rocks & Mehta [2022]). Alternatively, analyzing the model, Eq. (1) using various theoretical techniques (Borga et al. [1997], Chernoff [1952], Vogelstein et al. [2021], Potters & Bouchaud [2020]) is likely to offer even more insights into its properties. Collectively, these diverse approaches would aid our understanding of different DR methods under diverse conditions. A different possible future research direction is to explore the performance of nonlinear DR methods on data from generative models with a latent low-dimensional nonlinear structure. Autoencoders and their variational extensions are a natural extension of IDR to learn nonlinear reduced dimensional representations (Hinton & Salakhutdinov [2006], Kingma & Welling [2014], Higgins et al. [2016]). Meanwhile, Deep CCA and its variational extensions (Andrew et al. [2013], Wang et al. [2015], Chandar et al. [2016], Wang et al.) should be explored as a nonlinear version of SDR. Both of these types of methods can potentially capture more complex relationships between the modalities and improve the quality of the reduced representations, and while recent work suggests that (Anonymous [2023]), it is not clear if the SDR class of methods is always more efficient than the IDR one. Further, our analysis depends on the choice of metric used to quantify the performance of DR, and different choices should also be explored. For example, to capture nonlinear correlations, mutual information can be utilized to quantify the relationships between the reduced representations. Despite the aforementioned limitations, we believe that our work provides a compelling addition to the body of knowledge that SDR outperforms IDR in detecting shared signals quite generally. REFERENCES Galen Andrew, Raman Arora, Jeff Bilmes, and Karen Livescu. Deep canonical correlation analysis. In Sanjoy Dasgupta and David McAllester (eds.), Proceedings of the 30th International Conference on Machine Learning, volume 28 of Proceedings of Machine Learning Research, pp. 1247–1255, Atlanta, Georgia, USA, 17–19 Jun 2013. PMLR. Anonymous. Deep variational multivariate information bottleneck - a framework for variational losses. 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bTMMNT7IdW	According to my understanding, Eq.(7) aims at learn a set of parameters for $f_k$ and $g_k$ so that the learned models can mimic the approximated trajectory of data. In my opinion, such operation relies heavily on the quality of data representations, especicially at the early phase of training, since the feature extractor $\phi$ is not well trained yet. Thus, in such case, will the linear interpolation results in some	Latent Trajectory Learning for Limited Timestamps under Distribution Shift over Time Qiuhao Zeng¹, Changjian Shui², Long-Kai Huang, Peng Liu³, Xi Chen⁴ Charles X. Ling¹, Boyu Wang¹∗ ¹University of Western Ontario ²Vector Institute ³University of Toronto ⁴Noah’s Ark Lab Abstract Distribution shifts over time are common in real-world machine-learning applications. This scenario is formulated as Evolving Domain Generalization (EDG), where models aim to generalize well to unseen target domains in a time-varying system by learning and leveraging the underlying evolving pattern of the distribution shifts across domains. However, existing methods encounter challenges due to the limited number of timestamps (every domain corresponds to a timestamp) in EDG datasets, leading to difficulties in capturing evolving dynamics and risking overfitting to the sparse timestamps, which hampers their generalization and adaptability to new tasks. To address this limitation, we propose a novel approach SDE-EDG that collects the Infinitely Fine-Grid Evolving Trajectory (IFGET) of the data distribution with continuous-interpolated samples to bridge temporal gaps (intervals between two successive timestamps). Furthermore, by leveraging the inherent capacity of Stochastic Differential Equations (SDEs) to capture continuous trajectories, we propose their use to align SDE-modeled trajectories with IFGET across domains, thus enabling the capture of evolving distribution trends. We evaluate our approach on several benchmark datasets and demonstrate that it can achieve superior performance compared to existing state-of-the-art methods. 1 Introduction Domain Generalization (DG) is a fundamental problem in machine learning. It aims to learn a model that can perform well on unseen data based on the knowledge learned from multiple related domains (Arjovsky et al., 2019; Li et al., 2018b; Sagawa et al., 2019; Sun & Saenko, 2016; Chen et al., 2024). However, traditional DG techniques assume that the distribution in different domains remains stationary over time, which is often impractical in many real-world scenarios. In practice, the distribution of data may shift over time due to factors such as changes in the environment or the underlying system. For example, Age-related changes occur in all ocular tissues, including age-related structural changes in the optic nerve (Grossniklaus et al., 2013). Age-related ocular disease is the most prevalent condition associated with vision impairment and blindness in older adults worldwide (Flaxman et al., 2017). However, the data collected from individuals aged beyond 80 is lacking due to a very small sample size, privacy, and other factors. It’s necessary to build a prediction model based on the age-related pattern learned from the data collected from younger cases. To adapt to changing environments over time, recent research in the community has focused on the scenario of Evolving Domain Generalization (EDG) (Bai et al., 2023; Nasery et al., 2021b; Qin et al., 2022; Zeng et al., 2023; Yao et al., 2022), aimed to tackle such problems. Specifically, the goal of EDG is to learn and leverage the evolving patterns captured from source domains to achieve generalization capability on the unseen future target domains in a time-varying environment. However, one fundamental obstacle in existing EDG studies (Bai et al., 2023; Qin et al., 2022; Zeng et al., 2023) is that they suffer from a limited number of timestamps, resulting in overfitting to the sparse timestamp data. Consequently, they cannot properly capture the underlying evolving pattern and extrapolate into the future. In fact, recent research (Mariet & Kuznetsov, 2019) has revealed that ∗Corresponding authors: Boyu Wang, Charles X. Ling. the sample complexity of time series forecasting tasks scales as $O(\sqrt{1/M})$, where $M$ is the number of timestamps of the training time-series data. In this paper, we tackle this problem by constructing a continuously evolving trajectory. Specifically, we create the Infinitely Fined-Grid Evolving Trajectory (IFGET) in the latent representation space, with two steps: i) firstly, we develop sample-to-sample correspondence to collect the evolving trajectory of each individual sample; ii) Next, we generate continuous-interpolated samples by leveraging such correspondence, aimed to bridge the temporal gaps between timestamp intervals and avoid overfitting to sparse timestamps. It is denoted as IFGET, since it is a continuous trajectory and thereby can be subdivided into infinitely fine temporal grids. Nevertheless, dealing with continuous trajectories poses another challenge in EDG. Most existing EDG algorithms are designed for discrete timestamps and are not able to handle continuous timestamps, since they employ transition functions to predict data at the next timestamp based on current observation, inherently representing time discretely (Bai et al., 2023; Qin et al., 2022; Zeng et al., 2023; 2024). To address this issue, we propose to model the temporal dynamics of latent representations by employing Stochastic Differential Equations (SDEs) (Kong et al., 2020; Li et al., 2020; Xu et al., 2022; Kidger et al., 2021) to fit the IFGET, which provides a natural approach for characterizing continuous-time stochastic processes. Specifically, we propose a path alignment regularizer, which aligns the latent trajectories characterized by SDEs with the paths generated by IFGET, by maximizing the likelihood of the SDE trajectories based on the observations of IFGET. To summarize, our proposed algorithm, termed as SDE-EDG, has the following desirable properties: Capturing Evolving Patterns via Infinitely Fined-grid Evolving Trajectory (IFGET) To overcome the limitations of the small number of timestamps in current EDG data, we propose to learn the evolving dynamics by constructing the IFGET. To construct the evolving trajectory, we identify the sample-to-sample correspondence between successive domains and employ an interpolation function to generate a continuous-interpolated sample. IFGET alleviates overfitting to the limited timestamps and improves the generalization to distribution shifts over time. Modelling Trajectories of Latent Representations with Stochastic Differential Equations (SDEs) Leveraging the inherent capacity of SDE to model continuous temporal trajectory, our model aligns the depicted latent trajectories of SDE-EDG with IFGET during the training process. We show that SDE-EDG is capable of quantifying evolving stochastic processes, and theoretically demonstrate that SDE-EDG results in a lower generalization bound for downstream tasks. ## 2 RELATED WORKS Evolving Domain Adaptation / Generalization Evolving Domain Adaptation (EDA) (Hoffman et al., 2014; Kumagai & Iwata, 2017; Mancini et al., 2019; Liu et al., 2020a; Wang et al., 2020; Kumar et al., 2020; Wang et al., 2022) is a related field that focuses on scenarios where a single labeled domain is available alongside multiple unlabeled intermediate domains. The objective of EDA is to achieve generalization on unseen target domains. Recently, EDG has received considerable attention from researchers. Approaches to solving the EDG problem can be broadly categorized into two groups. The first group parameterizes the learning model with time-sensitive models (Qin et al., 2022; Nasery et al., 2021a; Bai et al., 2023). Qin et al., 2022 proposes to tackle the challenges of covariate shift and concept shift, with the probabilistic model incorporating variational inference. The second group (Zeng et al., 2023; 2024) maps the source data into future data leveraging evolving patterns. However, these methods still suffer from limited timestamps, which hinder the capture of temporal trends. In contrast, we propose to construct IFGET, where the temporal gaps are filled with interpolations. Ordinary Differential Equations (ODE)/ Stochastic Differential Equations (SDE) In recent years, Neural-ODEs (Chen et al., 2018b; Sun et al., 2020) have emerged as a powerful tool for continuous-time representation of neural networks. SDEs (Øksendal & Øksendal, 2003; Kong et al., 2020; Liu et al., 2020b) have incorporated stochastic terms into ODE solvers, injecting the model with slight random noise to improve generalization ability and noise robustness. To learn SDE neural networks’ parameters, Ryder et al., 2018; Xu et al., 2022; Li et al., 2020 use variational inference. --- 1 Our code is available at [https://github.com/HardworkingPearl/SDE-EDG-official](https://github.com/HardworkingPearl/SDE-EDG-official) technology and overcome the overfitting problem. SDE has shown excellent performance in machine learning applications, such as generative adversarial model (GAN) (Kidger et al., 2021; Park et al., 2021a), score-based diffusion model (Song et al., 2020). In this work, we propose a novel approach to modeling Evolving Domain Generalization (EDG) as a dynamical system by utilizing SDEs to effectively represent the continuously evolving trajectory of the data representations, and we apply maximum likelihood to efficiently fit the latent evolving trajectories utilizing SDEs. 3 PRELIMINARIES Ordinary / Stochastic Differential Equations Neural ordinary differential equations (Neural ODEs) (Chen et al., 2018a) approximate the evolving dynamics with the ordinary differential equation and are defined as \[ z_t = z_0 + \int_0^t f(z_s, s) ds, \] (1) where the hidden state $ z_t \in \mathbb{R}^d $ evolves with certain dynamics characterized by a neural network $ f : \mathbb{R}^d \to \mathbb{R}^d $, $ z_0 $ is the initial state, and $ s $ represents time in integrals. An SDE can be regarded as an ODE injected with noise over time: \[ z_t = z_0 + \int_0^t f(z_s, s) ds + \int_0^t g(z_s, s) dB_s, \] (2) where $ z_t $ is a latent state that evolves over time, $ f : \mathbb{R}^d \times \mathbb{R} \to \mathbb{R}^d $ is the drift function to capture the evolving dynamics, $ g : \mathbb{R}^d \times \mathbb{R} \to \mathbb{R}^{d \times \omega} $ is the diffusion function to reflect the uncertainties, and $ B_s $ is an $ d $-dimensional Brownian motion (Wiener Process). SDE has shown superior performance in modeling the dynamical system (Park et al., 2021a; Øksendal & Øksendal, 2003). Under the EDG settings, the drift function of SDE describes the trends of the distribution shift over time, and the diffusion function models the samples’ individual stochastics in their representation space. Evolving Domain Generalization Let $ D(x, y, t) $ be the probability distribution that characterizes temporal dynamics of an instance $ x \in X $ and its label $ y \in Y $, in which there exist underlying evolving patterns over $ t $. In Evolving Domain Generalization (EDG), we are given $ M $ source domains $ \{S_m\}_{m=1}^M $, where $ S_m = \{(x_{i\|m}, y_{i\|m})\}_{i=1}^N $ is the data of $ m $-th domain sampled from $ D(x, y\|t_m) $ at the timestamp $ t_m \in [0, T] $, and $ N $ is the sample size of the $ m $-th domain. Note that most existing works (Bai et al., 2023; Nasery et al., 2021b; Qin et al., 2022; Zeng et al., 2023) assume a constant interval $ \Delta t $ between two consecutive domains. In contrast, our approach exhibits flexibility in tackling the EDG problem even with irregular time intervals. In the proposed method, we will learn the evolving dynamics in a latent space $ Z $, given the practical advantages (Kirchmeyer et al., 2022), e.g., improved discrimination, dimension reduction, and computation resources savings compared to operations in the original input space $ X $. Specifically, for every instance $ x $, we encode it with a feature extractor $ \phi : X \to Z $, and we obtain the embedded feature $ z = \phi(x) \in Z $. We focus on the dynamics of $ D(z, y, t) $ throughout this work. The goal of EDG is to learn a robust and generalizable model from source domains by capturing and leveraging the evolving pattern learned from source domains so that it can perform well on the unseen target domains in the path space (Boué & Dupuis, 1998) at $ L $ timestamps $ t_{M+1}, \ldots, t_{M+L} \in (T, T + T^] $ ($ T^ = t_{M+L} - T $): \[ \min_\theta R_\nu(h_\theta) = \min_\theta \mathbb{E}_{(z,y) \sim D(z_{M+1:M+L}, y_{M+1:M+L})}[h_\theta(z) \neq y], \] (3) where $ \nu $ is the distribution of the stochastic path (Boué & Dupuis, 1998) of $ D $ along timestamps $ T $ to $ T + T^* $, $ z_{M+1:M+L} $ and $ y_{M+1:M+L} $ are short for $ z $ and $ y $ at the timestamps $ \{t_{M+1}, \ldots, t_{M+L}\} $, $ R_\nu $ is the risk of a learning model $ h_\theta $ parameterized by parameters $ \theta $. 4 METHODS With the prior knowledge of SDEs and EDG, we will now formally present our SDE-EDG approach: To build IFGET (section 4.1), we search sample-to-sample correspondence, which aids in generation of continuous interpolations; Neural SDEs models the trajectories of latent representations (Section 4.2); we construct IFGET and employ it as a regularization mechanism to promote the learning of evolving representations while avoiding the acquisition of invariant representations (Section 4.3). 4.1 Construct Infinitely Fined-Grid Evolving Trajectory In EDG, datasets have a considerably small size of domains/timestamps (e.g., at most 30 domains) \cite{Yao2022}. In contrast, models are trained on the historical data for time-series forecasting tasks spanning over at least hundreds of timestamps to predict future states \cite{Addison2020,Yu2016}. In light of this obstacle, we generate intermediate domains by applying interpolations between two consecutive domains. To ensure such interpolations reflect the evolving pattern of the underlying trajectory over domains, one should have the complete trajectory of each individual sample across domains. For example, in weather forecasting, one must have the historical meteorological data of each individual observation station to characterize the climate change trends. Unfortunately, such trajectories usually do not exist in EDG as there is no sample-to-sample correspondence across domains (e.g., we may not have images of the same person at different age stages), thereby preventing the model from tracking the complete trajectories and extracting the evolving trends. To address this issue, we propose to identify sample correspondence between timestamps, which is critical to better alignment of the data structure across domains \cite{Lu2023,Chen2022,Blitzer2006,Das2018}. Specifically, for each class $k$, we take the closest sample at $t_{m+1}$ to the datapoint $z^k_m$ at $t_m$, as its subsequent state at time $t_{m+1}$ to build sample-to-sample correspondence: $$\hat{z}^k_{i\|m+1} = \arg\min_{z^k_{j\|m+1} \in S^k_{m+1}} \text{Dist}(z^k_{i\|m}, z^k_{j\|m+1}),$$ where $\text{Dist}: Z \times Z \to [0, +\infty)$ is a distance metric defined over the embedding space, $S^k_{m+1}$ be the set of $N_B$ data points sampled from $D_{m+1}$ (short for $D(z, y\|t_{m+1})$) with class $y = k \in \{1, \ldots, K\}$ in a training iteration. The rationale here lies in the decomposition of latent variables into class-dependent and domain-dependent evolving components \cite{Qin2022}, resulting in samples from the same class in the $m$-th and $(m+1)$-th domains exhibiting smaller distances due to such sample pair’s shared class-dependent similarities, while the evolving difference maintains consistent magnitude. Utilizing sample-to-sample correspondence, we gather discrete samples within IFGET. To render it continuous, we generate continuously-interpolated samples bridging the temporal gaps. With the sample correspondence, we leverage the interpolation function to generate continuous-interpolated samples, such that the interpolation is generated along the approximated individual trajectory of a data point as shown in Figure 1: $$\hat{z}_{i\|m+\lambda} = \text{Interp}(z^k_{i\|m}, \hat{z}^k_{i\|m+1}, \lambda) = (1 - \lambda)z^k_{i\|m} + \lambda\hat{z}^k_{i\|m+1}, \forall z^k_{i\|m} \in S^k_m,$$ where the interpolation rate $\lambda \in (0, 1)$ is sampled from a Beta distribution $B(\beta_1, \beta_2)$. $\beta_1$ and $\beta_2$ are the parameters of the Beta distribution, and $S^k_m$ consists of instances sampled from $k$-th class of $m$-th domain. Here we apply a linear interpolation \cite{Yan2020,Zhang2018} as the interpolation function. The continuous-interpolated samples bridge temporal gaps in the discrete evolving trajectory, converting it into an infinitely fine-grained trajectory due to $\lambda \in (0, 1)$. Specifically, as $\lambda$ can be any value between $(0, 1)$, it enables us to approximate time moments between the $m$-th and $(m+1)$-th timestamps. We theoretically show that the sample complexity of EDG reduces with a smaller temporal interval in Theorem D.3, which leads to a reduction in error. We take interpolations as approximations of samples between time intervals, leading to a smaller time interval and thus a smaller sample complexity. Above all, we construct the Infinitely Fined-grid Evolving Trajectory $\{z^k_{i\|m}, \hat{z}^k_{i\|m+\lambda}, \hat{z}^k_{i\|m+1}\}_{m=1}^{M-1}$ by leveraging the sample-to-sample correspondence, and collecting the interpolations. 4.2 Modeling EDG with Stochastic Differential Equations The continuous trajectory in Section 4.1 significantly enhances the capability to capture evolving patterns, but existing EDG methods cannot handle the continuous timestamp data. Hence, we propose to model the data of EDG in the representation space with neural SDEs, since Neural SDEs naturally model continuous temporal trajectories. In contrast, traditional temporal models such as LSTM (Hochreiter & Schmidhuber, 1997) and Markov models (Bishop & Nasrabadi, 2006) are only able to model discrete timestamps. Here, SDE-EDG learns the temporal dynamics governing the semantic conditional distributions $ D(z\|y, t) $ over time. Specifically, SDE-EDG models the temporal trajectory of the data point from the domain at $ t_m $ to the arbitrary future timestamp $ t_{m'} : t_{m'} > t_m $ of each class $ k \in \{1, \ldots, K\} $: \[ \hat{z}_{m'}^k = z_m^k + \int_{t_m}^{t_{m'}} f_k(\hat{z}_s^k, s) ds + \int_{t_m}^{t_{m'}} g_k(\hat{z}_s^k, s) dB_s, \] where the latent variable $ \hat{z}_{m'}^k $ is transformed from $ m $-th domains latent variable $ z_m^k $, and $ f_k $ is the drift function of the $ k $-th class to capture the evolving patterns, and $ g_k $ is the diffusion function of the $ k $-th class to characterize the stochastics of the latent representations. Note that $ z $ is the latent variable (representation) induced by $ z = \phi(x) $, but $ \hat{z} $ is the synthetic feature generated by Eq. (6). Hence, SDE-EDG can generate the latent continuous trajectory by gradually transforming the sample representation from the current timestamp $ m $ to any desired future timestamp $ m' $. Thereby, our latent trajectories of SDE-EDG could effectively align with the collected continuous trajectories IFGET, which prevents overfitting to sparse timestamps. We design two objective functions to learn the drift functions $ f = \{f_k\}_{k=1}^K $ and diffusion functions $ g = \{g_k\}_{k=1}^K $ characterized by neural networks: one is aimed to impose Path Alignment Loss in Eq. (7), and another one is downstream classification loss in Eq. (10). By jointly optimizing $ \{\phi, f, g\} $ w.r.t these two losses, our approach achieves improved performance on EDG. ### 4.3 Align SDE-EDG with IFGET via Maximum Likelihood Neural SDEs are designed to capture the dynamics and evolution of data over time, particularly in continuous spaces. To fit the SDE-EDG into the evolving stochastic path given observations, we propose the path alignment regularizer by maximizing its likelihood of the IFGET $ \{z_{i\|m}, \hat{z}_{i\|m+\lambda}, \hat{z}_{i\|m+1}\}_{m=1}^{M-1} $. \[ J_{mle} = \sum_{m=1}^{M} \sum_{k=1}^{K} \sum_{i=1}^{N_B} - \frac{1}{MKN_B} \left( \log D(z = \hat{z}_{i\|m+1}\|z = z_{i\|m}) + \log D(z = \hat{z}_{i\|m+\lambda}\|z = z_{i\|m}) \right), \] Filling the gap between domains with continuous-interpolated samples results in a continuous and smooth evolving trajectory over time. Taking $ J_{mle} $ as a regularizer brings two advantages to EDG model training: 1) Empirically, the training process of neural SDEs converges faster with $ J_{mle} $, as shown in Figure 4; 2) $ J_{mle} $ regularizes the latent space to capture evolving patterns. This contributes to learning the evolving patterns in the EDG problem and improves the generalization capability to target domains as shown in Figure 2. On the other hand, in the absence of $ J_{mle} $, the model learns invariant representations across domains, leading to the occurrence of the Neural Collapse phenomenon (Han et al., 2022), where the latent representations of the same class across the domain collapse to a single point. Consequently, no evolving patterns manifest in the latent representation space as shown in Figure 3. Figure 2: The left and right images depict representations acquired for the Circle dataset through the SDE-EDG and IRM by $ \phi $. Distinct classes are distinguished by different shapes (triangles and circles), while various domains are denoted by different colors as indicated by the rainbow bar. SDE-EDG successfully learns representations with a discernible decision boundary, whereas IRM collapses towards a single direction, failing to depict a clear decision boundary. Figure 3: (a) Ground truth of the Sine dataset, and positive and negative labels are red and blue dots separately. (b-c) show prediction results to the Sine made by ERM, and SDE-EDG respectively, positive and negative predictions are red and blue dots separately. (d) Visualized learned evolving paths (synthetic latent variables) with the Path Alignment loss $ J_{mle} $ by SDE-EDG learned from the Sine dataset. (e) Synthetic latent variables without the Path Alignment loss $ J_{mle} $ by SDE-EDG learned from the Sine dataset. With $ J_{mle} $, the latent evolving dynamics can be correctly characterized, and SDE-EDG can capture the evolving patterns. 4.4 SDE-EDG FOR THE PREDICTION LOSS In this section, we formulate our approach for handling downstream classification tasks. With the Bayes rule, the predictive distribution is \[ D(y = k \| z, t = t_m) = \frac{D(z \| y = k, t = t_m) \times D(y = k \| t = t_m)}{\sum_{k'=1}^{K} D(z \| y = k', t = t_m) \times D(y = k' \| t = t_m)}, \] where we model $ D(z \| y = k, t = t_m) $ with non-parametric model, and $ D(y \| t = t_m) $ as a neural net with input as timestamp $ t $, function denoted as $ r(t) $. In each iteration, we first compute label distribution with respect to time $ D(y \| t = t_m) = [D(y = 1 \| t = t_m), \ldots, D(y = K \| t = t_m)] $. Specifically, $ D(y = k \| t = t_m) = \frac{\|S^k_m\|}{\sum_{k'=1}^{K} \|S^{k'}_m\|} $, where $ S^k_m $ consists of instances sampled from $ k $-th class of $ m $-th domain, $ \| \cdot \| $ denotes the size of the set. $ r $ is optimized by minimizing $ \\| \sum_{k'=1}^{K} \frac{\|S^{k'}_m\|}{\|S^k_m\|} - r(t_m) \\| $. The conditional distribution $ D(z \| y, t) $ modeled by SDEs lacks an analytic expression, and here we approximate it with a non-parametric model. Given that distributions characterized by SDEs may exhibit either uni-modal or multi-modal patterns, it’s also advantageous to model multi-modal representations with Neural SDEs (Min et al., 2023). In this context, we present the multi-modal classification loss here but leave the uni-modal loss in the appendix A due to space limitation. To preserve the multi-modal pattern of the latent variables, we employ the non-parametric distribution density method Parzen Window (Parzen, 1962) \[ D(z \| y = k, t = t_m) = \frac{\sum_{z_i \in S^k_m} - \exp(-\text{Dist}(z, z_i))}{\|S^k_m\|} \] where $ S^k_m $ includes instances sample from learned SDE-EDG belong to $ k $-th class of $ m $-th domain. By incorporating the estimations of label distribution ($ D(y \| t) $) and conditional distribution ($ D(z \| y, t) $), our predictions encompass the temporal evolution of both $ D(z \| y, t) $ and $ D(y \| t) $. Model optimization proceeds by minimizing the negative log probability: \[ J_{cls} = \sum_{m=1}^{M} \sum_{k=1}^{K} \sum_{i=1}^{N_B} - \frac{1}{MN_B} \log D(y = k \| z = z_i, t = t_m) \] The ultimate objective function is $ J = J_{cls} + \alpha J_{mle} $. The Maximum Likelihood Loss $ J_{mle} $ is a path alignment regularizer that aims to fit the stochastic evolving paths. The hyper-parameter $ \alpha > 0 $ is the weighting of $ J_{mle} $ to adjust its contribution to the overall loss. 5 EXPERIMENTS To evaluate the effectiveness of SDE-EDG, we verify our method on various datasets, Rotated Gaussian, Sine, and Circle datasets, Rotated MNIST, Portraits, Caltran, Power Supply, and Ocular Algorithm 1 SDE-EDG (an iteration during the training phase) 1: Input: $ \{S_1, S_2, ..., S_M\} $: M data sets from consecutive domains. $ N_B $: the number of instances sampled for each class in an iteration. RANDOMSAMPLE($ S, N $): a set of $ N $ instances sampled uniformly from the set $ S $ without replacement. $ J \leftarrow 0 $ 2: for $ m \in \{1, ..., M - 1\} $ do 3: for $ k \in \{1, ..., K\} $ do 4: $ S^k_m \leftarrow \text{RANDOMSAMPLE}(S^k_m, N_B) $ 5: $ z^k_{m+1} = z^k_m + \int_{t^k_m}^{t^k_{m+1}} f_k(z^k_s, s) ds + \int_{t^k_m}^{t^k_{m+1}} g_k(z^k_s, s) dB_s $ is computed for $ z^k_m \in S^k_m $ 6: $ S^k_{m+1} \leftarrow \text{RANDOMSAMPLE}(S^k_{m+1}, N_B) $ 7: Use Eq. (4) to search subsequent state $ z^k_{m+1} $. 8: Use Eq. (5) to generate continuous-interpolated samples $ \hat{z}_{m+\lambda} $. 9: $ J \leftarrow J + \alpha \cdot J_{mle} $, where $ J_{mle} $ is calculated w.r.t Eq. (7). 10: for $(z_i, y_i)$ in $ S^k_{m+1} $ do 11: $ J \leftarrow J + \frac{1}{MN_B} \log D(y = y_i \| z = z_i, t = t_m) $ 12: Optimize the loss w.r.t. $ J $ Disease. The objective of this section aims to answer the following key questions (1) What are the evolving trajectories that SDE-EDG learns, and what is the nature of its learning process? (Shown in Figure 3) (2) How does SDE-EDG compare to other methods in improving EDG performance? (such as Table 1, Figure 5 and Figure 4(a)-(b)) (3) What is the influence of the Maximum Likelihood Loss on the performance of SDE-EDG? (In Figure 3(d-e) and Figure 4(c)-(d)) 5.1 Experimental Setup Dataset: Rotated Gaussian (Zeng et al., 2023) consists of 30 domains where each domain has 500 instances generated by the same Gaussian distribution, but the decision boundary rotates from 0° to 338° with an interval of 12°. We split the domains into source domains (1-22 domains), intermediate domains (22-25 domains), and target domains (26-30 domains). The intermediate domains are utilized as the validation set. Circle (Pesaranghader & Viktor, 2016) contains evolving 30 domains where the instance are sampled from 50 2D Gaussian distributions. The label is assigned using a half-circle curve as the decision boundary (15 source domains, 5 validation domains, and 10 target domains). Sine (Pesaranghader & Viktor, 2016) The label is assigned using a sine curve as the decision boundary. We rearrange this dataset by extending it to 24 evolving domains. (12 source domains, 4 validation domains, and 8 target domains) Rotated MNIST (RMNIST) (RMNIST) (Ghifary et al., 2015) is composed of MNIST digits of various rotations. We follow Qin et al. (2022) and extend it to 19 evolving domains via applying the rotations with degree of $\{0°, 15°, 30°, ..., 180°\}$ in order (10 source domains, 3 validation domains, and 6 target domains). Portraits (Ginosar et al., 2015) (Yearbook (Yao et al., 2022)) is a real-world dataset that comprises photos of American high school seniors collected over 108 years (1905-2013) for gender classifications. The dataset is divided into 34 domains (19 source domains, 5 validation domains, and 10 target domains). Caltran (Hoffman et al., 2014) consists of real-world images captured by a fixed traffic camera deployed in an intersection over time. We divide it into 34 domains by time. The task of Caltran is to classify scenes to identify the presence of one or more vehicles in or approaching the intersection (19 source domains, 5 validation domains, and 10 target domains). PowerSupply (Dau et al., 2019) is created for the purpose of predicting the current power supply based on hourly records from an Italian electricity company. It includes 30 domains based on days and each data point is labeled as either morning or afternoon (15 source domains, 5 validation domains, and 10 target domains). Ocular Disease (Kaggle, 2020) Ocular Disease Intelligent Recognition (ODIR) is set with three classes: Normal, Diabetes and other diseases. Following the EDG setup, we sort the photographs in ascending order of the age of the patients (27 source domains, 2 validation domains, and 4 target domains). Baselines We compare with following baselines: (1) ERM (Vapnik, 1999); (2) Mixup (Yan et al., 2020); (3) MMD (Li et al., 2018b); (4) MLGD (Li et al., 2018a); (5) IRM (Arjovsky et al., 2019); (6) RSC (Huang et al., 2020); (7) MTL (Blanchard et al., 2021); (8) Fish (Shi et al., 2021); (9) CORAL (Sun & Saenko, 2016); (10) AndMask (Parascandolo et al., 2020); (11) DIVA (Ilse et al., 2020); (12) LSSAE (Qin et al., 2022); (13) GI (Nasery et al., 2021a) (14) DDA (Zeng et al., 2023) (15) DRAIN Table 1: The comparison of the classification accuracy (%) between SDE-EDG and other baseline methods across the synthetic and real-world datasets. The reported results are the average accuracy of the multiple target domains. ("RG" for RotatedGaussian, "Cir" for Circle, "RM" for Rotated MNIST, "Por" for Portraits, "Cal" for Caltran, "PS" for PowerSupply, "OD" for OcularDisease. GI fails to complete OD due to high time complexity.) \| ALGORITHM \| RG \| Cir \| SINE \| RM \| POR \| CAL \| PS \| OD \| AVG \| \|-----------\|----\|-----\|------\|----\|-----\|-----\|----\|----\|-----\| \| ERM \| 59.0 \| 49.9 \| 63.0 \| 43.6 \| 87.8 \| 66.3 \| 71.0 \| 57.9 \| 62.3 \| \| MIXUP \| 55.4 \| 48.4 \| 62.9 \| 44.9 \| 87.8 \| 66.0 \| 70.8 \| 59.7 \| 62.0 \| \| MMD \| 56.0 \| 50.7 \| 55.8 \| 44.8 \| 87.3 \| 57.1 \| 70.9 \| 57.6 \| 60.0 \| \| MLDG \| 59.9 \| 50.8 \| 63.2 \| 43.1 \| 88.5 \| 66.2 \| 70.8 \| 43.9 \| 60.8 \| \| IRM \| 47.5 \| 51.3 \| 63.2 \| 39.0 \| 85.4 \| 64.1 \| 70.8 \| 46.2 \| 58.4 \| \| RSC \| 32.8 \| 48.0 \| 61.5 \| 41.7 \| 87.3 \| 67.0 \| 70.9 \| 54.5 \| 58.0 \| \| MTL \| 59.0 \| 51.2 \| 62.9 \| 41.7 \| 89.0 \| 68.2 \| 70.7 \| 59.7 \| 62.8 \| \| FISH \| 41.6 \| 48.8 \| 62.3 \| 44.2 \| 88.8 \| 68.6 \| 70.8 \| 48.2 \| 59.2 \| \| CORAL \| 53.0 \| 53.9 \| 51.6 \| 44.5 \| 87.4 \| 65.7 \| 71.0 \| 60.1 \| 60.9 \| \| ANDMASK \| 76.3 \| 47.9 \| 69.3 \| 42.8 \| 70.3 \| 56.9 \| 70.7 \| 51.2 \| 60.7 \| \| DIVA \| 56.6 \| 67.9 \| 52.9 \| 42.7 \| 88.2 \| 69.2 \| 70.8 \| 53.1 \| 62.7 \| \| LSSAE \| 48.7 \| 73.8 \| 71.4 \| 46.4 \| 89.1 \| 70.6 \| 71.1 \| 52.3 \| 65.4 \| \| GI \| 50.8 \| 54.4 \| 65.2 \| 44.6 \| 88.1 \| 70.7 \| 71.4 \| - \| - \| \| DDA \| 66.8 \| 51.2 \| 66.6 \| 45.1 \| 87.9 \| 66.1 \| 70.9 \| 55.8 \| 63.8 \| \| DRAIN \| 61.0 \| 50.7 \| 71.3 \| 43.8 \| 89.4 \| 69.0 \| 71.0 \| 58.7 \| 64.4 \| \| SDE-EDG \| 97.7 \| 81.5 \| 72.2 \| 52.6 \| 89.6 \| 71.3 \| 75.7 \| 62.6 \| 75.4 \| (Bai et al., 2023). All experimental implementations are conducted using the PyTorch packages and are based on the DomainBed (Gulrajani & Lopez-Paz, 2020). To ensure a fair comparison, the neural network architecture (shown in Appendix C.2) of the encoding and classification parts are kept constant across all baselines used in different benchmarks. Five independent experiments with different random seeds are repeated to reduce the variances. 5.2 Experimental Results SDE-EDG Aligns with the Evolving Trajectories To find out what SDE-EDG learns, we visualize the temporal trajectories sampled by SDE-EDG in Fig 3d. It should be noted that SDE-EDG learns the evolving dynamics in the latent space $Z$, which is not directly interpretable. To address this issue, we use an Identity function as the encoding function, which enables us to learn the dynamics directly in raw data space $X$. Learning in original data space is more challenging but provides us with a more intuitive understanding of the learned dynamics. Fig 3 shows that the source domains data are between the range $\left[-\frac{\pi}{2}, 0\right]$, which means we only train machine learning models with the half sine. The trajectories of the same class are compact within this range, and the boundaries between them have a large margin to ensure good performance. However, the trajectories become looser and the margins become smaller as we move into unobserved domains with timings in the range $(0, \frac{\pi}{2}]$. With a much longer time gap from the source domains, SDE-EDG will eventually fail due to a larger discrepancy between learned paths and the ground truth. Quantitative Results The experimental results of SDE-EDG and other baselines are presented in Table 1, which shows the accuracy the average accuracy for all the target domains (complete results for each domain shown in Table 3f–l for space limitations). SDE-EDG surpasses other baselines’ average accuracy on all datasets. The results indicate a significant improvement over the compared traditional DG methods, which is a reasonable finding since traditional DG methods do not address evolving patterns in EDG. Furthermore, SDE-EDG outperforms LSSAE, DDA, and DRAIN (most recently EDG method) by 10.0%, 11.6%, and 11.0% on overall average accuracy, demonstrating the superior ability of our method to capture evolving patterns. The significant accuracy improvements observed in Table 1 indicate consistent enhancements in the performance of EDG tasks. In addition, EDG methods perform consistently better than DG methods, showing the importance of modeling evolving patterns to improve prediction performance in EDG tasks. In particular, the OcularDisease dataset is a challenging medical image classification task, significantly more complex than standard image classification. SDE-EDG demonstrates its ability to capture evolving patterns in demanding real-world scenarios. Figure 4: (a) Training accuracy convergence trajectories on Portraits (b)-(c) Accuracy with the domain index of RMNIST and Circle Datasets, respectively. (d)-(e) Effects of weighting $\alpha$ on the Maximum Likelihood Loss on RMNIST (RM) and PowerSupply (PS). Table 2: Rotated MNIST with different temporal gaps $\Delta t$ ($t$ here represents domain index). \| $\Delta t$ \| $130^\circ$ \| $140^\circ$ \| $150^\circ$ \| $160^\circ$ \| $170^\circ$ \| $180^\circ$ \| AVG \| \|-----------\|-------------\|-------------\|-------------\|-------------\|-------------\|-------------\|-----\| \| $\Delta t/2$ \| $75.6 \pm 0.8$ \| $61.8 \pm 0.8$ \| $49.9 \pm 0.8$ \| $50.0 \pm 0.9$ \| $45.1 \pm 0.7$ \| $44.1 \pm 0.9$ \| $54.4$ \| \| $\Delta t$ \| $75.1 \pm 0.8$ \| $61.3 \pm 0.9$ \| $49.8 \pm 0.8$ \| $49.8 \pm 0.8$ \| $39.7 \pm 0.7$ \| $39.7 \pm 0.9$ \| $52.6$ \| \| $\Delta t \times 2$ \| $58.6 \pm 0.8$ \| $49.1 \pm 0.7$ \| $45.6 \pm 0.7$ \| $42.4 \pm 0.8$ \| $36.9 \pm 0.8$ \| $36.1 \pm 0.8$ \| $44.8$ \| Figure 4b and figure 4c plot the accuracy trajectory of the baselines (ERM, MLDG, GI) and SDE-EDG across domains for RMNIST and Circle datasets, which show the superiority of SDE-EDG on the other 3 baselines with large margin improvements. SDE-EDG could keep large improvements initially, but eventually, as the distance between the target and source domains increases, all methods will achieve similar performance. Therefore, we conclude, that in EDG, models can achieve generalization in the relatively near future. Ablations: the Impact of Maximum Likelihood Loss on EDG Classifications We conducted an ablation study on the RMNIST and PowerSupply datasets to evaluate the effectiveness of the proposed Maximum Likelihood Loss $J_{mle}$, which aims to train SDE-EDG to fit IFGET. Figure 4d and Figure 4e show that SDE-EDG achieves the best performance with $\alpha = 1$ on the RMNIST dataset and $\alpha = 10$ on the PowerSupply dataset. The above empirical results suggest that $J_{mle}$ improves the performance of EDG by aligning the underlying evolving paths (optimizing the Path Alignment loss $J_{pa}$) and quantifying stochastic uncertainties (minimize Stochastic Uncertainty loss $J_{su}$), details proof in Appendix D.1. Specifically, when we only apply classification loss with $\alpha = 0$, the performance is the worst. On the other hand, with a much larger $\alpha = 200$, SDE-EDG focuses on aligning the evolving paths and giving lower importance to classification tasks. Thus, aligning stochastic evolving processes improves performance in EDG. Ablations: $\Delta t$ influence on EDG performance In Table 2, we set interval $\Delta t$ to $5^\circ$, and $20^\circ$ between source domains with SDE-EDG, where the $10^\circ$ interval is in the original setting. With smaller $\Delta t$ ($\Delta t \times 2 \rightarrow \Delta t \rightarrow \Delta t/2$), the accuracy experiences a consistent improvement, a finding that aligns with our motivation: a smaller temporal gap between domains reduces the generalization error. Therefore, interpolations between temporal gaps as an approximation to the sample at an arbitrary timestamp would lead to reduced $\Delta t$ and overcoming overfitting to available limited timestamps. ### 6 CONCLUSION This work presents a new approach SDE-EDG for modeling Evolving Domain Generalization (EDG). Our approach involves constructing IFGET by identifying sample-to-sample correspondence and generating continuous-interpolated samples via linear interpolations. Subsequently, we employ Stochastic Differential Equations (SDE) and train it in alignment with IFGET. Our contribution lies in revealing the importance of capturing the evolving patterns through the collected individual’s temporal trajectories, and of interpolating between time intervals to mitigate the issue of the limited number of source timestamps, which effectively prevents SDE-EDG from overfitting to the limited timestamps. We also provide a theoretical analysis demonstrating that our method can reduce the generalization risk. ACKNOWLEDGEMENTS We appreciate constructive feedback from anonymous reviewers and meta-reviewers. This work is supported by the Natural Sciences and Engineering Research Council of Canada (NSERC), Discovery Grants program. REFERENCES Spyros Makridakis vangelis Addison Howard inversion. M5 forecasting - accuracy, 2020. URL https://kaggle.com/competitions/m5-forecasting-accuracy Martin Arjovsky, Léon Bottou, Ishaan Gulrajani, and David Lopez-Paz. Invariant risk minimization. arXiv preprint arXiv:1907.02893, 2019. Guangji Bai, Chen Ling, and Liang Zhao. Temporal domain generalization with drift-aware dynamic neural networks. In The Eleventh International Conference on Learning Representations, 2023. URL https://openreview.net/forum?id=sWOSRj4nTln Christopher M Bishop and Nasser M Nasrabadi. Pattern recognition and machine learning, volume 4. Springer, 2006. Gilles Blanchard. 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Seth R Flaxman, Rupert RA Bourne, Serge Resnikoff, Peter Ackland, Tasanee Braithwaite, Maria V Cicinelli, Aditi Das, Jost B Jonas, Jill Keeffe, John H Kempen, et al. Global causes of blindness and distance vision impairment 1990–2020: a systematic review and meta-analysis. The Lancet Global Health, 5(12):e1221–e1234, 2017. Muhammad Ghifary, W Bastiaan Kleijn, Mengjie Zhang, and David Balduzzi. Domain generalization for object recognition with multi-task autoencoders. In Proceedings of the IEEE international conference on computer vision, pp. 2551–2559, 2015.
PdTe8S0Mkl	It was mentioned that the use of Roget’s thesaurus is to map words to related categories for a thematic-style analysis. But why are other, more common forms of thematic analysis not explored such as LDA, BERTopic, or Contextual Topic Models? These might even give better results on the differences between theme use as evidenced by the results presented in their corresponding papers (especially CTM).	Humans vs ChatGPT: Uncovering Non-trivial Distinctions by Evaluating Parallel Responses Anonymous authors Paper under double-blind review Abstract The advent of ChatGPT and similar Large Language Models has set the world in an uproar as it is able to generate human-like natural language. Due to the high similarity between the human text and ChatGPT text, it begs the question if the two are truly indistinguishable. In this study, the human-generated content is compared to ChatGPT-3.5, ChatGPT-4, and Davinci-3 using the same technical questions as found on StackOverflow and general questions found on Yahoo Answers. We leveraged Roget’s thesaurus to uncover thematic similarities and differences between the human corpora and GPT corpora. We performed a chi-square test on Roget’s 1034 categories and found a significant difference in the appearance of words for 365 of them. To uncover the differences in the neighborhoods of the word embedding we utilized the MIT Embedding Comparator to distinguish GloVe base vectors with respect to its trained version on human and ChatGPT corpora. Pre-trained BERT and Sentence-BERT were used to measure the semantic similarity in the answers (on the same questions) given by humans and ChatGPT, which came out highly similar. While that might indicate difficulty in distinguishing ChatGPT and human text, the significant differences in the appearance of words suggested a move towards classification using machine learning models. We observed that various machine learning models performed very well. In summary, we discern disparities and parallels that can be attributed to conceptual, contextual, or lexicographic factors. We endeavor to establish connections between each methodology and these respective categories. 1 Introduction Large Language Models (LLMs) have been propping up ever since OpenAI revealed ChatGPT. ChatGPT-3.5 and its more reasonable version, ChatGPT-4 have caused a major ripple not just in the tech industry, but all industries across the globe. The immense value of an AI capable of accurately and reliably comprehending and generating natural language has made LLMs enticing to organizations and individuals around the globe. To meet this demand, big tech companies have released their own LLMs: Github’s Copilot (Chen et al., 2021) and Google’s LaMDA - Language Model for Dialogue Applications (Thoppilan et al., 2022). In the brief time, LLMs such as ChatGPT have been accessible. ChatGPT-generated content has made its way into all aspects of life. It has already begun affecting education (Zhai, 2022) and professional occupation (Felten et al., 2023) where it masquerades as original, human-generated content. ChatGPT’s performance is also impressive even in specialized sectors, it has successfully passed the United States Medical Licensing Exam (USMLE) (Kung et al., 2023). OpenAI’s own study takes an early look at the impact such LLMs will have (Eloundou et al., 2023) where they found around 80% of the United States workforce would have at least Figure 1: This sunburst presents a view of the most significant Roget’s Theme and Categories based on the chi-square test performed on the words appearing per Roget’s Category. The top 20 categories with the highest chi-square score and p-value < 0.05 are shown. 10% of their tasks altered due to these LL.Ms. The study (Eloundou et al., 2023) finds that 19% of workers across different disciplines may have over 50% of their work affected, increasing the urgency for research. Recent literature has attempted to model machine learning and deep learning models to distinguish human text and text generated by ChatGPT (Guo et al., 2023; Mitrović et al., 2023; Shijaku & Canhasi). Based on their results, it is possible to differentiate the two using machine learning techniques. Roget’s Thesaurus, an English Language Thesaurus, written by Peter Mark Roget, a British Lexicographer (Roget & Roget, 1925) can also aid us. It is an excellent resource similar to WordNet (Fellbaum, 2010). It shines in its ability to measure semantic similarity (Jarmasz & Szpakowicz, 2003) and produce word clusters with higher correlation (Jarmasz, 2012) when compared to WordNet. Longer discussion regarding LLMs and methods to classify their text output from humans can be found in Appendix A. In addition, this paper delves deeper into non-trivial distinctions between using various evaluation metrics. We attempted to analyze syntactic, semantic, and lexicographic patterns between the two. Furthermore, we examined how the same words are used in diverse contexts by humans and ChatGPT by using Roget’s thesaurus as seen in Fig 1. In order to make the linguistic comparisons largely fair we collected our own dataset and structured it as a parallel corpus between humans and ChatGPT. The contributions made by this paper are: - Novel parallel datasets of text data where ChatGPT-3.5, ChatGPT-4 and Davinci-3 are prompted to answer questions and generate video scripts - We compare the Parts-Of-Speech tag distribution, BLEU score, stopword distribution, and semantic similarity using BERT models between parallel human responses and GPT-generated responses - Performing comparison of human-generated text to ChatGPT-generated text utilizing Roget’s Thesaurus as a human-grounded knowledge base - Compare lexicographic differences using Machine Learning approaches 2 DATASET 2.1 DATASET SUMMARY To accurately compare human-generated text and the text generated by ChatGPT, a novel dataset is constructed. A total of 13,496 data points have been collected. 6250 are human-generated content and the associated question or title from three platforms: StackOverflow, Yahoo Answers, and YouTube. 6144 data points are from GPT models answering the same questions or producing content for the same title. 1106 data points are of ChatGPT being told to pretend to be a user of StackOverflow or a YouTuber when generating content. Examples of how the dataset is structured are shown in Table 1. A summary of the data collection can be found in Table 2. The datasets have been given acronyms for the rest of this paper for brevity. Yahoo Answers by humans is shortened to YA-H and by Davinci-3 is YA-D3. StackOverflow answers by humans is shortened to SO-H, by ChatGPT-3.5 without context is SO-CGPT3, and with context is SO-CGPT3-C. YouTube transcripts, which are scripts read out by humans in videos, is shortened to YT-H and the scripts generated by ChatGPT-4 are YT-CGPT4. Table 1: Examples of responses given by a human and generated by a GPT model on the same topic \| QUESTION \| HUMAN RESPONSE \| GPT RESPONSE \| \|----------\|----------------\|--------------\| \| How do I most elegantly express left join with aggregate SQL as LINQ query? \| A solution, albeit one that defers handling of the null value to the code, could be: DateTime yesterday = DateTime.No... \| To express a left join with an aggregate function as a LINQ query, you can use the GroupJoin method... \| \| Death From Space — Gamma-Ray Bursts Explained \| Imagine if you could gather the energy from every star within a hundred million light years. From thousands of galaxies... \| Hello, space enthusiasts! Welcome back to our channel. Today, we’re going to delve into one of the most fascinating and powerful events... \| 2.2 DATASET CONSTRUCTION 2.2.1 COLLECTING HUMAN RESPONSES An existing dataset of Yahoo Answers (Soumikrakshit) was used in collecting human data. This is advantageous as it ensures that all the content was generated by a person before ChatGPT-generated content appeared in forums. The title and body of the question were also concatenated to produce the “prompt”. StackOverflow provides a snapshot of their database, which is exposed for research purposes. We queried it to acquire the titles, questions, and answers in HTML format. The data was cleaned using the BeautifulSoup library’s HTML parser. The title acts as the main question and the details of the question are given in the question body, therefore the title was concatenated with the body to create the “prompt” for ChatGPT. The top accepted answer per question was collected as human data. With the recent release of ChatGPT-4 and its ability to generate long-form content, it was necessary to compare that to human-generated content such as YouTube videos. OpenAI’s transcription model, Whisper (Radford et al., 2022), was utilized to extract transcriptions of the videos. The "medium" model of Whisper was used with 769M parameters. Table 2: The number of human-generated and GPT-generated data points collected from the platforms. \| Platform \| Human Datapoints \| GPT Datapoints \| Contextual GPT Datapoints \| \|-------------------\|------------------\|----------------\|--------------------------\| \| StackOverflow \| 1188 \| 1188 \| 1000 \| \| YouTube Transcripts\| 106 \| - \| 106 \| \| Yahoo Answers \| 4954 \| 4954 \| - \| \| Total \| 6250 \| 6144 \| 1106 \| 2.2.2 Collecting GPT responses ChatGPT-3 is OpenAI’s chatbot built on top GPT-3.5. In order to collect data from ChatGPT-3, the “prompts” created from the questions from StackOverflow and Yahoo Answers were fed into a new instance of chat. Afterward, the generated answer was collected. For contextual answers to StackOverflow questions, the prompt was modified by adding a phrase before the question "Answer as a StackOverflow user to the question ..." to the user input. ChatGPT-4 is the next iteration of ChatGPT-3. Its headlining features are more reasonable and generates long-form content. Instead of plainly using the title for the prompt, ChatGPT-4 was asked to write a YouTube script with the title. An example of the prompt is, “Write a YouTube script, only the content without scene direction or mentioning the host. Do not write out sections in your box brackets. with the title: …”. 3 METHODOLOGY 3.1 Evaluation Metrics 3.1.1 Parts-Of-Speech, Stop words and BLEU Score The Parts-Of-Speech (POS) distribution in human-generated text and ChatGPT-generated text were analyzed. Once tagged, the corpora containing human-generated text were compared to the corpora containing ChatGPT-generated text. The pairings for the comparisons are as follows: YT-H and YT-CGPT-4; YA-H and YA-D3; SO-H and SO-CGPT3; SO-CGPT3 and SO-CGPT3-C. These pairings are kept for all experiments. For each pairing, the most frequently occurring POS tags have been compared between human-text and ChatGPT-text. The second area to be looked into for potential differences is the occurrence of stop words. We observe if the usage of stopwords differs between humans and ChatGPT. The Bilingual Evaluation Understudy (BLEU) scores between human-generated text and ChatGPT-generated text have also been calculated. BLEU scores have been evaluated from BLEU-1 to BLEU-4. The number indicates the level of n-gram precision. For example, BLEU-1 represents unigram precision, BLEU-2 represents bigram precision, and so on. 3.2 Mapping to Roget’s Thesaurus As described by Roget, Roget’s thesaurus is “… a collection of the words it [the English language] contains and of the idiomatic combinations peculiar to it, arranged, not in alphabetical order as they are in a Dictionary, but according to the ideas which they express” (Roget 1852). Each of the answer pairs were fed through an algorithm that takes the words in the text and maps them to a corresponding category in Roget’s thesaurus. An example of a word being mapped is given in Fig. 2. The mapping starts out broad and gets increasingly more specific. More details are included in Appendix A. Figure 2: Example of mapping a word into Roget’s Thesaurus. The word "Expertise" is related to the category of "Knowledge". Which in turn is under various themes denoted in the diagram. 3.3 Comparing Word Neighborhoods Using Global Vectors for Word Representations (GloVe) (Pennington et al., 2014), we capture the contextual differences of word usage between humans and ChatGPT. We map words used by both humans and ChatGPT into a high-dimensional space and leverage GloVe to cluster all the words. Once the word embeddings are mapped, the Embedding Comparator introduced by MIT (Boggust et al., 2022) is used to perform a global comparison of embedding spaces and analyze local neighborhoods as well. For our experiment, we gather embeddings from three corpora: the GloVe algorithm trained on Wikipedia and Gigaword 5 which we take as the base, GloVe trained on a human corpus, and GloVe trained on a GPT corpus. 3.4 High-Level Textual Comparison Using BERT Models For a high-level comparison between human-generated text and ChatGPT-generated text, pre-trained BERT and Sentence-BERT models were deployed. The pre-trained BERT model assessed the same pairings of data, capturing the output and calculating the cosine similarity between human-text embedding $\hat{H}$ and GPT-text embedding $\hat{G}$ across 13 hidden layers. T-SNE plots visualized these embeddings for each layer. Concurrently, Sentence-BERT, specifically the "all-MiniLM-L6-v2" model from Hugging Face, was utilized for evaluating semantic similarity. The pairwise cosine similarity of these embedded data points was calculated and represented through stacked histograms, facilitating a detailed analysis of the similarity between human and GPT corpora. 3.5 Modelling with Machine Learning and Baseline BERT Each data pairing was cleaned through the removal of URLs, punctuation, and stop words. The data were encoded using the Term Frequency-Inverse Document Frequency (TF-IDF) vectorizer. The train test split was 75:25. Various classifiers were trained on a binary classification task using traditional machine-learning models. The models used were Support Vector Machine Classifier (SVM) (Hearst et al., 1998), Naive Bayes classifier (Rish et al., 2001), and eXtreme Gradient Boosting (XGB) (Chen et al., 2015). Afterward, an exhaustive feature reduction was performed on linear SVM to further see how distinct the lexicographic differences between the classes are. The same task was achieved through baseline BERT, with the BERT pre-processor "bert-en-uncased-preprocess" and encoder "bert-en-uncased-L-12-H-768-A-12" from TensorFlow being used. The encoder uses 12 hidden layers (i.e., Transformer blocks), a hidden size of 768, and 12 attention heads with the Adam optimizer is used and binary cross-entropy loss function. 4 RESULTS DISCUSSION 4.1 HUMAN vs CHATGPT CLASSIFICATION XGB performed the best on the datasets of SO-H vs SO-CGPT3 and YA-H vs YA-D3 with an accuracy of 92% and 78% respectively. SVM has performed the best in YT-H vs YT-CGPT4 and SO-CGPT3-C vs SO-H with accuracy of 96% and 94% respectively. XGB and SVM have tied at 83% in SO-CGPT3-C vs SO-CGPT-3. NB, while having a lower accuracy score than XGB and SVM, has also performed well in text classification. Our baseline BERT performs similarly. We believe further fine-tuning the parameters could improve the accuracy but our goal was not to optimize for classification. The complete list of results is found in Table 3. Performing feature reduction on linear SVM results in improvements across all data pairings except YouTube and Yahoo Answers as we see in Figure 3. The high performance of the statistical machine learning model lends credence to the idea that there are enough lexicographic differences between human text and GPT text to distinguish the two. Table 3: Model performance in classification task across datasets \| Dataset \| Model \| Accuracy \| ROC AUC \| F1 Score \| \|---------------\|-------\|----------\|---------\|----------\| \| SO-CGPT3 \| SVC \| 90% \| 0.90 \| 0.90 \| \| \| NB \| 86% \| 0.86 \| 0.86 \| \| SO-H \| XGB \| 92% \| 0.93 \| 0.93 \| \| \| BERT \| 77% \| 0.79 \| 0.77 \| \| YA-D3 \| SVC \| 77% \| 0.77 \| 0.77 \| \| \| NB \| 74% \| 0.74 \| 0.75 \| \| YA-H \| XGB \| 78% \| 0.78 \| 0.78 \| \| \| BERT \| 80% \| 0.84 \| 0.79 \| \| YT-CGPT4 \| SVC \| 96% \| 0.97 \| 0.96 \| \| \| NB \| 94% \| 0.93 \| 0.94 \| \| YT-H \| XGB \| 94% \| 0.94 \| 0.94 \| \| \| BERT \| 66% \| 0.69 \| 0.67 \| \| SO-CGPT3-C \| SVC \| 83% \| 0.83 \| 0.83 \| \| \| NB \| 80% \| 0.80 \| 0.80 \| \| SO-CGPT3 \| XGB \| 83% \| 0.83 \| 0.83 \| \| \| BERT \| 79% \| 0.86 \| 0.80 \| \| SO-CGPT3-C \| SVC \| 94% \| 0.94 \| 0.94 \| \| \| NB \| 90% \| 0.90 \| 0.90 \| \| SO-H \| XGB \| 90% \| 0.90 \| 0.90 \| \| \| BERT \| 75% \| 0.83 \| 0.75 \| 4.2 FINDINGS FROM ROGET’S CATEGORIES The mapping of the corpora to the 6 main Roget’s Themes has shown little difference between humans and ChatGPT. The high similarity of the themes means human-generated text and ChatGPT-text cannot be separated on a thematic basis when comparing the parent themes. When we compare the base Roget’s categories, we find that many of the categories show a strong relationship. The chi-square score is calculated for each of these base categories mapped to both humans and ChatGPT. The p-value is calculated for all of these mappings after calculating the chi-square value. For 365 of them, it is less than 0.05 which means that the observed data is extremely unlikely to have occurred by chance alone under the null hypothesis. This could convey the idea that the same concepts are used in different context by humans and ChatGPT which could be a distinguishing feature. The top 20 most significant Roget’s categories and their respective themes are illustrated in Figure 1. 4.3 POS TAG DISTRIBUTION, STOP WORDS AND BLEU The difference in POS-tag distribution for the 4 pairings has been normalized. Illustration can be found in Figure 2. The distribution of POS tags is observed to be largely similar between humans and ChatGPT. One difference is, GPT models tend to use more noun singular (NN) when generating qualitative content. When generating technical content, humans tend to use significantly more NN. However, the difference is slight at 0.04. When ChatGPT-3.5 was asked to mimic a human while answering StackOverflow questions, the difference in the distribution of POS tags was minimized. Indicating that if properly prompted, ChatGPT is capable of answering in a more human-like manner in one-shot answers. The results of the stopword analysis were determined to lack statistical significance and as such they have been included in Appendix A. Ultimately, it is evident that POS-tag distribution and stopword analysis cannot be a reliable indicator of discriminating between humans and GPT-generated content. BLEU score gives an interesting insight. The specific scores can be found in Appendix A under Table 4. BLEU-1 has the highest overall score with the highest being between YA-D3 and YA-H at 0.937. The high BLEU-1 score indicates that they use similar vocabulary. BLEU-3 and BLEU-4 have very poor scores which indicates sentence structures are possibly different. 4.4 BERT AND SENTENCE-BERT REPRESENTATION ANALYSIS The same pairings of human and GPT data have been propagated through BERT and SBERT. Afterward, the cosine similarity scores have been calculated and normalized. The scores are plotted as a stacked histogram in Figure 4 for SBERT and T-SNE plot in Figure 5 for BERT. YA-D3 vs YA-H have the least in common in terms of semantic textual similarity. It is the only pairing that has a significant portion of its cosine similarity in the negative. The other pairings all have high cosine similarity scores. The highest being between SO-CGPT3-C and SO-H. This is further evidence that discriminating between ChatGPT and humans at a high level is a challenging task. The high-level representations in the pre-trained BERT model appear to be insufficient in discriminating between the pairings. On a side note, the similarity between SO-CGPT3-C and SO-H being so high is evidence that when prompted to mimic a human, the answers produced by ChatGPT mirror closely those given by humans. Figure 4: SBERT cosine similarity, illustrating pairwise semantic similarity. Figure 5: T-SNE plots of layers of pre-trained BERT of StackOverflow embeddings. Red points are SO-H and blue points are SO-CGPT3. 4.5 Word Neighborhoods The embeddings from the base and trained GloVe models were obtained and analyzed using the Embedding Comparator. We find that when we compare the common words used by humans and ChatGPT and observe their local word neighborhood, the neighbors are different. An example are the words "comfort" and "terrorism", whose local neighborhoods are illustrated in Figure 6(a) and Figure 6(b) respectively. The neighborhoods in which the word "comfort" is found in a high-dimensional mapping of the human corpus are different than that of Davinci-3’s corpus. This is indicative that even if the words used by humans and GPT models are the same, the usage of these words differs in their contexts. Thus concepts they deliver are different. ![Local neighborhood for the word "comfort"](image1) ![Local neighborhood for the word "terrorism"](image2) Figure 6: Illustration of word neighbourhoods found using the Embedding Comparator (Boggust et al., 2022) on GloVe embeddings trained on human text (YA-H) and Davinci text (YA-D3). 5 Limitations and Future Work While the results are promising, it is important to note the limitations of this study. The data from the GPT models were analyzed as-is, unlike in real-world scenarios where ChatGPT’s output is often manually modified or rephrased using a tool. Future work can include collecting a larger parallel dataset with a larger variety - having ChatGPT answers with and without context. Another would be to perform the same experiments but across different languages to see whether the differences outlined in the paper hold true across various languages. 6 Conclusion Classification of human text and ChatGPT-generated text does not require a complex model. An observation made in (Ippolito et al., 2019) is that generation models produce high-likelihood words. This is in contrast with humans who are more likely to introduce statistical anomalies in their texts. Suggesting a significant difference in the vocabulary used by the two. This idea is reinforced by our findings. The analysis of the appearance of Roget’s categories reveals that there is a non-random pattern in the way humans use words that differ from ChatGPT. And we observe this trend again when analyzing the word neighborhoods for humans and ChatGPT in embedding space. In both texts, they appear to remain contextually relevant and as such there is high similarity when using BERT or SBERT models. In conclusion, syntactic and contextual differences are insignificant but conceptual differences appear to be significant. Lexicographic differences are also significant and can be picked up easily using machine learning approaches, which explains the high performance. REFERENCES Angie Boggust, Brandon Carter, and Arvind Satyanarayan. Embedding comparator: Visualizing differences in global structure and local neighborhoods via small multiples. In 27th international conference on intelligent user interfaces, pp. 746–766, 2022. 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Sy8upuD6Bw	- Page 5: - Would it be possible to give a sense of what the correlation between the presence of the noise token and the feedback request from the receiver is? It's not necessary for all of the experiments, maybe just the initial basic ones.	Emergent Communication with Conversational Repair Mitja Nikolaus * CerCo, CNRS mitja.nikolaus@cnrs.fr Abstract Research on conversation has put emphasis on the importance of a multi-level communication system, in which the interlocutors aim to establish and maintain common ground. In natural conversations, repair mechanisms such as clarification requests are frequently used to improve mutual understanding. Here we explore the effects of conversational repair on languages emerging in signaling games. We extend the basic Lewis signaling game setup with a feedback channel that allows for the transmission of messages backwards from the receiver to the sender. Further, we add noise to the communication channel so that repair mechanisms become necessary for optimal performance. We find that languages emerging in setups with feedback channel are less compositional. However, the models still achieve a substantially higher generalization performance in conditions with noise, putting to question the role of compositionality for generalization. These findings generalize also to a more realistic case involving a guessing game with naturalistic images. More broadly speaking, this study provides an important step towards the creation of signaling games that more closely resemble the conditions under which human languages emerged. 1 Introduction Conversation analysis has been describing human conversation as interactions between speaker and listener, in which the interlocutors are using multiple communicative levels to negotiate mutual understanding (Schegloff et al., 1977; Schegloff, 1982; Clark & Schaefer, 1989; Clark, 1996; Pickering & Garrod, 2021). Whenever speakers are verbalizing their communicative intent to a listener, thereby communicating some information, listeners can either acknowledge (explicitly or implicitly) the receipt of this information or initiate a repair routine (e.g., ask for clarification in case they did not understand the speaker correctly). While conversational repair mechanisms such as clarification requests (also known as other-initiated repairs) have been found to be present in a large range of human languages (Tabensky, 2001; Dingemanse & Enfield, 2015), most recent research on language evolution has focused on unidirectional communication channels, thus only allowing information flow from the sender to the receiver, and not backwards. However, for basic other-initiated repair to emerge, a feedback information flow from the receiver to the sender is necessary. In this work, we study the role of conversational repair for the nature of languages emerging in signaling games (Lewis, 1969). We extend a widely-used basic signaling game setup to allow for the flow of feedback messages from the receiver to the sender, thus implementing a bidirectional model of communication. By studying the languages emerging in this setup, we find that they generalize better to unseen test examples under noisy conditions, while showing a substantially lower degree of compositionality as measured by topographic similarity. We validate this result for a range of different noise levels, messages lengths, and input space sizes. * Work performed at Aix-Marseille University. Finally, we develop a more realistic guessing game setup with naturalistic scenes based on the Guess-What?! dataset (De Vries et al., 2017), in which the receiver needs to discriminate a target object from a set of distractor objects within the same visual scene. Our findings regarding the improved performance under noisy conditions generalize to this more realistic setup. 2 RELATED WORK 2.1 COMPUTATIONAL MODELING OF EMERGENT COMMUNICATION Computational models of emergent communication aim to implement aspects of human language evolution using communication games. While early attempts used Bayesian modeling to study the emergence of syntax using the so-called iterated learning model (Kirby & Hurford, 2002; Kirby et al., 2007), more recent approaches are leveraging deep reinforcement learning approaches to scale the models up to more realistic learning scenarios (Lazaridou et al., 2017; Lazaridou & Baroni, 2020; Guo et al., 2022; Chaabouni et al., 2020; Lazaridou et al., 2018; Chaabouni et al., 2022; Rodríguez Luna et al., 2020). In many studies, emergent communication is studied in a basic Lewis signaling game (Lewis, 1969), which involves a sender and a receiver. The sender is required to communicate some information to the receiver through a communication channel with limited capacity. Most models only consider a unidirectional communication channel, without any possibility for information flow backwards from the receiver to the sender, therefore not allowing for any conversational repair mechanisms to emerge. Exceptions are the game setups in Evtimova et al. (2018); Cao et al. (2018); Graesser et al. (2020), which allow for multi-directional flow of information. However, these studies did not consider communication channels with noise and consequently there exists no pressure for repair mechanisms to emerge. Jorge et al. (2016) analyzes languages emerging in a bidirectional signaling game with noise, but the noise is added to the communication channel in a way that it is not directly detectable by the message receiver. Here we focus on a bidirectional communication game setup, in which sender messages are replaced by a special noise token with a certain probability. Thereby, the receiver can in principle learn to detect the presence of the noise token and initiate a conversational repair routine. Compositionality and Generalization A range of computational studies has explored compositionality and generalization in emerging languages. Chaabouni et al. (2020) studies the phenomena in a principled approach and found that agents can succeed to communicate and generalize even to unseen objects without the emerged languages necessarily being compositional according to a range of measures. The authors find that generalization capabilities emerge if the input space is large enough. Rita et al. (2022a) looks into multi-agent game setups and finds that sufficiently heterogeneous populations produce more compositional languages with an increasing number of agents. These results are in line with research on experimental studies with human subjects (e.g., Raviv et al., 2019). Rita et al. (2022b) shows that the commonly used loss can be broken down into an information term and a co-adaptation term, and that controlling for overfitting on the co-adaptation loss increases compositionality and generalization performance. Other studies explore the role of template transfer (Korbak et al., 2021), communication channel capacity (Gupta et al., 2020), or communication over sets of objects (Mu & Goodman, 2021). In our work we directly compare the generalization performance and compositionality of models with unidirectional communication channel to those with an additional feedback channel. 2.2 CONVERSATIONAL REPAIR IN LANGUAGE EVOLUTION Historically, a large portion of research in linguistics has been dedicated to find universals in the syntax of human languages. While the existence of such a Universal Grammar is disputed, more recent trends highlight the possibility to describe universals with respect to the use rather than the structure of language. For example, it has been argued that certain communicative feedback devices such as other-initiated repair could be universally present in human languages (Dingemanse et al., 2013, 2015; Dingemanse & Enfield, 2015). Such universals of conversation are not explained by innateness, but rather by a selective pressure towards the evolution of common optimised forms that is exerted by the conversational environments (Dingemanse et al., 2013; Roberts & Mills, 2016). As such mechanisms form major building blocks of human communication, it is important to investigate how they impact the emergence of structure in language (Silva & Roberts, 2016). Healey et al. (2007) analyzes languages emerging between human interlocutors in a graphical language game and finds that repair is key for the emergence of complex symbol systems. Mills & Redeker (2022) suggests that self-repair increases the abstraction of emerging message systems. Lemon (2022) sketches out a framework for emergent communication with conversational grounding. Agents should be able to detect disagreements and resolve them, in order to maintain a common ground. Targeted feedback signals facilitate the coordination between communication partners. Related computational implementations can be found for example in Steel’s (1995), where a model for vocabulary formation within conversation that includes simple feedback mechanisms for responses and message acknowledgements is proposed. Other examples include Tria et al. (2012), which focuses on “blending repair”, a strategy that exploits the structure of the world to create new words, as well as de Ruiter & Cummins (2012), proposing a bayesian model of communication in which repair sequences are initiated if the entropy of the prior and posterior probability distributions over possible intentions surpass a certain threshold. Finally, van Arkel et al. (2020) compares pragmatic reasoning and other-initiated repair, using bayesian modeling and complexity analysis. In our work, we explicitly study the role of conversational repair by directly comparing models with and without feedback channel regarding the generalization performance and the compositionality of the emerging languages. Crucially, we leverage deep-learning based models that scale to more realistic input, instead of only small-scale toy language game setups. 3 METHODS 3.1 BASIC SIGNALING GAME We implement a signaling game (Lewis, 1969) following common practices in the literature (Kottur et al., 2017; Lazaridou et al., 2018; Chaabouni et al., 2020; Rita et al., 2022b). In the following, we will describe the details for the baseline used in all experiments. Two agents communicate using symbols in a discrimination game. A sender agent $S$ is provided with an input object $o_i$, sends a message token $m \in X$ using discrete symbols to the receiver agent $R$. The vocabulary of possible tokens is denoted as $X$. The receiver needs to discriminate the target object from a set of distractor objects $O$ by using the information provided in the message $M$. The input objects are defined by a number of attributes $A$ each with possible values $V$. An object is encoded using a concatenation of one-hot encodings for each attribute, i.e. the input dimensionality is $\|A\| \cdot \|V\|$. The capacity of the communication channel is defined by the number of symbols in the vocabulary $\|X\|$ and the message length $\|M\|$. Both sender and receiver are implemented as gated Recurrent Neural Networks (RNNs) using single-layer GRUs with layer normalization (Ba et al., 2016). In the basic setup, the number of distractor objects (including the target) $\|O\|$ is set to 2. The parameters $\theta_R$ of the receiver are optimized using a cross-entropy loss: $$L_{\text{receiver}}(\theta_R) = -\log(\pi_{\theta_R}(o_i\|O, M))$$ where $\pi_{\theta_R}$ is the current policy of the receiver. In parallel to the receiver, the sender agent is trained using REINFORCE (Williams, 1992): $$L_{\text{sender}}(\theta_S) = -\sum_{t=0}^{\|M\|} r \cdot \log(\pi_{\theta_S}(m_t\|o_i, m_{t-1}))$$ where $\pi_{\theta_S}$ is the current policy of the sender, $m_t$ is the message token at time step $t$, and $r$ is the reward ($r = 1$ if the receiver chooses the correct object from the set of distractor objects and $r = 0$ otherwise). We further use a running mean baseline to reduce the variance of the gradients as well as entropy regularization to encourage exploration. At training time, the messages from the speaker are sampled from the current policy, at test time we employ greedy decoding. We split the set of all possible objects into a training set (90%) and a test set (10%). Further hyper-parameters and implementation details can be found in Appendix A.1. The source code of the mod- els and all experiments is publicly available at https://github.com/mitjanikolaus/emergent_communication. 3.2 Basic Signaling Game with Noise and Feedback To explore the effects of feedback, we make two adjustments to the baseline model described in the preceding section. First, we introduce noise to the communication channel: With a probability of $ p_{\text{noise}} $, each token in the message $ M $ is replaced with a special noise token. $ M' $ denotes the message after manipulation with the noise. Secondly, we allow the receiver RNN to generate feedback messages. At each timestep, the receiver RNN consumes the sender message token and produces a feedback token $ n \in Y $. The sender RNN consumes this feedback token in addition to its last turn’s output (both tokens are embedded and afterwards concatenated). The loss functions for the agents with feedback are as follows: \[ L_{\text{receiver\_fb}}(\theta_R) = -\log(\pi_{\theta_R}(o_i\|O, M', N)) \] \[ L_{\text{sender\_fb}}(\theta_S) = -\sum_{t=0}^{\|M\|} r \cdot \log(\pi_{\theta_S}(m_t\|o_i, m_{t-1}, n_{t-1})) \] We set $ \|Y\| $ to 2, i.e. the receiver only produces binary feedback. This allows a receiver agent to use the feedback channel for example to send acknowledgements or open clarification requests (Dingemanse & Enfield [2015]). We leave the study of larger feedback channels for future work. The architecture of the model with feedback channel is displayed in Figure 1. ![Figure 1: Architecture of signaling game with feedback channel. Both the Sender RNN (RNN_S) and Receiver RNN (RNN_R) are unrolled in time.](image) 3.3 GuessWhat Signaling Game In order to test whether the results observed on the toy signaling game setup generalize to more realistic game setups, we develop another game setup in which agents communicate about objects in naturalistic images. In this game, the receiver has to discriminate a target object from a set of distractor objects that are all present in the same visual scene. This task resembles a common communicative task, in which a speaker is trying to refer to a single object within a visual scene. The proposed game is based on the GuessWhat?! dataset ([De Vries et al., 2017]), which was initially designed to create models of grounded task-oriented dialog. Here, we only use the annotated image data, which consists of images annotated with objects and their corresponding bounding boxes ([Lin et al., 2014]). For each image, we select one of the objects as the input object $ o_i $ and use the remaining objects as distractor objects. The remaining task procedure as well as the model implementation --- 1 See Section 4.1.4 for a discussion of alternative noise implementations. 2 Related work has proposed to study emergent communication using images from ImageNet ([Russakovsky et al., 2015]). Here, we propose a task which relies on discriminating objects within the same visual scene as opposed to different images, which is arguably harder and at the same time close to communication problems that humans are usually facing: Referring to an object in the shared visual environment. 3 We constrain the maximum number of distractor objects to 10. If there are more objects available, we randomly sample a subset of 10 objects. are identical to the basic signaling game (cf. Section 3.1). Two example images are shown in Appendix A.2. Following the procedure described in De Vries et al. (2017), we select all objects with bounding boxes of a minimal size (area ≥ 500 px²). We further discard all images that contain only one object. For each object, we extract features from the corresponding bounding box using Vision Transformer (vit-b-16; Dosovitskiy et al., 2020), which yields 768 dimensional vectors. We keep the original train and validation splits as defined in CoCo (Lin et al., 2014). In total, there are 70,702 images (385,961 objects; 5.5 per image on average) in the training split and 8,460 (45,541 objects; 5.4 per image on average) in the validation split (which we use as test set). 3.4 Evaluation For each setting, we start 3 different runs with varying random seed and report the mean and 95% confidence intervals for all metrics unless stated otherwise. We evaluate the models by measuring accuracy on a held-out test split (test_acc). We further report test accuracy in a separate forward pass for which the channel noise is disabled (test_acc_no_noise). This allows us to investigate how models are performing under optimal conditions even if they were trained with exposure to noise. Finally, we measure the compositionality of the emerged languages using topographic similarity (topsim; Brighton & Kirby, 2006), as it is common practice in the language emergence literature (Lazaridou et al., 2018; Chaabouni et al., 2020; Li & Bowling, 2019). For fair comparison, the compositionality metric is calculated in the separate forward pass during which the channel noise is disabled. 4 Results 4.1 Basic Signaling Game 4.1.1 Effect of Noise We start by investigating the case of (\|A\|, \|V\|) = (4, 4) for increasing amount of noise: $ p_{noise} \in \{0, 0.1, 0.3, 0.5, 0.7, 0.9\} $. To ensure convergence of the agents, following the results of Chaabouni et al. (2020), we employ them with a large enough channel capacity: A vocabulary size \|X\| of 2 and a message length \|M\| of 10⁴. ![Figure 2](image) As a first sanity check, we observe that without noise, both models perform optimally (test_acc ≈ 1). When comparing the test accuracy in settings with noise, we observe that for all settings the models with feedback outperform the baseline models. This suggests that the feedback channel allows the models to repair the communication under noisy conditions. Additionally, we find that higher noise increases the performance advantage of a feedback channel up to a noise level of $ p_{noise} = 0.7 $. --- In the case of (\|A\|, \|V\|) = (4, 4) the input space is $ \|V\|^{\|A\|} = 4^4 = 256 $. In that way the channel capacity is sufficiently larger than the input space: $ \|X\|^{\|M\|} = 2^{10} = 1024 \gg 256 $. $p_{noise} = 0.9$ the advantage decreases again and the model convergence becomes more unstable (as indicated by the increased variability of performance between runs). Under optimal conditions, if the channel noise is removed, both models perform approximately on par, suggesting that while the feedback models can repair communication under noise, this does not harm their performance when noise is absent. While the test accuracy of feedback models under noise is clearly superior, we observe a substantial drop in the topsim score for these models. This suggests that while the feedback allows the models to increase test accuracy in conditions with noise, this is coinciding with a decrease in compositionality (as measured by the topsim score). While Chaabouni et al. (2020) already observed that compositionality is not necessary to achieve generalization, here we even observe an opposing trend. Analysis of Feedback Messages In order to gain a better understanding of how the models employ the feedback channel to repair the communication, we analyze the messages of a converged model for the case $p_{noise} = 0.5$. To this end, we record the messages sent by the sender as well as the feedback messages sent by the receiver for the test set. Then we calculate the correlation (Matthew’s Correlation Coefficient; Matthews [1975]) of receiver messages with (1) the presence of noise in the sender messages, (2) the sender messages (excluding messages that contain noise), as well as (3) the one-hot encodings of the two input objects. Figure 3 visualizes the correlations using heatmaps. ![Heatmap](image) Figure 3: Matthew’s Correlation Coefficient between receiver messages and the presence of noise, the sender messages, and the one-hot encodings of the two input objects. The messages are recorded while the agents are playing the signaling game on the test set. When observing the response patterns we find that the feedback message tokens do not depend on the presence of a noise token in the previous turn (all correlation coefficients are close to 0 in the leftmost graph). This indicates that the feedback tokens are not used as open clarification requests, i.e. they are not simply signaling the presence of noise back to the sender. The second graph shows that there is however a positive correlation between the sender messages and receiver messages in the subsequent turn. Following a 1 sent by the sender, the receiver usually responds with 1 and vice versa. In this way, the feedback messages can function as an acknowledgement, signaling the received message back to the sender. For later messages (after message 5 approximately), we find a negative correlation that is slightly delayed. Finally, we find that there are also substantial correlations between the properties of the candidate objects (target and distractor) and the receiver messages. This hints that the feedback messages also serve to communicate certain aspects of the candidate objects to the sender (who does not have access to both objects). In this way, sender and receiver can be co-constructing meaning during the course of the interaction. Understanding the exact mechanisms of the feedback messages remains challenging, as the models could create any arbitrary messaging code. Still, we would like to estimate to which degree the models actually develop an efficient code to solve the signaling game. We implement an additional setup in which the receiver model is encouraged (using an additional loss term) to only signal the --- 5We also analyze the messages of 2 other runs with different seeds and observe highly similar patterns. presence of noise back to the listener. The details of this setup as well as result graphs can be found in Appendix A.3. We find that while in this case the receivers indeed signal the presence of noise, the generalization performance lacks behind that of models who develop their own feedback messaging code (but is still better than baseline performance without any feedback). The best performing models leverage the feedback message channel to exchange information more efficiently than models using the feedback channel for simple open clarification requests. 4.1.2 Effect of Input Space To ensure that the observed effects are not only a phenomenon of the specific input space, we experiment with multiple other configurations of larger and smaller input spaces. We keep the noise ratio at $p_{noise} = 0.5$ and vary the number of input attributes $\|A\|$ and values $\|V\|$: $(\|A\|, \|V\|) \in \{(2, 10), (4, 4), (3, 10), (2, 100), (2, 1000), (10, 1000)\}$. The results are depicted in Figure 4. We find that for all tested configurations, the feedback channel alleviates the detrimental effects of noise. The largest effects are observed for very small input sizes $(\|A\|, \|V\|) = (2, 10)$ or very large ones $(\|A\|, \|V\|) = (10, 1000)$. Notably, the input space is even surpassing the channel capacity in the three larger input space settings. In line with the findings of the previous section, we also observe a decrease in topsim scores for most settings. Also, the models’ generalization performances are comparable if the channel noise is removed. ![Figure 4: Results for different input space dimensions.](image) 4.1.3 Effect of Message Length Another important hyperparameter of the game setup is the message length of the communication channel. Here, we investigate the influence of this parameter on the performance advantage of a feedback channel. ![Figure 5: Results as a function of message length $\|M\|$.](image) We set $p_{\text{noise}} = 0.7$ and vary the message length: $\|M\| \in \{1, 3, 5, 10, 20, 30, 50\}$. As shown in Figure 5, we find that starting from $\|M\| = 5$, a performance advantage for the models with feedback emerges. The advantage increases until a length of 30, afterwards the gap between the performance of two model types decreases again. With a sufficiently high message length, the sender can simply repeat each message multiple times to increase chances of successful transmission without the need for any receiver feedback. When comparing the conditions $\|M\| = 10$ and $\|M\| = 20$, we find that models with an additional feedback channel and $\|M\| = 10$ even outperform models with a unidirectional message channel that is double in size ($\|M\| = 20$). This suggests that in this configuration it is more efficient to allow models for feedback communication than to increase the capacity of the unidirectional message channel. ### 4.1.4 Effect of Noise Implementation In our basic game setup the noise is implemented using a special token and is therefore simply detectable by the receiver agent. This relates to phenomena such as a listener not understanding a syllable or word because of some increased background noise. In order to model other phenomena, such as misunderstandings, the noise on the channel can be implemented as a random permutation of the message token with another token from the vocabulary. In this case, the presence of noise is not directly detectable by the listener and therefore more negotiation might be necessary in order to obtain a common ground. We therefore expect a lower generalization performance with this implementation of noise. We run the experiments described in Section 4.1.1 with this alternative implementation of noise. The results are shown in Appendix A.4. We find that for this kind of noise, the generalization performance drops more substantially with increasing noise level (e.g., mean test_acc of 0.70 vs. 0.89 for $p_{\text{noise}} = 0.7$), validating our hypothesis that this kind of noise is more challenging for communication. However, we still observe that feedback partially alleviates the effects of noise: The models with feedback outperform the baseline models. The compositionality of languages as measured by topsim is again lower for the models with feedback. ### 4.2 GuessWhat Signaling Game Based on the GuessWhat signaling game described in Section 3.3, we perform a set of experiments to investigate whether the findings on the basic signaling game also hold on more realistic communication game setups with naturalistic images. ![Figure 6: Generalization performance for models in the GuessWhat signaling game as a function of channel noise $p_{\text{noise}}$ (left) and message length $\|M\|$ (right).](image) We initially keep the same channel capacity as in the basic signaling game setup, a vocabulary size $\|X\|$ of 2 and a message length $\|M\|$ of 10. The left plot in Figure 6 shows the effect of increasing noise on models with and without feedback channel. In line with the previous findings, we find that the feedback channel alleviates the effects of noise, with a peak in performance difference that is again around $p_{\text{noise}} = 0.7$. Regarding the role of message length, the right plot in Figure 6 shows that the performance advantage increases with increasing $\|M\|$ (with a fixed channel noise of $p_{noise} = 0.5$). In contrast to the findings on the basic signaling game, this advantage does not decrease for the largest message length ($\|M\| = 50$). When evaluating the generalization capabilities without noise, both model types perform comparably (see Appendix A.5). 5 DISCUSSION AND CONCLUSION The findings of this work suggest that in signaling games with noisy conditions, a superior performance can be achieved when models are allowed to send feedback messages backwards from the receiver to the sender. While this increases the generalization performance of the models, the compositionality of the emerged languages decreases. This drop in compositionality might be explained by multiple factors. First, as already shown in Chaabouni et al. (2020), there is not always a direct link between compositionality and generalization performance. Secondly, natural languages are not perfectly compositional either, in many cases meaning is dependent on context (Goldberg, 2015). When allowing for a bidirectional information flow between sender and receiver, it is possible that both agents are jointly co-constructing mutual understanding and thereby creating contextualized meanings. Consequently, the sender messages become less compositional and more context-dependent (see also Section 4.1.1). Recently, Korbak et al. (2020); Conklin & Smith (2023) also highlighted the limitations of topsim as a measure of compositionality in emergent communication, to which our results add additional evidence. Lemon (2022) pointed to a lack of vision-and-language datasets that explicitly require conversational grounding in addition to symbol (visual) grounding. In this work we designed a simple referential signaling game that allows for the study of conversational repair in the context of a referential game within naturalistic scenes. In line with the findings from the basic signaling game, we find that a feedback channel allows models to improve their generalization performance under noise. With the development of models for an efficient generation of clarifying questions in dialog being an open challenge (Kiseleva et al., 2022), the proposed setup allows for the study of the emergence of crucial mechanisms for successful dialog, such as basic communicative grounding acts (Clark & Schaefer, 1989; Clark, 1996). So far, this work only investigated setups with binary message and feedback channels. To study the emergence of more advanced repair mechanisms such as restricted requests or restricted offers as opposed to open clarification requests (Dingemanse & Enfield, 2015), the capacity of the message channel should be increased in subsequent works. We experimented with two alternative implementations of noise (cf. Section 4.1.4), but even further setups should be investigated in the future and might trigger the emergence of more advanced repair mechanisms. This includes for example combining the two proposed noise implementations (special noise token for modeling non-understanding, and token permutations for modeling misunderstanding) within a single model, as well as non-uniform distributions of noise. Relatedly, we currently do not add any noise on the feedback messages from the receiver. While this design choice was taken to study the emergence of basic conversational repair, it is not realistic and will need to be adapted in the future to perform more extensive experiments on nested clarification requests (van de Braak et al., 2021). Other axes of future work could extend the model to explore the emergence of a preference for self-repair over other-initiated repair, which is typically found in human conversation (Schegloff et al., 1977). As indicated from these numerous opportunities for future work, the current work contributes another important step to the ongoing efforts on closing the gap between signaling games and realistic models of language evolution (Chaabouni et al., 2019; Rita et al., 2020; Galke et al., 2022). Kottur et al. (2017) also observe that agents exploit bidirectional communication channels to create non-compositional languages. They counteract by limiting the vocabulary size and removing one agent’s memory at every timestep, which prevents messages from being context-dependent. LaCroix (2019) questions compositionality as a target for language evolution research more generally. The author argues that focus should instead be put on reflexivity, as it is more consistent with a gradualist approach to language origins. Future work is required to operationalize measures of reflexivity and apply them to computational emergent communication experiments. ACKNOWLEDGMENTS Many thanks to Lukas Galke for fruitful and in-depth discussions related to this work. Further, this work was substantially improved thanks to the reviewers who provided very constructive feedback. This work, carried out within the Labex BLRI (ANR-11LABX-0036) and the Institut Convergence ILCB (ANR-16CONV-0002), has benefited from support from the French government, managed by the French National Agency for Research (ANR) and the Excellence Initiative of Aix-Marseille University (AMIDEX). The project leading to this publication has received funding from Excellence Initiative of Aix-Marseille - AMIDEX (Archimedes Institute AMX-19-IET-009), a French “Investissements d’Avenir” Programme. REFERENCES Jimmy Lei Ba, Jamie Ryan Kiros, and Geoffrey E. Hinton. Layer Normalization. NIPS, 2016. URL https://openreview.net/forum?id=BJLa_ZC9 Henry Brighton and Simon Kirby. Understanding Linguistic Evolution by Visualizing the Emergence of Topographic Mappings. Artificial Life, 12(2):229–242, 2006. ISSN 1064-5462. doi: 10.1162/artl.2006.12.2.229. URL https://ieeexplore.ieee.org/abstract/document/6791988 Kris Cao, Angeliki Lazaridou, Marc Lanctot, Joel Z. Leibo, Karl Tuyls, and Stephen Clark. 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KXOB15k1br	What is the significance of the improvements of TSAA when presenting the results with a critical diagram that includes a statistical test (Demšar, JMLR 2006, https://jmlr.org/papers/v7/demsar06a.html)?	TIME-SERIES AUTOAUGMENT: DATA AUGMENTATION POLICY SEARCH FOR LONG-TERM FORECASTING Anonymous authors Paper under double-blind review ABSTRACT Data augmentation is a popular regularization for addressing overfitting issues of neural networks. Recently, automatic augmentation showed strong results on image classification tasks. However, less attention had been given to automatic augmentation of time-series problems such as long-term forecasting. Toward bridging this gap, we propose an efficient, effective, and easy-to-code time-series automatic augmentation method we refer to as TSAA. We solve the associated bilevel optimization problem in two steps: a partial train of the non-augmented model for a few epochs and an iterative split process. The iterative process alternates between finding a good augmentation policy via Bayesian optimization and fine-tuning the model while pruning poor runs. Our method is evaluated extensively on challenging univariate and multivariate forecasting benchmark problems. Our results indicate that TSAA outperforms several strong baselines in most cases, suggesting it should be incorporated into prediction pipelines. 1 INTRODUCTION Modern machine learning tools require large volumes of data to effectively solve challenging tasks. However, high-quality labeled data is difficult to obtain as manual labeling is costly and it may require human expertise (Shorten & Khoshgoftaar, 2019). Small datasets may lead to overfitting in overparameterized models, a phenomenon in which the model struggles with examples it has not seen before (Allen-Zhu et al., 2019). One of the effective methods to alleviate poor generalization issues is via data augmentation (DA). Data augmentation aims to generate artificial new examples whose statistical features match the true distribution of the data (Simard et al., 1998). In practice, DA has been shown to achieve state-of-the-art (SOTA) results in e.g., vision (Krizhevsky et al., 2012) and natural language (Wei & Zou, 2019) tasks. Unfortunately, DA is not free from challenges. For instance, (Tian et al., 2020b) showed that the effectivity of augmented samples depends on the downstream task. To this end, recent approaches explored automatic augmentation tools, where a good DA policy is searched for (Lemley et al., 2017; Cubuk et al., 2019). While automatic frameworks achieved impressive results on image classification tasks (Zheng et al., 2022) and other data modalities, problems with time-series data received significantly less attention. Toward bridging this gap, we propose in this work a new automatic data augmentation method, designed for time-series forecasting problems. Time-series forecasting is a long-standing task in numerous scientific and engineering fields (Chatfield, 2000). While deep learning techniques achieved groundbreaking results on vision and NLP problems already a decade ago, time-series forecasting (TSF) was considered by many to be too challenging for deep models, up until recently (Oreshkin et al., 2020). While recent linear approaches showed interesting forecast results (Zeng et al., 2022), existing SOTA approaches for TSF are based on deep learning architectures that are structurally similar to vision models. In particular, current TSF deep models are overparameterized, and thus they may benefit from similar regularization techniques which were found effective for vision models, such as (automatic) data augmentation. Ultimately, our work is motivated by the limited availability of DA tools for time-series tasks (Wen et al., 2020). The main contributions of our work can be summarized as follows: 1) We develop a novel automatic data augmentation approach for long-term time-series forecasting tasks. Our approach is based on a carefully designed dictionary of time-series transformations, Bayesian optimization for policy search, and pruning tools that enforce early stopping of ineffective networks. While these components appear in existing work, their combination and adaptation to time-series forecasting was not done before, to the best of our knowledge. 2) We analyze the optimal policies our approach finds. Our analysis sheds light into the most effective transformations, and it may inspire others in designing effective data augmentation techniques for time-series data. 3) Our approach augments existing time-series forecasting baselines, and we extensively evaluated it on long-term forecasting univariate and multivariate TSF benchmarks with respect to several strong baseline architectures. We find that our framework enhances performance in most long-term forecast settings and across most datasets and baseline architectures. 2 RELATED WORK Time-series forecasting. Recently, several neural network approaches for TSF have been proposed. Based on recurrent neural networks, DeepAR (Salinas et al., 2020) produced probabilistic forecasts with uncertainty quantification. The N-BEATS (Oreshkin et al., 2020) model employs fully connected layers with skip connections, and subsequent work (Challu et al., 2022) improved long-term forecasting via pooling and interpolation. Another line of works based on the transformer architecture (Vaswani et al., 2017) used a sparse encoder and a generative decoder in the Informer (Zhou et al., 2021), trend-seasonality decomposition in the Autoformer (Wu et al., 2021), and Fourier and Wavelet transformations in the FEDformer (Zhou et al., 2022). Recently, Pyraformer (Liu et al., 2021) significantly reduced the complexity bottleneck of the attention mechanism, PatchTST (Nie et al., 2022) exchanges the point-wise attention input with a tokenized sub-series representation. Finally, Zeng et al. (2022) propose a single-layer MLP with a larger input lookback. Data augmentation. DA techniques have appeared since the early rise of modern deep learning to promote labeled image invariance to certain transformations (Krizhevsky et al., 2012). Typical image augmentations include rotation, scaling, crop, and color manipulations. Recent methods focused on modality-agnostic methods which blend linearly the inputs and labels (Zhang et al., 2018). Other works produce augmented views in the feature space (DeVries & Taylor, 2017; Verma et al., 2019). In contrast to image and text data, augmenting arbitrary time-series (TS) data have received less attention in the literature (Wen et al., 2020; Iwana & Uchida, 2021). In the review (Wen et al., 2020), the authors consider three different tasks: TS classification, TS anomaly detection, and TS forecasting. Their analysis is based on common time-series augmentation approaches such as scaling, adding noise (Um et al., 2017), window cropping or slicing, and stretching of time intervals (Le Guennec et al., 2016), dynamic time warping (Ismail Fawaz et al., 2019), perturbations of the frequency domain (Gao et al., 2020; Chen et al., 2023), and utilizing surrogate data (Lee et al., 2019). In Smyl & Kuber (2016), the authors discuss additional TS augmentation approaches including generating new TS using the residuals of a statistical TS (Bergmeir et al., 2016). Another technique would be to subsample the parameters, residuals, and forecasting from MCMC Bayesian models. The survey (Iwana & Uchida, 2021) further details a large list of TS augmentations such as jittering, rotation, time warping, time masking, interpolation and others in the context of time-series classification. Finally, the authors in (Wen et al., 2020) propose the selection and combination of augmentations using automatic approaches as a promising avenue for future research, which is the focus of the current work. We show in Fig. 1A two examples of DA policies we use. Automatic DA. To avoid hand-tailored DA, recent efforts aimed for automatic tools, motivated by similar advances in neural architecture search (NAS) approaches (Zoph & Le, 2017). AutoAugment (Cubuk et al., 2019) used a recurrent controller along with reinforcement learning for the search process, yielding a highly effective but computationally intensive framework. Following works such as Fast AutoAugment employed Bayesian optimization and density matching (Lim et al., 2019). RandAugment (Cubuk et al., 2020) reduces the search space significantly by introducing stochasticity. Tian et al. (2020a) suggested partial training using augmentation-wise weight sharing (AWS). Further, recent approaches utilize gradients for the search problem, including the differentiable automatic DA (DADA) (Li et al., 2020b) and Deep AutoAugment (Zheng et al., 2022). Cheung & Yeung (2020) developed automatic DA that does not depend on the data-modality as it exploits latent transformations. In (Fons et al., 2021), the authors propose adaptive-weighting strategies which favor a subset of time-series DA for classification, based on their effect on the training loss. 3 BACKGROUND Below, we briefly describe background information on Bayesian optimization and pruning approaches which we use to find the best augmentation policy and improve model training efficiency, respectively. Tree-structured Parzen Estimators and the Expected Improvement. Bayesian optimization relates to a family of techniques where an objective function $f(x) : \mathbb{R}^d \to \mathbb{R}^+$ is minimized, i.e., $$\min_x f(x).$$ (1) In the typical setting, $f$ is costly to evaluate, its gradients are not available, and $d \leq 20$. For instance, finding the hyperparameters $(x)$ of a neural network $(f)$ is a common use case for Bayesian optimization (Bergstra et al., 2013). Unlike grid/random search, Bayesian optimization methods utilize past evaluations of $f$ to maintain a surrogate model $p(y\|x)$ for the objective function $y = f(x)$. Thus, Bayesian optimization solves (1) while limiting the costly evaluations of $f$ to a minimum. A practical realization of Bayesian optimization is given by Sequential Model-Based Optimization (SMBO) (Hutter et al., 2011). SMBO iterates between model fitting with the existing parameters (exploitation) to parameter selection using the current model (exploration). SMBO constructs a surrogate model $p(y\|x)$, finds a set of parameters $x$ that performs best on the $p(y\|x)$ using an acquisition function, applies $x$ on the objective function $f$ to obtain the score $y$, updates the surrogate model, and repeats the last three steps until convergence. Most SMBO techniques differ in their choice of the surrogate model and acquisition function. We will focus on Tree-structured Parzen Estimator (TPE) for the surrogate model, combined with Expected Improvement for the acquisition function. The main idea behind TPE is to model the surrogate via two distributions, $l(x)$ and $g(x)$, corresponding to model evaluations that yield positive, and negative improvement. Formally, $$p(x\|y) = \begin{cases} l(x) & y < y^* \\ g(x) & y \geq y^* \end{cases},$$ (2) where $y^$ is a threshold score, and the surrogate model is obtained via Bayes rule. It can be shown that maximizing $l(x)/g(x)$ leads to an optimal Expected Improvement (EI) (Bergstra et al., 2011). Asynchronous Successive Halving. While Bayesian optimization uses a minimal number of evaluations of $f$, the overall minimization is computationally demanding due to the high cost of $f$, e.g., if $f$ is a neural network that needs to be trained. To alleviate some of these costs, Asynchronous Successive Halving (ASHA) (Jamieson & Talwalkar, 2016; Li et al., 2020a) enforces early stopping of poorly performing parameters $x$, whereas parameters with low $l(x)$, are trained to the fullest. In a fixed budget system, given a maximum resource $R$, minimum resource $r$, and a reduction factor $\eta$, ASHA works as follows. One creates model checkpoints during the training process at epochs $\eta^j$ where $j = 1, \ldots, \lfloor \log_\eta R/r \rfloor$. Each checkpoint is referred to as a rung, and at the end of each rung, one keeps only the best $\frac{1}{\eta}$ runs. To avoid waiting for all runs to reach the next rung, ASHA performs asynchronous evaluations to promote or halt runs on the go. We illustrate in Fig. 1B an example of a baseline model with multiple different runs, administered by the ASHA policy. Figure 1: A) Two examples of sub-policies applied on Electricity data. B) The above plot demonstrates the behavior of ASHA with respect to the baseline model (blue). Some of the poorly performing runs are discontinued at the end of rungs, whereas the other runs train to completion. 4 TIME-SERIES AUTOAUGMENT (TSAA) Automatic augmentation via bi-level optimization. The task of finding data augmentations automatically during the training of a deep neural network model can be formulated as a bi-level optimization problem, see e.g., (Li et al., 2020b). Namely, $$\min_{\theta} \mathcal{L}_{\text{val}}(\omega, \theta)$$ subject to $$\min_{\omega} \mathbb{E}_{p_\theta} [\mathcal{L}_{\text{tr}}(\omega, \theta)] ,$$ where $\mathcal{L}_{\text{tr}}$ and $\mathcal{L}_{\text{val}}$ denote the train and validation losses, respectively, typically mean squared error for TSF. The parameters $\omega$ and $\theta \sim p_\theta$ correspond to the network weights and the augmentation policy. The above minimization is difficult to solve, and thus we relax it as detailed next. TSAA overview. Our approach, which we call time-series automatic augmentation (TSAA), consists of two main steps, as illustrated in Fig. 2 and summarized in Alg. 1 in App. D. In the first step, we partially train the model for a few epochs, and construct a set of shared weights. The second step iterates between solving Eq. (3) in search of an augmentation policy using TPE and EI to solving Eq. (4) with fine-tuning and ASHA for an optimal model. A complexity analysis is given in App. E. Step 1: compute shared weights. Solving Eq. (4) iteratively requires repeated trainings of the deep model, which is computationally prohibitive. To reduce these costs, we propose to partially train the baseline model and generate a shared set of weights $\omega_{\text{shared}}$. Doing so, Step 2 is reduced to an iterative process of fine-tuning models for a small number of epochs, where $\omega_{\text{shared}}$ are shared across all augmentations policies. Beyond efficiency aspects, applying DA in the later stages of training is assumed to be more influential (Tian et al., 2020a). In practice, we partially train for $\lfloor \beta K \rfloor$ epochs, where $\beta = 0.5$ is a hyperparameter and $K$ is the active number of train epochs. In our tests, $K \leq 10$, and it may be strictly less due to an early stopping scheduler. $K$ is found by training the baseline model with no augmentation to completion and saving the weights after every epoch. Then, we define $$\omega_{\text{shared}} := \omega(\lfloor \beta K \rfloor), \quad R := K - \lfloor \beta K \rfloor,$$ where $R$ is the maximum resource parameter, and $r = 1$ is the minimum resource, see Sec. 3. Step 2: iterative split optimization. Given $\omega_{\text{shared}}$, it remains to solve Eqs. (3) and (4) to find the best augmentation policy $\theta^$ and final weights $\omega^$. In TSAA, we propose to split this problem to an iterative process, where we alternate between exploring augmentation policies $\theta$ via Eq. (3) to exploiting the current policy and produce model weights $\omega$ via Eq. (4). Namely, for a fixed set of weights $\omega$, the upper minimization finds the next policy $\theta$ to try by evaluating the validation set. Then, we fine-tune the model using a fixed $\theta$ with early stopping for a maximum of $R$ epochs to produce the next $\omega$. This procedure is repeated until a predefined number of trials $T_{\text{max}}$ is reached. The $k$ best-performing policies define $p_{\theta^}$ from which $\theta^$ is sampled, where we only allow policies that improve the baseline validation loss. Finally, we fine-tune the model again to obtain $\omega^$. Solving Eq. (3). Existing work solved the upper problem using reinforcement learning (Cubuk et al., 2019; Tian et al., 2020a), grid search (Cubuk et al., 2020; Fons et al., 2021), and one-pass optimization (Li et al., 2020b; Zheng et al., 2022). Inspired by (Lim et al., 2019), we propose to use Tree-structured Parzen Estimator (TPE) with Expected Improvement (EI), see Sec. 3. In the context of TSAA, the parameters $ x $ in Eq. (1) represent the policy $ \theta $ and $ f $ is $ L_{\text{val}} $. The Bayesian optimization is conducted over the policy search space and time-series augmentations we describe below. Policy search space. The augmentation policies $ \theta $ we consider are drawn from a distribution $ p_\theta $ over $ k $ sub-policies $ \Theta = \{\theta_1, \ldots, \theta_k\} $, i.e., \[ \theta \sim p_\theta := \{p(\theta_j)\|\theta_j \in \Theta\}. \] (5) Each sub-policy $ \theta_j $ is composed of $ n $ transformations $ T_{j,i} $, applied sequentially on the output data $ x_{i-1} $ of the previous transformation with $ x_0 $ being the input data and $ m_{j,i} $ being the magnitude of the transformation. That is, \[ \theta_j = T_{j,n}(x_{n-1}, m_{j,n}) \circ \cdots \circ T_{j,1}(x_0, m_{j,1}). \] (6) Time-series data augmentations. While natural images are invariant to geometric transformations as translation and rotation, arbitrary time-series data need not be invariant to a certain type of transformations. Moreover, capturing the invariance in regression problems such as TSF may be more challenging than in classification tasks including images. Finally, time-series data may include slow and fast phenomena such as bursts of electricity usage and seasonal peaks, for which some DA may be inapplicable. Thus, we propose to exploit DA that manipulate some features of the data and leave some features unchanged. For example, adjusting the trend while keeping the seasonality and noise components unaffected, or diversifying the time intervals in a way that the series mean and variance still stay the same. In particular, we suggest the following time-series transformations: identity, jittering, trend scaling, seasonality scaling, scaling, smoothing, noise scaling, flip, permutation, reverse, dynamic-time-stretching (DTS), window warping, and mixup. The magnitude of the augmentations can be controlled using a single parameter. The transformations are further elaborated in App. C and Tab. 2 in the appendix. Solving Eq. (4). Finally, solving the bottom minimization may be achieved in a straightforward fashion via fine-tuning. However, as motivated in Sec. 4, doing so iteratively is costly. To prune runs, we augment our approach with Asynchronous Successive Halving (ASHA). Our choice to use ASHA over other techniques such as Bayesian Optimization HyperBand (BOHB) (Falkner et al., 2018) is motivated by the following reasons. First, BOHB has shown to be slightly inferior to ASHA (Li et al., 2020a). Second, in our setting $ R \in \{1, 2, ..., 5\} $ and $ \eta $ is set to be more aggressive. As a result, only two SHA brackets at most can be exploited in the HyperBand, thus limiting its effectiveness. 5 RESULTS In what follows, we provide details regarding our experimental setup and we evaluate our approach. In the supplementary material, we give additional information on models and datasets (App. B), hyperparameters (App. E), and extended results (App. G.3). 5.1 IMPLEMENTATION DETAILS Baselines. We train all models based on the implementation and architecture details as they appear in (Oreshkin et al., 2020) for N-BEATS and (Zhou et al., 2021; Wu et al., 2021; Zhou et al., 2022) for the Transformer-based models. The model weights are optimized with respect to the mean squared error (MSE) using the ADAM optimizer (Kingma & Ba, 2015) with an initial learning rate of $ 10^{-3} $ for N-BEATS and $ 10^{-4} $ for Transformer-based models. The maximum number of epochs is set to 10 allowing early-stopping with a patience parameter of 3. The reported baseline results are obtained using our environment and hardware, and they may slightly differ from the reported values for the respective methods. Every experiment is run on three different seed numbers, and the results are averaged over the runs. The Pytorch library (Paszke et al., 2019) is used for all model implementations, and executed with NVIDIA GeForce RTX 3090 24GB. Method. We use Optuna (Akiba et al., 2019) for the implementations of TPE and ASHA. The number of trials $T_{\text{max}}$ is set to 100. For TPE, in order to guarantee aggressive exploration at the beginning, we run the first 30% of trials with random search. For ASHA, $\tau$ and $\eta$ are set globally to 1 and 3 respectively. The maximum resource parameter $R$, representing the epochs, is set differently for each experiment, due to the baseline’s early-stopping. After the augmentation policy search is finalized, a maximum of $k$ best policies are selected to obtain $p_{\theta^}$, where $k = 3$, and the final model is fine-tuned with $\theta^ \sim p_{\theta^}$ using the shared weights $\omega_{\text{share}}$. We opt to fine-tune the model and not re-train from random weights so that the final model training matches our optimization process as close as possible. Indeed, Cubuk et al. (Cubuk et al., 2020) discuss the potential differences between the final model behavior in comparison to the performance of the intermediate proxy tasks, i.e., the models obtained during optimization. As the similarity in performance between these models and the final model is not guaranteed, a natural choice is to similarly train the proxy tasks and the final model, as we propose to do. Augmentations. Each transformation includes a different increasing or decreasing magnitude range which are all mapped to the range $[0, 1]$. This way, $m = 0$ implies the identity and $m = 1$ is the maximum scale. To eliminate cases of the identity being repeatedly chosen, we replace the lower bound in the range with an $\epsilon > 0$ such that for all transformations in the search space only $m > 0$ is possible. The transformations Trend scale and Seasonality scale require computing the seasonality and trend components; we pre-compute these factors using the decomposition in STL (Cleveland et al., 1990) and treat it as part of the input data. Each augmentation is applied before the input is fed to the model, namely, on the input $x$ and the target $y$ of the train data batches. 5.2 Main results In our experiments, we employ a similar setup to (Wu et al., 2021; Zhou et al., 2022), where the input length is 96 and the evaluated forecast horizon corresponds to 96, 192, 336, or 720. For ILI, we use input length 36 and horizons 24, 36, 48, 60. For a fair comparison, we re-produce all baseline results. Table 1: Multivariate long-term time-series forecasting results on six datasets in comparison to five baseline models. Low MSE and MAE values are better, and high relative improvement MSE% and MAE% scores are better. Boldface text highlights the best performing models. \| Dataset \| Informer \| Autoformer-w \| FEDformer-w \| FEDformer-f \| TSAA \| \|---------\|----------\|--------------\|-------------\|-------------\|------\| \| \| MSE \| MAE \| MSE \| MAE \| MSE \| MAE \| MSE%↑ \| MAE%↑ \| \| ETM2 \| 96 \| 0.545 \| 0.588 \| 0.231 \| 0.310 \| 0.205 \| 0.290 \| 0.189 \| 0.282 \| 0.187 \| 0.274 \| 1.058 \| 2.837 \| \| \| 192 \| 1.054 \| 0.808 \| 0.289 \| 0.346 \| 0.270 \| 0.329 \| 0.258 \| 0.326 \| 0.255 \| 0.314 \| 1.163 \| 3.681 \| \| \| 336 \| 1.523 \| 0.948 \| 0.341 \| 0.375 \| 0.328 \| 0.364 \| 0.323 \| 0.363 \| 0.304 \| 0.350 \| 5.882 \| 3.581 \| \| \| 720 \| 3.878 \| 1.474 \| 0.444 \| 0.434 \| 0.433 \| 0.425 \| 0.425 \| 0.421 \| 0.398 \| 0.403 \| 6.353 \| 4.276 \| \| Electricity \| 96 \| 0.336 \| 0.416 \| 0.200 \| 0.316 \| 0.196 \| 0.310 \| 0.185 \| 0.300 \| 0.183 \| 0.297 \| 1.081 \| 1.000 \| \| \| 192 \| 0.360 \| 0.441 \| 0.217 \| 0.326 \| 0.199 \| 0.310 \| 0.201 \| 0.316 \| 0.195 \| 0.309 \| 2.010 \| 0.323 \| \| \| 336 \| 0.356 \| 0.439 \| 0.258 \| 0.356 \| 0.217 \| 0.334 \| 0.214 \| 0.329 \| 0.208 \| 0.323 \| 2.804 \| 1.824 \| \| \| 720 \| 0.386 \| 0.452 \| 0.261 \| 0.363 \| 0.248 \| 0.357 \| 0.246 \| 0.353 \| 0.238 \| 0.348 \| 3.252 \| 1.416 \| \| Exchange \| 96 \| 1.029 \| 0.809 \| 0.150 \| 0.281 \| 0.151 \| 0.282 \| 0.142 \| 0.271 \| 0.143 \| 0.272 \| -0.704 \| -0.369 \| \| \| 192 \| 1.155 \| 0.867 \| 0.318 \| 0.409 \| 0.284 \| 0.391 \| 0.278 \| 0.383 \| 0.270 \| 0.378 \| 2.878 \| 1.305 \| \| \| 336 \| 1.589 \| 1.011 \| 0.713 \| 0.616 \| 0.442 \| 0.493 \| 0.450 \| 0.497 \| 0.459 \| 0.504 \| -3.846 \| -2.231 \| \| \| 720 \| 3.011 \| 1.431 \| 1.246 \| 0.872 \| 1.227 \| 0.868 \| 1.181 \| 0.841 \| 1.213 \| 0.842 \| -2.710 \| -0.119 \| \| Traffic \| 96 \| 0.744 \| 0.420 \| 0.615 \| 0.384 \| 0.584 \| 0.368 \| 0.577 \| 0.361 \| 0.565 \| 0.352 \| 2.080 \| 2.493 \| \| \| 192 \| 0.753 \| 0.426 \| 0.670 \| 0.421 \| 0.596 \| 0.375 \| 0.610 \| 0.379 \| 0.571 \| 0.351 \| 4.195 \| 6.400 \| \| \| 336 \| 0.876 \| 0.495 \| 0.635 \| 0.392 \| 0.590 \| 0.365 \| 0.623 \| 0.385 \| 0.584 \| 0.359 \| 1.017 \| 1.644 \| \| \| 720 \| 1.011 \| 0.578 \| 0.658 \| 0.402 \| 0.613 \| 0.375 \| 0.632 \| 0.388 \| 0.607 \| 0.368 \| 0.979 \| 1.867 \| \| Weather \| 96 \| 0.315 \| 0.382 \| 0.259 \| 0.332 \| 0.269 \| 0.347 \| 0.236 \| 0.316 \| 0.180 \| 0.256 \| 23.729 \| 18.987 \| \| \| 192 \| 0.428 \| 0.449 \| 0.298 \| 0.356 \| 0.357 \| 0.412 \| 0.273 \| 0.333 \| 0.252 \| 0.311 \| 7.692 \| 6.607 \| \| \| 336 \| 0.620 \| 0.554 \| 0.357 \| 0.394 \| 0.422 \| 0.456 \| 0.332 \| 0.371 \| 0.296 \| 0.355 \| 10.843 \| 4.313 \| \| \| 720 \| 0.975 \| 0.722 \| 0.422 \| 0.431 \| 0.629 \| 0.570 \| 0.408 \| 0.418 \| 0.382 \| 0.395 \| 6.373 \| 5.502 \| \| ILI \| 24 \| 5.349 \| 1.582 \| 3.549 \| 1.305 \| 2.752 \| 1.125 \| 3.268 \| 1.257 \| 2.760 \| 1.123 \| -0.291 \| 0.178 \| \| \| 36 \| 5.203 \| 1.572 \| 2.834 \| 1.094 \| 2.318 \| 0.980 \| 2.648 \| 1.068 \| 2.362 \| 0.984 \| -1.898 \| -0.408 \| \| \| 48 \| 5.286 \| 1.594 \| 2.889 \| 1.122 \| 2.328 \| 1.006 \| 2.615 \| 1.072 \| 2.264 \| 0.988 \| 2.749 \| 1.789 \| \| \| 60 \| 5.419 \| 1.620 \| 2.818 \| 1.118 \| 2.574 \| 1.081 \| 2.866 \| 1.158 \| 2.520 \| 1.062 \| 2.098 \| 1.758 \| Table 2: Univariate long-term time-series forecasting results on five datasets in comparison to five baseline models. Low MSE and MAE values are better, and high relative improvement MSE% and MAE% scores are better. Boldface text highlights the best performing models. \| Dataset \| Informer \| Autoformer \| FEDformer-f \| N-BEATS-I \| N-BEATS-G \| TSAA \| \|---------\|----------\|------------\|-------------\|-----------\|-----------\|------\| \| \| MSE \| MAE \| MSE \| MAE \| MSE \| MAE \| MSE \| MAE \| MSE \| MAE \| MSE \| MAE \| \| ETM2 \| 96 \| 0.085 \| 0.225 \| 0.123 \| 0.270 \| 0.068 \| 0.198 \| 0.080 \| 0.213 \| 0.080 \| 0.210 \| 0.068 \| 0.192 \| 0.000 \| 3.030 \| \| \| 192 \| 0.130 \| 0.282 \| 0.141 \| 0.289 \| 0.096 \| 0.238 \| 0.103 \| 0.240 \| 0.110 \| 0.230 \| 0.068 \| 0.237 \| 0.000 \| 4.420 \| \| \| 336 \| 0.161 \| 0.314 \| 0.170 \| 0.319 \| 0.138 \| 0.286 \| 0.162 \| 0.312 \| 0.172 \| 0.320 \| 0.139 \| 0.290 \| -0.725 \| -1.399 \| \| \| 720 \| 0.221 \| 0.373 \| 0.206 \| 0.353 \| 0.189 \| 0.335 \| 0.199 \| 0.347 \| 0.201 \| 0.353 \| 0.187 \| 0.336 \| 1.058 \| -0.299 \| \| Electricity \| 96 \| 0.261 \| 0.367 \| 0.454 \| 0.508 \| 0.244 \| 0.364 \| 0.326 \| 0.402 \| 0.324 \| 0.397 \| 0.244 \| 0.354 \| 0.000 \| 2.747 \| \| \| 192 \| 0.285 \| 0.386 \| 0.511 \| 0.532 \| 0.276 \| 0.382 \| 0.350 \| 0.417 \| 0.363 \| 0.420 \| 0.277 \| 0.368 \| -0.362 \| 3.665 \| \| \| 336 \| 0.324 \| 0.417 \| 0.539 \| 0.651 \| 0.347 \| 0.432 \| 0.393 \| 0.440 \| 0.392 \| 0.443 \| 0.310 \| 0.394 \| -0.321 \| 5.516 \| \| \| 720 \| 0.632 \| 0.612 \| 0.673 \| 0.610 \| 0.408 \| 0.473 \| 0.458 \| 0.490 \| 0.489 \| 0.502 \| 0.378 \| 0.447 \| 7.353 \| 5.497 \| \| Exchange \| 96 \| 0.490 \| 0.554 \| 0.149 \| 0.308 \| 0.133 \| 0.284 \| 0.210 \| 0.344 \| 0.223 \| 0.351 \| 0.093 \| 0.236 \| 30.075 \| 16.901 \| \| \| 192 \| 0.790 \| 0.721 \| 0.290 \| 0.415 \| 0.292 \| 0.419 \| 0.130 \| 0.840 \| 0.783 \| 0.675 \| 0.215 \| 0.352 \| 15.862 \| 15.181 \| \| \| 336 \| 2.146 \| 1.223 \| 0.708 \| 0.662 \| 0.477 \| 0.532 \| 1.587 \| 1.047 \| 2.632 \| 1.266 \| 0.532 \| 0.572 \| 11.530 \| 7.519 \| \| \| 720 \| 1.447 \| 1.008 \| 1.324 \| 0.892 \| 1.304 \| 0.882 \| 0.870 \| 0.747 \| 2.588 \| 1.303 \| 0.527 \| 0.594 \| 39.425 \| 20.482 \| \| Traffic \| 96 \| 0.262 \| 0.348 \| 0.266 \| 0.372 \| 0.210 \| 0.318 \| 0.181 \| 0.268 \| 0.159 \| 0.240 \| 0.158 \| 0.239 \| 0.629 \| 0.417 \| \| \| 192 \| 0.294 \| 0.376 \| 0.272 \| 0.379 \| 0.206 \| 0.311 \| 0.177 \| 0.263 \| 0.181 \| 0.264 \| 0.160 \| 0.243 \| 6.605 \| 7.605 \| \| \| 336 \| 0.308 \| 0.390 \| 0.261 \| 0.374 \| 0.217 \| 0.322 \| 0.180 \| 0.271 \| 0.155 \| 0.239 \| 0.156 \| 0.244 \| -0.645 \| -2.092 \| \| \| 720 \| 0.364 \| 0.440 \| 0.269 \| 0.372 \| 0.243 \| 0.342 \| 0.229 \| 0.316 \| 0.212 \| 0.304 \| 0.189 \| 0.279 \| 10.849 \| 8.224 \| \| Weather \| 96 \| 0.005 \| 0.048 \| 0.009 \| 0.078 \| 0.009 \| 0.073 \| 0.003 \| 0.044 \| 0.003 \| 0.043 \| 0.001 \| 0.024 \| 66.667 \| 44.186 \| \| \| 192 \| 0.004 \| 0.051 \| 0.009 \| 0.068 \| 0.007 \| 0.067 \| 0.004 \| 0.046 \| 0.004 \| 0.047 \| 0.001 \| 0.027 \| 75.000 \| 41.304 \| \| \| 336 \| 0.003 \| 0.043 \| 0.006 \| 0.058 \| 0.006 \| 0.062 \| 0.004 \| 0.048 \| 0.005 \| 0.054 \| 0.002 \| 0.035 \| 33.333 \| 18.605 \| \| \| 720 \| 0.004 \| 0.049 \| 0.007 \| 0.063 \| 0.006 \| 0.060 \| 0.004 \| 0.049 \| 0.004 \| 0.048 \| 0.002 \| 0.034 \| 50.000 \| 29.167 \| on our system, and the augmentations are applied on the same generated batches as the baseline. Our main results are summarized in Tab.1 and Tab.2 including all the baseline results and TSAA. For TSAA, we include the best performing model trained on all baseline architectures. The full results for every architecture with and without TSAA are provided in the appendix spanning tables 6-14. We detail the mean absolute error (MAE) and mean squared error (MSE) (Oreshkin et al., 2020). Lower values are better, and boldface text highlights the best performing model for each dataset and metric. For TSAA, we also include the relative improvement percentage, i.e., $100 \cdot \frac{(e_b - e_n)}{e_b}$, where $e_b$ is the best baseline error and $e_n$ is our result. We denote by MSE% and MAE% the relative improvement of MSE and MAE, respectively. A higher improvement is better. Multivariate time-series forecasting results.* Based on the results in Tab.1, we observe that most datasets benefit from automatic augmentation, where in the vast majority of cases, TSAA improves the baseline scores. It is apparent that TSAA yields stronger performance in particular in the long-horizon settings with 6.35% $(0.425 \rightarrow 0.398)$ reduction in ETM2, 3.25% $(0.246 \rightarrow 0.238)$ reduction in Electricity, and 2.1% $(2.328 \rightarrow 2.264)$ reduction in ILI. One of the more prominent results appears for Weather 96 and 336 with reductions in MSE of 23.73% $(0.236 \rightarrow 0.180)$ and 10.84% $(0.332 \rightarrow 0.296)$, respectively. For the Exchange dataset, TSAA obtains slightly higher errors with respect to the FEDformer-w baseline. Overall, TSAA achieves the best results in 39 error metrics, in comparison to FEDformer-f and FEDformer-w with 4 and 5 best models, respectively. Univariate time-series forecasting results. Similar to the multivariate results, most long horizon settings benefit from TSAA. With a 21.74% average reduction across all datasets with a horizon of 720. Furthermore, the results that stand out the most are the MSE and MAE reductions in Weather, with a 66%, 75%, 33.50%%, and respectively 44.2%, 41.3%, 17.6%, 29.2% performance improvements corresponding to the 96, 192, 336 .and 720 horizons. Further, it is evident in Tabs.10-14, that the improvements in the Weather dataset are not limited to a specific baseline architecture. In contrast to the multivariate setting, TSAA achieves significantly better scores on the Exchange dataset with average improvements of 21% and 11.27% for the MSE and MAE metrics. Notably, the results in the univariate case are slightly more involved than the multivariate setting such that that only Weather always benefits from TSAA, whereas the results for other datasets are mixed. Still, TSAA shows a positive advantage over all baseline models. In particular, TSAA obtained the best models for 32 error metrics, whereas FEDformer-f and N-BEATS-G improved 9 and 2 measures. Policy analysis. The most noticeable selected transformations are illustrated in Fig.3. It is evident that the transformations Trend Downscale, Jittering, Mixup, and Smoothing are some of the prominent Figure 3: The best five performing transformations per dataset attained with TSAA, measured with the percentage proportion of the selected operations (%ops). Each colored bar represents a transformation and the y-axis represents the percentage proportion the given transformation accounts for. selections in the overall setup. Trend Downscale accounts for more than 30% of the operations in ETTm2, Weather and Electricity; this may indicate that the deep models tend to overestimate the trend, and thus it requires downscaling. Jittering and Smoothing on the other hand, do not violate time-series characteristics such as trend or seasonality but still promote diversity within the given dataset, where Smoothing is approximately the opposite of Jittering. Notably, Mixup appeared as one of the five most important transformations for four and three datasets in the multivariate and univariate settings, respectively. We believe that Mixup is beneficial to TSF since it samples from a vicinal distribution whose variability is higher than the original train set. We show in Fig. 4 the outcome with and without TSAA compared to the ground truth, showing that employing custom policies per signal may significantly improve forecasting. 6 ABLATION AND ANALYSIS 6.1 PARAMETER SELECTION Choice of $\beta$. In what follows, we motivate our choice for the $\beta$ hyperparameter which dictates for how many epochs we pre-train the baseline architecture to obtain $\omega_{\text{shared}}$. To this end, we investigate the effect of utilizing different values of $\beta$. We consider four different settings: 1) full training with augmentation, i.e., $\beta = 0.0$, 2) half training with augmentation, i.e., $\beta = 0.5$, 3) augmentation applied only in the last epoch, 4) baseline training with no augmentation, i.e., $\beta = 1.0$. We used TSAA on the ILI with respect to N-BEATS-G in the univariate setting, and Informer, Autoformer and FEDformer-f in the multivariate case, as well as on multivariate ETTm2 with Autofomer and FEDformer-f. We plot the averaged results of these architectures in Fig. App. 6A, showing four colored curves corresponding to the various forecasting horizons 24/96, 36/192, 48/336 and 60/720 with colors blue, orange, green and red, respectively. The best models are obtained for $\beta = 0.0$ and $\beta = 0.5$, that is, full- and half-augmented training. Somewhat surprisingly, two of the four best forecasting horizons (36/192, 48/336) are obtained for $\beta = 0.5$. Overall, the fully augmented model (i.e., $\beta = 1.0$) attains a 5.1% average improvement over the baseline, whereas using $\beta = 0.5$ yields a 5.3% average improvement. Thus, fully training with augmentation provides no improvement to the overall performance, while requiring significantly more resources. Indeed, Tian et al. (2020a) employs a similar strategy, and thus, we propose to generate $\omega_{\text{shared}}$ after training for half of the active epochs, and we fine-tune the model using optimal augmentation policies. Reduction factor and linked operations. In Sec. 3 we introduced the reduction factor $\eta$ that controls the number of kept runs in ASHA. Additionally, we discuss in Sec. 5 that every sub-policy is composed of $n$ linked operations of time-series augmentations. Here, we would like to empirically justify our choices for these two hyperparameters. Our ablation study uses the ILI dataset on the 36, 48, 60 forecasting tasks, with N-BEATS-G for the univariate case, and Informer, Autoformer, and FEDformer-f for the multivariate configuration. We test the values $\eta \in \{2, 3\}$ and $n \in \{1, 2\}$. Every experiment is repeated three times, and we analyze the average results. Figure 4: The ground truth, prediction, and prediction with augmentation attained with TSAA applied to the same forecast target in ETTm2 with Informer, Autoformer multivariate, and Weather with FEDformer-f and NBEATS-G univariate. It is shown that augmentation can assist the different models to achieve more accurate predictions. The attained policies are given underneath each plot. Overall, we propose to use the values $\eta = 3$ and $n = 2$ due to the following observations arising from our experiments. The improvement difference between $\eta = 2$ and $\eta = 3$ is only 0.12% in favor of $\eta = 2$, thus suggesting that neither exhibits a statistically-dominant performance advantage. Nevertheless, $\eta = 3$ is resource efficient as it reduces the amount of kept runs $1/\eta$ by 16.67%. Moreover, a single operation $n = 1$ attains a 6.4% average improvement compared to the baseline, whereas two linked operations $n = 2$ yield a 7.4% average improvement. Convergence of TSAA. In our experiments, we look for good augmentation policies for $T_{\text{max}} = 100$ iterations. Here, we explore the effect of this value on the performance of the resulting models. We evaluate our framework on the ILI dataset with the architectures Informer, Autoformer, FEDformer-f, N-BEATS-G and N-BBEATS-I using varying values for $T_{\text{max}} \in \{100, 150, 200, 250\}$. Intuitively, greater $T_{\text{max}}$ values may result in an improved convergence and a better overall performance as the framework can explore and exploit a larger variety of configurations from the search space. Indeed, we show in Fig. App. B the normalized average MSE values obtained for the various tests. We observe an MSE reduction of 1% for the transformer-based models when increasing $T_{\text{max}} = 100$ to $T_{\text{max}} = 250$. The N-BEATS architecture benefited more and achieved a 7.25% reduction. In conclusion, the hyperparameter $T_{\text{max}}$ presents a natural trade-off to the practitioner: higher $T_{\text{max}}$ values generally lead to better performance at a higher computational cost, whereas lower values are less demanding computationally but present inferior performance. 7 DISCUSSION In this work, we study the task of data augmentation in the setting of time-series forecasting. While recent approaches based on automatic augmentation achieved state-of-the-art results in image classification tasks, problems involving arbitrary time-series information received less attention. Thus, we propose a novel time-series automatic augmentation (TSAA) method that relaxes a difficult bilevel optimization. In practice, our framework performs a partial training of the baseline architecture, followed by an iterative split process. Our iterations alternate between finding the best DA policy for a given set of model weights, to fine-tuning the model based on a specific policy. In comparison to several strong methods on multiple univariate and multivariate benchmarks, our framework improves the baseline results in the majority of prediction settings. In the future, we would like to explore better ways for relaxing the bilevel optimization, allowing to train an end-to-end model (Li et al., 2020b; Zheng et al., 2022). Further, we believe that our approach would benefit from stronger time-series augmentation transformations. Thus, one possible direction forward is to incorporate learnable DA modules, similar in spirit to filters of convolutional models. REFERENCES Takuya Akiba, Shotaro Sano, Toshihiko Yanase, Takero Ohta, and Masanori Koyama. Optuna: A next-generation hyperparameter optimization framework. In Proceedings of the 25th ACM SIGKDD international conference on knowledge discovery & data mining, pp. 2623–2631, 2019. Zeyuan Allen-Zhu, Yuanzhi Li, and Yingyu Liang. Learning and generalization in overparameterized neural networks, going beyond two layers. Advances in neural information processing systems, 32, 2019. Christoph Bergmeir, Rob J Hyndman, and José M Benítez. Bagging exponential smoothing methods using stl decomposition and box–cox transformation. International journal of forecasting, 32(2): 303–312, 2016. James Bergstra, Rémi Bardenet, Yoshua Bengio, and Balázs Kégl. 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l1U6sEgYkb	- It appears that the 3D DV deformable attention mechanism lifts PV features to 3D using known camera parameters. Also, when the authors concatenate DV features, they depend on these camera parameters. Have the authors attempted to test the tolerance of calibration parameters to noise?	DV-3DLane: End-to-end Multi-modal 3D Lane Detection with Dual-view Representation Yueru Luo\textsuperscript{1,2}, Shuguang Cui\textsuperscript{2,1}, Zhen Li\textsuperscript{2,1} \textsuperscript{1} FNii, CUHK-Shenzhen \textsuperscript{2} School of Science and Engineering, CUHK-Shenzhen \{222010057@link., shuguangcui@, lizhen@\}cuhk.edu.cn Abstract Accurate 3D lane estimation is crucial for ensuring safety in autonomous driving. However, prevailing monocular techniques suffer from depth loss and lighting variations, hampering accurate 3D lane detection. In contrast, LiDAR points offer geometric cues and enable precise localization. In this paper, we present DV-3DLane, a novel end-to-end Dual-View multi-modal 3D Lane detection framework that synergizes the strengths of both images and LiDAR points. We propose to learn multi-modal features in dual-view spaces, i.e., perspective view (PV) and bird’s-eye-view (BEV), effectively leveraging the modal-specific information. To achieve this, we introduce three designs: 1) A bidirectional feature fusion strategy that integrates multi-modal features into each view space, exploiting their unique strengths. 2) A unified query generation approach that leverages lane-aware knowledge from both PV and BEV spaces to generate queries. 3) A 3D dual-view deformable attention mechanism, which aggregates discriminative features from both PV and BEV spaces into queries for accurate 3D lane detection. Extensive experiments on the public benchmark, OpenLane, demonstrate the efficacy and efficiency of DV-3DLane. It achieves state-of-the-art performance, with a remarkable 11.2 gain in F1 score and a substantial 53.5% reduction in errors. The code is available at https://github.com/JMoonr/dv-3dlane. 1 Introduction Autonomous driving (AD) technology in recent years has made remarkable strides, bringing us closer to the realization of fully self-driving vehicles. Within this field, one of the key challenges is the accurate detection of 3D lanes, a critical component for ensuring safe and reliable navigation. 3D lane detection entails identifying the 3D positions of lane boundaries in the environment, providing essential data for tasks like path planning and vehicle control. 3D lane detection is proposed to mitigate the limitations posed by the absence of depth information in 2D prediction. Currently, the majority of 3D lane detection methods rely on vision-centric approaches, i.e., monocular solutions, where some designs are naturally borrowed and benefit from advances in 2D lane methods. Taking the perspective-view (PV) image as input, these monocular methods mainly utilize the inverse perspective mapping (IPM) technique to warp the PV features into BEV. However, there are misalignment issues in the IPM-based methods when encountering non-flat roads, due to the rigid flat assumption of IPM [Nedevschi et al., 2004; Yan et al., 2022]. While some recent efforts have been made to address this issue and... have shown promising results by directly predicting 3D lanes in PV [Bai et al., 2022b; Huang et al., 2023; Luo et al., 2023]. These monocular 3D approaches, as vision-centric solutions, inevitably get stuck in capturing the complexity of real-world driving scenarios, when encountering adverse weather and lighting conditions. In contrast, as an active sensor, LiDAR excels in spatial localization and 3D structure perception, complementing the capabilities of passive sensor cameras, and it gets more widely used thanks to hardware advancements. A bunch of recent works in 3D object detection have demonstrated the power of LiDARs [Zhou & Tuzel, 2018; Lang et al., 2019; Yin et al., 2021a] and multiple modalities [Liang et al., 2019; Wang et al., 2021; Yang et al., 2022; Li et al., 2022b; Chen et al., 2023] in autonomous driving scenarios. Whereas, fewer endeavors [Bai et al., 2018; Luo et al., 2022] have been made to exploit multi-modal strength for 3D lane detection. Albeit using extra LiDAR data, M²-3DLane [Luo et al., 2022] failed to make full use of features in image space which is crucial to 3D lane performance. Besides, M²-3DLane employs a naive fusion to aggregate multi-modal features, resulting in inferior performance to the camera-only methods (e.g., Luo et al., 2023). Given the rich semantics inherent in images and the accurate positional information afforded by the BEV representation [Philion & Fidler, 2020; Li et al., 2022d], we strive to exploit the multi-modal features to enhance the performance of 3D lane detection. Existing methods tend to fuse two modalities into a single space [Liang et al., 2022; Liu et al., 2023b], e.g., BEV, for feature extraction and subsequent prediction. However, this approach constrains the model’s capacity to harness modality-specific features. We contend that features represented in both PV space and BEV space bear significance, facilitating improved representation learning. Motivated by the above observation, we introduce DV-3DLane, a novel end-to-end multi-modal 3D lane detection framework. To maintain a dual-view space representation, we adopt a symmetric backbone consisting of a PV branch and a BEV branch to extract features in PV and BEV spaces, respectively. To leverage the merits of both images and points for comprehensive feature learning in each view, we design a bidirectional feature fusion (BFF) strategy. Subsequently, to effectively facilitate query-based detection using the retained dual-view features, we devise a unified query generator (UQG). This generator initially produces two sets of lane-aware queries: one from the PV space and the other from the BEV space. These two query sets are compelled to capture lane knowledge regarding semantics and spatiality, guided by auxiliary 2D segmentation supervision. Further, these two sets are then combined into a unified set that serves the decoder. To achieve the unification of dual-view queries, we propose a lane-centric clustering technique. Besides, we employ a Transformer decoder to aggressively integrate discriminative features from both views into the unified queries. For effective feature aggregation across different view spaces, we introduce a 3D dual-view deformable attention mechanism that considers the inherent properties of 3D space, resulting in deformed 3D sample points. These 3D sample points are then projected onto the PV and BEV planes, yielding 2D sample points in each respective view space. These projected 2D points are utilized for feature sampling within their respective view spaces. In summary, our contributions are threefold: • We introduce DV-3DLane, an end-to-end multi-modal 3D lane detection framework that harnesses the power of dual-view representation. • We devise the BFF strategy to mutually fuse features across modalities, and design the UQG to merge lane-aware queries from dual views, yielding a unified query set. Further, a 3D dual-view deformation attention mechanism is introduced to aggregate dual-view features effectively. • We conduct thorough experiments on the OpenLane benchmark to validate the effectiveness of our method. Experimental results show that DV-3DLane surpasses previous methods significantly, achieving an impressive 11.2 gain in F1 score and a remarkable 53.5% reduction in errors. Moreover, a 3D dual-view deformation attention mechanism is introduced to aggregate dual-view features effectively. 2 RELATED WORK 2.1 2D LANE DETECTION Recent works in 2D lane detection can be broadly categorized into four main approaches: 1) Segmentation-based methods [Lee et al., 2017; Pan et al., 2017; Neven et al., 2018; Hou et al., Xu et al. (2020); Zheng et al. (2021) devote to classifying pixels into lanes or the background, necessitating further post-processing steps (e.g., grouping and curve fitting) to produce lane instances. 2) Anchor-based methods, inspired by region-based object detectors such as Faster-RCNN (Ren et al., 2015), employ line-like anchors to localize lanes (Wang et al., 2018; Li et al., 2019; Tabelini et al., 2021a). To overcome the limitations of straight-line constraints, Jin et al. (2022) employ eigenlane space to produce diverse lane shape candidates. 3) Point-based methods (Ko et al., 2021; Qu et al., 2021; Wang et al., 2022; Xu et al., 2022) attempt to flexibly localize key points along each lane instance and subsequently group the points belonging to the same lane. 4) Parametric methods (Van Gansbeke et al., 2019; Tabelini et al., 2021b; Liu et al., 2021; Feng et al., 2022) formulate lane detection as a curve fitting problem, leveraging prior knowledge about lane shapes by representing them using various parametric forms, such as polynomials and splines. Figure 2: Overview of DV-3DLane. First, images and point clouds undergo separate processing by the image backbone and point backbone. In the middle stage of backbones, we introduce Bidirectional Feature Fusion (BFF) to fuse multi-modal features across views. Subsequently, the instance activation map (IAM) is utilized to produce lane-aware queries $Q_{pv}$ and $Q_{bev}$. These queries are then subjected to Dual-view Query Clustering, which aggregates dual-view query sets $Q_{pv}$ and $Q_{bev}$ into a unified query set $C$, further augmented with learnable point embeddings $E_{points}$ to form query $Q$. Additionally, we introduce 3D Dual-view Deformable Attention to consistently aggregate point features from both view features $F_{pv}$ and $F_{bev}$ into $Q$. $\oplus$ denotes broadcast summation. Notably, the $\oplus E_{points}$ operation is performed only in the first layer, while in the following layer, $\oplus Q$ is utilized. Different colored boxes denote queries targeting different lanes; dashed boxes represent the background, and box texture indicates features. 2.2 3D LANE DETECTION Existing methods center on vision-centric solutions and draw inspiration from the 2D task. Typically, monocular approaches (Garnett et al., 2019; Efrat et al., 2020; Guo et al., 2020; Chen et al., 2022; Wang et al., 2023; Liu et al., 2022; Li et al., 2022a; Ai et al., 2023; Yao et al., 2023) construct surrogate representations using inverse perspective mapping (IPM), and perform predictions in this surrogate space. Nonetheless, IPM inherently introduces discrepancies between the perspective and the surrogate view in non-flat areas due to its planar assumption. To address this limitation, recent efforts have endeavored to predict 3D lanes from the perspective view (Yan et al., 2022; Bai et al., 2022b; Huang et al., 2023; Luo et al., 2023), or employ a depth-aware projection to enhance lane perception by incorporating LiDAR information (Luo et al., 2022). 2.3 MULTI-MODAL DETECTION Despite advancements in lane detection, multi-modal methods remain relatively underexplored. Previous works typically utilize either BEV (Bai et al., 2018; Yin et al., 2020; Luo et al., 2022) or PV (Zhang et al., 2021b) as representation spaces for performing 2D lane segmentation (Yin et al., 2020; Zhang et al., 2021b) or 3D lane detection (Bai et al., 2018; Luo et al., 2022). For BEV-based methods, Bai et al. (2018) rasterizes LiDAR points to create a BEV image and transforms PV images into BEV using the estimated ground height derived from the LiDAR data. Similarly, M$^2$-3DLane (Luo et al., 2022) utilizes the BEV space to fuse multi-modal features. To project PV features into BEV space, they lift compact 2D features into 3D space guided by the depth map and... further employ a pillar-based method \cite{lang2019pv} to splat them into BEV. While these methods primarily focus on 3D tasks, \cite{yin2020pv} leverages BEV space for fusing camera and LiDAR features, serving for 2D BEV lane segmentation. Conversely, \cite{zhang2021pv} adopts PV to fuse multi-modal features for 2D lane segmentation. In contrast to lane detection, multi-modal methods have been extensively studied in 3D object detection, with most previous multi-modal methods attempting to fuse image features into BEV space due to its compactness and interoperability for ambient perception \cite{ma2022pv}. These methods either adopt point-level fusion \cite{sindagi2019pv,wang2021pv,yin2021pv} to paint points, instance-level fusion to project 3D proposals to image space \cite{yoo2020pv,bai2022pv}, or feature-level fusion to transform features from PV space into BEV space \cite{liu2023pv,liang2022pv}. However, few works consider both the perspective view and BEV simultaneously. 3 METHODOLOGY The overall framework of our DV-3DLane is depicted in Figure 2. Section 3.1 describes the bidirectional feature fusion module, which merges different modalities bidirectionally and constructs multi-modal features in both PV and BEV spaces. In Section 3.2, we present the unified query generator, which generates two lane-aware query sets from dual views and unifies them into a shared space in a lane-centric manner. Section 3.3 introduces the 3D dual-view deformable attention module, which effectively aggregates dual-view features into unified queries, serving for prediction. 3.1 BIDIRECTIONAL FEATURE FUSION Instead of merging different views into one single space \cite{bai2018pv,luo2022pv,liang2022pv,li2022pv,liu2023pv}, we propose to retain features in both PV and BEV spaces while incorporating multi-modal features for each view. To achieve this, we employ a dual branch to extract features for each view, using images and points as input, respectively. Intermediately, we conduct bidirectional feature fusion between the symmetric branches to enhance each view with multiple modalities, as shown in Figure 3 and summarized in Algorithm 1. Algorithm 1 Bidirectional Feature Fusion (BFF)* Input: LiDAR points $P_{pt}$, image $I$, camera parameters $T$ Output: mm-aware PV features $F_{pv}$, BEV features $F_{bev}$, “mm” denotes multi-modal. $F_{pv}^{s1} = \text{PillarNet-S1}(P_{pt})$, $F_{pv}^{s1} = \text{ResNet-S1}(I)$ $\triangleright S1$: stage one. $P_{pt2pv} = \{(u_i,v_i)\|i \in P\} = \text{Project}(T,P_{pt})$ $F_{pt2pv} = \text{Scatter}(idx = P_{pt2pv}, src = F_{pv}^{s1})$ $\triangleright$ points $\rightarrow$ pixels. $F_{pv2pt} = \text{Grid_Sample}(src = F_{pv}^{s1}, coords = P_{pt2pv})$ $\triangleright$ pixels $\rightarrow$ points. $F_{pv} = \text{ResNet}(\text{Concat}(F_{pv}^{s1}, F_{pv2pt}))$ $F_{bev} = \text{PillarNet}(\text{Concat}(F_{pv}^{s1}, F_{pv2pt}))$ $\triangleright$ dual-view multi-modal feature extraction. Concretely, we place points and images in their designated branches. After obtaining low-level features within each branch, we perform bidirectional feature fusion. By projecting 3D points $P_{pt} = \{(x_i,y_i,z_i)\|i \in P\}$ onto the PV plane, we obtain their corresponding 2D coordinates $P_{pt2pv} = \{(u_i,v_i)\|i \in P\}$, where $P$ is the cardinality of the point set. 1) For points-to-pixels fusion, we utilize a Scatter operation to construct dense point feature grids $F_{pt2pv}$, (depicted in the upper part of Figure 3 with blue cells denoting positions hit by the projected 3D points). 2) For pixels-to-points fusion, we employ bilinear interpolation to sample features at 2D positions hit by the projection of 3D points, yielding $F_{pv2pt}$ (shown in the lower part of Figure 3). The resulting cross-modal features in PV and BEV are concatenated with their respective original modal features. The fused multi-modal features in each view, i.e., PV and BEV, are then fed into subsequent modules in the corresponding branch, generating $F_{pv}$ and $F_{bev}$, respectively. Notably, $F_{pv}$ and $F_{bev}$ encapsulate multi-modal information represented in distinct spaces. 3.2 Unified Query Generator We introduce a unified query generator for end-to-end 3D lane detection. To this end, we first generate two distinct lane-aware query sets, termed dual-view queries, from the previously obtained multi-modal features, $F_{pv}$ and $F_{bev}$. Then, we present a lane-centric clustering strategy to unify these dual-view queries into a cohesive set of queries. Dual-view Query Generation. To effectively capture semantic and spatial features related to lanes, which are termed as “lane-aware” knowledge, we utilize an instance activation map (IAM) \cite{cheng2022} assisted method to generate lane-aware queries in PV and BEV spaces. Taking PV branch as an example, we produce a set of IAMs, denoted as $A_{pv}$, via the following equation: $$A_{pv} = \sigma(F(\text{Concat}(F_{pv}, S_{pv})))$$ where $A_{pv} \in \mathbb{R}^{N \times H_{pv} \times W_{pv}}$, $F_{pv} \in \mathbb{R}^{C \times H_{pv} \times W_{pv}}$, $N$ denotes query number, $\sigma$ is the sigmoid function, Concat represents concatenation operation, and $S_{pv}$ comprises two-channel spatial localization features for each pixel \cite{liu2018}. The lane-aware query $Q_{pv}$ assisted by IAMs is generated via: $$Q_{pv} = A_{pv} \otimes F_{pv}^T,$$ where $Q_{pv} \in \mathbb{R}^{N \times C}$, $\otimes$ denotes the matrix product. Similarly, lane-aware BEV query $Q_{bev} \in \mathbb{R}^{N \times C}$ is formed using: $$Q_{bev} = \sigma(F([F_{bev}, S_{bev}])) \otimes F_{bev}^T.$$ To force the query sets to learn lane-aware features, during training, we employ an auxiliary instance segmentation for each branch on top of the query set. Labels for the auxiliary segmentation are generated in pairs for these two branches, which are further assigned to predictions using mask-based bipartite matching \cite{cheng2022}, as illustrated in Figure 4(a) and (b). Dual-view Query Clustering. Given dual-view query sets $Q_{pv}$ and $Q_{bev}$, we propose employing a lane-centric clustering technique to generate a unified query set for end-to-end lane detection. While kMax-DeepLab \cite{yu2022} previously used k-means cross-attention to group pixels into distinct clusters, i.e., instance masks, our approach focuses on unifying queries from different views. Queries from $Q_{pv}$ and $Q_{bev}$ targeting the same lane are merged within the same cluster. Specifically, we initiate lane cluster centers $C \in \mathbb{R}^{N \times C}$ with $Q_{pv}$, and assign each query in $Q_{bev}$ to its nearest cluster center among $C$. Notably, cluster centers can be chosen from either $Q_{pv}$ or $Q_{bev}$. Empirically, we found that using $Q_{pv}$ produces better results. To achieve clustering, we perform attention between $C$ (query) and $Q_{bev}$ (key), while applying argmax along the cluster center (query) dimension \cite{yu2022} as follows: $$A = \arg\max_N(C \times Q_{bev}), \quad \hat{C} = A \cdot Q_{bev} + C,$$ where $\hat{C} \in \mathbb{R}^{N \times C}$ refers to updated centers unifying queries from dual views. In practise, we use gumbel-softmax \cite{jang2016, liang2023} to substitute $\arg\max$. Considering the variation and slenderness of lanes, we employ a refined point query scheme \cite{luo2023} to enhance lane detection. Instead of using a single query for each lane, multiple-point queries are employed for more precise capture \cite{luo2023, liao2022, zhang2021a, liu2023a}. Consequently, in the first layer, we construct point-based queries $Q \in \mathbb{R}^{N \times M \times C}$ with $Q = C \oplus E_{points}$, where $\oplus$ denotes broadcast sum, $E_{points} \in \mathbb{R}^{M \times C}$ is the learnable point embedding, and in the subsequent layer, we update $Q$ by $Q = \hat{C} \oplus Q$. Supervision on Query Clustering. Given the critical importance of deep supervision for the clustering \cite{yu2022}, we leverage the InfoNCE loss \cite{oord2018} to supervise the query... clustering in a lane-centric manner, as illustrated in Figure 4(c) and formulated as: $$L_{NCE} = -\log \frac{\exp(q \cdot k^+/\tau)}{\exp(q \cdot k^+/\tau) + \sum_{k^- \in N} \exp(q \cdot k^-/\tau)},$$ where $\tau$ is a temperature hyper-parameter [Wu et al., 2018], $q$ denotes one query, $k^+$ indicates the positive sample w.r.t. $q$, and $N$ denotes the collection of all negative samples from the different query set relative to the one containing $q$. Notably, queries assigned to the background do not incur penalties in the clustering learning process. With this supervision, queries from different views are grouped together when matched to the same ground truth lane. Consequently, lane-aware knowledge residing in two view spaces is synergized into the unified query. ### 3.3 3D Dual-View Deformable Attention Apart from informative query generation, feature aggregation plays a crucial role in DV-3DLane. Instead of projecting points from densely sampled grids [Chen et al., 2022] or their lifted pillars [Li et al., 2022d] onto the PV plane for feature sampling, as shown in Figure 5(a), we adopt sparse queries to sample features from different views. Moreover, our approach distinguishes itself from several existing sparse query methods, as depicted in Figure 5(b) and (c). For instance, DeepInteraction [Yang et al., 2022] (Figure 5(b)) employs a sequential method to sample PV and BEV features, while FUTR3D [Chen et al., 2023] (Figure 5(c)) projects 3D points into different spaces, sampling features individually for each space. In contrast, as outlined in Algorithm 2, we leverage the inherent properties of 3D space by predicting both 3D reference points and their 3D offsets using queries, forming 3D deformed points. These 3D deformed points are then projected into each space, establishing a consistent feature sampling strategy across spaces, as depicted in Figure 5. Consequently, features corresponding to the same 3D points from different views are effectively sampled and integrated into the query. Algorithm 2 3D DV Deformable Attention Input: unified query set $Q$, PV features $F_{pv}$, BEV features $F_{bev}$, camera parameters $T$. Output: updated unified query $Q$. Ref$_{3d} = \text{MLP}_1(Q)$ ▷ 3D reference points. $\Delta \text{Ref}_{3d} = \text{MLP}_2(Q)$ $S_{3d} = \{(x_i, y_i, z_i)\|i \in N\} = \Delta \text{Ref}_{3d} + \text{Ref}_{3d}$ ▷ deformed 3D positions. $D_{pv} = \text{DeformAttn}(\text{Project}_{pv}(S_{3d}, T), F_{pv})$ ▷ project 3D deformed points to PV. $D_{bev} = \text{DeformAttn}(\text{Project}_{bev}(S_{3d}), F_{bev})$ $Q = \text{SE}(D_{pv}, D_{bev})$ ![Figure 5](image-url) Figure 5: Illustration comparing 3D dual-view deformable attention with other approaches. ### 3.4 Prediction and Loss Auxiliary Tasks. During training, we incorporate two auxiliary tasks: 1) 2D instance segmentation [Luo et al., 2023; Cheng et al., 2022] loss $L_{seg}$ for both PV and BEV branches, aiding in extracting discriminative lane features in each view; 2) Depth estimation for the PV branch, which guides effective 3D structure-aware feature extraction of $F_{pv}$. Depth labels are generated from LiDAR points, and the loss $L_{depth}$ is calculated following BEVDepth [Li et al., 2022c]. 3D Lane Prediction and Loss. As we adopt point-based queries $Q \in \mathbb{R}^{(N \times M) \times C}$, each query naturally corresponds to a 3D point, and every group of $M$ points constructs a complete 3D lane. Thus, we predict $x$, $z$, and visibility for each point query on the predefined $y$ coordinates [Chen et al., 2022; Luo et al., 2023] and a classification probability for each lane. Overall, the total loss is: $$L_{lane} = w_x L_x + w_z L_z + w_v L_v + w_c L_c,$$ $$L_{aux} = w_{seg} L_{seg} + w_{depth} L_{depth},$$ $$L_{total} = L_{lane} + L_{aux},$$ where $w_s$ denotes different loss weights. We adopt the L1 loss $L_x$ and $L_z$ to learn the $x$, $z$ positions, focal loss [Lin et al., 2017] $L_c$ to learn the lane category, and BCELoss $L_v$ to learn visibility. 4 EXPERIMENTS 4.1 DATASETS We evaluate our method on OpenLane [Chen et al., 2022], the sole public 3D lane dataset featuring multi-modal sources. OpenLane is a large-scale dataset built on Waymo Open Dataset [Sun et al., 2020], comprising 200K frames and 880K lanes across six driving scenarios and 14 lane categories. The LiDAR data, collected using 64-beam LiDARs, is sampled at 10Hz. This extensive dataset provides a solid foundation for evaluating 3D lane algorithms comprehensively. 4.2 METRICS We adopt the evaluation metrics established by OpenLane [Chen et al., 2022], framing 3D lane detection evaluation as a matching problem based on the edit distance between predictions and ground truth. Successful matching results in computed metrics, including F-Score, category accuracy, and error in X/Z-axes. A successful match for each predicted 3D lane is defined when at least 75% of its points have a distance to the ground truth below the predefined threshold $D_{thre}$. 4.3 IMPLEMENTATION DETAILS Models. In the base version of DV-3DLane, we employ ResNet34 [He et al., 2016] and PillarNet34 [Shi et al., 2022] as the backbones for our camera and LiDAR branches, respectively. For the lite version, we utilize ResNet18 and PillarNet18. The base version features two decoder layers, while the lite version employs a single decoder layer. Following LATR [Luo et al., 2023], we set the number of lane queries to 40, and we employ deformable attention with 4 heads, 8 sample points, and 256 embedding dimensions. Training. We use the Adam optimizer [Kingma & Ba, 2014] with a weight decay of 0.01. The learning rate is set to $2e^{-4}$, and our models undergo training for 24 epochs with a batch size of 32. We employ the cosine annealing scheduler [Loshchilov & Hutter, 2016] with $T_{max} = 8$. Our input images are of resolution $720 \times 960$, and we adopt a voxel size of (0.2m, 0.4m) for the X and Y axes. 4.4 MAIN RESULTS It’s important to note that the existing metrics use a rather lenient distance threshold of $D_{thre}=1.5$m. However, in the context of ensuring safety in AD, this value, although commonly used for assessment purposes, may be considered overly permissive. Following M$^2$-3DLaneNet [Luo et al., 2022], we extend our evaluation to include a more stringent threshold, $D_{thre}=0.5$m. Further, we illustrate the relationship between the F1 score performance and different distance thresholds for various models, as shown in Figure 6. Notably, our method consistently achieves superior results, even when evaluated under a much more stringent criterion of $D_{thre}=0.1$m. In contrast, other approaches experience a noticeable decline in performance as the distance threshold decreases. These findings confirm the robustness of our method across varying distance thresholds, particularly highlighting its advantage in precise localization. We present the main results in Table 1 obtained from experiments conducted on the OpenLane-1K dataset. The evaluation uses both $D_{thre}=1.5$m and $D_{thre}=0.5$m criteria, allowing for a comprehensive and insightful comparison. It is evident that DV-3DLane consistently outperforms previous state-of-the-art (SoTA) methods across all metrics. Notably, when applying a more strict 0.5m threshold, DV-3DLane demonstrates a substantial 11.2% improvement in the F1 score. It is noteworthy that our method excels in localization accuracy, leading to significant performance improvements. Specifically, our method achieves remarkable reductions in localization errors: 52%/50% for... Table 1: Comprehensive 3D Lane evaluation comparison on OpenLane with variable metrics. † denotes the results obtained using their provided models. “Image-Branch” and “LiDAR-Branch” refer to our image and LiDAR branches, respectively. “LATR + LiDAR” denotes the model that combines the SOTA method LATR with LiDAR input, projecting all points into the image space and using them as additional features in the network. \| Dist. \| Methods \| Backbone \| Modality \| F1 ↑ \| Acc. ↑ \| X error (m) ↓ near \| X error (m) ↓ far \| Z error (m) ↓ near \| Z error (m) ↓ far \| \|-------\|---------\|----------\|----------\|------\|-------\|------------------\|------------------\|------------------\|------------------\| \| 1.5 m \| 3DLaneNet \| VGG-16 \| C \| 44.1 \| - \| 0.593 \| 0.494 \| 0.140 \| 0.195 \| \| \| GenLaneNet \| ERPN \| C \| 32.3 \| - \| 0.591 \| 0.684 \| 0.411 \| 0.521 \| \| \| PersFormer \| EffNet-B7 \| C \| 50.5 \| 89.5 \| 0.319 \| 0.325 \| 0.112 \| 0.141 \| \| \| Anchor3DLane \| EffNet-B3 \| C \| 52.8 \| 89.6 \| 0.408 \| 0.349 \| 0.186 \| 0.143 \| \| \| M²-3DLaneNet \| EffNet-B7 \| C+L \| 55.5 \| 88.2 \| 0.283 \| 0.256 \| 0.078 \| 0.106 \| \| \| Anchor3DLane \| ResNet-18 \| C \| 50.7 \| 89.3 \| 0.422 \| 0.349 \| 0.188 \| 0.146 \| \| \| PersFormer \| ResNet-50 \| C \| 52.7 \| 88.4 \| 0.307 \| 0.319 \| 0.083 \| 0.117 \| \| \| LATR \| ResNet-50 \| C \| 61.9 \| 92.0 \| 0.219 \| 0.259 \| 0.075 \| 0.104 \| \| \| DV-3DLane-Tiny (Ours) \| ResNet-18 \| C+L \| 63.4 \| 91.6 \| 0.137 \| 0.159 \| 0.034 \| 0.063 \| \| \| DV-3DLane-Base (Ours) \| ResNet-34 \| C+L \| 65.4 \| 92.4 \| 0.118 \| 0.131 \| 0.032 \| 0.053 \| \| \| DV-3DLane-Large (Ours) \| ResNet-50 \| C+L \| 66.8 \| 93.3 \| 0.115 \| 0.134 \| 0.029 \| 0.049 \| \| \| Improvement \| - \| - \| 74.9 \| 71.3 \| 0.04 \| 0.122 \| 0.046 \| 0.055 \| \| 0.5 m \| PersFormer \| EffNet-B7 \| C \| 36.5 \| 87.8 \| 0.343 \| 0.263 \| 0.161 \| 0.115 \| \| \| Anchor3DLane \| EffNet-B3 \| C \| 34.9 \| 88.5 \| 0.344 \| 0.264 \| 0.181 \| 0.134 \| \| \| M²-3DLaneNet \| EffNet-B7 \| C+L \| 48.2 \| 88.1 \| 0.217 \| 0.203 \| 0.076 \| 0.103 \| \| \| Anchor3DLane \| ResNet-18 \| C \| 32.8 \| 87.9 \| 0.350 \| 0.266 \| 0.183 \| 0.137 \| \| \| PersFormer \| ResNet-50 \| C \| 43.2 \| 87.8 \| 0.229 \| 0.245 \| 0.078 \| 0.106 \| \| \| LATR \| ResNet-50 \| C \| 54.0 \| 91.7 \| 0.171 \| 0.201 \| 0.072 \| 0.099 \| \| \| LATR + LiDAR \| ResNet-50 \| C+L \| 57.4 \| 92.1 \| 0.167 \| 0.185 \| 0.071 \| 0.088 \| \| \| Image-Branch (Ours) \| ResNet-34 \| C \| 52.9 \| 90.3 \| 0.173 \| 0.212 \| 0.069 \| 0.098 \| \| \| LiDAR-Branch (Ours) \| PillarN-34 \| L \| 54.1 \| 84.4 \| 0.282 \| 0.191 \| 0.096 \| 0.124 \| \| \| DV-3DLane-Tiny (Ours) \| ResNet-18 \| C+L \| 60.9 \| 91.8 \| 0.097 \| 0.124 \| 0.033 \| 0.062 \| \| \| DV-3DLane-Base (Ours) \| ResNet-34 \| C+L \| 63.5 \| 92.4 \| 0.090 \| 0.102 \| 0.031 \| 0.053 \| \| \| DV-3DLane-Large (Ours) \| ResNet-50 \| C+L \| 65.2 \| 93.4 \| 0.082 \| 0.101 \| 0.028 \| 0.048 \| \| \| Improvement \| - \| - \| 71.2 \| 71.7 \| 0.089 \| 0.100 \| 0.044 \| 0.051 \| X near/far, and 61%/52% for Z near/far. Due to space limitations, results in various scenarios and studies about robustness concerning calibration noise are included in our Appendix. Effect of Multiple Modalities. To explore the impact of individual modalities, we conduct experiments using single modalities, as outlined in the “Image-Branch” and “LiDAR-Branch” rows of Table 1. The results illustrate that DV-3DLane significantly enhances performance compared to using images alone or relying solely on LiDAR data. Notably, our method significantly surpasses configurations that simply equip LATR with LiDAR input across all metrics, underscoring the substantial improvements achieved by DV-3DLane in leveraging information from both modalities. Moreover, to evaluate the effect of dual-view, we conduct experiments using single-modality input but transforming features extracted from the backbone into another view, yielding single-modal dual-view features. Then, our dual-view decoder is applied, and the results are detailed in our Appendix. Additionally, we conduct experiments using our “Image-Branch” on the ApolloGuo et al. (2020) dataset, which exclusively contains image data. The results are provided in our Appendix. Qualitative Results. We present a qualitative comparison between DV-3DLane and LATR Luo et al. (2023) in Figure 7, demonstrating that our method achieves more robust and accurate predictions across various scenarios. More visualization results are included in our Appendix. ### 4.5 Ablation Studies We conduct all ablation studies on OpenLane-300 following established practices Chen et al. (2022); Luo et al. (2023); Huang et al. (2023), while adopting a 0.5m threshold $D_{thre}$ for evaluation. Effect of Bidirectional Feature Fusion. The corresponding experiments are included in our Appendix, due to space limitations. We kindly direct the readers to refer to the Appendix for details. The results confirm the effectiveness of the proposed bidirectional feature fusion approach. Effect of Unified Query. We study the effect of our unified queries generation strategy in Table 2, where “Random” means random initialization using nn.Embedding, “Qpv” denotes using only PV queries, and “Qbev” refers to using only BEV queries. Replacing our unified queries with randomly initialized ones Carion et al. (2020); Zhu et al. (2020); Li et al. (2022d) results in a decrease of 1.0 in the F1 score compared to our approach. Interestingly, employing a single space instance- Figure 7: Qualitative Results. We present the projection of 3D lanes from ground truth, predictions of DV-3DLane and the SOTA method LART [Luo et al., (2023)] in rows (a), (b), (c), respectively. Row (d) depicts the comparison between ground truth (red) and ours (green) in 3D space. We highlight the differences with colored arrows. Best viewed in color and zoom in for details. aware query yields even lower F1 scores of 69.6%/69.1% for PV/BEV, respectively, than random initialization. This underscores the inadequacy of a single-space lane-aware query in capturing complex 3D lane features comprehensively existing in both PV and BEV spaces. However, our dual-view strategy, generating lane-aware queries w.r.t. both views, improves overall performance to 70.7, achieving the best result. This demonstrates that our method effectively integrates the strengths of features from two spaces, forming a cohesive query set. Effect of 3D Dual-view Deformable Attention. To evaluate the efficacy of our proposed Dual-view Deformable Attention, we conduct ablation studies in Table 3, where “PV space” and “BEV space” mean using single space in the decoder. “DeepInteration” [Yang et al., (2022)] denotes sequential fusion of features from different spaces, and “FUTR3D” [Chen et al., (2023)] refer to a modality-agnostic approach where sampling locations differ across views. We compare DV-3DLane against alternative approaches, including single-view fused method, as well as methods proposed in DeepInteration and FUTR3D, as described in Section 3.3. The results underscore the significance of our approach. In detail, sampling only PV space features leads to a notable drop (70.7 → 63.6) in performance, showing the importance of BEV space due to its advantages in localization. Besides, our method outperforms the sequential approach of DeepInteration with a substantial 2.0 gain in F1 score. Furthermore, compared to the modality-agnostic approach proposed in FUTR3D, our method achieves a 0.5 improvement, emphasizing the importance of consistent sampling locations in deformable attention across different spaces. \| Methods \| F1 \| X error (m) near \| X error (m) far \| Z error (m) near \| Z error (m) far \| \|---------\|------\|------------------\|-----------------\|------------------\|-----------------\| \| Random \| 69.7 \| 0.123 \| 0.151 \| 0.059 \| 0.081 \| \| Qpv \| 69.6 \| 0.124 \| 0.155 \| 0.059 \| 0.079 \| \| Qbest \| 69.1 \| 0.122 \| 0.145 \| 0.058 \| 0.077 \| \| Ours \| 70.7 \| 0.123 \| 0.146 \| 0.058 \| 0.078 \| \| Methods \| F1 \| X error (m) near \| X error (m) far \| Z error (m) near \| Z error (m) far \| \|---------------\|------\|------------------\|-----------------\|------------------\|-----------------\| \| PV space \| 63.6 \| 0.150 \| 0.202 \| 0.060 \| 0.081 \| \| BEV space \| 68.5 \| 0.127 \| 0.151 \| 0.064 \| 0.087 \| \| DeepInteration\| 68.7 \| 0.126 \| 0.157 \| 0.059 \| 0.081 \| \| FUTR3D \| 70.2 \| 0.118 \| 0.145 \| 0.057 \| 0.077 \| \| Ours \| 70.7 \| 0.123 \| 0.146 \| 0.058 \| 0.078 \| 5 CONCLUSION In this work, we introduce DV-3DLane, a novel end-to-end multi-modal 3D lane detection framework that leverages the strengths of both PV and BEV spaces. To this end, we propose three novel modules that effectively utilize dual-view representation on different levels, consistently enhancing performance. Extensive experiments substantiate the outstanding advancements achieved by DV-3DLane, establishing a new state of the art on OpenLane. ACKNOWLEDGMENTS This work was supported by NSFC with Grant No. 62293482, by the Basic Research Project No. HZQB-KCZYZ-2021067 of Hetao Shenzhen HK S&T Cooperation Zone, by Shenzhen General Program No. JCYJ20220530143600001, by Shenzhen-Hong Kong Joint Funding No. SGDX20211123112401002, by the National Key R&D Program of China with grant No. 2018YFB1800800, by the Shenzhen Outstanding Talents Training Fund 202002, by Guangdong Research Project No. 2017ZT07X152 and No. 2019CX01X104, by the Guangdong Provincial Key Laboratory of Future Networks of Intelligence (Grant No. 2022B1212010001), by the Guangdong Provincial Key Laboratory of Big Data Computing. The Chinese University of Hong Kong, Shenzhen, by the NSFC 61931024&12326610, by the Shenzhen Key Laboratory of Big Data and Artificial Intelligence (Grant No. ZDSYS201707251409055), and the Key Area R&D Program of Guangdong Province with grant No. 2018B03033800, by Tencent&Huawei Open Fund. REFERENCES Jianyong Ai, Wenbo Ding, Jiuhua Zhao, and Jiachen Zhong. Ws-3d-lane: Weakly supervised 3d lane detection with 2d lane labels. In 2023 IEEE International Conference on Robotics and Automation (ICRA), pp. 5595–5601. IEEE, 2023. Min Bai, Gellert Mattyus, Namdar Homayounfar, Shenlong Wang, Shrinidhi Kowshika Lakshmikanth, and Raquel Urtasun. Deep multi-sensor lane detection. In 2018 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 3102–3109. IEEE, 2018. Xuyang Bai, Zeyu Hu, Xinge Zhu, Qingqiu Huang, Yilun Chen, Hongbo Fu, and Chiew-Lan Tai. Transfusion: Robust lidar-camera fusion for 3d object detection with transformers. 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oTRwljRgiv	What is the comparison in FLOPs between the ExeDec models and the baseline models (including the beam search, etc)? Even if the parameter count is the same, the loop in ExeDec could give these models significantly more power than the Transformer / Latent Programmer baselines.	ExeDec: Execution Decomposition for Compositional Generalization in Neural Program Synthesis Kensen Shi Google DeepMind kshi@google.com Joey Hong * UC Berkeley joey.hong@berkeley.edu Yinlin Deng * University of Illinois Urbana-Champaign yinlind2@illinois.edu Pengcheng Yin Google DeepMind pcyin@google.com Manzil Zaheer Google DeepMind manzilaheer@google.com Charles Sutton Google DeepMind charlessutton@google.com Abstract When writing programs, people have the ability to tackle a new complex task by decomposing it into smaller and more familiar subtasks. While it is difficult to measure whether neural program synthesis methods have similar capabilities, we can measure whether they compositionally generalize, that is, whether a model that has been trained on the simpler subtasks is subsequently able to solve more complex tasks. In this paper, we characterize several different forms of compositional generalization that are desirable in program synthesis, forming a meta-benchmark which we use to create generalization tasks for two popular datasets, RobustFill and DeepCoder. We then propose ExeDec, a novel decomposition-based synthesis strategy that predicts execution subgoals to solve problems step-by-step informed by program execution at each step. When used with Transformer models trained from scratch, ExeDec has better synthesis performance and greatly improved compositional generalization ability compared to baselines. Finally, we use our benchmarks to demonstrate that LLMs struggle to compositionally generalize when asked to do programming-by-example in a few-shot setting, but an ExeDec-style prompting approach can improve the generalization ability and overall performance. 1 Introduction Program synthesis aims to assist programmers by automatically producing code according to a user’s specification of what the code should do (Gulwani et al., 2017). Program synthesis systems, such as programming by example (PBE) systems, have been effective for tasks such as string manipulation (Gulwani, 2011; Devlin et al., 2017; Shi et al., 2022b), writing short Java functions (Shi et al., 2019), and tensor manipulation (Shi et al., 2022a). Neural program synthesizers, especially those based on large language models (Chen et al., 2021a; Austin et al., 2021; Li et al., 2022), have been particularly successful at generating code functions and blocks across a variety of general-purpose programming languages. An essential capability of human programmers is their ability to generalize by recombining parts of prior knowledge to solve new tasks. For example, a capable programmer can quickly adapt to new concepts and APIs, and compose different code idioms in unseen ways to solve novel problems. These skills are instances of compositional generalization, which is the ability to generalize to test examples consisting of different compositions of components individually seen during training (Keysers et al., 2020). While compositionality has been studied in natural language processing (Chomsky, 1957; Lake & Baroni, 2018; Gu et al., 2021), it has not been studied deeply in the context of programming by example. This problem is potentially fruitful not only because it might help to build more robust program synthesizers, but also as an example of how more general problem-solving is compositional. These authors contributed during internships at Google DeepMind. To build neural synthesis systems that are better at compositional generalization, we propose designing systems that learn to decompose a complex task into a list of simpler subtasks. Each subtask is defined by a goal, so the process of decomposing a task is essentially planning. Indeed, decomposition is a skill so fundamental to software engineering that the first programming course at Stanford University introduces decomposition within the first week (Parlante, 2022). This can enable compositional generalization because subtasks seen during training can be combined in different ways at test time. Based on this intuition, we propose ExeDec, a novel search method for neural program synthesis that performs decomposition within the execution space. A PBE task defines a program by pairs of program inputs with their desired outputs. Thus, it is natural to describe a subgoal by the desired intermediate state, i.e., values of local variables, for the next subtask. To describe the intuition in another way, we imagine that a human programmer does not decide on what code to write one token at a time, but rather thinks about what the result of the next code block should be, and then writes code to accomplish that. Specifically, ExeDec uses two neural models, a subgoal model that predicts the desired program state for the next part of the program, and a synthesizer model that attempts to generate a program that reaches that subgoal from the prior state. We interleave neural prediction with program execution within a beam search that enables exploring different predicted decompositions. To evaluate this approach, we introduce a new meta-benchmark for measuring the compositional generalization abilities of program synthesizers. Given a standard program synthesis benchmark containing a domain-specific language and a distribution over target programs, our meta-benchmark describes train-test splits for 5 different types of compositional generalization, such as length generalization or composing API functions in different combinations in the training and test sets. While ExeDec has slightly better performance than a Transformer baseline in the i.i.d. setting, ExeDec also achieves a $2\times$ to $4\times$ accuracy increase in the compositional generalization setting. Additionally, ExeDec improves upon an ablation that does not explicitly propose subgoals, showing the importance of reasoning about execution subgoals instead of directly predicting code. Interestingly, a similar approach can be applied to explore compositional generalization in large language models (LLMs). We explore whether the LLM can solve PBE tasks that compositionally generalize beyond those in a few-shot prompt. We similarly find that the LLM performs significantly worse when compositional generalization is required, and that an adaptation of ExeDec to the few-shot prompting setup increases the LLM’s performance overall, including in compositional generalization. Even so, compositional generalization during program generation in LLMs remains a challenge. ## 2 COMPOSITIONAL GENERALIZATION IN PROGRAMMING The goal in program synthesis is to find a program in a given language that is consistent with a specification. Formally, we are given a domain specific language (DSL) which defines a set $\mathcal{P}$ of programs. Elements in the DSL include functions (which we call operations), identifiers, constants, and so on. In programming by example (PBE), the desired program is specified by a set of input/output (I/O) examples denoted $X = \{(I_1, O_1), \ldots, (I_n, O_n)\}$. Then, solving specification $X$ means finding a program $P \in \mathcal{P}$ that correctly solves all of the examples; $P(I_i) = O_i$, $\forall i$. A robust program synthesizer should generalize to programs not in the training set. Regardless of the programming language or DSL, programs are nearly always built from smaller parts, which we call subprograms, such as lines and blocks of code, functions, and so on. For compositional generalization, we are interested in whether the synthesizer can combine subprograms in new ways from the training set. We design our benchmark around five compositional generalization tasks applicable to program synthesis (Figure 1). These tasks measure whether synthesizers can generalize to longer programs or to programs that use concepts, such as API methods, in different compositional ways. These concepts partition the DSL operations into groups.\footnote{Ideally, operations within a group should have meaningful commonalities that form one concept, and each concept should have roughly equal semantic complexity, but these are not strictly required.} In this section, we describe the generalization tasks abstractly, forming a meta-benchmark that can be applied in future work to construct new compositional generalization benchmarks using existing datasets or DSLs. Then, in Section 3, we concretize the tasks for specific DSLs for our experiments. The five generalization tasks are: 1. Length-Generalization: Can a model produce longer code than seen in training, when necessary? Here, “length” counts the number of subprograms and not the number of tokens, so there is more emphasis on generalizing to more complex compositional patterns. For this task, we train on problems of lengths 1 to $ n $ and test on lengths $ n + 1 $ to $ m $ (where $ m > n $). 2. Compose-Different-Concepts: Can a model use concepts in different combinations than seen in training? Specifically, train the model on compositions of operations from the same concept, and test on compositions from different concepts. For example, if two concepts consist of operations $ \{A_1, A_2, \ldots\} $ and $ \{B_1, B_2, \ldots\} $, then the training programs have the form $ A_i \circ A_j $ and $ B_i \circ B_j $, and the testing programs have the form $ A_i \circ B_j $ and $ B_i \circ A_j $ (and similarly for compositions of 3 or more operations). A real-world example might be training on program containing only TensorFlow or only NumPy, but synthesizing code at test time using both libraries. 3. Switch-Concept-Order: Can a model compose concepts in different orders than seen in training? We train on compositions of operations drawn from one sequence of concepts and test on a different sequence of concepts, e.g., train on $ A_i \circ B_j $ and test on $ B_i \circ A_j $. As a real-world example, in the training data a function might be validating inputs at the beginning of the code, but we want to use the function in a different context, e.g., to validate results at the end. 4. Compose-New-Operation: Can a model learn to use a new isolated operation within a larger composition? In this task, we train on the isolated operation and compositions without the operation, and test on compositions using the operation. A real-world example of this kind of generalization would be composing a new function with others in a larger solution, after seeing examples of the function used in isolation. 5. Add-Operation-Functionality: Can a model extend its understanding of an operation by drawing on parallels to other operations? We omit from the training data some functionality of an operation that could be inferred from other context, and test on programs using that functionality. This task can occur when a library function is upgraded with a new parameter whose behavior can be inferred from analogous parameters in other functions. These five tasks can be grouped into three themes: (a) length generalization; (b) mix and match concepts (tasks 2 and 3): compose concepts in ways that were not seen during training; and (c) apply general principles (tasks 4 and 5): adapt to new, updated, or custom APIs. ### 3 Benchmark Creation While Section 2 focused on the meta-benchmark describing five compositional generalization tasks, this section describes our instantiation of those tasks into compositional generalization datasets for two popular synthesis domains, RobustFill (Devlin et al., 2017) and DeepCoder (Balog et al., 2017). RobustFill.* In the RobustFill domain, the objective is to synthesize a sequence of string manipulation operations from I/O examples, where each example’s input is a single string. A RobustFill program is a concatenation of expressions. There are 4 categories of expressions: operations that extract a substring from the input (e.g., `GetToken(regex, index)`), operations that return a modified version of the input (e.g., `ToCase(case)`), a special `Compose` operation (applying a modification operation to the result of another operation), or a constant string character. For example, the program `GetFrom(' ') \| Const('.') \| Compose(ToCase(PROPER), GetToken(WORD, 1))` is a Algorithm 1 ExeDec: synthesis via decomposition in the execution space. Note, $ \{x_i\} $ is short for $[x_1, \ldots, x_n]$ throughout, where $ n $ is the number of I/O examples. ``` 1: function ExeDec(\{(I_i, O_i)\}) 2: t ← 1 3: $(I^{(1)}, O^{(1)}) ← (I_i, O_i), \forall i$ 4: while True do 5: $S^{(t)}$ ← SubgoalModel(\{(I^{(t)}, O^{(t)})\}) ▷ Predict the next execution subgoals 6: $P^{(t)} ← SynthesizerModel(\{(I^{(t)}, S^{(t)})\})$ ▷ Predict the next subprogram 7: $E^{(t)}_i ← Execute(P^{(t)}, I^{(t)}_i), \forall i$ 8: if $∀i. E^{(t)}_i = O^{(t)}_i$ then ▷ Is this the last subprogram? 9: return CombineProgramParts($P^{(1)}, \ldots, P^{(t)}$) 10: ▷ Update $\{(I^{(t)}, O^{(t)})\}$ to represent work that is left to be done (domain-specific). 11: $(I^{(t+1)}, O^{(t+1)}) ← UpdateSpecification(I^{(t)}, O^{(t)}, E^{(t)}), \forall i$ 12: t ← t + 1 ``` concatenation of 3 expressions and transforms the input string “TURING, Alan” into the output string “Alan.Turing”. See Appendix A for the full RobustFill DSL, which we extended from the original RobustFill paper (Devlin et al., 2017) by adding more operations. Appendix B contains further details about our constructed datasets, including the different compositional generalization splits and the process for generating synthetic programming tasks according to those splits. DeepCoder. The DeepCoder domain involves manipulation of integer lists in a line-by-line programming style. Tasks have one or more inputs which may be integers or integer lists. Each line of a DeepCoder program applies one DSL operation to inputs or previous variables and assigns the result to a new variable. The result of the last line is the program’s output. Operations include first-order list operations (Sort, Reverse, and various forms of indexing, slicing, and aggregating) and higher-order operations (Haskell-inspired Map, Filter, Count, ZipWith, and Scanl) which manipulate lists using one of several hardcoded lambda functions. As an example, the program $ x0 = INPUT \| x1 = Map (+2) x0 \| x2 = Sort x1 $ (where “\|” denotes a new line) transforms the input list $[5, 3, -4]$ into the output list $[9, 16, 25]$. See Appendix A for the full DeepCoder DSL and Appendix B for more details about our instantiation in the DeepCoder domain. Choice of Domains. Both domains allow us to generate a large amount of synthetic training data with ground-truth decompositions into subprograms. For more realistic code in general-purpose programming languages, such data collection requires more effort, especially if “natural” decompositions are desired. Beyond the difference in string versus list manipulation, RobustFill and DeepCoder are quite different in other important ways, allowing us to study the compositional generalization of various approaches in different scenarios. First, RobustFill gradually builds an output by combining results of subprograms that are mostly independent, while DeepCoder applies operations repeatedly to the same few objects until the output is reached. In this sense, RobustFill is closer to inverse CAD (Ellis et al., 2019), instantiating complex objects with many fields like dataclasses, or other tasks involving several independent analyses, while DeepCoder is closer to tensor manipulation (Shi et al., 2022a), dynamic programming, or other tasks involving sequences of manipulations or updates applied to the same objects. Second, RobustFill uses the same input for each subprogram while DeepCoder involves program states that change due to the new variable bindings on each line, making DeepCoder more complex and closer to realistic programs with execution states changing over time. 4 Program Synthesis via Decomposition In this section we describe our proposed program synthesis method based on execution decomposition, where the model predicts step-by-step execution subgoals and synthesizes subprograms for each step. Execution Decomposition (ExeDec). The ExeDec strategy outlined in Algorithm 1 aims to reason about the step-by-step execution behavior of a program rather than the code tokens. As in Section 2, we assume that the program is a sequence of one or more subprograms that may be combined later. At each step, to synthesize the next subprogram, we first call a SubgoalModel that takes I/O examples and predicts the next execution subgoals, i.e., the output of the next subprogram for each example. Because the subgoal is the desired output at this step, predicting the next subprogram is itself a PBE task. Thus, we provide the inputs and subgoals to a SynthesizerModel which predicts the corresponding subprogram. Finally, we execute the predicted subprogram and compute an updated specification that describes the work that remains to be done by the rest of the program. This updated specification is maintained throughout the step-by-step synthesis process. Because the overall program is specified by I/O examples, we use I/O examples for the updated specification as well. Intuitively, the inputs in the updated specification will be the current program state, and the outputs will be the output of the overall task, but the details are slightly different because of specifics of the DSLs. We begin with the original I/O examples for the overall synthesis task, and we update them in a domain-specific way as subprograms are synthesized (line 10). For instance, in RobustFill the input for each subprogram is the same as the original input, while the output becomes smaller as we remove already-synthesized prefixes of the output: $(I_i^{(t+1)}, O_i^{(t+1)}) \leftarrow (I_i^{(t)}, \text{REMOVEPREFIX}(O_i^{(t)}, E_i^{(t)}))$; this is because the top level operation in RobustFill programs is always concatenation.$^2$ For DeepCoder, the input is the full program state (i.e., the set of variables and their values for each example) which is expanded with new variables as subprograms are synthesized, while the output remains constant for each example: $(I_i^{(t+1)}, O_i^{(t+1)}) \leftarrow (I_i^{(t)} \cup E_i^{(t)}, O_i^{(t)})$. If ExeDec synthesizes a subprogram that executes to the entire remaining output, there are no more subprograms to synthesize, so the subprograms are combined to form the full synthesized program. Algorithm 1 describes a single synthesis attempt, but we actually perform a search comprising multiple synthesis attempts running efficiently in parallel using a modified beam search where each beam state is a partial rollout of the step-by-step synthesis algorithm. Appendix C has more details. Model Architecture.* Recall from Algorithm 1 that ExeDec relies on two models, the SubgoalModel and SynthesizerModel. We let both be sequence-to-sequence (seq2seq) models, which have been shown to be successful on various natural language (Bahdanau et al., 2016; Vaswani et al., 2017) and program synthesis tasks (Devlin et al., 2017). We choose our seq2seq model to be a Transformer due to its impressive performance on natural language tasks over traditional RNNs (Vaswani et al., 2017). We modify the baseline Transformer architecture to account for the fact that we operate on sets of inputs due to having multiple I/O examples. We call our model a Specification-Transformer. For consistent notation for the two models, we let $\{X_i\}$ be the multi-example input to the transformer and $Y$ its output. Formally, $X_i = (I_i, O_i)$ for SubgoalModel and $(I_i, S_i)$ for SynthesizerModel, and $Y = [S_1, \text{Sep}, S_2, \text{Sep}, \ldots, S_n]$ for SubgoalModel and $Y = P$ for SynthesizerModel, where Sep is a new token added to our vocabulary to partition the subgoals across examples. Note that subgoals $S_i$ and subprogram $P$ are sequences of tokens. Our Specification-Transformer consists of two modules. A Transformer encoder receives the specification $\{X_i\}$ and produces an encoding $\phi$. Following Devlin et al. (2017), our encoder performs double attention on the specification. That is, for each example $X_i$, the encoder performs the operation $\phi_i \leftarrow \text{TransformerEncoder}(X_i)$, where the encoder performs self-attention on input $I_i$ followed by cross-attention from the output (either $O_i$ or $S_i$) to $I_i$. Then, the encoding $\phi$ is simply the concatenation across examples $\phi \leftarrow \text{Concat}(\{\phi_i\})$. Next, a Transformer decoder takes the encoding and autoregressively generates the output token-by-token. Formally, let $Y_{\ell-1} = [y_1, y_2, \ldots, y_{\ell-1}]$ be the output (subgoals or subprogram) generated so far. The decoder predicts the next output token as $y_\ell \leftarrow \text{TransformerDecoder}(Y_{\ell-1}, \phi)$. As described by Vaswani et al. (2017), the Transformer encoder and decoder both apply a stack of self-attention and feed-forward units. For the SubgoalModel, we use Aligned Relative Attention (ARA), a new technique that helps the model output a sequence of sequences (a subgoal for each I/O example, concatenated together); see Appendix D for details. No-Subgoal Ablation. We also experiment with an ablation of ExeDec that performs step-by-step decomposition but without predicting execution subgoals first, instead directly predicting the next subprogram from the I/O examples. In Algorithm 1, this ablation is achieved by replacing lines 5 and 6 with a single line, $P^{(t)} \leftarrow \text{COMBINEDMODEL}(\{(I_i^{(t)}, O_i^{(t)})\})$, thus skipping the step of predicting execution subgoals. This ablation uses the same model architecture (without ARA) and an analogous $^2$If the synthesized subprogram does not execute to a prefix of the current output for all examples, this synthesis attempt cannot succeed due to RobustFill’s concatenation of subprograms. Such “invalid” subprograms are detected and handled during a beam search. beam search. Several prior works (Zohar & Wolf, 2018; Ellis et al., 2019; Chen et al., 2019) perform synthesis step-by-step, providing execution feedback to the synthesizer after each step to inform future predictions. Our ablation captures the essence of those approaches adapted to our setting. Model Training. We generate training problems as described in Section 3. We train the ExeDec and ablation models using decomposed data, that is, based on teacher forcing using Algorithm 1. Specifically, for each subprogram in the ground-truth solution, we collect (A) the updated specification based on executing the previous ground-truth subprograms, (B) the subprogram’s execution result on all examples, and (C) the subprogram itself. Then, we train the SubgoalModel to predict (B) given (A), the SynthesizerModel to predict (C) given (B) and the example inputs from (A), and the CombinedModel to predict (C) given (A). Each model type is trained separately for each generalization task. Appendix E contains more training details, including model sizes and hyperparameters. ## 5 EXPERIMENTS We experiment with Transformers trained from scratch and with LLMs using few-shot prompting. ### 5.1 TRANSFORMERS TRAINED FROM SCRATCH These experiments compare ExeDec, a version with smaller models called ExeDec-Small, the no-subgoal ablation, a Transformer baseline without any decomposition, and Latent Programmer (Hong et al., 2021). All models use the same hyperparameters and architecture except: (1) ExeDec-Small and Latent Programmer use smaller models (details and reasoning in Appendix E), (2) ARA only applies to the SubgoalModel, and (3) because the baseline Transformer and Latent Programmer are trained on entire programs instead of subprograms, but the number of training examples is held constant, they actually see more subprograms during training than our models. Using our compositional generalization datasets (Section 3) and models (Section 4), we ran the different approaches and measured their overall success rate on 1000 test examples per generalization task. We repeated the experiments using 5 different random initializations for model training. Figure 2 shows the results when using a beam size of 10. Appendix F contains results with beam size 1, and Appendix G analyzes the accuracy of individual steps. Discussion. On both domains, ExeDec significantly outperforms the Transformer baseline on every generalization task and in the i.i.d. setting (testing on the training distribution without any compositional generalization). Specifically, ExeDec achieves +44% higher average compositional generalization than the Transformer baseline on RobustFill and +18% on DeepCoder, a $4.4 \times$ higher success rate. But despite the notable improvements, DeepCoder in particular remains a difficult domain with deeply nested operation compositions that obscure the intended computation, while RobustFill has a more flat compositional structure that is easier to learn. Our step-by-step decomposition approach introduces important inductive biases into the approach. By training models on the decomposed data, we teach the models that subprograms can be reasoned about separately, regardless of the compositional patterns present in other subprograms. The SubgoalModel does not see any code tokens and is only affected by compositional generalization patterns indirectly (since the distribution over programs affects the distribution over execution traces), and the SynthesizerModel only sees code tokens for the current subprogram and cannot reference any compositional patterns that appear when comparing to other subprograms. In contrast, the Transformer baseline sees all compositional patterns in the full programs, making it more likely to overfit to those patterns. The decomposition strategy also encourages our models to understand intermediate program states while the Transformer baseline is not trained with such execution information. Compared to the no-subgoal ablation, ExeDec achieves higher compositional generalization performance on a majority of generalization tasks across the two domains, averaging +7% improvement on RobustFill (a 34% reduction in failures) and +5% on DeepCoder (a $1.28 \times$ multiplicative improvement). This supports our hypothesis that predicting execution states is more robust than predicting code in the compositional generalization setting. ExeDec-Small performs slightly worse than ExeDec (1.4% worse on average and up to 3% worse on any individual generalization task) but ExeDec-Small still significantly outperforms the other approaches overall. Even though ExeDec performs the best in most situations, the no-subgoal variation is slightly better on DeepCoder’s training distribution and Length-Generalization. Appendix H provides some intuition on “spurious patterns” related to this result. In theory, one could combine the two decomposition variations in an ensemble to get the best of both approaches on unknown test distributions. Finally, we observe that in most cases ExeDec has smaller variance across random initializations than the no-subgoal variation, i.e., ExeDec might be more consistent in practice. As a case study, we compare ExeDec, the no-subgoal ablation, and the Transformer baseline on example RobustFill and DeepCoder problems in Appendix I. Through these examples, we discuss some behaviors and observations that clarify the advantages to ExeDec’s approach. 5.2 LLM Experiments It is fundamentally difficult to measure compositional generalization in LLMs, because compositional generalization is a function of the relationship between the training and test distributions, but in LLMs it is not easy to control the pretraining data. However, we have more control in a few-shot prompting setup, as long as we focus on program concepts that cannot have occurred in the pretraining data set. Based on this insight, in these experiments, we used our benchmarks to measure the compositional generalization ability of PaLM 2 Unicorn (Google et al., 2023) during few-shot prompting for PBE. We use the same compositional generalization splits for DeepCoder and RobustFill, except that the few-shot examples and test problems have length at most 3. We make the problems easier because LLMs in general perform poorly on program synthesis tasks specified only through I/O examples, compared to natural language specifications. Within each split we balance the distribution of program lengths as much as possible, and we use 200 test problems per generalization task. Each prompt contains a description of the DSL including the available functionality, followed by 4 few-shot examples of PBE tasks and solutions drawn from the training split (different tasks are randomly chosen for different test problems), followed by the specification for a test problem (see Appendix J). To make the tasks better suited to LLMs, we transform our DSL programs into Python functions that call a hypothetical dsl library to access the DSL functionality. The RobustFill subprogram `GetToken(WORD, 1)` becomes `dsl.GetToken(x, dsl.Type.Word, 1)`, and the DeepCoder --- 3For example, Compose-Different-Concepts, Switch-Concept-Order, and Compose-New-Operation all require programs of length at least 2, so these tasks have a 50/50 split between programs of lengths 2 and 3. Table 1: Compositional generalization results for the LLM experiments. Each cell contains the number of solved tasks out of 200 test problems. For approaches, @1 means 1 greedy decoding and @5 means using 5 samples with temperature 0.4. For columns, “None” means no generalization, “Gen. Tasks” refers to the 5 compositional generalization tasks in the order given in Section 2 (consistent with the other figures), and “Avg” is the average across the 5 generalization tasks. \| Approach \| RobustFill \| DeepCoder \| DeepCoder-Pythonic \| \|--------------\|------------\|-----------\|-------------------\| \| \| None \| Gen. Tasks\| Avg \| None \| Gen. Tasks\| Avg \| \| Baseline @1 \| 4 \| 4 \| 0 \| 1 \| 0 \| 0 \| 1.0 \| 23 \| 0 \| 1 \| 6 \| 0 \| 5 \| 2.4 \| 25 \| 1 \| 0 \| 11 \| 0 \| 4 \| 3.2 \| \| Ablation @1 \| 16 \| 4 \| 0 \| 1 \| 6 \| 2.2 \| 31 \| 1 \| 2 \| 13 \| 3 \| 5 \| 4.8 \| 30 \| 4 \| 0 \| 12 \| 4 \| 4 \| 4.8 \| \| ExeDec @1 \| 21 \| 5 \| 0 \| 0 \| 1 \| 6 \| 2.4 \| 36 \| 2 \| 3 \| 11 \| 4 \| 10 \| 6.0 \| 46 \| 3 \| 3 \| 15 \| 5 \| 16 \| 8.4 \| \| Baseline @5 \| 15 \| 1 \| 0 \| 1 \| 2 \| 5 \| 1.8 \| 36 \| 0 \| 1 \| 11 \| 5 \| 13 \| 6.0 \| 34 \| 2 \| 1 \| 15 \| 5 \| 8 \| 6.2 \| \| Ablation @5 \| 29 \| 7 \| 1 \| 0 \| 4 \| 7 \| 3.8 \| 51 \| 1 \| 4 \| 18 \| 8 \| 21 \| 10.4 \| 42 \| 4 \| 8 \| 19 \| 10 \| 11 \| 10.4 \| \| ExeDec @5 \| 32 \| 5 \| 0 \| 0 \| 5 \| 10 \| 4.0 \| 56 \| 2 \| 5 \| 17 \| 8 \| 17 \| 9.8 \| 59 \| 5 \| 9 \| 25 \| 12 \| 30 \| 16.2 \| subprogram $x_2 = \text{Map} \ ((\ast + 2) \ x_1$ becomes $x_2 = \text{dsl.Map(dsl.SQUARE, } x_1)$. For DeepCoder, we alternatively try using Pythonic expressions for all DSL functionality except the $\text{Scanl}$ operation, which is difficult to inline; the previous example then becomes $x_2 = [x \ast \ast 2 \text{ for } x \text{ in } x_1]$. By representing DSL programs as Python functions in this way, we enable the LLM to draw upon its general understanding of Python from its pretraining data, while requiring the LLM to use a new Python library from only a description of the library along with 4 few-shot examples. This setting mirrors realistic use-cases where a user asks about a new, custom, or proprietary library that the LLM was not trained on. Appendix J contains examples of our prompts and Python-style programs. The LLM is allowed to use arbitrary Python, although it usually follows the style in the examples. We experimented with three prompting approaches analogous to the other experiments: 1. The baseline approach is to predict the entire solution program in one decoding. 2. The Ablation-style approach predicts the program step-by-step. Given the problem specification and history of previous steps, the LLM predicts the next line of code. We then execute the program-so-far and concatenate the predicted line of code along with its execution results into the history portion of the prompt, which will influence future steps. This stepwise process continues until the desired outputs are reached, the program fails to execute, or a budget of 3 steps is exhausted. 3. The ExeDec-style approach is similar, except that at each step, the LLM predicts the next execution subgoal followed by a line of code for that step (analogous to calling the SubgoalModel and SynthesizerModel). Note that the LLM’s subgoal prediction might be inconsistent with the predicted code, so in the history of previous steps, we replace the predicted subgoals with the actual execution results (analogous to how the specification is updated in ExeDec). Over multiple steps, this process creates a prompt almost identical to that of the Ablation-style approach, except that ExeDec-style has the execution results of a step before the code for that step, while the Ablation-style has the execution results after the code. The results are in Table 1. The ExeDec-style prompting strategy leads to the best performance for all no-generalization cases, and all but one case for the generalization average. Also, the ExeDec-style approach significantly improves when programs are written in a more natural form (going from DeepCoder to DeepCoder-Pythonic), which is a promising sign for its general applicability. For DeepCoder-Pythonic, the ExeDec-style approach solves between 40% and 75% more tasks than the next-best approach, considering each combination of no-generalization vs. generalization average and greedy decoding vs. pass@5 sampling. But despite these improvements, compositional generalization remains difficult for LLMs. Appendix K discusses common failure modes in the LLM experiments. 6 RELATED WORK Compositional Generalization. Compositional generalization is well-studied in NLP, with established benchmarks evaluating the understanding of natural language sentences with compositionally novel structures, either constructed by synthesizing examples based on predefined generalization patterns similar to this work (Lake & Baroni, 2018; Bahdanau et al., 2019), or by partitioning i.i.d. samples into splits with disjoint compositional structures (Finegan-Dollak et al., 2018; Keysers Our benchmark takes inspiration from SCAN (Lake & Baroni, 2018) and COGS (Kim & Linzen, 2020), which define a taxonomy of compositional patterns in natural language. While some generalization concepts are similar to those in Section 2, we focus on measuring compositional generalization of computer programs using I/O examples without natural language utterances, whose compositional structures are quite different from those in natural language. To improve compositional generalization in natural language understanding, earlier works have proposed specialized task-dependent neural architectures (Rusin et al., 2019; Li et al., 2019; Liu et al., 2020; Chen et al., 2020; Herzig & Berant, 2020). More generalized approaches include meta-learning (Lake, 2019; Wang et al., 2021a; Conklin et al., 2021) and data augmentation (Andreas, 2020; Oren et al., 2021; Akyürek et al., 2021; Wang et al., 2021b; Qiu et al., 2022). There have also been recent attempts in improving the compositional generalization capabilities of large language models via representation learning (Furrer et al., 2020; Herzig et al., 2021) and in-context learning (Zhou et al., 2023; Drozdov et al., 2023). In machine learning for code, some works include length generalization results (Bieber et al., 2020; Balog et al., 2017; Ellis et al., 2019), and Nye et al. (2021) use compositional generalization in some experiments, but we study compositional generalization in a much more systematic manner. Programming by Example. Various techniques have been applied to program synthesis (Gulwani et al., 2017), and recently much attention has focused on machine learning for programming by example (Devlin et al., 2017; Parisotto et al., 2017; Ellis et al., 2021). Many methods incorporate learning to guide the search over programs, such as using learned premise selection (Balog et al., 2017; Odena & Sutton, 2020), syntax-guided search (Yin & Neubig, 2017; Lee et al., 2018), bottom-up search (Shi et al., 2022a; Barke et al., 2020), two-level search (Nye et al., 2019), and execution-guided synthesis methods (Odena et al., 2020; Shi et al., 2022b). Multi-step Program Synthesis. ExeDec is an instance of multi-step program synthesis, which broadly refers to methods involving multiple calls to (potentially different) models. Execution-guided synthesis is a popular form of this, iteratively generating and refining partial programs using execution information (Zohar & Wolf, 2018; Ellis et al., 2019; Chen et al., 2019; Shrivastava et al., 2021), and some approaches do this with latent representations of the program state (Chen et al., 2021b) or execution traces (Shin et al., 2018). Planning is another form of multi-step synthesis that first generates high-level plans of what the program should do (Nye et al., 2019; Murali et al., 2018; Zhang et al., 2023), sometimes with latent representations of plans (Hong et al., 2021). Our method, ExeDec, draws ideas from both avenues of multi-step synthesis, making plans by predicting subgoals and using step-by-step program execution to guide the search. ### 7 Conclusion We explored the important aspect of compositional generalization in neural program synthesis. The ability to decompose complex tasks into smaller subtasks is a fundamental skill employed by human programmers, and measuring whether neural program synthesis methods exhibit similar capabilities is crucial for assessing their potential. We introduced a meta-benchmark that characterizes 5 forms of compositional generalization in program synthesis, and we instantiated these generalization tasks in the RobustFill and DeepCoder domains. The findings demonstrate that the ExeDec approach of predicting decompositions of program execution, rather than solely focusing on program syntax, leads to significantly improved compositional generalization for both Transformers trained from scratch and LLMs in a few-shot setting. This suggests that incorporating information about the step-by-step decomposition and leveraging it in the synthesis of programs can enhance the ability of neural models to tackle more complex tasks. Even so, compositional generalization remains challenging for neural program synthesizers, and our meta-benchmark can help measure continued progress in this area. Limitations. One limitation of ExeDec is its need for a training dataset with ground-truth decompositions. Our experiments used synthetic programs with line-by-line decomposition, but perhaps better results could be obtained with a dataset containing more natural decompositions. Furthermore, the line-by-line decomposition could be a limitation as programmers often think in larger chunks or hierarchically; Appendix L discusses a potential hierarchical formulation of ExeDec to address this limitation in future work. Lastly, our SubgoalModel predicts tokenizations of objects, but to handle more complex objects, a more general SubgoalModel might instead predict abstractions of objects. REPRODUCIBILITY STATEMENT Our code, datasets, and checkpoints for the Transformer models trained from scratch are available at https://github.com/google-deepmind/exedec. Additionally, Appendix E contains details about model hyperparameters and sizes for the models we trained. ACKNOWLEDGEMENTS The authors would like to thank Xinyun Chen, Martin Abadi, Rif Saurous, and the anonymous reviewers for their helpful comments. REFERENCES Ekin Akyürek, Afra Feyza Akyürek, and Jacob Andreas. Learning to recombine and resample data for compositional generalization. In International Conference on Learning Representations (ICLR), 2021. Jacob Andreas. Good-enough compositional data augmentation. In Association for Computational Linguistics (ACL), 2020. Jacob Austin, Augustus Odena, Maxwell Nye, Maarten Bosma, Henryk Michalewski, David Dohan, Ellen Jiang, Carrie Cai, Michael Terry, Quoc Le, and Charles Sutton. Program synthesis with large language models. arXiv preprint arXiv:2108.07732, 2021. 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fH2wf2w2Ss	Is it that unconditional sampling of CLIP image embeddings is somehow important or easier than sampling an image directly? Or is it the two-stage pipeline itself that is the important part? Could the condition generation and subsequent image generation be done in a single pipeline with end-to-end training? What exactly is the interaction between the first and second stage models?	Two-Stage Diffusion Models: Better Image Synthesis by Explicitly Modeling Semantics Anonymous authors Paper under double-blind review Abstract Recent progress with conditional image diffusion models has been stunning, and this holds true whether we are speaking about models conditioned on a text description, a scene layout, or a sketch. Unconditional image diffusion models are also improving but lag behind, as do diffusion models which are conditioned on lower-dimensional features like class labels. We advocate for a simple method that leverages this phenomenon for better unconditional generative modeling. In particular, we suggest a two-stage sampling procedure. In the first stage we sample an embedding describing the semantic content of the image. In the second stage we use a conditional image diffusion model to sample the image conditioned on this embedding, and then discard the embedding. The combined model can therefore leverage the power of conditional diffusion models on the unconditional generation task, achieving large improvements in unconditional image generation. The same method can be generalized to yield similar improvements for image generation conditioned on a low-dimensional signal like a class label. 1 Introduction Recent text-to-image diffusion generative models (DGMs) have exhibited stunning sample quality (Saharia et al., 2022) to the point that they are now being used to create art (Oppenlaender, 2022). Further work has explored conditioning on scene layouts (Zhang & Agrawala, 2023), segmentation masks (Zhang & Agrawala, 2023; Hu et al., 2022), or the appearance of a particular object (Ma et al., 2023). We broadly lump these methods together as “conditional” DGMs to contrast them with “unconditional” image DGMs which sample an image without dependence on text or any other information. Relative to unconditional DGMs, conditional DGMs typically produce more realistic samples (Ho & Salimans, 2022; Bao et al., 2022; Hu et al., 2022) and work better with few sampling steps (Meng et al., 2022). Furthermore our results suggest that sample realism grows with “how much” information the DGM is conditioned on. We therefore distinguish between “strongly-conditional” generation, where we condition on a high-dimensional feature like a long text prompt, and “lightly-conditional” generation, where Figure 1: Class-conditional ImageNet-256 samples from our method, 2SDM, and a diffusion model baseline, EDM (Karras et al., 2022), both trained for 12 GPU days. Samples within the same column are generated with the same random seed and class label. In most columns the samples from 2SDM are visibly better, agreeing with the FIDs reported in Section 5. we condition on a lower dimensional feature like a class label or short text prompt. As hinted at in Fig. 3 an image is likely to be more realistic if conditioned on being “an aerial photograph of a road between green fields” (strongly-conditional generation) than if it is if simply conditioned on being “an aerial photograph” (lightly-conditional generation). This gap in performance is problematic. Imagine you need to sample a dataset of synthetic aerial photos. A researcher doing so would currently have to either (a) make up a scene description before generating each dataset image, and ensure these cover the entirety of the desired distribution, or (b) accept the inferior image quality gleaned by conditioning just on each image being “an aerial photograph”. Figure 3 shows that the difference in quality can be stark. We argue that a solution to this problem comes from revisiting the methodology of DALL-E 2, also known as unCLIP (Ramesh et al., 2022). UnCLIP is a method for text-conditional image generation which we describe in detail in Section 2. It was originally proposed as a way to “invert” a pretrained CLIP embedder and thereby map from text to image space but, perhaps due to improved text embeddings and a desire for methodological simplicity, we are not aware of future work building on the two-stage unCLIP approach (Rombach et al., 2022; Chang et al., 2023; Hoogeboom et al., 2023). We hope to counter this trend, arguing that, while unCLIP may provide little benefit for “strongly-conditional” text-to-image generation (especially when the text prompt is long or heavily “prompt-engineered”), its benefits are in fact much greater than previously acknowledged when applied to unconditional or “lightly-conditional” generation. Our final approach, based on unCLIP, is depicted in Fig. 2. A first “auxiliary DGM” samples vectors within an embedding space, with any vector describing a particular set of semantic characteristics of an image. The second stage, a “conditional image DGM”, takes such a vector as input and samples an image with these semantic characteristics. The vector embedding is informative, as evidenced by the fact that all images within each row on the right of Fig. 2, which are all conditioned on the same embedding, look very similar. The conditional image DGM therefore inherits all the previously-described advantages of strongly-conditional DGMs even though our overall generative model is --- 1We used the prompt “Aerial photography of a patchwork of small green fields separated by brown dirt tracks between them. A large tarmac road passes through the scene from left to right.” 2This may be done to, e.g., later train state-of-the-art classification system (Azizi et al., 2023). unconditional (or, with the generalization in Section 4, lightly-conditional). We call the resulting model a Two-Stage Diffusion Model (2SDM). Contributions In Sections 2 and 3 we revisit unCLIP and then provide a novel explanation for why it is well-suited to the unconditional and lightly-conditional setting which was not explored by Ramesh et al. (2022). We then demonstrate empirically that our lightly-conditional variant, 2SDM, yields large improvements on a variety of image datasets, tasks, and metrics in Section 5. 2 BACKGROUND Conditional DGMs We provide a high-level overview of conditional DGMs that is sufficient to understand our contributions, referring to Karras et al. for a more complete description and derivation. A conditional image DGM (Tashiro et al., 2021) samples an image $ x $ given a conditioning input $ y $, where $ y $ can be, for example, a class label, a text description, or both of these in a tuple. We can recover an unconditional DGM by setting $ y $ to a null variable in the below. Given a dataset of $(x, y)$ pairs sampled from $ p_{\text{data}}(\cdot, \cdot) $, a conditional DGM $ p_\theta(x\|y) $ is fit to approximate $ p_{\text{data}}(x\|y) $. It is parameterized by a neural network $ \hat{x}_\theta(\cdot) $ trained to optimize \[ \mathbb{E}_{u(\sigma)p_\sigma(x_\sigma\|x, \sigma)p_{\text{data}}(x, y)} \left[ \lambda(\sigma)\|\|x - \hat{x}_\theta(x_\sigma, y, \sigma)\|\|^2 \right] \] where $ x_\sigma \sim p_\sigma(\cdot\|x, \sigma) $ is a copy of $ x $ corrupted by Gaussian noise with standard deviation $ \sigma $; $ u(\sigma) $ is a broad distribution over noise standard deviations; and $ \lambda(\sigma) $ is a weighting function. If $ \lambda $ and $ u $ are chosen appropriately, Eq. (1) is a lower bound on the data likelihood. It is common to instead set $ \lambda $ and $ u $ to values that maximize perceptual quality of the generated images but there remains a close relationship to the ELBO (Kingma & Gao, 2023). During inference, samples from $ p_\theta(x\|y) $ are drawn via a stochastic differential equation with dynamics dependent on $ \hat{x}_\theta(\cdot) $. CLIP embeddings CLIP (contrastive language-image pre-training) (Radford et al., 2021) consists of two neural networks, an image embedder $ e_i(\cdot) $ and a text embedder $ e_t(\cdot) $, trained on a large captioned-image dataset. Given an image $ x $ and a caption $ y $, the training objective encourages the cosine similarity between $ e_i(x) $ and $ e_t(y) $ to be large if $ x $ and $ y $ are a matching image-caption pair and small if not. The image embedder therefore learns to map from an image to a semantically-meaningful embedding capturing any features that may be included in a caption. We use a CLIP image embedder with the ViT-B/32 architecture and weights released by Radford et al. (2021). We can visualize the information captured by the CLIP embedding by showing the distribution of images produced by our conditional DGM given a single CLIP embedding; see Fig. 2. UnCLIP for text-to-image UnCLIP (Ramesh et al., 2022) uses the following text-to-image procedure: given a text prompt, it is embedded by a CLIP text embedder. A diffusion model then samples a plausible CLIP image embedding with high cosine similarity to this text image embedding. Finally, a conditional image diffusion model samples an image conditioned on CLIP image embedding and text prompt. This is described as “inverting” the CLIP embedder framework to map from image to text, hence the name unCLIP. In the next section we investigate when and why the quality of images produced by a CLIP-conditional image DGM may be greater than those generated by an unconditional image DGM. 3 CONDITIONAL VS. UNCONDITIONAL DGMs What does it mean to say that conditional DGMs beat unconditional DGMs? A standard procedure to evaluate unconditional DGMs is to start by sampling a set of $ N $ images independently from the model: $ x^{(1)}, \ldots, x^{(N)} \sim p_\theta(\cdot) $. We can then compute the Fréchet Inception distance (FID) (Heusel et al., 2017) between this set and the dataset. If the generative model matches the data distribution well, the FID will be low. For conditional DGMs the standard procedure has one extra step: we first independently sample $ y^{(1)}, \ldots, y^{(N)} \sim p_{\text{data}}(\cdot) $. We then sample each image given the corresponding $ y^{(i)} $ as $ x^{(i)} \sim p_\theta(\cdot\|y^{(i)}) $. Then, as in the unconditional case, we compute the FID between the set of images $ x_1, \ldots, x_N $ and the dataset, without reference to $ y_1, \ldots, y_N $. Even though it does not measure alignment between $ x, y $ pairs, conditional DGMs beat comparable unconditional DGMs on this metric in many settings: class-conditional CIFAR-10 generation (Karras et al., 2022), segmentation-conditional generation (Hu et al., 2022), or bounding box-conditional generation (Hu et al., 2022). Why do conditional DGMs beat unconditional DGMs? Conditional DGMs “see” more data during training than their unconditional counterparts because updates involve $y$ as well as $x$. Bao et al. (2022); Hu et al. (2022) prove that this is not the sole reason for their successes because the effect holds up even when $y$ is derived from an unconditional dataset through self-supervised learning. To our knowledge, the best explanation for their success is, as stated by Bao et al. (2022), that conditional distributions typically have “fewer modes and [are] easier to fit than the original data distribution.” When do conditional DGMs beat unconditional DGMs? We present results in Fig. 4 to answer this question. We show FID scores for conditional DGMs trained to condition on embeddings of varying information content. We produce $y$ by starting from the CLIP embedding of each image in our dataset and using either principal component analysis to reduce their dimensionality (left two panels) or K-means clustering to discretize them (right two panels) (Hu et al., 2022). We see that, given a small training budget, it is best to condition on little information. With a larger training budget, performance appears to improve steadily as the dimensionality of $y$ is expanded. We hypothesize that (1) conditioning on higher-dimensional $y$ slows down training because it means that points close to any given value of $y$ will be seen less frequently and (2) with a large enough compute budget, any $y$ correlated with $x$ will be useful to condition on. This suggests that, as compute budgets grow, making unconditional DGM performance match conditional DGM performance will be increasingly useful. A perspective on unCLIP Recall that unCLIP leverages a CLIP-conditional generative model even when the original task calls for only a text-conditional image generative model. In light of this section, it makes sense that this should provide a benefit as long as the combination of text and CLIP embedding contains “more” information than the text prompt alone, which will always be the case. However, the disparity is even larger if we compare the CLIP-conditional generative model with an unconditional generative model (i.e. one conditioned on zero bits of information). The unCLIP approach can therefore be expected to provide larger benefits for unconditional (or lightly-conditional) generation than for the text-conditional setting in which it was developed. 4 Method We now formally introduce 2SDM, a variant of unCLIP for the unconditional setting. Recall that, for unconditional generation, the user does not wish to specify any input to condition on and, for the lightly-conditional setting, any such input is low-dimensional. We will denote any input $a$ (letting $a$ be a null variable in the unconditional setting) and from now on always use $y := e_i(x)$ to refer to a CLIP embedding. To make this deterministic encoding compatible with a probabilistic generative modeling perspective, we consider a joint distribution $p_{data}(x, y, a) = p_{data}(x, a)\delta_{e_i(x)}(y)$, where $p_{data}(x, a)$ is described by a dataset and $\delta_{e_i(x)}(y)$ is a Dirac conditional distribution enforcing that $y$ is the CLIP embedding of $x$. From now on all distributions denoted with $p_{data}$ should be understood as marginals and/or conditionals of this joint distribution, including our target distribution. Figure 5: FID throughout training. We show results for each method trained from scratch and, on AFHQ and FFHQ, for finetuning a pretrained EDM model (which was trained for the equivalent of 32 GPU days). 2SDM quickly outperforms EDM when trained from scratch and quickly improves on the pretrained model when used for finetuning. \[ p_{\text{data}}(x\|a) \] 2SDM approximates this target distribution as \[ p_{\text{data}}(x\|a) = E_{p_{\text{data}}(y\|a)}[p_{\text{data}}(x\|y,a)] \approx E_{p_{\phi}(y\|a)}[p_{\theta}(x\|y,a)] \] where $ p_{\phi}(y\|a) $ is a second DGM modeling the CLIP embeddings. We can sample from this distribution by sampling $ y \sim p_{\phi}(\cdot\|a) $ and then leveraging the conditional image DGM to sample $ x \sim p_{\theta}(\cdot\|y,a) $. We then return $ x $ and make no further use of $ y $. From now on we will call $ p_{\theta}(x\|y,a) $ the conditional image model and $ p_{\phi}(y\|a) $ the auxiliary model. In our experiments the auxiliary model uses a small architecture relative to the conditional image model and so adds little extra cost.$^3$ Auxiliary model Our auxiliary model is a conditional DGM targeting $ p_{\text{data}}(y\|a) $, where $ y $ is a 512-dimensional CLIP embedding. Following Eq. (1), we train it by minimizing \[ E_{u(\sigma)p_{\sigma}(y_\sigma\|y,\sigma)p_{\text{data}}(y,a)}[\lambda(\sigma)\|\|y - \hat{y}_\theta(y_\sigma,a,\sigma)\|\|^2]. \] Analogously to Eq. (1), $ y_\sigma \sim p_{\sigma}(\cdot\|y,\sigma) $ is a copy of the CLIP embedding $ y $ corrupted with Gaussian noise, and $ u $ and $ y $ are the training distribution over noise standard deviations and weighting function respectively. We follow the architectural choice of Ramesh et al. (2022) and use a DGM with a transformer architecture. It takes as input a series of 512-dimensional input tokens: an embedding of $ \sigma $; an embedding of $ a $ if this is not null; an embedding of $ y_\sigma $; and a learned query. These are passed through six transformer layers and then the output corresponding to the learned query token is used as the output. Like Ramesh et al. (2022), we parameterize the DGM to output an estimate of the denoised $ a $ instead of estimating the added noise as is more common in the diffusion literature. On AFHQ and FFHQ we find that data augmentation is helpful to prevent the auxiliary model overfitting. We perform augmentations (including rotation, flipping and color jitter) in image space and feed the augmented image through $ e_i(\cdot) $ to obtain an augmented CLIP embedding. Following Karras et al. (2022), we pass a label describing the augmentation into the transformer as an additional input token so that we can condition on there being no augmentation at test-time. Conditional image model Including the additional conditioning input $ a $, the conditional image model’s training objective is \[ E_{u(\sigma)p_{\sigma}(x_\sigma\|x,\sigma)p_{\text{data}}(x,y,a)}[\lambda(\sigma)\|\|x - \hat{x}_\theta(x_\sigma,y \oplus a,\sigma)\|\|^2]. \] where $ y \oplus a $ is the concatenation of $ y $ and $ a $ to form a single vector which the image model is conditioned on. We match our diffusion process hyperparameters, including $ u $ and $ \lambda $, to those of Karras et al. (2022), and also use their proposed Heun sampler. For AFHQ and FFHQ, we use the U-Net architecture originally proposed by Song et al. (2020). For ImageNet, we use the slightly larger --- $^3$For our ImageNet experiments, sampling from our auxiliary model takes 35ms per batch item. Sampling from our image model takes 862ms and so 2SDM has inference time only 4% greater than our baselines. Table 1: Comparison of 2SDM and EDM on a suite of metrics. Best performance for each metric and dataset is shown in bold. Higher is better for metrics marked ↑; lower is better for ↓. Results reported for EDM on FFHQ and AFHQ are computed with the pretrained checkpoints released by Karras et al. (2022). Results reported for 2SDM on FFHQ are with finetuning from this pretrained checkpoint. All others are trained from scratch. \| Dataset \| Method \| Inception Score ↑ \| Precision ↑ \| Recall ↑ \| FID ↓ \| sFID ↓ \| \|---------------\|--------\|-------------------\|-------------\|----------\|-------\|--------\| \| AFHQ-64 \| 2SDM \| 10.00 \| 0.844 \| 0.619 \| 1.56 \| 13.7 \| \| \| EDM \| 8.91 \| 0.752 \| 0.614 \| 2.04 \| 13.7 \| \| FFHQ-64 \| 2SDM \| 3.47 \| 0.721 \| 0.697 \| 2.32 \| 4.98 \| \| \| EDM \| 3.33 \| 0.697 \| 0.569 \| 2.46 \| 4.90 \| \| Class-cond. \| 2SDM \| 17.3 \| 0.541 \| 0.573 \| 17.4 \| 4.63 \| \| ImageNet-64 \| EDM \| 13.6 \| 0.530 \| 0.532 \| 25.4 \| 6.50 \| \| Uncond. \| 2SDM \| 15.6 \| 0.614 \| 0.526 \| 21.0 \| 5.59 \| \| ImageNet-64 \| EDM \| 11.3 \| 0.523 \| 0.524 \| 35.1 \| 9.14 \| \| Class-cond. \| 2SDM \| 52.1 \| 0.590 \| 0.603 \| 24.3 \| 7.36 \| \| latent \| EDM \| 40.4 \| 0.532 \| 0.610 \| 34.2 \| 9.59 \| U-Net architecture proposed by Dhariwal & Nichol (2021). We match the data augmentation scheme to be the same as that of Karras et al. (2022) on each dataset. There are established conditional variants of both architectures (Dhariwal & Nichol, 2021; Karras et al., 2022) that add a learned linear projection to the embedding of the noise standard deviation $\sigma$. We use the same technique to incorporate the concatenated conditioning inputs $y \oplus a$. 5 EXPERIMENTS Experimental setup and results overview We perform experiments in five settings: unconditional AFHQ modeling at $64 \times 64$ resolution (Choi et al., 2020); unconditional FFHQ modeling at $64 \times 64$ resolution (Karras et al., 2018); unconditional ImageNet modeling at $64 \times 64$ resolution (Deng et al., 2009); class-conditional ImageNet modeling at $64 \times 64$ resolution; and finally class-conditional latent ImageNet modeling at $256 \times 256$ resolution, in which we train the diffusion models in the latent space of the pretrained VAE used by Stable Diffusion (Rombach et al., 2022). In every setting, we compare against EDM (Karras et al., 2022), a standard DGM directly modeling $p_{\text{data}}(x\|a)$, with an identical architecture to 2SDM. We match the training compute of our conditional image model with that of EDM in every case. The auxiliary model is trained for one day on a single V100 GPU so adds little additional cost. On AFHQ and FFHQ, we match the EDM parameters to those of Karras et al. (2022). On ImageNet-64, we have a smaller training budget and so decrease the batch size to 128 and the learning rate to $1 \times 10^{-4}$. For simplicity we match 2SDM to use the same learning rate and batch size. For the first three of our listed settings, Fig. 5 reports the FID throughout the training of the conditional image diffusion model (or image DGM baseline). In each case, the auxiliary model is trained for one day on one V100 GPU. We consider training the conditional image model from scratch (for up to 4 GPU days on AFHQ and FFHQ, or up to 11 GPU days on ImageNet-64), and see that it improves upon our EDM baseline for any training budgets over 1-2 GPU days. For AFHQ, this improvement is so substantial that 2SDM’s FID after two GPU days is better than that of the pretrained EDM model released by Karras et al. (2022), which was trained for the equivalent of 32 V100 GPU days. In addition to training from scratch, on AFHQ and FFHQ we consider initializing 2SDM’s training from the pretrained EDM checkpoints. To do so, we simply add a learnable linear projection of the CLIP embedding and initialize its --- Each FID in Fig. 5 is estimated using 20,000 images, each sampled with the SDE solver proposed by Karras et al. (2022) using 40 steps, $S_{\text{churn}} = 50$, $S_{\text{noise}} = 1.007$, and other parameters set to their default values. Our other reported FID scores use 50,000 samples, as is standard, and the same sampler hyperparameters. weights to zero. We see that this allows for a fast and significant improvement in FID over the baseline in each case. We note, though, that training 2SDM from scratch for 4 GPU days outperforms 4 GPU days of finetuning on AFHQ and so recommend training 2SDM from scratch when sufficient compute is available. Figure 4 also compares against “2SDM + oracle”, which is a supposed upper bound on 2SDM’s performance given by sampling a CLIP image embedding from an oracle (in practice, the dataset) and then using 2SDM’s conditional image model to sample an image conditioned on it. It therefore describes the performance that 2SDM would achieve with a perfect auxiliary model. On AFHQ-64, 2SDM with an oracle achieves a FID 56% lower than EDM. Without an oracle, 2SDM still achieves a FID 48% lower than 2SDM. We therefore say that 2SDM yields an improvement 87% as large as can be gleaned by using a purely conditional DGM. Similarly for FFHQ, 2SDM obtains an improvement 81% as large as is possible with a purely conditional DGM.$^5$ We can therefore say that our cheaply-trained auxiliary model is good enough to allow us to capture the majority of the benefits of conditional generation for the unconditional generation task. Intriguingly, on ImageNet-64, 2SDM achieves better FID without an oracle. This suggests that imperfections in the distribution learned by the auxiliary model improve the visual quality of the generated images. We observed this trend consistently on ImageNet, and believe that characterizing exactly when and why it occurs is an intriguing direction for future work. Finally, Fig. 5 also compares against “Class-cond”, which is an ablation of 2SDM in which we replace the CLIP embedding $y$ with a single discrete label obtained by K-means clustering of the CLIP embedding (as on the right of Fig. 4). For unconditional generation tasks, we can then replace our auxiliary model with a simple categorical distribution modeling $p_{\text{data}}(y\|a) = p_{\text{data}}(y)$ similarly to Hu et al. (2022), simplifying the generative procedure. We see that this baseline is outperformed by 2SDM, justifying our choice to use a continuous $y$. We report our final FIDs on AFHQ and FFHQ alongside the state-of-the-art in Table 2. Despite our limited training budget, our results on AFHQ beat the state-of-the-art and our results on FFHQ come second to EDM-G++ (Kim et al., 2022), a potentially orthogonal approach to improving EDM. ### Latent diffusion on ImageNet-256 We combine 2SDM and the latent diffusion modeling framework (Rombach et al., 2022) on the ImageNet-256 dataset as follows. We take the pretrained Stable Diffusion VAE encoder and decoder released by Rombach et al. (2022). We feed a $256 \times 256 \times 3$ dataset image through the VAE encoder to create $64 \times 64 \times 4$ tensors, which we use as the training targets $x$ for our conditional image model. The training targets for the CLIP embeddings $y$ are created by embedding the $256 \times 256 \times 3$ images with the standard CLIP image embedder. We use the ImageNet class labels as additional inputs $a$. At test time, we take $a$ as an input; we then sample $y$ given $a$ from our auxiliary model; we then sample $x$ given $y$ and $a$ from our conditional image model; we finally use the Stable Diffusion VAE decoder to produce an image given $x$. Samples from this version of 2SDM, as well as our EDM baseline operating in the same latent space, are shown in Fig. 1. While the compute used for each (12 GPU days) is far from that of the state-of-the-art for this dataset, the samples from 2SDM are noticeably better, supporting the FID scores in Table 1. ### Diverse metrics In Table 1 we show a comparison of 2SDM and EDM on a variety of metrics. The Inception Score (Salimans et al., 2016; Barratt & Sharma, 2018) measures the diversity of the output from an image classifier when run on sampled images. The Precision and Recall metrics (Kynkäänniemi et al., 2019) estimate, roughly speaking, the proportion of generated images that lie on the data manifold (Precision) and the proportion of dataset images that can be found within the $^5$See Table 3 for the FIDs used in these calculations. Table 2: A comparison of FID with the state-of-the-art (SOTA) in bold. EDM (single seed) is our re-computation of the EDM’s reported results using a single seed instead of taking the best of three. \| Dataset \| AFHQ-64 \| FFHQ-64 \| \|--------------------------\|---------\|---------\| \| PFGM++ (Xu et al., 2023) \| — \| 2.43 \| \| EDM (Karras et al., 2022)\| 1.96 \| 2.39 \| \| EDM (single seed) \| 2.04 \| 2.46 \| \| EDM-G++ (Kim et al., 2022)\| — \| 1.77 \| \| 2SDM \| 1.56\| 2.31 \| manifold of generated images (Recall). The FID approximates the distance between the distribution of embeddings of dataset images and that of embeddings of generated images. The sFID is similar but uses an embedding with more spatial information. 2SDM outperforms EDM on 22 of the 25 metric-dataset combinations, and is outperformed on only 2. Comparison of relative improvements between tasks In terms of FID, and for the networks trained from scratch and matched for training compute, the percentage improvement of 2SDM over EDM is 48.2% on AFHQ-64; 26.0% on FFHQ-64; 31.5% on class-conditional ImageNet-64; 40.2% on unconditional ImageNet-64; and 28.9% on class-conditional ImageNet-256. While these are all substantial improvements, we point out two comparisons in particular. First, the gain from using 2SDM on unconditional ImageNet-64 (40.2%) is greater than that on class-conditional modeling of the same dataset (31.5%). This supports our argument that two-stage diffusion techniques like 2SDM can have even greater impact in unconditional (or lightly-conditional) generation than in the text-conditional (or strongly-conditional) setting in which they were originally introduced with unCLIP (Ramesh et al., 2022). Noting that the class label already contains some of the information stored in a CLIP embedding, this finding also fits with our discussion of the effects of conditioning in Section 3. The performance of an image model conditioned on just a class label (EDM on class-cond. ImageNet) should therefore be somewhere in between that of an unconditional image model (EDM on uncond. ImageNet) and that of a CLIP-conditional image model (2SDM, assuming the auxiliary model is good), leading to this finding. Second, the 28.9% improvement in performance for the latent diffusion model on ImageNet-256 is only slightly less than the 31.5% improvement for pixel-space diffusion on class-conditional ImageNet-64. This confirms that 2SDM can be readily combined with the widely used latent diffusion framework. Inference speed Sampling from 2SDM does impose a small additional cost relative to EDM, since we must begin by sampling from the auxiliary model. In all experiments, when we use 40 diffusion steps, sampling from our auxiliary model takes 8.8s with batch size 256. This corresponds to 35ms per batch item. Our conditional image model and our EDM baseline use identical architecture (other than the projection of $ y $) and we could not detect a difference between their sampling times which were 862ms per batch item on our ImageNet architecture and 789ms per batch item on our AFHQ and FFHQ architecture. This means that the increase in time due to using 2SDM instead of EDM is less than 4%. Furthermore, we can negate this increase by using two less sampling steps for the conditional image model. Table 5 in the appendix shows that this lets us make 2SDM faster than EDM with almost no effect on sample quality. Overfitting analysis We test for overfitting on AFHQ and FFHQ in the appendix through interpolation plots and nearest neighbour searches. We summarize these results in Fig. 6 by sampling 100 images from each method; computing the LPIPS distance of each one to every training set image and taking the minimum over all training set images; and then plotting the histogram of these minima. To create the black line, we use 100 training set images and take the minima over non-zero LPIPS distances to training set images to avoid them being reported as their own nearest neighbours. We can be confident that a method is overfitting if its curve is further to the left than the black curve. We see that both 2SDM and EDM overfit slightly on AFHQ (which contains only 15,000 images) but no overfitting is visible on FFHQ (which has 70,000 images). Seeing as these plots are similar for 2SDM and EDM, and given that ImageNet is a much larger dataset than AFHQ and FFHQ, we are confident that 2SDM’s gains do not come from overfitting. We do, however, include another method, Overfit-2SDM, as a point of Figure 6: Distribution of LPIPS (Zhang et al., 2018) distances to the nearest neighbour in the training set for sampled images from EDM, 2SDM, and Overfit-2SDM. We see clear signs of overfitting for Overfit-2SDM on AFHQ but not for any other methods or datasets. interest and note of warning for future work on this topic. Overfit-2SDM is a variation of 2SDM in which we train the CLIP parameters jointly with the auxiliary and conditional image DGMs. It achieves state-of-the-art FID on AFHQ but, as we see in Fig. 6, only through near-total overfitting to the training set. See the appendix for more details. 6 RELATED WORK Intermediate variables in diffusion models Our work takes inspiration from Weilbach et al. (2022), who show improved performance in various approximate inference settings by modeling problem-specific auxiliary variables (like $ y $) in addition to the variables of interest ($ x $) and observed variables ($ a $). We apply these techniques to the image domain and incorporate pretrained CLIP embedders to obtain auxiliary variables. Latent diffusion 2SDM also relates to methods which perform diffusion in a learned latent space (Rombach et al., 2022): our auxiliary model $ p_\phi(y\|a) $ is analogous to a “prior” in a latent space and our conditional image model $ p_\theta(x\|a,y) $ to a “decoder” Such methods typically use a near-deterministic decoder and so their latent variables must summarize all information about the image. Our conditional DGM decoder, on the other hand, is a DGM that will function reasonably however little information is stored in $ y $. This means that 2SDM provides an additional degree of freedom in terms of what to store. Furthermore, as we showed in Section 5, 2SDM can be fruitfully combined with latent diffusion. Self-supervised representations Bao et al. (2022); Hu et al. (2022) both use self-supervised learning to obtain auxiliary variables and then training a diffusion model $ p(x\|a) $. However, they do not model $ a $ and therefore are not able to sample $ x $ without an oracle that can provide $ a $. Their success when given an oracle, however, provides reason to believe that our approach is likely to yield benefits even if the embedder that produces $ a $ is obtained through self-supervised learning and without access to additional (or multi-modal) data as our CLIP embedder was trained with. Integrating additional data Our method can be understood as a means to leverage the “world knowledge” inside a CLIP embedder for improved performance on the image generation task. Another way in which additional knowledge, or data, could be leveraged is by training a multi-headed diffusion model which simultaneously approximates the score function and makes predictions of side information like class labels. Deja et al. (2023) propose a method for doing so but do not demonstrate improved performance on the unconditional generation task. 7 DISCUSSION AND CONCLUSION We have demonstrated 2SDM, a variant of unCLIP for unconditional or lightly-conditional image generation, and argued that it has more benefits in this setting that in the text-conditional setting in which unCLIP was originally proposed. Therefore, even if the trend towards simple single-stage architectures continues for large-scale text-to-image models (Rombach et al., 2022; Chang et al., 2023; Hoogeboom et al., 2023), unCLIP-style approaches could offer large jumps in performance for lightly-conditional image generation tasks. 2SDM also holds promise for improving video generation. This is a domain for which CLIP could be readily applied, and being able to learn relationships in the relatively low-dimensional CLIP embedding space could significantly increase training throughput relative to working purely in pixel (or VAE embedding) space. A massive unexplored design space remains. For pedagogical purposes we intentionally kept 2SDM simple, using known diffusion architectures and objectives. It is likely that optimizing these design choices for the lightly-conditional 2SDM use-case would improve performance. In addition, there are almost certainly more useful quantities that we could condition on than CLIP embeddings. Bao et al. (2022); Hu et al. (2022) have shown that self-supervised learning techniques provide a promising avenue for obtaining useful “latent” representations. Exactly the properties that an embedding should have to be beneficial for techniques like 2SDM is another open question that is ripe for future work to tackle. Such a line of work may also fix one limitation of 2SDM, namely that it relies on the availability of a pretrained CLIP embedder. While this is freely available for natural images, it could be a barrier to other applications. Improvements may also be gleaned by conditioning on multiple quantities, or “chaining” a series of conditional DGMs together. An alternative direction is to sim- plify 2SDM’s architecture by, for example, learning a single diffusion model over the joint space of x and y instead of generating them sequentially. We did not use classifier-free guidance (Ho & Salimans, 2022) in this work, which can improve visual fidelity at the cost of losing the mass-covering behavior that diffusion models are known for. Conditioning on the CLIP embedding with a high guidance scale could help to optimize for visual quality in future work. 8 ETHICS STATEMENT Like much foundational research in modern generative modeling, this work carries risks like aiding the generation of deepfakes for dis- or misinformation campaigns. This leads to a second negative consequence: that trust in various forms of visual evidence, such as photographs, videos, and audio recordings, may no longer be possible. One avenue with which to address these consequences is research towards developing robust and effective methods for detecting and mitigating the harmful effects of deepfakes and synthetic media manipulation. Furthermore, increasing public awareness about the existence and potential impact of deepfakes can empower people to critically evaluate information and be more resilient to manipulation attempts. 2SDM has a potential risk on top of this: it leverages a publicly available “foundation model” in the form of a CLIP embedder to enhance the quality of generated content. Biases present in the foundation model may influence the outputs of 2SDM even if they are not present in the image dataset used for training. Stringent evaluation of foundation models may mitigate potential harms arising from this. 9 REPRODUCIBILITY STATEMENT We release source code at https://anonymous.4open.science/r/2sdm. We will additionally release trained checkpoints on acceptance. REFERENCES Shekoofeh Azizi, Simon Kornblith, Chitwan Saharia, Mohammad Norouzi, and David J Fleet. Synthetic data from diffusion models improves imagenet classification. arXiv preprint arXiv:2304.08466, 2023. 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V3j5d0GQgH	Why the mean and variance of class means are evaluated in Figure 1? I cannot follow the logic in the discussion quoted below. > However, preliminary experiments benchmarking ETF with CIFAR-100 in Fed-LT suggest that only a few features have relatively large means, while most of the small-mean features are contaminated by severe noise, as shown in Fig. 1(a). Such observations are inconsistent with the feature collapse property, and we coin it as feature degeneration. How do the mean and variance of features relate to the feature collapse?	FedLoGe: Joint Local and Generic Federated Learning under Long-tailed Data Zikai Xiao1∗, Zihan Chen2∗, Liyinglan Liu3, Yang Feng4, Jian Wu1, Wanlu Liu1, Joey Tianyi Zhou5,6, Howard Hao Yang1, Zuozhu Liu1† 1Zhejiang University, 2Singapore University of Technology and Design, 3University of Electronic Science and Technology of China, 4Angelalign Technology Inc, 5IHPC, Agency for Science, Technology and Research, Singapore, 6CFAR, Agency for Science, Technology and Research, Singapore zikai@zju.edu.cn Abstract Federated Long-Tailed Learning (Fed-LT), a paradigm wherein data collected from decentralized local clients manifests a globally prevalent long-tailed distribution, has garnered considerable attention in recent times. In the context of Fed-LT, existing works have predominantly centered on addressing the data imbalance issue to enhance the efficacy of the generic global model while neglecting the performance at the local level. In contrast, conventional Personalized Federated Learning (pFL) techniques are primarily devised to optimize personalized local models under the presumption of a balanced global data distribution. This paper introduces an approach termed Federated Local and Generic Model Training in Fed-LT (FedLoGe), which enhances both local and generic model performance through the integration of representation learning and classifier alignment within a neural collapse framework. Our investigation reveals the feasibility of employing a shared backbone as a foundational framework for capturing overarching global trends, while concurrently employing individualized classifiers to encapsulate distinct refinements stemming from each client’s local features. Building upon this discovery, we establish the Static Sparse Equiangular Tight Frame Classifier (SSE-C), inspired by neural collapse principles that naturally prune extraneous noisy features and foster the acquisition of potent data representations. Furthermore, leveraging insights from imbalance neural collapse’s classifier norm patterns, we develop Global and Local Adaptive Feature Realignment (GLA-FR) via an auxiliary global classifier and personalized Euclidean norm transfer to align global features with client preferences. Extensive experimental results on CIFAR-10/100-LT, ImageNet-LT, and iNaturalist demonstrate the advantage of our method over state-of-the-art pFL and Fed-LT approaches. Our codes are available at https://github.com/ZackZikaiXiao/FedLoGe. 1 Introduction Federated learning (FL) enables collaborative model training across decentralized clients without exposing local private data [McMahan et al., 2017; Kairouz et al., 2021]. Recent work further investigates the federated long-tailed learning (Fed-LT) task, where the global data exhibits long-tailed distributions and local clients hold heterogeneous distributions [Chen et al., 2022b; Shang et al., 2022b]. They usually learn a well-trained generic global model, whose performance might degrade when universally applied to all clients with diverse data, degrading its practical applicability. For example, in the realm of smart healthcare, as demonstrated by Lee & Shin (2020), Chen et al. (2022a), ∗Co-first author. †Corresponding author. and Elbatel et al. (2023), the capacity of the global model to deliver high-quality diagnostics is limited, as patient distributions vary across specialized hospitals. Additionally, in cross-institutional financial applications like credit scoring (Dastile et al., 2020) and fraud detection (Awoyemi et al., 2017), individuals from different regions or age groups may exhibit dissimilar credit patterns. A high-quality global model with balanced performance could speed up local adaptation and attract new clients, while personalized models aim to provide enhanced local performance by considering local data characteristics. Nevertheless, existing works on Fed-LT have primarily focused on addressing the imbalance in the context of the global long-tailed data (Yang et al., 2023a; Qian et al., 2023; Xiao et al., 2023), neglecting the tailoring of models to the needs of individual clients, since local data statistics could be diverse and not necessarily long-tailed. Personalized federated learning (pFL) (Tan et al., 2022), which trains customized local models for a single or a group of clients, offers an alternative solution to prioritize each client’s (or group’s) distinct data statistics and preferences, in which the global generic model is deemed as a bridge for training local personalized models and boosting the local performance with expressive representations (Li et al., 2021b; Collins et al., 2021; Li et al., 2021a). However, conventional pFL approaches are not supposed to attain a superior global generic model in Fed-LT. ![Figure 1](image) Figure 1: (a): The mean (sorted in descending order) and variance of class means unveil feature degeneration: The feature collapse property ceases to prevail, and features with diminished means exhibit substantial variance rather than being zero; (b): After training the backbone with SSE-C, in which noisy features with bigger variance are partially pruned (gray shaded vertical lines), and enhance the quality (smaller variance) of dominant features. Designing a framework to simultaneously train global and local models in the presence of Fed-LT remains a critical challenge. Inspired by Kang et al. (2019), we find that the generality and transferability of feature extractors are significantly superior to classifiers (Kim et al., 2022; Vasconcelos et al., 2022). On the contrary, adjusting the classifier has proven to be quite effective in addressing the imbalance and heterogeneity issues (Li et al., 2022a; Zhang et al., 2022a). In other words, the feature extractor can serve as the cornerstone to reflect global trends, while adjusting classifiers can induce the model to adaptively achieve superior personalized performance across the server and heterogeneous clients. Consequently, we conceptualize our model learning with two intertwined processes: global representation learning and imbalanced/heterogeneous classifier adaptation. Adopting this viewpoint, we identify two key challenges for personalized Fed-LT: C1: How to learn effective representations under heterogeneous and imbalanced data? Due to heterogeneity, each client captures different feature distributions, leading to divergence during model aggregation and inferior global performance. Recent studies show that training with a fixed classifier can reduce divergence among heterogeneous clients to improve performance (Oh et al., 2022; Dong et al., 2022). The fixed classifier serves as a consistent criterion for learning representations across clients over time, rather than being an optimal choice itself. For instance, Yang et al. (2022) proposes to initialize the classifier as a simplex equiangular tight frame (ETF) with maximal pairwise angles under imbalanced learning. In general, these fixed classifiers force the feature prototypes to converge to an optimal structure to improve representation learning. However, their effectiveness in resolving Fed-LT regarding both global and local models is not investigated. We examine the effectiveness of training with fixed classifiers in Fed-LT from the perspective of neural collapse (NC) (Papyan et al., 2020). The NC identifies a salient property of the feature space, showing that all within-same-class features tend to collapse to their respective class means. However, as shown in Fig. 1(a), preliminary experiments benchmarking ETF with CIFAR-100 in Fed-LT suggest that only a few features have relatively large means, while most of the small-mean features are contaminated by severe noise. Such observations are inconsistent with the feature collapse property, and we coin it as feature degeneration. More details of the computation process for the data in Fig. 1 as well as the necessary explanation can be found in Appendix A.2. To resolve the feature degeneration for improved representation learning, we propose the Static Sparse Equiangular Tight Frame Classifier (SSE-C) inspired by the sparse coding theories (Frankle & Carbin, 2019; Glorot et al., 2011). The assumption of SSE-C is that the small-mean degenerated features contribute little to model performance while forcing sparsity on them would help learn more expressive representations. We refer to the small-mean features as negligible features and the large-mean features as dominant features. The SSE-C dynamically prunes the classifier weights of those small-mean noisy features, while holding more expressive dominant features. Probing into weights trained with SSE-C validates our assumption, as shown in Fig. 1(b) and experiments. C2: How to conduct effective feature realignment to improve the performance of both the generic and personalized models based on data preferences? In Fed-LT, the long-tailed global data distribution and heterogeneous local distributions raise the requirements to learn different global and personalized local models for satisfactory performance. Thus, the feature extractor trained with a fixed classifier needs to be realigned to both global and local models. For the global model, it is necessary to realign the model to improve its performance on global tail classes. For the personalized model, the classifier needs to align features with respective heterogeneous local data preferences. In this work, we unify the feature realignment for both the server and clients under the neural collapse framework. The key idea is to align both global and local classifiers based on the weight norm of the classifiers. Previous works show that classifier weight norms are closely correlated with the corresponding class cardinalities (Kang et al., 2019; Tan et al., 2021; Li et al., 2020b). Further research provides both empirical explanation and theoretical justification that the classifier weight norm is larger for majority classes while smaller for minorities (Kim & Kim, 2020; Dang et al., 2023; Thrampoulidis et al., 2022). We propose the Global and Local Adaptive Feature realignment (GLA-FR) module to align the backbone trained with SSE-C to the server and clients. In particular, we devise auxiliary classifier heads for the global ($\psi$) and $K$ local classifiers ($\{\phi_k\}_{k=1}^K$), which are trained alternately with SSE-C in each epoch, see Algorithm 1. The alignment includes two stages: global alignment and local alignment. The global realignment is simple yet effective, adjusting the weights based on the norms of $\psi$ (Eq[6]) to tackle with global balanced test set. The alignment for personalized models is a bit different, as their data distributions differ greatly from the global distribution. We integrate the global trends with each local client’s preference by adjusting the global classifiers $\psi$ with the norms of local classifier $\phi_k$ (Eq[7]). Our work represents pioneering efforts to achieve a harmonious integration of global and personalized model learning under Fed-LT, thereby facilitating each participating institution in obtaining a model that is more adeptly tailored to its inherent characteristics and preferences. Comprehensive --- 1 Our $\psi_{SSE-C}$ leads to two notable improvements in Fig. 1 (b) over Fig. 1 (a): First, it masks noisy features with large variances, now in sparse areas marked by grey. Second, the role of dominant features is enhanced, as shown by reduced variances among those with larger means, reflecting their increased precision and efficacy. experiments on representative datasets CIFAR-10/100-LT, ImageNet, and iNaturalist demonstrate the outperformance and efficacy of both global generic and local personalized Fed-LT models. 2 RELATED WORK 2.1 FEDERATED LEARNING Federated Long-tailed Learning Recent research launched attempts to resolve the Fed-LT task. Model decoupling methods explore frameworks such as classifier retraining (Shang et al., 2022b) and prototype-based classifier rebalancing (Yang et al., 2023a; Dai et al., 2023) for Fed-LT. Shang et al. (2022a) and Wang et al. (2022) investigate calibration and distillation methods to improve the model performance. Many efforts also have been made from the perspective of meta-learning (Qian et al., 2023; Shen et al., 2021), client selection (Zhang et al., 2023; Yang et al., 2021), re-weighting (Wang et al., 2021; Shen et al., 2021), and aggregation (Chou et al., 2022). Personalized Federated Learning (pFL) To deal with the poor generalization performance of the single generic global at local data, pFL has been vastly investigated. A group of works seeks to train local personalized models via transferring knowledge from the generic global model (Li & Wang, 2019; T Dinh et al., 2020; Fallah et al., 2020; Chen et al., 2023). Multi-task learning-based methods are also been explored with client clustering (Sattler et al., 2020; Briggs et al., 2020; Ghosh et al., 2020) and model interpolation (Deng et al., 2020; Li et al., 2021a; Diao et al., 2020). For neural network-based FL framework, parameter decoupling methods have gained popularity due to their simplicity. Parameter decoupling aims to achieve personalization by decoupling the local private model parameters from the global model parameters. For horizontal decoupling, Li et al. (2021b) personalizes the batch normalization layers, Pillutla et al. (2022) explores decoupling different parts, and personalizing the last layer is adopted in Arivazhagan et al. (2019), Collins et al. (2021), and Briggs et al. (2020). For vertical decoupling, Shen et al. (2022) personalizes channels. 2.2 NEURAL COLLAPSE FOR REPRESENTATION LEARNING Neural collapse refers to a set of four interconnected phenomena that demonstrate a pervasive inductive bias in the terminal phase of training, as shown by Papyan et al. (2020). Subsequently, several works have sought to explain the neural collapse phenomena from the perspective of peeled models (Ji et al., 2021; Fang et al., 2021), unconstrained feature models (Tirer & Bruna, 2022; Mixon et al., 2020; Zhu et al., 2021), and Riemannian manifolds (Yaras et al., 2022). Building upon the findings on neural collapse, Yang et al. (2022) first proposed fixing the classifier to an ETF structure and introduced dot regression loss. The ETF structure was later utilized for semantic segmentation (Zhong et al., 2023), handling heterogeneity in federated learning (Li et al., 2023), transfer learning (Li et al., 2022b), incremental learning (Yang et al., 2023b), and object detection (Ma et al., 2023). In summary, our method stands out by introducing personalization within the neural collapse framework, effectively overcoming global imbalanced data and enhancing local model personalization through tailored feature distribution alignment. Contrary to FedETF’s fixed classifier approach and FedRod’s dual classifier structure, our method ensures superior handling of global long-tail bias and precise local model tuning, marking a novel advancement in Fed-LT. For detailed comparisons, please see Section A.13 in the Appendix. 3 PROPOSED METHOD In this section, we introduce our proposed Fed-LoGe (see Algorithm 1), a simple yet effective framework to achieve joint personalized and generic model learning for Fed-LT. To boost representation learning and address the global-local inconsistency, we introduce a training paradigm consisting of a sparsified ETF module and global-local feature alignment modules. 3.1 PRELIMINARIES We consider an FL system with $K$ clients and a server. The overall objective is to train $1 + K$ models: 1 generic and $K$ personalized models. Specifically, the generic model is parameterized by $ w = \{\theta, \psi\} $, whereas the $ k $th personalized model for each client $ k \in [K] $ is denoted as $ w_k $. We decoupled the neural network models into a feature extractor $ f(x, \theta) $ and a set of classifiers. The feature extractor, parameterized by $ \theta $, transforms input $ x $ into features $ h $. The generic classifier $ g(h, \psi) $ and personalized classifiers $ g(h, \phi_k) $ then map these features to the output labels. The overall global and local objective functions can be respectively expressed as: \[ \text{Global: } \min_w \sum_{k=1}^{K} \frac{\|D_k\|}{\|D\|} L_k(w_k; D_k), \quad \text{Local: } \min_{\{\theta, \phi_k\}} L(\theta, \phi_k; D_k), \] where $ D = \{D_k\}_{k=1}^{K} $ is the global long-tailed dataset composed of $ K $ heterogeneous local datasets, each with size $ \|D_k\| $. The training is executed over $ T $ rounds. In each round $ t $, the server distributes the current global model $ w^{(t)} $ to all clients for local updates. Furthermore, given that $ c $ represents the class index, for $ \forall c \in C $, the classifier vector is $ \psi_c $, and the corresponding features are $ h_c $. ### 3.2 Static Sparse Equiangular Tight Frame Classifier (SSE-C) Initializing the classifier as ETF and subsequently freezing it during training has proven to be an effective strategy in federated learning, attributed to ETF’s inherent structure, which ideally exhibits commendable properties of feature collapse under balanced data. However, we found that in the context of Fed-LT, the feature collapse property was not satisfied when initializing the fixed classifiers due to feature degeneracy (the existence of a high-noise feature with a small norm), which is illustrated in Fig. 1(a). Accordingly, we propose the Static Sparse Equiangular Tight Frame Classifier (SSE-C) via fixing classifier to learn higher-quality features and reduce the impact of negligible features so as to achieve effective representation learning. The server will obtain SSE-C with $ L_{\text{SSE-C}} $ prior to the local training, and all clients will fix SSE-C throughout the training. We first initialize the classifier as a conventional ETF matrix by \[ \psi = \sqrt{\frac{C}{C-1}} U \left( I_C - \frac{1}{C} 1_C 1_C^T \right) \] where $ \psi = [\psi_{:,1}, \cdots, \psi_{:,C}] \in \mathbb{R}^{d \times C}, U \in \mathbb{R}^{d \times C} $ allows any rotation and satisfies $ U^T U = I_C, I_C $ is the identity matrix. $ d $ is the dimension of the classifier vector, and $ 1_C $ is an all-ones vector. We can deduce the important property that all class vector has the equal $ \ell_2 $ norm and maximal pair-wise angle $ -\frac{1}{C-1} $ in $ \mathbb{R}^d $. Note that randomly assigning $ \beta $ proportion of weights in the ETF matrix to 0 will disrupt the ETF condition: the class vector angles cease to be maximal and equal, and the norms of the classifier vector become unequal. As such, it is necessary to train a sparse ETF structure that satisfies the ETF geometric conditions. We introduce a sparse indicator matrix $ S $ with the same dimensions as $ \psi $, where $ \beta $ proportion of the elements are randomly set to 0. Then, the sparsified matrix $ \psi' $ can be represented as $ \psi' = \psi \odot S $. We design the Equal Norm Loss and Maximal Angle Loss to optimize the geometric structure to meet the conditions of ETF. First, the $ \ell_2 $ norm of class vectors should be equal. We constrain all norms of class vector values to a predetermined $ \gamma $: \[ l_{\text{norm}}(\psi', \gamma, S) = \sum_{i=1}^{C} \left( \\| \psi'_{:,i} \odot S_{:,i} \\|_2 - \gamma \right)^2. \] Second, we maximize the minimum angle between class vector pairs. Following MMA (Wang et al., 2020), we normalize the classifier vector by $ \hat{\psi}'_{:,i} = \frac{\psi'_{:,i}}{\\|\psi'_{:,i}\\|_2} $ and maximize only the minimum angle with the formula: \[ l_{\text{angle}}(\hat{\psi}', S) = -\frac{1}{C} \sum_{i=1}^{C} \cos^{-1} \left( \max_{j \in \{1,2,...,C\} \setminus \{i\}} \left( (\hat{\psi}'_{:,i} \odot S_{:,i})^T (\hat{\psi}'_{:,j} \odot S_{:,j}) \right) \right). \] By integrating the $ l_{\text{norm}} $ and $ l_{\text{angle}} $, we obtain the $ L_{\text{SSE-C}} $ for the following training: \[ L_{\text{SSE-C}} = l_{\text{norm}}(\psi', \gamma, S) + l_{\text{angle}}(\hat{\psi}', S). \] Then we solve the objective $ \psi_{\text{SSE-C}} = \arg \min_{\psi} L_{\text{SSE-C}} $ by SGD. By regularizing class vector norms and maximizing their minimum angle, the classifier exhibits sparsity while maintaining ETF properties, effectively guiding the model in learning robust features. Algorithm 1 An overview of FedLoGe framework Input: $ w^{(0)} = \{ \theta^{(0)}, \psi^{(0)} \}, \{ \phi_k^{(0)} \}_{k=1}^K, K, E, T, C $ Output: $ w^{(T)} = \{ \theta^{(T)}, \psi^{(T)} \}, \{ w_k \}_{k=1}^K = \{ \theta^{(T)}, \phi_k^{(T)} \}_{k=1}^K $ Stage 1: Representation Learning with SSE-C 1: Obtain $ \psi_{\text{SSE-C}} $ by Equation 5 2: for each round $ t = 1 $ to $ T $ do 3: $ S_t \leftarrow $ subset of selected clients 4: for each client $ k \in S_t $ in parallel do 5: $ \theta_k^{(t)}, \psi_k^{(t)} \leftarrow \text{CLIENTUPDATE}(k, \theta_k^{(t-1)}, \psi_k^{(t-1)}, \psi_{\text{SSE-C}}) $ 6: $ \theta^{(t)}, \psi^{(t)} = \text{AGGREGATION}(\theta_k^{(t-1)}, \psi_k^{(t-1)}) $, for all $ k \in S_t $ Stage 2: Global Feature Realignment 7: for each classifier vector $ c = 1 $ to $ C $ do 8: $ \psi_c^{(T)} \leftarrow \psi_c^{(T)} / \\| \psi_c^{(T)} \\|_2 $ (Equation 6) \\ Align the long-tailed norm to balanced norm Stage 3: Local Feature Realignment 9: for each client $ k = 1 $ to $ K $ do 10: for each classifier vector $ c = 1 $ to $ C $ do 11: $ \phi_{k,c}^{(T)} \leftarrow \psi_c^{(T)} * \\| \phi_{k,c}^{(T)} \\|_2 $ (Equation 7) \\ Incorporate Global Classifier with local statistics 12: $ \phi_k^{(T)} \leftarrow \phi_k^{(T)} - \eta \nabla_\phi(L(\theta^{(T)}, \phi_k^{(T)}); x) $ \\ Finetune $ \phi_k^{(T)} $ 13: return $ w^{(T)} = \{ \theta^{(T)}, \psi^{(T)} \}, \{ w_k \}_{k=1}^K = \{ \theta^{(T)}, \phi_k^{(T)} \}_{k=1}^K $ function CLIENTUPDATE(k, $ \theta^{(t)}, \psi^{(t)}, \psi_{\text{SSE-C}} $) 1: $ \theta_k^{(t)}, \psi_k^{(t)} = \theta^{(t)}, \psi^{(t)} $ 2: for each local epoch $ i = 1 $ to $ E $ do 3: Compute features $ h_i \leftarrow f(x_i, \theta^{(t)}) $ 4: $ \theta_k^{(i+1)} \leftarrow \theta_k^{(i)} - \eta \nabla_\theta(L(\psi_{\text{SSE-C}}; x_i)) $ \\ Fix $ \psi_k^{(t)}, \phi_k^{(t)} $, Update $ \theta_k^{(t)} $ with $ \psi_{\text{SSE-C}} $ 5: $ \psi_k^{(i+1)} \leftarrow \psi_k^{(i)} - \eta \nabla_\psi(L(\theta^{(t)}, \psi_k^{(i)}); h_i) $ \\ Fix $ \theta_k^{(t)}, \phi_k^{(t)} $, Update $ \psi_k^{(t)} $ 6: $ \phi_k^{(i+1)} \leftarrow \phi_k^{(i)} - \eta \nabla_\phi(L(\theta^{(t)}, \phi_k^{(i)}); h_i) $ \\ Fix $ \theta_k^{(t)}, \psi_k^{(t)} $, Update $ \phi_k^{(t)} $ 6: return $ \theta_k^{(i+1)}, \psi_k^{(i+1)} $ 3.3 Global and Local Adaptive Feature Realignment (GLA-FR) To be adapted in both global and local models, the feature extractor trained with a fixed classifier needs to be realigned to address the imbalance and heterogeneity. Hence, we conduct feature realignment to both global and personalized models after training the SSE-C guided feature extractor, where the realignment should be consistent with the local data statistics/class cardinality. To obtain a good estimation of class cardinality, in prior work such as Kang et al. (2019), Tan et al. (2021), Li et al. (2020b), the classifier weight norms $ \\| \psi_c \\| $ are found to be correlated with the corresponding class cardinalities $ n_c $, in which $ \psi_c $ is the classifier weight vector for the $ c $-th class. Kim & Kim (2020) provides an explanation from the perspective of decision boundaries - the weight vector norm for more frequent classes is larger, biasing the decision boundary towards less frequent classes. Also, for the neural collapse framework with imbalanced data, the relations between the weight norm of classifiers and the class cardinality also exist (Thrampoulidis et al., 2022; Dang et al., 2023). These findings are consistent, which motivates us to measure/estimate local data statistics based on the norm weight of the classifier. The frozen ETF classifier is not suitable for feature alignment, attributed to the lack of valid norms to estimate class cardinality. We design a new auxiliary global head $ \psi $ to obtain valid norms, which participate in gradient updates and weight aggregation alongside the backbone. After $ T $ rounds of training, we get the global weight $ w^{(T)} = \{ \theta^{(T)}, \psi^{(T)} \} $. The $ \theta^{(T)} $ is well trained with $ \psi_{\text{SSE-C}} $. For the global adaptive feature distribution process (GA-FR), let $ \psi_c $ denote a classifier vector corresponding to the $ c $-th class, where $ c \in C $. The aligned classifier vector $ \psi_c' $ can be obtained by dividing... Table 1: Test accuracies of our and SOTA methods on CIFAR-10/100-LT with diverse imbalanced and heterogeneous data settings. GM/PM denotes Global/Personalized model. \[ \psi_c' = \frac{\psi_c}{\\|\psi_c\\|_2} \] (6) Here, $ \\|\psi_c\\|_2 $ represents the $ \ell_2 $ norm of $ \psi_c $. Each $ \psi_c' $ will be a unit vector, preserving the direction of $ \psi_c $ and possessing the magnitude of 1. For personalized adaptive feature realignment (LA-FR), we adapt the global auxiliary classifier $ \psi $ to the personalized classifier by multiplying the norm of $ \phi_k $, which implies that clients will leverage information from categories with a larger sample size while omitting information pertaining to the rare categories. For local classifier vector $ \phi_{k,c} $ at client $ k $, the process of LA-FR is as follows: \[ \phi_{k,c}' = \psi_c \ast \\|\phi_{k,c}\\|_2 \] (7) 3.4 Algorithms Overall, our framework Fed-LoGe consists of three critical stages: representation learning with SSE-C, global feature realignment, and local feature realignment, for the training of the shared backbone $ \theta $, global auxiliary classifier $ \psi $, and $ K $ local classifiers ($ \{\phi_k\}_{k=1}^K $), respectively. In the first stage, the server first constructs the SSE-C with Eq. 5 before the training and then distributes it to all clients. Upon receiving SSE-C, each client fixes it as the classifier to train the backbone $ \theta $, global classifier $ \psi $, and local classifier $ \phi_k $ alternately. Specifically, we update $ \theta $ with fixed $ \psi_{SSE-C} $. Subsequently, the $ \theta $ is frozen to update the global head $ \psi $ and each local classifier $ \phi_k $. At the end of each round, the $ \theta $ and $ \psi $ are aggregated at the server, while $ \phi_k $ is retained locally. Global adaptive feature realignment (GA-FR) is performed in the second stage, where each class vector is redistributed by the server according to its individual norms, as outlined by Eq. 6. Subsequently, in the third phase, personalized adaptive feature realignment (LA-FR) for the class vectors of the global auxiliary head $ \psi $ is performed. Following LA-FR, local finetuning could be further conducted to boost the model performance. A summary of Fed-LoGe is given in Algorithm 1. 4 Experiments 4.1 Experimental Setup Dataset, Models and Metrics: We consider image classification tasks for performance evaluation on benchmark long-tailed datasets: CIFAR-10/100-LT, ImageNet-LT, and iNaturalist-User- \| Dataset \| ImageNet \| Inaturalist \| \|---------\|----------\|-------------\| \| Method/Model \| Many \| Med \| Few \| GM \| PM \| Many \| Med \| Few \| GM \| PM \| \| FedAvg \| 0.481 \| 0.307 \| 0.159 \| 0.329 \| 0.528 \| 0.591 \| 0.418 \| 0.238 \| 0.425 \| 0.590 \| \| FedProx \| 0.493 \| 0.318 \| 0.180 \| 0.343 \| 0.500 \| 0.525 \| 0.484 \| 0.223 \| 0.432 \| 0.596 \| \| FedBN \| 0.471 \| 0.300 \| 0.168 \| 0.319 \| 0.504 \| 0.573 \| 0.396 \| 0.221 \| 0.413 \| 0.563 \| \| FedPer \| - \| - \| - \| 0.653 \| - \| - \| - \| - \| 0.638 \| - \| \| FedRep \| 0.460 \| 0.309 \| 0.187 \| 0.330 \| 0.574 \| 0.571 \| 0.453 \| 0.237 \| 0.429 \| 0.627 \| \| Ditto \| 0.492 \| 0.319 \| 0.176 \| 0.342 \| 0.674 \| 0.508 \| 0.452 \| 0.245 \| 0.437 \| 0.584 \| \| FedROD \| 0.483 \| 0.305 \| 0.165 \| 0.331 \| 0.7033 \| 0.585 \| 0.416 \| 0.243 \| 0.421 \| 0.699 \| \| FedBABU \| 0.443 \| 0.240 \| 0.055 \| 0.230 \| 0.425 \| 0.561 \| 0.401 \| 0.199 \| 0.377 \| 0.696 \| \| FedETF \| 0.425 \| 0.239 \| 0.05 \| 0.222 \| 0.418 \| 0.587 \| 0.431 \| 0.245 \| 0.437 \| 0.713 \| \| Ratio Loss \| 0.495 \| 0.337 \| 0.189 \| 0.351 \| 0.521 \| 0.587 \| 0.454 \| 0.290 \| 0.452 \| 0.589 \| \| FedLoGe \| 0.430 \| 0.373 \| 0.285 \| 0.356 \| 0.726 \| 0.519 \| 0.508 \| 0.473 \| 0.503 \| 0.759 \| Table 2: Test accuracies of our and SOTA methods on ImageNet-LT and iNaturalist-160k with diverse heterogeneous data settings. 160k (Van Horn et al., 2018). The CIFAR-10/100-LT datasets are sampled into a long-tailed distribution employing an exponential distribution governed by the Imbalance Factor (IF) in Cao et al. (2019). All experiments are conducted with non-IID data partitions, implemented by the Dirichlet distributions-based approach with parameter $\alpha$ to control the non-IIDness (Chen & Chao, 2022). ResNet-18 is trained over $K = 40$ clients on CIFAR-10-LT, while ResNet-34 and ResNet-50 are implemented on CIFAR-100-LT and ImageNet-LT, respectively, with $K = 20$ clients. The configurations for iNaturalist-160k align with those utilized for ImageNet-LT. We use $\alpha = 1, 0.5$ and IF = 50, 100 in CIFAR-10/100-LT. We use $\alpha = 0.1, 0.5$ in ImageNet-LT and iNaturalist, respectively. A global balanced dataset is used for the calculation of test accuracy to evaluate global model (GM) performance. We also report the accuracy across many, medium, and few classes. The detailed categorization for many/med/few classes can be found in the Appendix. For personalized model (PM) evaluation, we use local test accuracy, and the local test set is sampled from the global test set. Each local test set has an identical distribution to the local training set. The accuracy of the PM is the arithmetic mean of local test accuracy across all clients. Compared Methods: In addition to FedAvg and FedProx (Li et al., 2020a) which are included for reference, we consider two types of state-of-the-art baselines: (1) pFL methods, including FedBN (Li et al., 2021b), FedPer (Arivazhagan et al., 2019), FedRep (Collins et al., 2021), Ditto (Li et al., 2021a), and FedROD (Chen & Chao, 2022). (2) Federated (Long-tailed) Representation learning, including FedBABU (Oh et al., 2022), FedETF (Li et al., 2023) and Ratio Loss (Wang et al., 2021). ### 4.2 Performance Comparison For evaluation on CIFAR-10-LT and CIFAR-100-LT, FedLoGe consistently outperforms all baselines over all settings, achieving the highest overall accuracies in all settings for both GM and PMs; see Tab. 1. For ImageNet-LT and iNaturalist-160k, Tab. 2 highlights that FedLoGe consistently surpasses all baselines, marking significant accuracy improvements, particularly in middle and tail classes. Overall, benefiting from the enhanced representation learning and classifier realignments, FedLoGe attains superior GM performance, attributed to the SSE-C design for representation learning, and simultaneously obtains impressive PM performance together with refined feature alignment. ### 4.3 Ablation Study and Sensitivity Analysis Ablations of SSE-C and GLA-FR: In the ablation study, we evaluate the SSE-C and GA-FR/LA-FR individual impacts with respect to both GM and PM performance, over CIFAR-100-LT (IF=100, $\alpha = 0.5$), as given in Tab. 3. The results lead to the following conclusions: Compared to dense ETF, SSE-C can train a superior backbone, enhancing both GM and PM performance. GA-FR can be combined with any backbone to boost GM performance, while LA-FR can be used with any backbone to boost PM performance. Employing SSE-C and GLA-FR simultaneously yields significant enhancements in both GM and PM performance. Negligible and dominant features with SSE-C: We investigated the effects of pruning on different features over CIFAR-100-LT ($\alpha=0.5$, IF=100). We computed the class mean for each category. Within each class mean, we ordered the means and pruned the corresponding classifier vector weights in descending (from dominant features) and ascending order (from negligible features) with pruning ratios from 0 to 100%. After training with SSE-C, pruning dominant features drastically decreases performance, while pruning negligible features barely affects performance, which is visualized in Fig. 3(a, b). This observation indicates that SSE-C can adeptly learn more effective features while autonomously disregarding high-noise features. Sensitivity analysis for $\gamma$ in SSE-C: We evaluate the model performance with various values of $\gamma$ (which is assigned as the norm of SSE-C in norm equal loss) over CIFAR-100-LT (IF=100, $\alpha = 0.5$), as depicted in Fig. 4(a). Our observations indicate that a smaller norm leads to a reduced gradient during backpropagation, producing subpar performance. Conversely, a larger norm enables faster convergence. For CIFAR-10/100-LT, the optimal value is approximately 0.1, whereas, for ImageNet and iNaturalist, it is approximately 1.6. Sensitivity analysis for Sparse Ratio $\beta$ in SSE-C: The $\beta$ indicates the pruning proportion in SSE-C. We evaluate the performance with the sparsity from 0 to 90% at intervals of 10% on CIFAR10/100-LT (IF = 100, $\alpha = 0.5$). As shown in Fig. 4(b) and (c), minor sparsification yields a slight model performance enhancement, and the performance is obtained around 60% sparsity. Surprisingly, a large sparsity ratio still remains superior compared to models without sparsification. CONCLUSION This paper presented FedLoGe, a model training framework that enhances the performance of both local and generic models in Fed-LT settings in the unified perspective of neural collapse. The proposed framework is comprised of SSE-C, a component developed inspired by the feature collapse phenomenon to enhance representation learning, and GLA-FR, which enables fast adaptive feature realignment for both global and local models. As a result, FedLoGe attains significant performance gains over current methods in personalized and long-tail federated learning. Future research will explore adaptive sparsity and expand the framework to diverse loss functions and tasks. ACKNOWLEDGEMENTS This work is supported by the National Natural Science Foundation of China (Grant No. 62106222, No. 62201504), the Natural Science Foundation of Zhejiang Province, China (Grant No. LZ23F020008, No. LGJ22F010001), Zhejiang Lab Open Research Project (No. K2022PD0AB05) and the Zhejiang University-Angelalign Inc. R&D Center for Intelligent Healthcare. REFERENCES Manoj Ghuhan Arivazhagan, Vinay Aggarwal, Aaditya Kumar Singh, and Sunav Choudhary. 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btpgDo4u4j	The paper repeatedly uses terms like the latent action representation and latent action planning, without a carefully derived definition of it. For self consistency, it would be helpful to define these terms more concretely; otherwise, in its current format, the contributions of the paper can be hard to follow.	Efficient Planning with Latent Diffusion Wenhao Li School of Software Engineering, Tongji University Shanghai, 201804, China liwenhao@cuhk.edu.cn Abstract Temporal abstraction and efficient planning pose significant challenges in offline reinforcement learning, mainly when dealing with domains that involve temporally extended tasks and delayed sparse rewards. Existing methods typically plan in the raw action space and can be inefficient and inflexible. Latent action spaces offer a more flexible paradigm, capturing only possible actions within the behavior policy support and decoupling the temporal structure between planning and modeling. However, current latent-action-based methods are limited to discrete spaces and require expensive planning steps. This paper presents a unified framework for continuous latent action space representation learning and planning by leveraging latent, score-based diffusion models. We establish the theoretical equivalence between planning in the latent action space and energy-guided sampling with a pretrained diffusion model and incorporate a novel sequence-level exact sampling method. Our proposed method, LatentDiffuser, demonstrates competitive performance on low-dimensional locomotion control tasks and surpasses existing methods in higher-dimensional tasks. 1 Introduction A considerable volume of samples gathered by operational systems gives rise to the issue of offline reinforcement learning (RL), specifically, the recovery of high-performing policies without additional environmental exploration (Wu et al., 2019; Kumar et al., 2020; Kostrikov et al., 2021; 2022; Ghosh et al., 2022). However, domains that encompass temporally extended tasks and severely delayed sparse rewards can present a formidable challenge for standard offline approaches (Li et al., 2015; Ren et al., 2021; Li et al., 2023). Analogous to the online setting, an emergent objective in offline RL involves the development of efficacious hierarchy methodologies that can obtain temporally extended lower-level primitives, subsequently facilitating the construction of a higher-level policy operating at a more abstract temporal scale (Ajay et al., 2021; Pertsch et al., 2021; Villegas et al., 2022; Rosete-Beas et al., 2022; Rao et al., 2022; Yang et al., 2023). Within the hierarchical framework, current offline RL approaches can be broadly categorized into model-free and model-based. The former conceptualizes the higher-level policy optimization as a auxiliary offline RL issue (Liu et al., 2020; Liu & Sun, 2022; Ma et al., 2022; Kipf et al., 2019; Ajay et al., 2021; Rosete-Beas et al., 2022). In contrast, the latter encompasses planning in the higher-level policy space by generating future trajectories through a dynamics model of the environment, either predefined or learned (Li et al., 2022; Co-Reyes et al., 2018; Lynch et al., 2020; Lee et al., 2022; Venkatraman, 2023). Concerning lower-level primitive learning, these two methods exhibit similarities and are typically modeled as goal-conditioned or skill-based imitation learning or offline RL problems. Conversely, the instabilities arising from offline hierarchical RL methodologies due to the “deadly triad” (Sutton & Barto, 2018; Van Hasselt et al., 2018), restricted data access (Fujimoto et al., 2019; Kumar et al., 2020), and sparse rewards (Andrychowicz et al., 2017; Ma et al., 2022) remain unaddressed. This spawns another subset of model-based approaches along with more effective hierarchical variants that endeavor to resolve problems from a sequence modeling viewpoint Chen et al. (2021); Janner et al. (2021; 2022); Ajay et al. (2023). Irrespective of whether a method is model-free or model-based, it adheres to the traditional settings, wherein planning occurs in the raw action space of the Markov Decision Process (MDP). Although seemingly intuitive, planning in raw action space can be inefficient and inflexible (Wang et al., 2020; Yang et al., 2021; Jiang et al., 2023). Challenges include ensuring model accuracy across the entire space and the constraint of being tied to the environment’s temporal structure. Conversely, human planning offers enhanced flexibility through temporal abstractions, high-level actions, backward planning, and incremental refinement. Drawing motivation from TAP (Jiang et al., 2023), we put forth the notion of the latent action. Planning within the domain of latent actions delivers a twofold advantage compared to planning with raw actions. Primarily, it encompasses only plausible actions under behavior policy support, yielding a reduced space despite the raw action space’s dimensionality and preventing the exploitation of model frailties. Secondarily, it permits the separation of the temporal structure between planning and modeling, thus enabling a more adaptable and efficient planning process unconstrained by specific transitions. These dual benefits render latent-action-based approaches naturally superior to extant methodologies when handling temporally extended offline tasks. Nevertheless, two shortcomings of TAP inhibit its ability to serve as a general and practical framework. Initially, TAP is confined to discrete latent action spaces. In real-world contexts, agents are likely to carry out a narrow, discrete assortment of tasks and a broader spectrum of behaviors (Co-Reyes et al., 2018). This introduces a predicament — should a minor skill modification be necessary, such as opening a drawer by seizing the handle from top to bottom instead of bottom to top, a completely novel set of demonstrations or reward functions might be mandated for behavior acquisition. Subsequently, once the latent action space has been ascertained, TAP necessitates a distinct, resource-intensive planning phase for generating reward-maximizing policies. The price of planning consequently restricts latent actions to discrete domains. To tackle these limitations, this paper proposes a novel framework, LatentDiffuser, by concurrently modeling continuous latent action space representation learning and latent action-based planning as a conditional generative problem within the latent domain. Specifically, LatentDiffuser employs unsupervised techniques to discern the latent action space by utilizing score-based diffusion models (SDMs) (Song et al., 2021; Nichol & Dhariwal, 2021; Ho & Salimans, 2022) within the latent sphere in conjunction with a variational autoencoder (VAE) framework (Kingma & Welling, 2014; Rezende et al., 2014; Vahdat et al., 2021). We first segment the input trajectories, map each slice to latent action space (which needs to be learned), and apply the SDM to the latent sequence. Subsequently, the SDM is entrusted with approximating the distribution over the offline trajectory embeddings, conditioned on the related return values. Planning—or reward-maximizing trajectory synthesis—is realized by initially producing latent actions through sampling from a simple base distribution, followed by iterative, conditional denoising, and eventually translating latent actions into the trajectory space using a decoder. In other words, LatentDiffuser can be regarded as a VAE equipped with an SDM prior (Vahdat et al., 2021). Theoretically, we demonstrate that planning in the domain of latent actions is tantamount to energy-guided sampling using a pre-trained diffusion behavior model. Exact energy-guided sampling is essential to carry out high-quality and efficient planning. To achieve this objective, we modify QGPO (Lu et al., 2023) to realize exact sampling at the sequence level. Comprehensive numerical results on low-dimensional locomotion control tasks reveal that LatentDiffuser exhibits competitive performance against robust baselines and outperforms them on tasks of greater dimensionality. Our main contributions encompass: 1) Developing a unified framework for continuous latent action space representation learning and planning that delivers flexibility and efficiency in temporally extended offline decision-making. 2) Our theoretical derivation confirms the equivalence between planning in the latent action space and energy-guided sampling with a pretrained diffusion model. It introduces an innovative sequence-level exact sampling technique. 3) Numerical experiments exhibit the competitive performance of LatentDiffuser and its applicability across a range of low- and high-dimensional continuous control tasks. 2 RELATED WORK Owing to spatial constraints, this section will briefly present the most pertinent domain of LatentDiffuser: offline RL or imitation learning (IL) based on a hierarchical structure. In terms of algorithmic specificity, existing techniques can be broadly classified into goal-based and skill-based methods (Pateria et al., 2021). For further related literature, including but not limited to model-based RL, action representation learning, offline RL, and RL as sequence modeling, kindly refer to Appendix C and the appropriate citations within the papers. Goal-based approaches primarily concentrate on attaining a designated state. The vital aspect of such techniques concerns the selection or creation of subgoals, which reside in the raw state space. Once the higher-level subgoal is ascertained, the lower-level policy is generally acquired through standard IL methods or offline RL based on subgoal-augmented/conditioned policy, a universal value function (Schaal et al., 2015), or their combination. In extant methods, the subgoal is either predefined (Zhou et al., 2019; Xie et al., 2021; Ma et al., 2021), chosen based on heuristics (Ding et al., 2014; Guo & Zhai, 2016; Pateria et al., 2020; Mandlekar et al., 2020), or generated via planning or an additional offline RL technique (Liu et al., 2020; Liu & Sun, 2022; Li et al., 2022; Ma et al., 2022). Moreover, some methods (Eysenbach et al., 2019; Paul et al., 2019; Lai et al., 2020; Kujanpää et al., 2023) are solely offline during the subgoal selection or generation process. This paper also pertains to the options framework (Sutton et al., 1999; Stolle & Precup, 2002; Bacon et al., 2017; Wulfmeier et al., 2021; Salter et al., 2022; Villecroze et al., 2022)), as both the (continuous) latent actions of LatentDiffuser and (discrete) options introduce a mechanism for temporal abstraction. Skill-based methods embody higher-level skills as low-dimensional latent codes. In this context, a skill signifies a subtask’s policy, semantically representing “the capability to perform something adeptly” (Pateria et al., 2021). Analogous to goal-based approaches, once the higher-level skill is identified, the lower-level skill-conditioned policy is generally acquired through standard IL or offline RL methods. More precisely, few works utilize predefined skills (Nasiriany et al., 2022; Fatemi et al., 2022). The majority of studies employ a two- or multi-phase training framework: initially, state sequences are projected into continuous latent variables (i.e., skills) via unsupervised learning; next, optimal skills are generated based on offline RL (Kipf et al., 2019; Pertsch et al., 2021; Ajay et al., 2021; Rosete-Beas et al., 2022; Lee et al., 2022; Venkatraman, 2023) or planning1 (Co-Reyes et al., 2018; Lynch et al., 2020; Lee et al., 2022; Venkatraman, 2023) in the skill space. ![Figure 1](image.png) Figure 1: The physical meaning of the goal-conditioned policy, skill and latent action (corresponding to 2 timesteps in the raw MDP). The red diamond represents a particular (goal) state, the gray, dotted diamond is a placeholder, and the red circle denotes any state. In contrast with the aforementioned hierarchical methodologies, LatentDiffuser initially learns a more compact latent action space and subsequently employs the latent actions to make decisions. As demonstrated in Figure 1, latent action not only differs from the goal-conditioned policy, which pertains to the trajectory of reaching a particular state, but also from the skill, which relates to the trajectory of completing a specific (multi-step) state transition. The latent action also corresponds to the agent’s received reward and the subsequent expected return. The unique physical implications of latent action and the methodology utilized by LatentDiffuser render the proposed method advantageous in several ways. 1) The future information in the latent action allows the algorithm to execute more efficient planning. 2) Unlike existing works wherein multiple optimization objectives and the fully coupling or separating of representation learning and decision making (RL or planning) lead to intricate training processes and reduced training efficiency, LatentDiffuser exhibits end-to-end training and unifies representation learning, sampling, and planning. 3 Problem Formulation In this paper, we approach the offline RL problem as a sequence modeling task, in alignment with previous work (Janner et al., 2022; Ajay et al., 2023; Li et al., 2023). The following subsection delineates the specificities of sequence modeling, or more accurately, the conditional generative modeling paradigm. We examine a trajectory, $\tau$, of length $T$, which is sampled from a MDP that features a --- 1It is important to note that planning is only feasible when the environment model is known or can be sampled from the environment model. Consequently, some of these works focus on online RL tasks, while others first learn an additional environment model from the offline dataset and then plan in the skill space. fixed stochastic behavior policy. This trajectory comprises (refer to Appendix for more modeling selections of $\tau$) a series of states, actions, rewards, and reward-to-go values, $G_t := \sum_{i=t}^{\infty} \gamma^{t-i} r_i$, as proxies for future cumulative rewards: $\tau := (s_1, a_1, r_1, G_1, s_2, a_2, r_2, G_2, \ldots, s_T, a_T, r_T, G_T)$. It is crucial to note that the definition of $\tau$ diverges from that in prior studies (Janner et al., 2022; Ajay et al., 2023; Li et al., 2023), as each timestep now contains both the reward and reward-to-go values. This modification has been specifically engineered to facilitate the subsequent learning of latent action spaces. Sequential decision-making is subsequently formulated as the standard problem of conditional generative modeling: $$\max_\theta \mathbb{E}_{\tau \sim D} [\log p_\theta (\tau_0 \| y(\tau_0))],$$ where $\tau_0 := \tau$. The objective is to estimate the conditional trajectory distribution using $p_\theta$ to enable planning or generating the desired trajectory $\tau_0$ based on the information $y(\tau_k)$. Existing instances of $y$ may encompass the return (Janner et al., 2022; Li et al., 2023), the constraints met by the trajectory (Ajay et al., 2023; Li et al., 2023), or the skill demonstrated in the trajectory (Ajay et al., 2023). The generative model is constructed in accordance with the conditional diffusion process: $$q(\tau_{k+1} \| \tau_k), \quad p_\theta(\tau_{k-1} \| \tau_k, y(\tau_0)).$$ As per standard convention, $q$ signifies the forward noising process while $p_\theta$ represents the reverse denoising process (Ajay et al., 2023). Latent Actions We introduce the concept of latent action (Figure 1) proposed in TAP (Jiang et al., 2023). TAP specifically models the optimal conditional trajectory distribution $p^(\tau \| s_1, z)$ using a series of latent variables, $z := (z_1, \ldots, z_M)$. Assuming that the state and latent variables $(s_1, z)$ can be deterministically mapped to trajectory $\tau$, $p^(\tau \| s_1, z) := p(s_1)1(\tau = h(s_1, z))\pi^(z \| s_1)$ is obtained. The terms $z$ and $\pi^(z \| s_1)$ are subsequently referred to as the latent actions and the optimal latent policy, respectively. In a deterministic MDP, the trajectory corresponding to an arbitrary function $h(s_1, z)$ with $\pi^(z \| s_1) > 0$ will constitute an optimal executable plan, implying that the optimal trajectory can be recovered by following the latent actions $z$, beginning from the initial state $s_1$. Consequently, planning within the latent action space $Z$ facilitates the discovery of an desired, optimal trajectory. TAP, however, remain restricted to discrete latent action spaces and necessitate independent, resource-intensive planning. Motivated by these limitations, we present a unified framework that integrates representation learning and planning for continuous latent action via latent, score-based diffusion models. ### 4 Algorithm Framework This section provides a comprehensive elaboration of the model components and design choices, such as the network architecture, loss functions, as well as the details of training and planning. By unifying the representation learning and planning of latent action through the incorporation of a latent diffusion model and the exact energy-guided sampling technique, LatentDiffuser achieves effective decision-making capabilities for temporally-extended, sparse reward tasks. Specifically, we first explore the representation learning for latent action in Section 4, followed by a detailed discussion on planning using energy-guided sampling in Section 4.2, and provide a algorithm summary in Section 4.3 to close this section. ![Figure 2: Representation learning for latent action with the latent score-based diffusion model.](image-url) 4.1 Representation Learning for Latent Action The latent action space allows for a more compact, efficient, and adaptable method by effectively capturing behavior policy support and detaching the temporal structure, thus providing innate benefits in handling temporally extended offline tasks. As indicated in Section 3, before proceeding to planning, we must first learn a continuous latent action space. For this purpose, we propose the LatentDiffuser based on a latent diffusion model (LDM) (Vahdat et al., 2021), as depicted in Figure 2. LatentDiffuser is constituted by an encoder $ q_\phi(z_0 \| s_1, \tau) $, a score-based prior $ p_\theta(z_0 \| s_1) $, and a decoder $ p_\psi(\tau \| s_1, z_0) $. In accordance with Vahdat et al. (2021), we train LatentDiffuser by minimizing the variational upper bound on the negative trajectory log-likelihood $ \log p(\tau \| s_1) $, meaning that the information $ y(\tau) $ in Equation (1) is instantiated as the initial state $ s_1 $: \[ \mathcal{L}(s_1, \tau, \phi, \theta, \psi) = \mathbb{E}_{q_\phi(z_0 \| s_1, \tau)} \left[ -\log p_\psi(\tau \| s_1, z_0) \right] + \text{KL} \left( q_\phi(z_0 \| s_1, \tau) \\| p_\theta(z_0 \| s_1) \right) \] \[ = \mathbb{E}_{q_\phi(z_0 \| s_1, \tau)} \left[ -\log p_\psi(\tau \| s_1, z_0) \right] + \mathbb{E}_{q_\phi(z_0 \| s_1, \tau)} \left[ \log q_\phi(z_0 \| s_1, \tau) \right] \] \[ + \mathbb{E}_{q_\phi(z_0 \| s_1, \tau)} \left[ -\log p_\theta(z_0 \| s_1) \right] \] utilizing a VAE approach (Kingma & Welling, 2014; Rezende et al., 2014), wherein the $ q_\phi(z_0 \| s_0, \tau) $ approximates the true posterior $ p(z_0 \| s_0, \tau) $. This paper employs Equation (3) with a decomposed KL divergence into entropy and cross entropy terms. The reconstruction and entropy terms are easily estimated for any explicit encoder as long as the reparameterization trick is applicable (Kingma & Welling, 2014). The challenging aspect of training LatentDiffuser pertains to training the cross entropy term, which involves the score-based prior. Unlike Vahdat et al. (2021), which addresses this challenge by simultaneously learning an encoder/decoder architecture alongside a score-based prior, we adopt a simpler yet efficacious approach (Rombach et al., 2022) by training a VAE $ \{q_\phi, p_\psi\} $ and a score-based diffusion model $ \{q_\theta\} $ consecutively based on the offline dataset $ D_\tau $. This does not necessitate a delicate balancing of reconstruction and generative capabilities. Encoder $ q_\phi $ and Decoder $ p_\psi $ We use the almost consistent encoder design with TAP (Jiang et al., 2023). Specifically, we handle $ x_t := (s_t, a_t, r_t, G_t) $ as a single token. The encoder $ \phi $ processes token $ x_t $ using a GPT-2 style Transformer$^2$, yielding $ T $ feature vectors, where $ T $ is the episode horizon. Subsequently, we apply a 1-dimensional max pooling with a kernel size and stride of $ L $, followed by a linear layer, and generate $ T/L $ latent actions. Moreover, different from the TAP Decoder architecture, we use a modular design idea. More concretely, each latent action is tiled $ L $ times to match the number of input/output tokens $ T $. We then concatenate the initial state $ s_1 $ and the latent action, and apply a linear projection to provide state information to the decoder. After adding positional embedding, the decoder reconstructs the trajectory $ \hat{\tau} := (\hat{x}_1, \hat{x}_2, \ldots, \hat{x}_T) $, with $ \hat{x}_t := (\hat{s}_t, \hat{a}_t, \hat{r}_t, \hat{G}_t) $. To enhance the decoder’s representation ability, we design the decoder modularly for different elements in $ x_t $, as shown in Figure 2. Noting that the action decoder is designed based on the inverse dynamics model (Agrawal et al., 2015; Pathak et al., 2017) in a manner similar to (Ajay et al., 2023; Li et al., 2023), with the aim of generating raw action sequences founded on the state sequences. The training of the encoder and decoders finally entails the use of a reconstruction loss computed as the mean squared error between input trajectories $ \{\tau\} $ and reconstructed trajectories $ \{\hat{\tau}\} $, coupled with a low-weighted ($ \approx 10^{-6} $) Kullback-Leibler penalty towards a standard normal on the learned latent actions, akin to VAE approaches (Kingma & Welling, 2014; Rezende et al., 2014). This prevents the arbitrary scaling of latent action space. Score-based Prior $ \theta $ Having trained the VAE $ \{q_\phi, p_\psi\} $, we now have access to a compact latent action space. Distinct from VAE’s adoption of a uniform prior or TAP’s utilization of an autoregressive, parameterized prior over latent actions, LatentDiffuser employs a score-based one. Thus, by harnessing the “diffusion-sampling-as-planning” framework, we seamlessly transform planning into conditional diffusion sampling, ultimately circumventing the need for an independent, costly planning stage. Concretely, the score-based prior is modeled as a conditional, score-based diffusion probabilistic model, which is parameterized using a temporal U-Net architecture (Janner et al., 2022; Ajay et al., 2023). This architecture effectively treats a sequence of noised latent action $ x_k(z) $ as an image, where the height represents a single latent action’s dimension and the width signifies --- $^2$Different from the casual Transformer used in TAP, see Appendix for more discussion. the number of the latent actions. Conditioning information $ y(z) := s_1 $ is then projected using a multi-layer perceptron (MLP). The training of the score-based prior is formulated as a standard score-matching problem detailed in Appendix B.2. ### 4.2 Planning with Energy-Guided Sampling Upon acquiring the latent action space, we are able to effectively address temporally-extended offline tasks using planning. Intriguingly, when examined from a probabilistic standpoint, the optimal latent action sequence sampling coincides with a guided diffusion sampling problem (Lu et al., 2023), wherein the guidance is shaped by an (unnormalized) energy function. By adopting a “diffusion-sampling-as-planning” framework (Janner et al., 2022), we can perform planning through conditional sampling using the pretrained LatentDiffuser, without necessitating further costly planning steps (Janner et al., 2021; Jiang et al., 2023). This renders LatentDiffuser a holistic framework that seamlessly consolidates representation learning and planning within the latent action space. In the subsequent sections, the equivalence between optimal latent actions sampling and energy-guided diffusion sampling is demonstrated, followed by the introduction of a practical sampling algorithm to facilitate efficient planning. Planning is Energy-Guided Diffusion Sampling* Considering a deterministic mapping from $ \tau $ to $ z $, achieved by the learned encoder $ q_\phi $, the following theorem (refer to Appendix I.1 for the proof) is derived for the optimal latent policy defined in Section 3: Theorem 1 (Optimal latent policy). Given an initial state $ s_1 $, the optimal latent policy satisfies: $ \pi^(z \| s_1) \propto \mu(z \| s_1) e^{\beta \sum_{t=1}^{T} Q_\zeta(s_t, a_t)} $, wherein $ \mu(z \| s_1) $ represents the behavior latent policy and $ Q_\zeta(\cdot, \cdot) $ refers to the estimated Q-value function. $ \beta \geq 0 $ signifies the inverse temperature controlling the energy strength. By rewriting $ p_0 := \pi^, q_0 = \mu $ and $ z_0 = z $, we can reformulate the optimal planning into the following diffusion sampling problem: \[ p_0(z_0 \| s_1) \propto q_0(z_0 \| s_1) \exp(-\beta E(h(z_0, s_1))), \] where $ E(h(z_0, s_1)) := -\sum_{t=1}^{T} Q_\zeta(s_t, a_t) $ and $ h(z_0, s_1) $ denotes the pretrained decoder $ p_\psi $. The behavior latent policy $ q_0(z_0 \| s_1) $ is modeled by the pretrained LatentDiffuser. We then adopt the “diffusion-sampling-as-planning” to generate desired (e.g., reward-maximizing) latent actions $ z_0 $. Concretely, we employ $ q_0 := q, p_0 = p $ at diffusion timestep $ k = 0 $. Then a forward diffusion process is constructed to simultaneously diffuse $ q_0 $ and $ p_0 $ into an identical noise distribution, where $ p_{k\|0}(z_k \| z_0, s_1) := q_{k\|0}(z_k \| z_0, s_1) = N(z_k \| \alpha_k z_0, \sigma_k^2 I) $. Based on (Lu et al., 2023, Theorem 3.1), the marginal distribution $ q_k $ and $ p_k $ of the noised latent actions $ z_k $ at the diffusion timestep $ k $ adhere to: \[ p_k(z_k \| s_1) \propto q_k(z_k \| s_1) \exp(E_k(h(z_k, s_1))), \] where $ E_k(h(z_k, s_1)) $ is $ \beta E(h(z_0, s_1)) $ when $ k = 0 $ and $ -\log E_{q_{0:k}(z_0 \| s_1)}[\exp(-\beta E(h(z_0, s_1)))] $ when $ k > 0 $. We then need to estimate the score function of $ p_k(z_k \| s_1) $. Quoting the derivation of Lu et al. (2023), the score function satisfies: $ \nabla_{z_k} \log p_k(z_k \| s_1) = \nabla_{z_k} \log q_k(z_k \| s_1) + \nabla_{z_k} E_k(h(z_k, s_1)) $. Consequently, the optimal planning has been formulated as energy-guided sampling within the latent action space, with $ \nabla_{z_k} E(h(z_k, s_1)) $ as the desired guidance. Practical Sampling Method Estimating the target score function $ \nabla_{z_k} \log p_k(z_k \| s_1) $ is non-trivial because of the intractable energy guidance $ \nabla_{z_k} E(h(z_k, s_1)) $. We borrow the energy-guided sampling method proposed in (Lu et al., 2023) and propose a sequence-level, exact sampling methods by training a total of three neural networks: (1) a diffusion model to model the behavior latent policy $ q_0(z_0 \| s_1) $; (2) a state-action value function $ Q_\zeta(s, a) $ to define the intermediate energy function $ E(h(z_0, s_1)) $; and (3) an time-dependent energy model $ f_\eta(z_k, s_1, k) $ to estimate $ E_k(h(z_k, s_1)) $ and guide the diffusion sampling process. Recall that we already have (1) a diffusion model, i.e., the score-based prior $ p_\theta(z_0 \| s_1) $ and (2) a state-action value function $ Q_\zeta(s, a) $, i.e., the return decoder. According to Lu et al. (2023, Theorem 3.2), the only remained time-dependent energy model, $ f_\eta(z_k, s_1, k) $, can be trained by minimizing the following contrastive loss: $$\min_{\eta} \mathbb{E}_{p(k,s_1)} \mathbb{E}_{\prod_{i=1}^{M} q(z^{(i)}_0 \| s_1) p(\epsilon^{(i)})} \left[ - \sum_{i=1}^{M} \frac{e^{-\beta E_0(h(z^{(i)}_0, s_1))}}{\sum_{j=1}^{M} e^{-\beta E_0(h(z^{(j)}_0, s_1))}} \log \frac{e^{f_\eta(z^{(i)}_k, s_1, k)}}{\sum_{j=1}^{M} e^{f_\eta(z^{(j)}_k, s_1, k)}} \right],$$ where $k \sim \mathcal{U}(0, K)$, $z_k = \alpha_k z_0 + \sigma_k \epsilon$, and $\epsilon \sim \mathcal{N}(0, I)$. To estimate true latent actions distribution $q(z_0 \| s_1)$ in Equation 6, we utilize the pretrained encoder $q_\phi$ and score-based prior $p_\theta$ to generate $M$ support latent actions $\{z^{(i)}_0\}_M$ for each initial state $s_1$ by diffusion sampling. The contrastive loss in Equation (6) is then estimated by: $$\min_{\eta} \mathbb{E}_{k, s_1, \epsilon} - \sum_{i=1}^{M} \frac{e^{-\beta E_0(h(z^{(i)}_0, s_1))}}{\sum_{j=1}^{M} e^{-\beta E_0(h(z^{(j)}_0, s_1))}} \log \frac{e^{f_\eta(z^{(i)}_k, s_1, k)}}{\sum_{j=1}^{M} e^{f_\eta(z^{(j)}_k, s_1, k)}},$$ where $z^{(i)}_0, z^{(j)}_0$ correspond to the support latent actions for each initial state $s_1$. ### 4.3 Algorithm Summary In general, the training phase of LatentDiffuse is composed of three parts, corresponding to the training of encoder and decoders $\{q_\phi, p_\psi\}$, score-based prior $p_\theta$, and intermediated energy model $f_\eta$, as shown in Algorithm 1. Throughout the training process, it is imperative to employ two distinct datasets: the first being a standard offline RL dataset, $\mathcal{D}$, which encompasses trajectories sampled from behavior policies, whereas the second dataset consists of support latent actions for each initial state $s_1 \in \mathcal{D}$, generated by the pre-trained VAE, i.e., the encoder, score-based prior and decoders. Algorithm 1 LatentDiffuser: Efficient Planning with Latent Diffusion - Initialize the latent diffusion model, i.e., the encoder $q_\phi$, the score-based prior $p_\theta$ and the decoder $p_\psi$; the intermediate energy model $f_\eta$ - for each gradient step do - Sample $B_1$ trajectories $\tau$ from offline dataset $\mathcal{D}$ - Generate reconstructed trajectories $\hat{\tau}$ with the encoder $q_\phi$ and decoder $p_\psi$ - Update $\{\phi, \psi\}$ based on the standard VAE loss - end for - for each gradient step do - Sample $B_2$ trajectories $\tau$ from offline dataset $\mathcal{D}$ - Sample $B_2$ Gaussian noises $\epsilon$ from $\mathcal{N}(0, I)$ and $B_2$ time $k$ from $\mathcal{U}(0, K)$ - Generate latent actions $z_0$ with the pretrained encoder $q_\phi$ and decoder $p_\psi$ - Perturb $z_0$ according to $z_k := \alpha_k z_0 + \sigma_k \epsilon$ - Update $\{\theta\}$ with the standard score-matching loss in Appendix B.2 - end for - for each initial state $s_1$ in offline dataset $\mathcal{D}$ do - Sample $M$ support latent actions $\{z^{(i)}_0\}_M$ from the pretrained score-based prior $p_\theta$ - end for - for each gradient step do - Sample $B_3$ initial state $s_1$ from offline dataset $\mathcal{D}$ - Sample $B_3$ Gaussian noises $\epsilon$ from $\mathcal{N}(0, I)$ and $B_3$ time $k$ from $\mathcal{U}(0, K)$ - Retrieve support latent actions $\{z^{(i)}_0\}_M$ for each $s_1$ - Perturb $z^{(i)}_0$ according to $z^{(i)}_k := \alpha_k z^{(i)}_0 + \sigma_k \epsilon$ - Update $\{\eta\}$ based on the contrastive loss in Equation (7) - end for Moreover, the optimal planning is tantamount to conducting conditional diffusion sampling based on the score-based prior and the intermediate energy model. Formally, the generation employs reverse denoising process at each diffusion timestep $k$ by utilizing the score function $\nabla_{z_k} \log p_k(z_k \| s_1)$ based on the score function of the score-based prior $\nabla_{z_k} \log q_k(z_k \| s_1)$ and intermediate energy model $\nabla_{z_k} E_k(h(z_k, s_1))$, along with the state and action decoder $p_\psi(\tau \| s_1, z_0)$ to map the sampled latent actions $z_0$ back to the original trajectory space. Explicitly, the generative process is \[ p(s_1, z_0, \tau) = p_0(z_0 \| s_1) p_\psi(\tau \| s_1, z_0). \] To avoid the accumulation of errors during sampling, we adopt the receding horizon control used in the existing methods (Ajay et al., 2023; Li et al., 2023). ## 5 EXPERIMENTS This section aims to assess the efficacy of the LatentDiffuser for extended temporal offline tasks in comparison to current SOTA offline RL methods, which integrate hierarchical structures, and conditional generation models. The empirical evaluation encompasses three task categories derived from D4RL (Fu et al., 2020): namely, Gym locomotion control, Adroit, and AntMaze. Gym locomotion tasks function as a proof-of-concept in the lower-dimensional realm, in order to ascertain whether LatentDiffuser is capable of accurately reconstructing trajectories for decision-making and control purposes. Subsequently, LatentDiffuser is evaluated on Adroit—a task with significant state and action dimensionality—as well as on LatentDiffuser within the AntMaze environment, which represents a sparse-reward continuous-control challenge in a series of extensive long-horizon maps (Li et al., 2023). The subsequent sections will describe and examine the performance of these tasks and their respective baselines individually. Scores within 5% of the maximum per task will be emphasized in bold (Kostrikov et al., 2022). ### 5.1 PROOF-OF-CONCEPT: GYM LOCOMOTION CONTROL Baselines Initially, an outline of the baselines is provided: CQL (Kumar et al., 2020), IQL (Kostrikov et al., 2022), D-QL (Wang et al., 2023), and QGPO (Lu et al., 2023) are all model-free offline RL methods. MoReL (Kidambi et al., 2020) is a model-based offline RL method. DT (Chen et al., 2021), TT (Janner et al., 2021), Diffuser (Janner et al., 2022), and DD (Ajay et al., 2023) address offline RL tasks via conditional generative modeling. Finally, TAP (Jiang et al., 2023) and HDMI (Li et al., 2023) employ a hierarchical framework grounded in generative modeling. Due to spatial constraints, only algorithms with the highest performance rankings are displayed herein; for a comprehensive comparison, please refer to the appendix. Table 1: The performance in Gym locomotion control in terms of normalized average returns. Results correspond to the mean and standard error over 5 planning seeds. \| Dataset \| Environment \| CQL \| TT \| DD \| D-QL \| TAP \| QGPO \| HDMI \| LD \| \|---------------\|--------------\|---------\|---------\|---------\|---------\|---------\|---------\|---------\|---------\| \| Med-Expert \| HalfCheetah \| 91.6 \| 95 \| 90.6±1.3\| 96.8±0.3\| 91.8 ± 0.8\| 93.5±0.3\| 92.1±1.4\| 95.2±0.2\| \| Med-Expert \| Hopper \| 105.4 \| 110.0 \| 111.8±1.8\| 111.1±1.3\| 105.5 ± 1.7\| 108.0±2.5\| 113.5±0.9\| 112.9±0.3\| \| Med-Expert \| Walker2d \| 108.8 \| 101.9 \| 108.8±1.7\| 110.1±0.3\| 107.4 ± 0.9\| 110.7 ± 0.6\| 107.9±1.2\| 111.3±0.2\| \| Medium \| HalfCheetah \| 44.0 \| 46.9 \| 49.1±1.0\| 51.1±0.5\| 45.0 ± 0.1\| 54.1 ± 0.4\| 48.0±0.9\| 53.6±0.4\| \| Medium \| Hopper \| 58.5 \| 61.1 \| 79.3±3.6\| 90.5±4.6\| 63.4 ± 1.4\| 98.0 ± 2.6\| 76.4±2.6\| 98.5±0.7\| \| Medium \| Walker2d \| 72.5 \| 79 \| 82.5±1.4\| 87.0±0.9\| 64.9 ± 2.1\| 86.0 ± 0.7\| 79.9±1.8\| 86.3±0.9\| \| Med-Replay \| HalfCheetah \| 45.5 \| 41.9 \| 39.3±4.1\| 47.8±0.3\| 40.8 ± 0.6\| 47.6 ± 1.4\| 44.9±2.0\| 47.3±1.2\| \| Med-Replay \| Hopper \| 95 \| 91 \| 100±0.7\| 101.3±0.6\| 87.3 ± 2.3\| 96.9 ± 2.6\| 99.6±1.5\| 100.4±0.5\| \| Med-Replay \| Walker2d \| 77.2 \| 82.6 \| 75±4.3\| 95.5±1.5\| 66.8 ± 3.1\| 84.4 ± 4.1\| 80.7±2.1\| 82.6 ± 2.1\| \| Average \| \| 77.6 \| 78.9 \| 81.8 \| 88.0 \| 82.5 \| 86.6 \| 82.6 \| 87.5 \| Table 1 shows that LatentDiffuser surpasses specifically designed offline RL methods in the majority of tasks. Furthermore, the performance discrepancy between LatentDiffuser and two-stage algorithms, such as TAP and HDMI, underscores the benefits provided by the proposed framework, which unifies learning of latent action space representation and planning. ### 5.2 HIGH-DIMENSIONAL MDP: ADROIT Baselines Taking into account the large dimensions characterizing the Adroit task actions, only baselines that perform well in the previous task are evaluated. Additionally, D-QL necessitates 50 repeated samplings by default for action generation (Wang et al., 2023). This requirement would result in a substantial training overhead for high-dimensional action tasks. Consequently, to ensure a fair comparison, D-QL is configured to allow only 1 sampling, akin to QGPO (Lu et al., 2023). Table 2 demonstrates the advantages of LatentDiffuser become even more pronounced in high-dimensional tasks. Furthermore, a marked decrease in sequence modeling method performance is Table 2: Adroit results. These tasks have high action dimensionality (24 degrees of freedom) \| Dataset \| Environment \| CQL \| TT \| DD \| D-DL@1 \| TAP \| QGPO \| HDMI \| LD \| \|-----------\|-------------\|-----\|----\|-----\|--------\|-----\|------\|------\|----\| \| Human \| Pen \| 37.5\| 36.4\| 64.1±9.0\| 66.0±8.3\| 76.5±8.5\| 73.9±8.6\| 66.2±8.8\| 79.0±8.1\| \| Human \| Hammer \| 4.4 \| 0.8 \| 1.0±0.1\| 1.3±0.1\| 1.4±0.1\| 1.4±0.1\| 1.2±0.1\| 4.6±0.1\| \| Human \| Door \| 9.9 \| 0.1 \| 6.9±1.2\| 8.0±1.2\| 8.8±1.1\| 8.5±1.2\| 7.1±1.1\| 9.8±1.0\| \| Human \| Relocate \| 0.2 \| 0.0 \| 0.2±0.1\| 0.2±0.1\| 0.2±0.1\| 0.2±0.1\| 0.1±0.1\| 0.2±0.1\| \| Cloned \| Pen \| 39.2\| 11.4\| 47.7±9.2\| 49.3±8.0\| 57.4±8.7\| 54.2±9.0\| 48.3±8.9\| 60.7±9.1\| \| Cloned \| Hammer \| 2.1 \| 0.5 \| 0.9±0.1\| 1.1±0.1\| 1.2±0.1\| 1.1±0.1\| 1.0±0.1\| 4.2±0.1\| \| Cloned \| Door \| 0.4 \| -0.1\| 9.0±1.6\| 10.6±1.7\| 11.7±1.5\| 11.2±1.4\| 9.3±1.6\| 12.0±1.6\| \| Cloned \| Relocate \| -0.1\| -0.1\| -0.2±0.0\| -0.2±0.0\| -0.2±0.0\| -0.2±0.0\| -0.1±0.0\| -0.1±0.0\| \| Expert \| Pen \| 107.0\| 72.0\| 107.6±7.6\| 112.6±8.1\| 127.4±7.7\| 119.1±8.1\| 109.5±8.0\| 131.2±7.3\| \| Expert \| Hammer \| 86.7\| 15.5\| 106.7±1.8\| 114.8±1.7\| 127.6±1.7\| 123.2±1.8\| 111.8±1.7\| 132.5±1.8\| \| Expert \| Door \| 101.5\| 94.1\| 87.0±0.8\| 93.7±0.8\| 104.8±0.8\| 98.8±0.8\| 85.9±0.9\| 111.9±0.8\| \| Expert \| Relocate \| 95.0\| 10.3\| 87.5±2.8\| 95.2±2.8\| 105.8±2.7\| 102.5±2.8\| 91.3±2.6\| 109.5±2.8\| \| Average (w/o expert) \| \| 11.7\| 6.1\| 16.2\| 17.1\| 19.6\| 18.79\| 16.6\| 21.3\| \| Average (w/ expert) \| \| 40.3\| 20.1\| 43.2\| 46.1\| 51.9\| 49.5\| 44.3\| 54.6\| observed. Two primary factors are identified: first, larger action dimensions necessitate tokenization- and autoregression-based techniques (such as TT) to process increasingly lengthy sequences; second, DD and HDMI employ an inverse dynamic model to generate actions independently, while the expansion in action dimension renders the model fitting process more challenging. 5.3 Long-horizon Continuous Control: AntMaze Baselines To validate the benefits of latent actions in longer-horizon tasks, an additional comparison is made with hierarchical offline RL methods designed explicitly for long-horizon tasks: CompILE (Kipf et al., 2019), GoFAR (Ma et al., 2022), and HiGoC (Li et al., 2022). Concurrently, CQL and TT are removed due to their inability to perform well in high-dimensional Adroit. Table 3: AntMaze performance correspond to the mean and standard error over 5 planning seeds. \| Environment \| CompILE \| GoFAR \| HiGoC \| DD \| D-DL@1 \| TAP \| QGPO \| HDMI \| LD \| \|----------------------\|---------\|-------\|-------\|----\|--------\|-----\|------\|------\|----\| \| AntMaze-Play U-Maze-3\| 41.2±3.6\| 38.5±2.2\| 31.2±3.2\| 73.1±2.5\| 52.9±4.1\| 82.2±2.1\| 59.3±1.3\| 86.1±2.4\| 85.4±1.9\| \| AntMaze-Diverse U-Maze-3\| 23.5±1.8\| 25.1±3.1\| 25.5±1.6\| 49.2±3.1\| 32.5±5.9\| 69.8±0.5\| 38.5±2.6\| 73.7±1.1\| 75.6±2.1\| \| AntMaze-Diverse Large-2\| - \| - \| - \| - \| 46.8±4.4\| - \| 69.2±3.2\| - \| 71.5±3.5\| 75.8±2.0\| \| Single-task Average \| 32.4\| 31.8\| 28.4\| 50.4\| 39.0\| 73.7\| 45.4\| 77.1\| 78.9\| \| Multi-Aut-Diverse Large-2\| - \| - \| - \| - \| 45.2±4.9\| - \| 71.6±3.3\| - \| 73.6±3.8\| 73.3±2.6\| \| Multi-task Average \| - \| - \| - \| - \| 45.2\| 71.6\| - \| 73.6\| 73.3\| Table 3 highlights that sequence modeling-based hierarchical methods significantly surpass RL-based approaches. Moreover, LatentDiffuser demonstrates performance comparable to two-stage techniques such as TAP and HDMI through end-to-end training. 6 Conclusions In this work, we present a novel approach, LatentDiffuser, for tackling temporal-extended offline tasks, addressing the limitations of previous state-of-the-art offline reinforcement learning methods and conditional generation models in handling high-dimensional, long-horizon tasks. LatentDiffuser is capable of end-to-end learning for both representation of and planning with latent action, delivering a unified, comprehensive solution for offline decision-making and control. Numerical results on Gym locomotion control, Adroit, and AntMaze, demonstrate the effectiveness of LatentDiffuser in comparison with existing hierarchical- and planning-based offline methods. The performance gains are particularly noticeable in high-dimensional and long-horizon tasks, illustrating the advantages of LatentDiffuser in addressing these challenging scenarios. Acknowledgments We extend our heartfelt gratitude to Professor Hongyuan Zha for his enlightening discussions. This work was supported in part by Postdoctoral Science Foundation of China (2022M723039). REFERENCES Pulkit Agrawal, Joao Carreira, and Jitendra Malik. Learning to see by moving. In ICCV, 2015. Anurag Ajay, Aviral Kumar, Pulkit Agrawal, Sergey Levine, and Ofir Nachum. Opal: Offline primitive discovery for accelerating offline reinforcement learning. In ICLR, 2021. Anurag Ajay, Yilun Du, Abhi Gupta, Joshua B Tenenbaum, Tommi S Jaakkola, and Pulkit Agrawal. Is conditional generative modeling all you need for decision-making? In ICLR, 2023. Marcin Andrychowicz, Filip Wolski, Alex Ray, Jonas Schneider, Rachel Fong, Peter Welinder, Bob McGrew, Josh Tobin, OpenAI Pieter Abbeel, and Wojciech Zaremba. 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QXCjvHnDmu	how is the success rate computed? the paper mentions that the attack succeeds if the target string is produced exactly. I imagine this has some limitations. Examples in fig. 4 do not show the output starting from	Open Sesame! Universal Black Box Jailbreaking of Large Language Models Anonymous authors Paper under double-blind review This paper contains unfiltered, possibly offensive content generated by LLMs Abstract Large language models (LLMs), designed to provide helpful and safe responses, often rely on alignment techniques to align with user intent and social guidelines. Unfortunately, this alignment can be exploited by malicious actors seeking to manipulate an LLM’s outputs for unintended purposes. In this paper we introduce a novel approach that employs a genetic algorithm (GA) to manipulate LLMs when model architecture and parameters are inaccessible. The GA attack works by optimizing a universal adversarial prompt that—when combined with a user’s query—disrupts the attacked model’s alignment, resulting in unintended and potentially harmful outputs. Our novel approach systematically reveals a model’s limitations and vulnerabilities by uncovering instances where its responses deviate from expected behavior. Through extensive experiments we demonstrate the efficacy of our technique, thus contributing to the ongoing discussion on responsible AI development by providing a diagnostic tool for evaluating and enhancing alignment of LLMs with human intent. To our knowledge this is the first automated universal black box jailbreak attack. 1 Introduction Large language models (LLMs) are generally trained using extensive text datasets gathered from the internet, which have been shown to encompass a considerable volume of objectionable material. As a result, contemporary LLM developers have adopted the practice of “aligning” such models through a variety of fine-tuning mechanisms. Various techniques are employed for this purpose (Ouyang et al., 2022; Glaese et al., 2022; Bai et al., 2022) with the overall objective being that of preventing LLMs from producing harmful or objectionable outputs in response to user queries. At least superficially these endeavors appear to be successful: public chatbots refrain from generating overtly inappropriate content when directly questioned. Recent research has raised increasing concerns about the vulnerability of machine learning models to adversarial attacks (Madry et al., 2018; Carlini & Wagner, 2017; Goodfellow et al., 2014; Lapid & Sipper, 2023b). Such attacks manipulate input data with imperceptible perturbations to mislead models into producing incorrect outputs. LLMs, being widely adopted for various tasks, are by no means immune to such attacks. In the context of LLMs, “jailbreaks” (Liu et al., 2023) refer to the careful engineering of prompts to exploit model biases and generate outputs that may not align with their intended purpose. These prompts are strategically designed to trigger unintended responses from the model (Wei et al., 2023), demonstrating the challenges in maintaining robustness and ethical behavior in advanced language technologies. These prompts are human crafted and take time to design. Automating the process of jailbreaking LLMs presents a significant challenge, due to the intricate nature of the task, involving carefully engineering prompts that exploit model biases to generate unintended outputs. Several factors contribute to the difficulty of automating this process: Figure 1: Our attack strategy involves constructing a single adversarial prompt that consistently undermines the alignment of leading commercial models, using only the model’s output—i.e., black box (BB) access. The instances shown are outputs from these systems. Notably, the universal adversarial prompt is proficient at inducing a variety of potentially detrimental behaviors from these models, underscoring their susceptibility to misuse. - Complexity of bias exploitation. Jailbreaking relies on identifying and capitalizing on small biases within the LLM. These biases might not be easily discernible or quantifiable, rendering their systematic exploitation non-trivial. - Dynamic model behavior. LLMs can exhibit diverse responses to slightly different inputs due to their probabilistic nature. Jailbreaking prompts may need constant refinement to adapt to the model’s shifting behavior, requiring ongoing manual intervention. - Adversarial adaptation. As models evolve to defend against adversarial attacks, automated jailbreaking techniques may become obsolete quickly. Adapting automated methods to keep up with the defense mechanisms of LLMs (Alon & Kamfonas, 2023; Chen et al., 2023; Robey et al., 2023) adds another layer of complexity. Given these challenges, automating the jailbreaking process for LLMs remains an open research problem. Researchers continue to explore methods that combine manual curation, human oversight, and algorithmic approaches to create more-sophisticated and nuanced jailbreak prompts. In this paper we propose a universal, black box jailbreak attack that can cause aligned language models to produce unintended content. In particular, when presented with a user prompt that might have preventable harmful intent, our approach involves affixing an adversarial suffix to the query, with the intention of eliciting unfavorable model responses. In this process the user’s initial query remains unaltered, while supplementary tokens are appended to elicit woeful model behavior (Figure 1). The construction of these adversarial tokens constitutes the core component of our method, and while each of these components has been separately discussed in prior literature, our innovation lies in their meticulous integration, resulting in consistently effective practical attack strategies without the use of gradients or any other model internals. To our knowledge this is the first automated universal black box jailbreak attack. In the next section we present previous work. Section 3 defines the threat model. Section 4 delineates our method. Section 5 describes the experiments we conducted and the results thereof. Our findings are discussed in Section 6, followed by conclusions in Section 7. ## 2 Previous Work Adversarial examples—inputs intentionally crafted to provoke errors or undesired behavior from machine learning models—have been studied extensively (Goodfellow et al., 2014; Carlini & Wagner, 2017; Vitrack Tamam et al., 2023; Madry et al., 2018; Lapid & Sipper, 2023a; Biggio et al., 2013; Lapid et al., 2022). Research efforts have focused both on devising adversarial attacks and on developing defense strategies against such attacks (Wong et al., 2018; Cohen et al., 2019; Li et al., 2019; Carlini et al., 2022). Effective defenses remain a challenge, often leading to reduced model accuracy (Tsipras et al., 2018). While originally explored in the domain of image classification (Goodfellow et al., 2014; Szegedy et al., 2013), the application of adversarial attacks to language models has recently been gathering momentum, extending to diverse tasks, such as question answering (Jia & Liang, 2017; Zang et al., sentiment analysis (Jin et al., 2020; Alzantot et al., 2018), and document classification (Fatehi et al., 2022; Yadollahi et al., 2021). Nonetheless, the success of these attacks on the aligned models under scrutiny has proven to be somewhat limited (Kaddour et al., 2023). This limitation stems from the intricacies of optimizing discrete tokens for language-model attacks, as well as from the fundamental distinction that—unlike in image-based attacks—subtle textual perturbations are rarely imperceptible nor well-defined. In numerous classification tasks, e.g., sentiment analysis, this necessitates modifications to the attack to guarantee that token substitutions do not alter the underlying text class. For example, given a prompt “The movie was amazing!”, an attack that modifies “amazing” to “bad” is of little value as it has changed the semantics of the prompt. Herein, we focus on a threat model that is considerably clearer, searching for a prompt suffix, which, when added to a given instruction, will provoke undesirable model behavior. Chat (2023) holds a list of hand-crafted jailbreaks that were found by humans. (Zou et al., 2023) recently presented a white-box attack causing LLMs to behave offensively. Though successful, the attack is limited because due to its white-box nature, meaning full access to the targeted model, including architecture, gradients and more. Such access is often not granted in real life. (Shin et al., 2020) has also shown another gradient-based approach, which is quite similar to (Zou et al., 2023). They focused on different NLP tasks like sentiment analysis, natural language inference, fact retrieval, and more. In (Guo et al., 2021), they proposed the first gradient-based attack on transformer models. They also evaluated their attack on classification tasks, sentiment classification and natural language inference. Another problem with a white-box attack involves the enormous number of LLM parameters, resulting in very high GPU and memory consumption. Thus, a white-box approach is extremely costly. Moreover, due to the tokens’ discrete nature, it is impossible to use standard gradient descent directly on the tokens and the algorithm needs to be modified. Maus et al. (2023) proposed a black-box framework for generating adversarial prompts that fool text-to-image models and text generators, using both the Square Attack (Andriushchenko et al., 2020) algorithm and Bayesian optimization Eriksson & Jankowiak (2021). Our black box approach does not rely on a model’s internals, and thus we do not need to deal with these kinds of difficulties. 3 Threat Model In this section we delineate the threat model for the proposed research, which concerns the exploitation of LLMs in a universal jailbreak scenario. The objective of this attack is to induce the LLM to generate harmful and undesirable behaviors by leveraging only the textual outputs it produces, thereby adhering to a black box paradigm. - Limited access. The adversary’s access to the target LLM is restricted solely to the textual outputs it generates. No access to the model’s internal architecture, parameters, or training data is granted. This constraint engenders a real-world scenario, wherein external access to model internals is often not feasible. Consequently, the attack methodology must rely exclusively on crafting input prompts and interpreting resulting text to manipulate the model’s responses. - Universal jailbreak. The focus of the attack is on achieving a universal jailbreak: an exploit that can be applied to a wide range of textual instances without prompt modification. This approach maximizes the practicality and real-world relevance of the threat. - Attack goal. The primary goal of the attack is to coerce the LLM into generating harmful and malicious behaviors, i.e., generating text that contains offensive, violent, or otherwise socially unacceptable content. 4 Our Method In this section, we present the main technical innovation of our paper: a novel technique for exploiting vulnerabilities within a language model, to elicit undesirable responses. Our approach works under black box conditions, which means we can only query the model and receive its raw output. We use neither gradients nor any model internals. 4.1 Genetic Algorithm A genetic algorithm (GA) is a search heuristic that mimics the process of natural evolution (Algorithm 1) (Sipper et al., 2017; Sipper, 2002). It is commonly used to find approximate solutions to optimization and search problems. We will now elaborate on the different components of the GA, adapted to our jailbreaking task. Algorithm 1: Standard genetic algorithm (GA) Input: problem to solve Output: Solution to problem Generate initial population of candidate solutions to problem; while termination condition not satisfied do Compute fitness value of each individual in population; Perform parent selection; Perform crossover between parents to derive offspring; Perform mutation on resultant offspring; end return Best individual found; 4.2 Population Encoding The GA begins with the creation of an initial population of individuals (Algorithm 2), each representing a potential solution to the problem at hand. Our individuals are prompts—a set of tokens—thus we chose to encode each individual as a vector of integers, representing tokens. More formally, let $P$ be a population of $n$ prompts, each prompt being of length $m$: $$P = \{(x_1, x_2, \ldots, x_m) \mid x_i \in T \text{ for } i = 1, 2, \ldots, m\}_1^n,$$ where $T$ is a vocabulary of tokens. We experimented with 3 different $n$ values, $n \in \{10, 20, 30\}$, and 3 different $m$ values, $m \in \{20, 40, 60\}$, for each given model. Algorithm 2: Initialization Input: population size $n$, prompt length $m$, tokens vocabulary $T$ Output: initialized population $P$ $P \leftarrow []$; for $i \leftarrow 1$ to $n$ do $I \leftarrow \text{random.choices}(T, m)$; $P \leftarrow P + I$; end return $P$; 4.3 Fitness One of the primary considerations in formulating the attack involved defining the goal, i.e., determining the fitness (loss) function to be used for optimizing the universal adversarial prompt. There are several potential loss functions to choose from. As stated by (Zou et al., 2023), we could seek to maximize the probability of the model generating a highly specific designated sequence (such as a string containing instructions for stealing one’s identity). While this approach effectively evaluates the model’s capacity to produce relevant content, it lacks in its suitability as the attack’s objective because in many cases there is no single, good, accurate output for a given input prompt—there are likely multiple valid responses that align with the intended purpose and may be more plausible for the target language model. Instead, for each given input prompt, we contextualized the LLM to answer with “Sure, here is a...” if it knows the answer and “No, I don’t...” otherwise. This approach is used in manual jailbreaking techniques and herein we leverage it for automated optimization. Formally, given: a vocabulary $T$ of $v$ tokens, $T = \{T_1, T_2, \ldots, T_v\}$; a sequence of tokens $x_{1:t}$, where each $x_i \in V$: a large language model LLM. LLM is a function that maps the token sequence to a probability distribution for the next token, $x_{t+1}$: $$\text{LLM}(x_{t+1}\|x_{1:t}),$$ such that $x_{t+1} \in V$. The input prompt consists of the user-generated instruction $x_{1:t} = x_{\text{user}}$, sampled from a given dataset $D$, and an adversarial suffix $x_{\text{adv}}$: $$x = x_{\text{user}} \\| x_{\text{adv}},$$ where $\\|$ is the concatenation operator. $D$ is a dataset of harmful behaviors, elaborated upon in Section 5. For a given instruction $x_{\text{user}}$ and a target output $y_{\text{target}}$ (“Sure, here is a...”), we wish to find an adversarial suffix, $x_{\text{adv}}$, such that the loss of $x_{\text{user}}$ is: $$L_{\text{white-box}}(x_{\text{user}} \\| x_{\text{adv}}) = -\log \text{LLM}(y_{\text{target}}\|x_{\text{user}} \\| x_{\text{adv}}).$$ Hence, the universal attack optimization finds $x^_{\text{adv}}$ such that it minimizes the loss $L_{\text{white-box}}$ for any given $x_{\text{user}}$: $$x^_{\text{adv}} = \arg\min_{x_{\text{adv}}} \mathbb{E}_{x_{\text{user}} \in D} L_{\text{white-box}}(x_{\text{user}} \\| x_{\text{adv}}).$$ By minimizing the negative log-likelihood we encourage the adversarial suffix to guide the language model to generate responses that align with the user’s intent. Under our threat model we cannot access a model’s confidence scores and so must define a fitness function that does not rely on these. Given the output generated by the model and a target output, the fitness function aims to quantify the alignment between these two elements in the embedding space. To achieve this, a text embedder is employed to convert both the model’s output and the target output into their respective embedding representations. Then, the cosine similarity between these embeddings is computed, reflecting the semantic alignment between the generated output and the target output. The loss is then defined as the negative of this cosine similarity, incentivizing the model to generate outputs that exhibit a high degree of semantic similarity with the target output. Formally, the fitness function $L_{\text{black-box}}$ can be expressed as: $$L_{\text{black-box}}(x_{\text{user}} \\| x_{\text{adv}}) = -C_S(f_{\text{embed}}(\text{LLM}(x_{\text{user}} \\| x_{\text{adv}})), f_{\text{embed}}(y_{\text{target}})),$$ where $f_{\text{embed}}(\cdot)$ represents the text embedder, and $C_S(\cdot,\cdot)$ represents the cosine similarity between two embedding vectors. This loss formulation guides the model towards producing outputs that align closely with the intended semantic content specified by the target output in the embedding space. Fitness approximation through random subset sampling. To alleviate computational complexity in evaluating the algorithm’s fitness across the dataset during each GA iteration, we adopt fitness approximation through random subset sampling Jin (2005); Yu & Kim (2018). Instead of assessing the universal attack on the entire training set, we randomly select a subset of size $f$. This subset approximates the input distribution of the complete training set, allowing us to efficiently estimate the universal attack’s impact on a wide range of inputs. Importantly, the random subset sampling is performed anew in each iteration, guiding the optimization process with diverse and representative inputs. Throughout the experiments, we used $f = 50$. Algorithm 3 presents the pseudocode of the fitness-evaluation procedure. Algorithm 3: Fitness evaluation Input: individual $I$, loss $L_{\text{black-box}}$, fitness approximation size $f$, embedder $f_{\text{embed}}$ Output: fitness of individual $I$ 1. $\{x_{\text{train}}, y_{\text{train}}\}_{i=1}^{f} \leftarrow$ randomly pick $f$ instances from training set; 2. $L_{\text{total}} \leftarrow 0$; 3. for $x_i \in \{x_{\text{train}}\}_{i=1}^{f}$ do 4. $x_{\text{adv}, i} \leftarrow x_i \|\| I$; 5. $y_{\text{output}, i} \leftarrow \text{LLM}(x_{\text{adv}, i})$; 6. $L_{\text{total}} \leftarrow L_{\text{total}} + L_{\text{black-box}}(f_{\text{embed}}(y_{\text{output}, i}), f_{\text{embed}}(y_{\text{train}, i}))$; 7. end 8. return $L_{\text{total}}/f$; ### 4.4 Selection A selection process is used to choose individuals from the current population, to become parents for the next generation. Selection is typically biased towards individuals with higher fitness values. This increases the likelihood of passing favorable traits to the next generation. We used tournament selection Bickle (2000) with $k = 2$, meaning we randomly pick 2 individuals from the population and choose the fitter as parent to undergo crossover and mutation. ### 4.5 Crossover and Mutation Crossover involves combining genetic material from two parent individuals to create one or more offspring. This process simulates genetic recombination and introduces diversity into the population. It allows the algorithm to explore new regions of the search space by recombining existing information. Conversely, mutation introduces small random changes in an individual’s genetic material (Figure 2). Crossover is usually perceived as an exploration mechanism, which is balanced by the exploitation mechanism of mutation Lim et al. (2017). ![Figure 2: One-point crossover (left), wherein two parent individuals exchange parts of their genomes at a randomly selected point in their vectors to create two offspring. Mutation (right), wherein a single parent individual modifies its genome by randomly choosing indexes and replacing the tokens there with randomly chosen ones.] ### 4.6 Elitism Elitism is a strategy commonly used in GAs and other evolutionary algorithms to preserve the best-performing individuals throughout the generations, ensuring that the overall quality of the population does not deteriorate over time. This strategy helps maintain progress towards finding optimal or near-optimal solutions in optimization and search problems. Herein we chose the elitism value as a function of population size $n$: $\lambda = \frac{n}{5}$. 4.7 Assembling the Pieces Algorithm 4 presents the GA, combining all the pieces discussed above. Algorithm 4: GA for generating LLM universal adversarial prompt Input: dataset of prompts $D$, population size $n$, prompt length $m$, tokens vocabulary $T$, generations $g$, loss $L_{\text{black-box}}$, fitness approximation $f$, tournament size $k$, elitism $e$ Output: optimized prompt 1. $P \leftarrow$ Initialization (Algorithm 2); 2. for $i \leftarrow 1$ to $g$ do 3. $F \leftarrow$ fitness evaluation (Algorithm 3); 4. $E \leftarrow$ elitism (save $e$ elitist individuals); 5. $S \leftarrow$ selection (parents for reproduction); 6. $O \leftarrow$ crossover and mutation (to create offspring); 7. $P \leftarrow E + O$; 3. end 4. return Best individual found; 5 Experiments and Results Dataset. The experimental dataset, Harmful Behavior, released by (Zou et al., 2023), denoted as $D$, comprises instances of harmful behaviors specifically designed to challenge the capabilities of LLMs. This dataset is carefully curated to encompass a diverse range of harmful inputs. These instances are aimed at triggering vulnerabilities in LLMs’ understanding and generation of language. The dataset’s design ensures a comprehensive assessment of model responses to harmful stimuli. To ensure robust evaluation of our proposed universal jailbreaker we partition dataset $D$ into a training set (70%) and a test set (30%). The training set is utilized for the optimization by the GA, while the test set serves as an independent evaluation set to measure the algorithm’s effectiveness and generalizability post-factum. We used two different seeds for the splitting and the results are the average of these two. We used a generation count of 100 for all experiments and 3 different population sizes, $n \in \{10, 20, 30\}$. As mentioned above, for each of the individuals we randomly chose a subset of size $f = 50$ and evaluated its fitness, resulting in 50000, 100000, 150000 queries to the target models, respectively. Models. Our study involved two prominent LLMs: - LLaMA2–7b-chat Touvron et al. (2023). A model trained to chat with users, which was aligned through reinforcement learning with human feedback (RLHF), utilizing a blend of 1,418,091 meta instances along with seven smaller datasets. - Vicuna–7b Chiang et al. (2023). A model that was fine-tuned through supervised instruction fine-tuning, using approximately 125,000 conversations gathered from ShareGPT.com as the training dataset (for more details see Zheng et al. (2023)). These models are recognized for their advanced language generation capabilities and are widely adopted in various natural language processing applications. Embedder. Aiming to obtain a universal LLM jailbreak in a black box manner—where the internal workings of the models are inaccessible—a pivotal component of our experimental setup is the embedder. The primary objective of the embedder is to bridge the gap between the textual outputs generated by the LLMs and the intended target outputs, enabling a quantitative comparison of their semantic congruence. Our methodology involves encoding both the target output and the generated output into the same embedding space. This embedded representation serves as a reference point for the desired semantics. Formally, let $y_{\text{target}}$ represent the target output and $E_{\text{target}}$ denote its embedded representation. Then: $$E_{\text{target}} = f_{\text{embed}}(y_{\text{target}}).$$ (7) Table 1: Results: Best evolved jailbreaker’s attack performance over Harmful Behavior dataset. Each table shows the results in terms of the text-embedder used in that specific experiment. Each line represents one experimental setting. $ n $: population size; $ m $: prompt length; SR: success rate of prompt without attack, as percent of test set prompts; ASR: attack success rate of evolved adversarial prompt, as percent of test set prompts. Best results are boldfaced. The penultimate row shows the average score across all experiments. The last row in each table shows the very low success rates for no attack (this is per model, regardless of embedder, but was added to each table for clarity). \| \| BGE \| MPNet \| MiniLM \| \|-------\|--------------\|--------------\|--------------\| \| $ n $ \| $ m $ \| Vicuna-7b \| LLaMA-7b-chat \| $ n $ \| $ m $ \| Vicuna-7b \| LLaMA-7b-chat \| $ n $ \| $ m $ \| Vicuna-7b \| LLaMA-7b-chat \| \| 20 \| 94.8% \| 97.8% \| \| 20 \| 95.5% \| 99.4% \| \| 20 \| 94.5% \| 99.0% \| \| \| 10 \| 94.6% \| 98.4% \| \| 10 \| 97.4% \| 98.4% \| \| 10 \| 94.2% \| 95.5% \| \| \| 60 \| 94.7% \| 98.4% \| \| 60 \| 97.1% \| 98.4% \| \| 60 \| 90.7% \| 98.4% \| \| \| 20 \| 98.4% \| 99.7% \| \| 20 \| 97.1% \| 99.4% \| \| 20 \| 95.5% \| 98.1% \| \| \| 40 \| 96.5% \| 98.1% \| \| 40 \| 93.9% \| 98.4% \| \| 40 \| 95.5% \| 95.8% \| \| \| 60 \| 94.2% \| 99.4% \| \| 60 \| 95.5% \| 98.0% \| \| 60 \| 92.0% \| 98.1% \| \| \| 20 \| 95.2% \| 98.7% \| \| 20 \| 96.5% \| 99.4% \| \| 20 \| 98.1% \| 98.4% \| \| \| 30 \| 92.3% \| 97.8% \| \| 30 \| 92.3% \| 98.7% \| \| 30 \| 92.0% \| 97.8% \| \| \| 40 \| 94.6% \| 99.0% \| \| 40 \| 94.4% \| 97.8% \| \| 40 \| 96.5% \| 99.0% \| \| \| 60 \| \| \| \| 60 \| \| \| \| 60 \| \| \| \| \| average \| 94.0% \| 98.6% \| \| average \| 95.5% \| 98.7% \| \| average \| 95.0% \| 97.8% \| \| \| no attack \| 0.6% \| 16.3% \| \| no attack \| 0.6% \| 16.3% \| \| no attack \| 0.6% \| 16.3% \| \| For each generated output $ y_{\text{output}} $ by the LLM in response to a different input, the embedder is employed to encode $ y_{\text{output}} $ into its corresponding embedded representation $ E_{\text{output}} $: \[ E_{\text{output}} = f_{\text{embed}}(y_{\text{output}}). \] By employing suitable embedding techniques, such as pretrained neural networks or semantic similarity measures, we can quantitatively measure the distance between embeddings $ E_{\text{target}} $ and $ E_{\text{output}} $. This distance serves as a metric of semantic similarity between the generated output and the desired target output. Herein, we used three different text embedders, including: bge-large-en HuggingFace (a), all-mpnet-base-v2 HuggingFace (b), and all-MiniLM-L6-v2 HuggingFace (c), which are general text embedders that can map any given text to a low-dimensional ($\mathbb{R}^{1024}$) dense vector encapsulating its semantics. We will denote them BGE, MPNet and MiniLM respectively. Similarity. We used cosine similarity as the distance measure, i.e., we computed cosine similarity between the embeddings of the model-generated outputs and the corresponding target outputs. This metric quantifies the semantic alignment between the generated content and the intended prompt. In our scenario, we want the values to be as low as possible. In addition, in order to compute the attack success rate (ASR), we checked for each instance if the model produced the precise target string as its output (“Sure, here is a …”) and it did not contain any “rejections” on the test set (“I’m not sure“, “I’m sorry, I can’t“, “No, I don’t“, “I cannot“, “As an AI“). Results. Table 1 presents a summary of our main results. The MPNet text embedder consistently achieved the highest average ASR on both Vicuna-7b and LLaMA-7b-chat. In addition, Appendix B shows results of a transferability study, demonstrating that prompts generated from one model can be successfully transferred to another. In Appendix D, Table 4 shows qualitative samples for Vicuna-7b. In Appendix C, Table 3 shows qualitative samples for LLaMA2-7b-chat. The samples showcase evolved universal jailbreaking prompts. To avoid sensitive text we only show the LLM’s start of output, evidencing it has indeed been breached. 6 DISCUSSION In this study, we investigated the effectiveness of employing GAs for the purpose of black box jailbreaking of LLMs. In this section we discuss a number of major points. Transferability. Our experimentation with transferring an attack evolved for one model to another model sheds light on transferability dynamics (Appendix B). Interestingly, the outcomes demonstrated enhanced transferability when transitioning from either Vicuna-7b or LLaMA-7b-chat to the more advanced LLaMA-13b-chat model. The efficacy of the transferred attack from LLaMA-7b-chat to LLaMA-13b-chat was particularly noteworthy, showcasing a robust compatibility within the LLaMA family of models. Results also indicated a surprising degree of adaptability when moving from LLaMa-7b-chat or Vicuna-7b to Vicuna-13b. These findings suggest a relationship between model architectures, revealing potential opportunities for leveraging pre-existing knowledge from earlier jailbreaks to enhance the capabilities of newer iterations, albeit with varying degrees of success. Further, it underscores that optimizing a suffix involves more than just the addition of random tokens. Overall, LLaMA models seem to be less robust than Vicuna models. Implications and potential countermeasures. The implications of our findings are noteworthy both for the research community and for practitioners. The success of the black box jailbreaking attack underscores the need for continuous evaluation and fortification of LLMs against adversarial techniques. Developers and organizations relying on these models for various applications should be aware of their vulnerabilities and explore potential mitigation strategies. One possible countermeasure could involve dynamically adjusting the model’s sensitivity to longer prompts, which might limit the extent to which the GA can exploit its internal processes. Additionally, the added prompts involve “garbage” tokens that might be detected by another LLM or by using perplexity (e.g., as in Alon & Kamfonas (2023)). Limitations and future work. As with any research undertaking, this study has its limitations. Our experiments were conducted under specific conditions, and the robustness of the attack may vary across different LLM architectures and prompt types. Furthermore, this attack adds perceptible perturbations, which is a limitation. The ethical implications of employing such attacks should be carefully considered, as adversarial techniques could be used for malicious purposes. Appendix A discusses ethical considerations. Future research could involve exploring the interaction between prompt construction and GA parameters in more detail. We plan to test our approach on additional LLMs, such as Guanaco Dettmers et al. (2023), Orca Mukherjee et al. (2023), and more. Further, investigating the generalizability of these findings to other AI systems beyond LLMs would provide a broader perspective on the effectiveness of GAs in black box attacks. 7 Conclusions This paper introduced the novel concept of a universal black-box jailbreak attack on LLMs. Throughout our exploration we have underscored the intricate challenges involved in developing robust and reliable LLMs. The complexity of language and the potential for adversarial manipulations highlight the need for reassessing the security mechanisms underpinning these systems. The question of aligning LLMs more effectively speaks to a fundamental concern in the field. While adversarial training holds promise, it is evident that a comprehensive solution requires a holistic approach. This involves interdisciplinary collaboration among researchers, developers, and policymakers to establish a framework that fuses performance with ethical considerations. Adversarial training, combined with innovative regularization techniques and rigorous testing, could lay the groundwork for mitigating universal jailbreak attacks. In conclusion, the journey to enhance the security of LLMs is a multifaceted one. Our findings serve as an (urgent) call for a paradigm shift towards creating not only powerful but also ethically sound LLMs. As the field advances, the onus is on us, as a community, to shape the future of AI-driven language understanding, ensuring it aligns with human values and societal well-being. Acknowledgments This research was supported by [removed for anonymity]. REFERENCES Gabriel Alon and Michael Kamfonas. Detecting language model attacks with perplexity. arXiv preprint arXiv:2308.14132, 2023. 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gENfMmUIkT	Since the approach is worked on resource-constrained IoT devices, to compare, the authors did not bring any such metric (For instance, FLOPs, Parameters, Energy Consumptions) to compute the computational power needed by the models.	A PIPELINE-BASED APPROACH FOR OBJECT DETECTION ON RESOURCE CONSTRAINED INTERNET OF THINGS DEVICES Anonymous authors Paper under double-blind review ABSTRACT Object detection with computer vision and convolutional neural networks on resources constrained devices can be challenging. The limited power and processing capacity of these devices complicates the use of deep neural networks and other object detection methods. To address this problem, we propose a pipeline-based approach. We introduce a multi-step detection pipeline considering the size of the objects to be detected and the correlation among them. To evaluate the performance of this approach, we test it in a collaborative smart surveillance system employing edge computing and the internet of things paradigm. Additionally, field testing was conducted considering real world surveillance scenarios. Results showed that the introduction of the pipeline-based processing improved the execution time by a factor of 3 and produced a significant improvement on the mean average precision. 1 INTRODUCTION Ensuring safety is a major concern in metropolitan areas across the world. With the growth of smart cities, an innovative solution for combatting crime is the implementation of smart video-surveillance systems. They have the ability to automatically detect potential threats and quickly notify law enforcement agencies, providing a more effective approach to crime prevention. Smart surveillance systems commonly employ the internet of things (IoT) paradigm and are embedded both on edge and cloud devices. Therefore, the major challenges faced by such systems are: resources management at the edge, bandwidth management and latency of critical messages. In the present work we propose a pipeline-based approach for object detection on resources constrained IoT devices and apply the methods in a surveillance system. In our work, we define “pipeline” as the sequence of all linked steps taken by the systems for the capture and processing of all necessary information for its operation. A pipeline was carefully designed aiming processing efficiency and classification accuracy, implemented and evaluated with real world testing in partnership with the São Paulo Police Department. Leveraging our approach with a surveillance system results in advantages like mobility, amount of detected objects/threats and automatic notifications. Moreover, the main contributions of our work are listed below: • A pipeline-based approach to enable multi-class object detection on resource constrained edge devices. • AI Pipeline: a sequence of inference steps optimized to satisfy the trade-off between processing time and precision. • Real world tests (TRL 6): demonstration of the system working in the field, installed in police vehicles. 2 RELATED WORK The existing IoT smart video-surveillance systems are capable of object and event identification by employing state of the art methods in computer vision (CV) and artificial intelligence (AI). Many of them are proposed for edge computing, cloud computing or, more commonly, for a hybrid edge-cloud collaboration. Recurrent challenges of these types of systems are: resources management at the edge, bandwidth management and latency of critical messages. One strategy to tackle these problems is to carefully design the pipeline steps. In this section we analyse the pipeline flow of recent systems, specifically the steps taken to increase the efficiency, detection accuracy and bandwidth saving. Panganiban et al. (2022) propose an IoT license plate recognition system based on three different pipeline approaches: edge-heavy, cloud-heavy and hybrid. The general flow for all three pipeline is to firstly detect, in the video feed images, regions of possible license plate and forward them to the next step, where the recognition of characters is performed. In the edge-heavy pipeline both steps are performed on the edge devices and the results are stored in the cloud. On the contrary, in the cloud-heavy pipeline all steps are performed in the cloud. This approach requires a larger bandwidth (BW), since all video frames are streamed to the cloud. The hybrid pipeline strategy is to perform plate detection in the edge and sent a cropped image containing the region of interest to the cloud, where plate recognition is performed. The evaluation metric adopted to evaluate all pipeline approaches was the capture-to-result time in seconds (CTR). For a low BW (<1600 kbps) and less than 4 edges nodes, the hybrid pipeline performed better with a CTR of 15 s. However, for more than 4 edge instances, the edge-heavy pipeline performed better with a CTR of 10 s. The cloud-heavy pipeline achieved the best CTR of 5 s, but required a BW of 2,500 kbps and less than 4 nodes. Authors Ke et al. (2021) describe an IoT parking occupancy estimation system based on CV and AI. The strategy of the proposed system is to split the computational load between edge devices and servers, targeting optimal system performance. The first step of the proposed pipeline is to manually label parking spaces on the server side and then apply a matching algorithm according to the vehicle positions, which are detected on the edge side. The pipeline also considers normal or low visibility conditions, e.g. foggy or rainy weather. In the second case, the pipeline approach is to combine two detection methods: a Mobilenet CNN model with single shot multibox detector (SSD) and a Background modeling detector (BG). Instead of images, the forwarded results are the detected bounding boxes and position, reducing data volume in the network. The system evaluation was done in real world scenarios at parking garage. Considering several weather conditions, the achieved average accuracy was 95.6%. The work proposed by Pathy & Saleh (2022) describes an IoT smart video-surveillance system, capable weapon detection and automatic notifications of events. Object detection and classification (e.g. firearm, knife, phone, card) is performed by edge devices with light YoLov5 models (v5n, v5-lite e and v5-lite s). Among the system’s pipeline steps, it is proposed a Software-Defined Networking (SDN) for more efficient network usage, controlling bandwidth and speeding up critical notifications. Therefore, in this case, the pipeline approach is to control the network rather than modifying the detection steps. The evaluation of the proposed adaptive QoS model revealed improvements in performance in terms of jitter, packet loss, and average throughput. The evaluation of the light CNN models employed at the edge devices revealed the YoLoV5n to perform the best, achieving a mAP of 95% for pistol detection. However, the system was not evaluated in real world scenarios. Sultana & Wahid (2019) propose a smart surveillance system for home usage. The architecture of the system include edge nodes installed throughout a house and fog-nodes, which can control several edge instances. The first step of the pipeline flow is to detect motion on the edge side with pixel-based background subtraction techniques, which can be performed nearly instantly. In case motion is detected, the video stream is forwarded to a fog node, where firearm and knife detection are performed with VGGNet. Lastly, upon a detection of a crime, automatic notifications are sent by the fog servers to the authorities. By employing this pipeline design the systems saves energy, BW and CPU. Object detection is performed in 15 s at the fog and the system total operation time is 18 s. The system entitled Hawk-Eye (Ahmed & Echi, 2021) can detect multiple classes of threats, such as weapons, vehicles and masked people. Two different pipeline flows are proposed: the first is evaluated in edge devices and the second in the cloud. In both cases, the pipeline initial flow is to detect motion with a background subtraction method. Next, objects are detected and classified with a neural network. At the cloud, a Mask R-CNN model was built, enabling the system to make a high-quality segmentation mask for each object in the images. A lighter CNN was employed at the edge, enabling the system to detect and classify objects locally, without relying network availability. Regarding object classification, no further steps are taken in the pipeline. The achieved results were different for the cloud and edge-based pipelines: the prediction time for pistols was 4.1 s with the R-CNN (cloud) and 5 ms for CNN (edge). As can be seen above, among the recent investigations in smart surveillance, many are interested in weapon detection, since firearms and knives can indicate a severe security threat. Some of these works propose interesting pipeline approaches to increase the recall and reduce the false positive rate of weapon detection. Ruiz-Santaquiteria et al. (2021) propose an AI-based method for weapon detection, which combines both object appearance and human body pose. Similarly, Castillo Lamas et al. (2022) describe a weapon detection system based on human pose estimation, which aims to mitigates false positives that can arise in systems based on weapon appearance exclusively. In both systems the additional step in the pipeline flow led to an increase in the weapon detection mAP. Cob-Parro et al. (2021) proposed a IoT smart video surveillance system specifically for edge computing. The AI-application uses a MobileNet-SSD architecture and is capable of detecting, tracking and counting people. Due to the limited processing power of the edge nodes, the researches were interested in the relation between performance and energy consumption of the system. The pipeline-based strategy to increase the performance was to use parallel processing of multiple video streams, which was possible due to the VPUs present in the edge devices. The inference computational cost for the algorithm using a CPU and a VPU was, respectively, 13.93 ms and 8.71 ms. Regarding people detection, no specific steps are taken in the pipeline, other than the standard MobileNet-SSD methods. Chen et al. (2022) describe a video surveillance system for smart cities. The authors propose an IoT edge-cloud collaboration system, capable of classifying multiples classes of large objects, e.g. vehicles and bicycles. The first step in the pipeline flow is to perform, at the edge, object classification with YoLo and foreground estimation (image matting). Secondly, the extracted foreground objects are then compared with those classified by the CNN. Next, the objects that can not be automatically classified are sent to a cloud AI system, where they are manually labeled and used to retrain the CNN. Lastly, the final step in the pipeline is to update the model at the edge devices, increasing the object classification capability of the system. Across all classes, the achieved mAP was 0.983 with YOLOv4. In addition, the industry has also shown interest in mobile smart surveillance system. For instance, Neto et al. (2018) proposed a fog-computing based system capable of crime detection in public bus services. The system can classify events in real-time and generate automatic notifications upon the detection of predetermined threats, warning the competent authorities. The first step considered in the pipeline is to pre-process the images with edge devices in-vehicles. Next, object classification is done in the cloud, which is also responsible to notify authorities upon detection of crimes. Similarly, De Biase et al. (2020) propose a collaborative and mobile surveillance system embedded in normal vehicles. The system uses edge computing for threat identification and automatic warning notifications. We noticed from the aforementioned works that systems often consider strategic steps in the pipeline design to tackle recurring challenges of IoT-based smart surveillance. Most systems split the computational load between edge and cloud and some systems reduce data volume, optimizing the network usage. However, none is focused on overcoming such challenges by a pipeline-approach uniquely, specially when considering multi-class object detection on edge devices without GPUs and without cloud assistance. Therefore, we propose a pipeline-based approach for object detection on resource constrained IoT devices and evaluate it in a surveillance system embedded in single-board computers, which will be described in the following sections. 3 PROPOSED ARCHITECTURE 3.1 TRAINING PROCESS In this section we describe the training process phase, where convolutional neural networks (CNNs) were trained to classify objects from regular RGB images. We trained the CNNs YOLOv3 (Redmon & Farhadi [2018]) and YOLOv4 (Bochkovskiy et al. [2020]) to classify the following objects: 1. People. 2. Firearms. 3. Vehicles. 4. License plates. 5. License plate’s characters. We used our own built dataset, composed by 135,000 labelled images, which includes real images of vehicles (cars, motorcycles and bicycles), real and synthetic images of weapons, real and synthetic images of license plates. Moreover, real images of firearms were labeled in a manner to reduce the interference of human body parts. Since the events classifications should be done with edge computing, we evaluated the following light models: 1. YOLOv3 Tiny (2 detectors YOLO) 2. YOLOv3 Tiny 3L (2 detectors YOLO) 3. YOLOv4 Tiny (2 detectors YOLO) 4. YOLOv4 Tiny 3L (3 detectors YOLO) The metrics employed for the models performance evaluation are based on the classic confusion matrix. More specifically, we employed the precision for each class ($P$), the accuracy or total number of correct prediction ($A$) and the recall ($R$). These metrics are defined as follow: 1. $P = \frac{TP}{TP+FP}$ 2. $A = \frac{TP+TN}{TP+FP+TN+FN}$ 3. $R = \frac{TP}{TP+FN}$ Where, - TP: True Positive - FP: False Positive - TN: True Negative - FN: False Negative The goal for firearm detection was to achieve a mean average precision of (mAP) of at least 80%. Initially, a first round of training was performed with standard parameters. Based of these first results, more refined training rounds were conducted. Data augmentation methods like mosaic, mixup and blur were applied and evaluated. Finally, for designing the detection pipeline, different input images dimensions were evaluated according to object distance from the camera. 3.2 EMBEDDED INFERENCE PIPELINE The mechanism for embedded event detection is based on a pipeline of object recognition steps, which optimizes both processing time and accuracy. There are two main principles behind the pipeline-based approach: 1. Large objects are easier to detect. This makes it possible to re-scale the input images to significantly low resolutions, which in turn reduces inference time. Specifically, reducing resolution to half of the original cuts the overall amount of pixels to 1/4, resulting in a much smaller inference time. 2. The presence of some objects are correlated. For example, since license plates are always attached to a vehicle, there is no need to try to detect license plates in the absence of a vehicle. This enables rapidly dropping uninteresting frames, thus improving the rate of frame processing. A step of the pipeline can be defined by a function of the following form: \[ P_{\text{step}}(f, p) \rightarrow r \] Where: - $ f $ is the captured frame from the camera, in its original resolution. Using the original frame in every step enables extraction of the bounding box for detected objects while preventing build-up from consecutive resizing operations. - $ p $ contains the result of the previous step, such as detected items and respective bounding boxes. This item will be $ \text{nil} $ (empty) in the first step of the pipeline. - $ r $ contains the result of a particular step. It can be logged, transmitted, or passed to a subsequent step of the pipeline. Finally, the connection between the steps is done via an external orchestrator module, which is application-specific. ### 3.3 Use Case: Detection Pipeline for a Surveillance System Considering a surveillance system, a four-step pipeline was assembled for the detection of three types of events: crowds, guns, and license plates. The detection pipeline is defined below, as well as shown visually in Figure 1. ![Figure 1: The proposed detection pipeline for a mobile surveillance system.](image) \[ P_{\text{scan}}(f, \text{nil}) \rightarrow r_{\text{scan}} \] \[ P_{\text{gun}}(f, r_{\text{scan}}) \rightarrow r_{\text{gun}} \] \[ P_{\text{plate}}(f, r_{\text{scan}}) \rightarrow r_{\text{plate}} \] \[ P_{\text{chars}}(f, r_{\text{plate}}) \rightarrow r_{\text{chars}} \] In the first step, $ P_{\text{scan}} $, the original frame is down-scaled and fed into a network which searches only for two categories: person and car. If at least one person is detected, the system will verify whether there is a crowd, and in that case, generate a crowd event; it will also zoom into the largest person box (closer to the camera), and feed it to the $P_{\text{gun}}$ step. If the latter finds a gun within the cropped person image, it will generate a gun event. Returning the $P_{\text{scan}}$ step, if it finds a vehicle, it will zoom into the vehicle box and feed it to the $P_{\text{plate}}$ step, which will look for license plates. If a plate is found, it will again be zoomed into, and passed to the final $P_{\text{chars}}$ step. In this step, the characters are extracted and submitted to a heuristic which verifies whether they constitute a license plate, in which case a license plate event is generated. 4 IMPLEMENTATION The proposed system was implemented in real hardware and tested in the field, reaching a technology readiness level of prototype demonstration in relevant environment (TRL 6). 4.1 HARDWARE The main component of the hardware solution is the Labrador 64\footnote{https://caninosloucos.org/en/labrador-64-en/}, a 1.3GHz quad-core arm-based single-board computer with 2 GB of volatile memory and 16 GB of flash. Connected to the Labrador, are: - USB Action Cam: camera used to capture high-resolution frames in a mobile environment. - Pulga Stack: a modular subsystem composed of a cortex-M4F-based microprocessor, a GPS module for time and location, and a LoRaWAN module for event transmission. The latter is also connected to a dipole 915 MHz antenna. The Labrador and the Pulga Stack are connected via a four-wire flat cable, as shown in Figure 2a. - Power cable: a cable for powering the system with a 12 V power supply. On the Labrador end, it features a P4 connector; on the other end, it features an automobile auxiliary power plug\footnote{Commonly called a “car cigarette lighter”.} ![Boards and antenna of the implemented prototype.](image1) (a) Boards and antenna of the implemented prototype. ![The complete assembled prototype.](image2) (b) The complete assembled prototype. Figure 2: The implemented prototype used for the evaluation of the detection pipeline. All the pulga boards (base, core, LoRa and GPS) are assembled as a stack. The Labrador, Pulga Stack, flat cable, and antenna are enclosed in a 3D-printed box with openings for interface connection and passive ventilation, as shown in Figure 2. The USB Camera is mounted on top... of the box, and fixed with a double-coated tape. The box is mounted on the middle of the top side of the vehicle panel, and the power cable is connected to the vehicular power outlet. 4.2 SOFTWARE The detection Pipeline was implemented in Python and integrated within the `event_notifier`, a program responsible for managing the connection with the camera and other sensors, as well as starting and stopping the pipeline, and transmitting events and saving evidences. Each step of the pipeline was implemented as a separate class, all of them implementing the method `run`, which receives the original frame and the result of the previous step. The pipeline was implemented as a separate class, which was in turn instantiated and called by a thread that manages the camera input, evidence storage, and the pipeline itself. All steps use a YOLO object detection approach to perform inference, with the help of the `darknet` library. Each step uses a different `darknet` configuration composed of a network model file and a specific input image size. Varying the image size allows tuning the accuracy and execution time of the whole pipeline. 5 EVALUATION The evaluation was conducted on the hardware specified in Section 4, namely the Labrador, a 1.3GHz quad-core single-board computer with 2 GB of volatile memory. \| \| Scan \| Gun \| Plate \| Char \| \|----------------------\|------\|-----\|-------\|------\| \| Input image (pixels) \| 256 \| 320 \| 256 \| 192 \| \| Execution time (s) \| 1.5 \| 2.5 \| 0.5 \| 6 \| Table 1: The detection pipeline configuration and execution time. Figure 3: Deployment in Police patrol vehicles. The Sentinel device is powered by the car’s battery and it is kept on whenever the vehicle’s engine is also on. In addition, the evaluation of the prototype devices employing our detection pipeline was conducted in partnership with the police department. Twelve devices were deployed in patrol vehicles for pilot testing rounds, where we mostly analysed license plate reading performance. From the preliminary results, we noticed that 78 % of the notifications received by the Police server were correct. With further analysis, we determined that this was due to a confusion between characters “5” and “6”. After fixing this issue, detection accuracy increased to approximately 90 %. Examples of the geolocation data received are shown in [4a]. Image evidences of detected events are saved in the sentinel internal storage and can be latter retrieved manually. An example of detection frame is shown in [46], where three categories of objects were correctly detected. 6 CONCLUSION We introduced our pipeline-based approach for object detection in IoT devices. It was designed to leverage CV and AI methods with objects size and correlation to overcome the constrained resources of edge computing. We trained YÖLO CNNs with our own built dataset and we evaluated our approach in a mobile surveillance system. Pilot testing was conducted in real world scenarios. The results shown that our approach is appropriate for IoT devices and edge computing, improving both the inference time and the mean average precision in object detection. Future works could include the evaluation of our approach in other object detection systems with even more constrained resources. In addition, it would be interesting to test the approach in a even larger quantity of smart IoT devices, as in a swarm paradigm. REFERENCES Ahmed Abdelmoamen Ahmed and Mathias Echi. 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cZo6pDtDZr	What specific properties of the hash family do you need? The reason I ask is that communicating a random hash function requires a large number of bits; however, this can be reduced drastically if one can settle for $\ell$-wise independence in the analysis using standard techniques from the pseudorandomness literature (see, e.g. Vadhan's monograph).	NEAR-OPTIMAL ALGORITHMS FOR PRIVATE ESTIMATION AND SEQUENTIAL TESTING OF COLLISION PROBABILITY Anonymous authors Paper under double-blind review ABSTRACT We present new algorithms for estimating and testing collision probability, a fundamental measure of the spread of a discrete distribution that is widely used in many scientific fields. We describe an algorithm that satisfies $(\alpha, \beta)$-local differential privacy and estimates collision probability with error at most $\varepsilon$ using $\tilde{O}\left(\frac{\log(1/\beta)}{\alpha^2\varepsilon^2}\right)$ samples for $\alpha \leq 1$, which improves over previous work by a factor of $\frac{1}{\alpha^2}$. We also present the first sequential testing algorithm for collision probability, which can distinguish between collision probability values that are separated by $\varepsilon$ using $\tilde{O}\left(\frac{1}{\varepsilon^2}\right)$ samples, even when $\varepsilon$ is unknown. Our algorithms have nearly the optimal sample complexity and in experiments we show that they require significantly fewer samples than previous methods. 1 INTRODUCTION A key property of a discrete distribution is how widely its probability mass is dispersed over its support. One of the most common measures of this dispersal is collision probability. Let $p = (p_1, \ldots, p_k)$ be a discrete distribution. The collision probability of $p$ is defined $$C(p) = \sum_{i=1}^{k} p_i^2.$$ Collision probability takes its name from the following observation. If $X$ and $X'$ are independent random variables with distribution $p$ then $C(p) = \Pr[X = X']$, the probability that the values of $X$ and $X'$ coincide. If a distribution is highly concentrated then its collision probability will be close to 1, while the collision probability of the uniform distribution is $1/k$. Collision probability has played an important role in many scientific fields, although each time it is rediscovered it is typically given a different name. In ecology, collision probability is called the Simpson index and serves as a metric for species diversity (Simpson [1949]; Lemster [2021]). In economics, collision probability is known as the Herfindahl–Hirschman index, which quantifies market competition among firms (Herfindahl [1997]), and also the Gini diversity index, a measure of income and wealth inequality (Gini [1912]). Collision probability is also known as the second frequency moment, and is used in database optimization engines to estimate self join size (Cormode & Garofalakis [2016]). In statistical mechanics, collision probability is equivalent to Tsallis entropy of second order, which is closely related to Boltzmann–Gibbs entropy (Tsallis [1988]). The negative logarithm of collision probability is Rényi entropy of second order, which has many applications, including assessing the quality of random number generators (Skorski [2017]) and determining the number of reads needed to reconstruct a DNA sequence (Motahari et al., 2013). Collision probability has also been used by political scientists to determine the effective size of political parties (Laakso & Taagepera [1979]). Collision probability is not equivalent to Shannon entropy, the central concept in information theory and another common measure of the spread of a distribution. However, collision probability has a much more intuitive interpretation, and is also easier to estimate. Specifically, estimating the Shannon entropy of a distribution with support size $k$ requires $\Omega\left(\frac{k}{\log k}\right)$ samples (Valiant & Valiant... while the sample complexity of estimating collision probability is independent of $ k $. Additionally, the negative logarithm of the collision probability of a distribution is a lower bound on its Shannon entropy, and this lower bound becomes an equality for the uniform distribution. 1.1 Our contributions We present novel algorithms for estimating and testing the collision probability of a distribution. Private estimation: We give an algorithm for estimating collision probability that satisfies $(\alpha, \beta)$-local differential privacy.$^1$ As in previous work, our algorithm is non-interactive, which means that there is only a single round of communication between users and a central server, and communication-efficient, in the sense that each user sends $ O(1) $ bits to the server (in fact, just 1 bit). If $ \alpha \leq 1 $ then our algorithm needs $ \tilde{O}\left(\frac{\log(1/\beta)}{\alpha^2 \varepsilon^2}\right) $ samples to output an estimate that has $ \varepsilon $ additive error, which nearly matches the optimal sample complexity and improves on previous work by an $ O\left(\frac{1}{\varepsilon^2}\right) $ factor (Bravo-Hermsdorff et al., 2022). Sequential testing: We give an algorithm for determining whether collision probability is equal to a given value $ c_0 $ or differs from $ c_0 $ by at least $ \varepsilon > 0 $, assuming that one of those conditions holds. Our algorithm needs $ O\left(\frac{1}{\varepsilon^2}\right) $ samples to make a correct determination, which nearly matches the optimal sample complexity. Importantly, $ \varepsilon $ is not known to the algorithm. In other words, the algorithm automatically adapts to easy cases by drawing fewer samples. While sequential testing algorithms have been developed for many distributional properties, such as total variation distance (Daskalakis & Kawase, 2017), as far as we know there is no existing sequential testing algorithm for collision probability. Instead, previous work has focused on the batch setting, in which the number of samples is specified in advance (Canonne, 2022a). All of our theoretical guarantees hold with high probability, and we present numerical simulations showing that our algorithms use significantly fewer samples than existing methods. For simplicity, in the main body of this paper we state all theorems using big-$ O $ notation and argue for their correctness with proof sketches only, reserving more detailed theorem statements and proofs for the Appendix. 2 Related work The collision probability of a distribution is equal to its second frequency moment, and frequency moment estimation has been widely studied in the literature on data streams, beginning with the seminal work of Alon et al. (1999). Locally differentially private estimation of frequency moments was first studied by Butucea & Issartel (2021), who gave a non-interactive mechanism for estimating any positive frequency moment. The sample complexity of their mechanism depends on the support size of the distribution, and they asked whether this dependence could be removed. Their conjecture was affirmatively resolved for collision probability by Bravo-Hermsdorff et al. (2022), but removing the dependence on support size led to a much worse dependence on the privacy parameter. It has remained an open question until now whether this trade-off is necessary. Property and closeness testing has a rich literature (Acharya et al., 2019a; 2013; Diakonikolas et al., 2015; Goldreich & Ron, 2000; Canonne, 2022b), but the sequential setting is studied much less intensively. Existing algorithms for sequential testing almost always define closeness in terms of total variation distance, which leads to sample complexities on the order $ O(\sqrt{k}/\epsilon^2) $, where $ k $ is the support size of the distribution and the distribution is separated from the null hypothesis by $ \epsilon $ in terms of total variation distance (Daskalakis & Kawase, 2017; Oukhn et al., 2021). By contrast, all of our results are entirely independent of $ k $, making our approach more suitable when the support size is very large. There are several batch testing approaches which are based on collision statistics. Most notably, the optimal uniform testing algorithm of Paninski (2003) distinguishes the uniform distribution from a distribution that is $ \epsilon $ far from uniform in terms of total variation distance with a sample complexity $ \Theta(\sqrt{k}/\epsilon^2) $. However, in the batch setting, the parameter $ \epsilon $ is given to the testing algorithm as input. $^1$Instead of denoting the privacy parameters by $ \varepsilon $ and $ \delta $, as is common in the privacy literature, we will use them to denote error and probability, as is common in the statistics literature. 3 PRELIMINARIES We study two problems related to learning the collision probability $ C(p) = \sum_i p_i^2 $ of an unknown distribution $ p = (p_1, \ldots, p_k) $. In the private estimation problem, a set of $ n $ users each possess a single sample drawn independently from distribution $ p $. We are given an error bound $ \varepsilon \geq 0 $ and confidence level $ \delta \in [0, 1] $. A central server must compute an estimate $ \hat{C} $ that satisfies $ \|\hat{C} - C(p)\| \leq \varepsilon $ with probability at least $ 1 - \delta $ while preserving the privacy of the users’ samples. A mechanism is a distributed protocol between the server and the users that privately computes this estimate. The execution of a mechanism can depend on the samples, and the output of a mechanism is the entire communication transcript between the server and the users. Mechanism $ M $ satisfies $(\alpha, \beta)$-local differential privacy if for each user $ i $ and all possible samples $ x_1, \ldots, x_n, x'_i $, we have \[ \Pr[M(x_1, \ldots, x_n) \in O] \leq e^\alpha \Pr[M(x_1, \ldots, x_{i-1}, x'_i, x_{i+1}, \ldots, x_n) \in O] + \beta, \] where $ O $ is any set of possible transcripts between the server and the users. In other words, if the privacy parameters $ \alpha $ and $ \beta $ are small then changing the sample of a single user does not significantly alter the distribution of the mechanism’s output. Local differential privacy is the strongest version of differential privacy, and is suitable for a setting where the server is untrusted (Dwork et al., 2014). The sample complexity of the mechanism is the number of users $ n $. In the sequential testing problem, we are given a confidence level $ \delta \in [0, 1] $ and the promise that exactly one of the following two hypotheses hold: The null hypothesis is that $ C(p) = c_0 $, while the alternative hypothesis is that $ \|C(p) - c_0\| \geq \varepsilon > 0 $. An algorithm must decide which hypothesis is correct based on samples from $ p $. Instead of fixing the number of samples in advance, the algorithm draws independent samples from $ p $ one at a time, and after observing each sample decides to either reject the null hypothesis or to continue sampling. If the null hypothesis is false then the algorithm must reject it, and if the null hypothesis is true then the algorithm must not stop sampling, and each of these events must occur with probability at least $ 1 - \delta $. Importantly, while $ c_0 $ is known to the algorithm, $ \varepsilon $ is not known, and thus the algorithm must adapt to the difficulty of the problem. The sample complexity of the algorithm is the number of observed samples $ N $ if the null hypothesis is false, a random variable. 4 PRIVATE ESTIMATION In this section we describe a distributed protocol for privately estimating the collision probability of a distribution. In our protocol, a set of users each draw a sample from the distribution, and then share limited information about their samples with a central server, who computes an estimate of the collision probability while preserving the privacy of each user’s sample. As discussed in Section 1, the collision probability of a distribution is the probability that two independent samples from the distribution will coincide. Therefore the most straightforward strategy the server could employ would be to collect all the users’ samples and count the number of pairs of samples containing a collision. However, this approach would not be privacy-preserving. Instead, in Mechanism 1 below, each user applies a one-bit hash function to their private sample and shares only their hash value with the server. The server counts the number of collisions among all pairs of hash values and then applies a bias correction to form an estimate of the collision probability. To increase the robustness of this estimate, the server first partitions the hash values into groups and uses the median estimate from among the groups. The hashing procedure in Mechanism 1 is carefully designed to both preserve user privacy and also yield an accurate estimate. On the one hand, if each user privately chose an independent hash function, then their hash values would be entirely uncorrelated and contain no useful information about the underlying distribution. On the other hand, if every user applied the same hash function to their sample, then the server could invert this function and potentially learn some user’s sample. Instead, in Mechanism 1, the server sends the same hash function to all users, but each user prepends their sample with a independently chosen salt, or random integer, before applying the hash function. Salts are commonly used in cryptographic protocols to enhance security, and they play a similar role in our mechanism. The number of possible salts serves as a trade-off parameter between the privacy and accuracy of our mechanism, with more salts implying a stronger privacy guarantee. Mechanism 1 Private estimation for collision probability Given: Number of users $ n $, confidence level $ \delta \in [0, 1] $, privacy parameters $ \alpha \geq 0, \beta \in [0, 1] $. 1: Server transmits random hash function $ h : \{0, 1\}^* \mapsto \{0, 1\} $ to each user. 2: Each user $ i $ chooses salt $ s_i $ uniformly at random from $ \{1, \ldots, r\} $, where $ r = 6 \left( \frac{e^\alpha + 1}{e^\alpha - 1} \right)^2 \log \frac{4}{\beta} $. 3: Each user $ i $ draws sample $ x_i $ from distribution $ p $. 4: Each user $ i $ sends hash value $ v_i = h(s_i, x_i) $ to the server, where $ s_i, x_i $ is the binary encoding of $ s_i $ prepended to $ x_i $ and separated by a delimiter. 5: Server partitions users into $ k = 8 \log \frac{1}{\delta} $ groups of size $ m = \frac{n}{k} $ each. 6: Server computes the all-pairs hash value collision frequency \[ \bar{c}_g = \frac{2}{m(m-1)} \sum_{i,j \in I_g, i<j} 1 \{v_i = v_j\} \] for each group $ g $, where $ I_g $ is the set of users in group $ g $. 7: Server lets \[ \hat{c}_g = r(2\bar{c}_g - 1) \] be the bias-corrected estimate for each group $ g $. 8: Server outputs $ \hat{C} $, the median of the $ \hat{c}_g $'s. The theorems in this section provide guarantees about the privacy and accuracy of Mechanism 1. Theorem 1. Mechanism 1 satisfies $ (\alpha, \beta) $-local differential privacy. Proof sketch. We show that the communication transcript between the server and the users is not very likely to be different if a single user changes their sample. Note that the communication transcript consists of the random hash function chosen by the server and the users’ hash values. Suppose for now that the hash function is fixed. Each user’s choice of a random salt induces a distribution on their hash value, and this distribution can change if the user changes their sample. If the distribution changes too drastically then the mechanism will not be private. However, in expectation over the choice of the hash function, the distribution is always uniform, and deviations from this expectation will be small with high probability if the number of possible salts is sufficiently large. More concretely, note that the number of possible salts $ r $ in Mechanism 1 increases as the privacy parameters $ \alpha $ and $ \beta $ decrease. Finally, since the hash function is chosen independently of the samples, the hash function reveals no information about the samples by itself. Theorem 2. If the number of samples $ n $ satisfies \[ n \geq \Omega \left( \left( \frac{e^\alpha + 1}{e^\alpha - 1} \right)^2 \log \frac{4}{\beta} \log \frac{1}{\delta} \right) \] then the estimate $ \hat{C} $ output by Mechanism 1 satisfies $ \|\hat{C} - C(p)\| \leq \varepsilon $ with probability $ 1 - \delta $. Additionally, if $ \alpha \leq 1 $ then it suffices that \[ n \geq \Omega \left( \frac{\log \frac{1}{\beta} \log \frac{1}{\delta}}{\alpha^2 \varepsilon^2} \right). \] Proof sketch. The first step of the argument is to relate the likelihood of a hash collision to that of the underlying sample collision. It is not hard to see that if $ x_i \neq x_j $ then $ \Pr[v_i = v_j] = \frac{1}{2} $, while if $ x_i = x_j $ then $ \Pr[v_i = v_j] = \frac{1}{2} + \frac{1}{r} $, because two users with the same sample and same salt are guaranteed to produce the same hash value. This discrepancy allows us to use the number of hash collisions as an estimator of the number of sample collisions. In particular, it implies that each group estimate $ \hat{c}_g $ is an unbiased estimate of $ C(p) $. Next we bound the variance of each $ \hat{c}_g $. Clearly $ \text{Var}[\hat{c}_g] = O(r^2) \text{Var}[c_g] $. Bounding the variance $ \text{Var}[c_g] $ is non-trivial, because the $ v_i $'s are not independent, since they are correlated by the random choice of the hash function. By the law of total variance we have \[ \text{Var}[c_g] = E[\text{Var}[c_g \| h]] + \text{Var}[E[c_g \| h]]. \] Since the $ v_i $'s are independent for a given hash function, the first term can be bounded by applying Hoeffding’s theorem for U-statistics. The second term can be bounded by a fairly direct calculation. Having shown that the $ \hat{c}_g $'s are unbiased estimates of collision probability, and also having shown that each of their variances is bounded, it remains to show that their median is concentrated about their mean. This concentration follows from the analysis of the median-of-means estimator (Lugosi & Mendelson 2019). 4.1 LOWER BOUND The next theorem proves that the sample complexity bound in Theorem 2 is tight for small $ \alpha $ up to logarithmic factors. Theorem 3. Let $ \hat{C}_{\alpha,n}(p) $ be a collision probability estimate returned by an $ (\alpha, 0) $-locally differentially private mechanism that draws $ n $ samples from distribution $ p $. If $ \alpha \leq 1 $ and $ n \in o\left(\frac{1}{\alpha^2\varepsilon^2}\right) $ then there exists a distribution $ p $ such that \[ E\left[\|\hat{C}_{\alpha,n}(p) - C(p)\|\right] \geq \varepsilon. \] Proof sketch. We apply a technique due to Duchi et al. (2016) for proving minimax lower bounds for locally differentially private estimation. Their technique is a private version of Le Cam’s two-point method (LeCam 1973). It follows from Proposition 1 due to Duchi et al. (2016) that for all distributions $ p_0, p_1 $ there exists distribution $ p $ such that \[ E\left[\|\hat{C}_{\alpha,n}(p) - C(p)\|\right] \geq \frac{\|C(p_0) - C(p_1)\|}{2} \left(1 - \sqrt{2\alpha^2 n D_{KL}(p_0 \|\| p_1)}\right). \] Thus if there exist $ p_0 $ and $ p_1 $ such that $ D_{KL}(p_0 \|\| p_1) \leq O\left(\frac{1}{\alpha^2 n}\right) $ and $ \|C(p_0) - C(p_1)\| \geq \Omega\left(\frac{1}{\alpha \sqrt{n}}\right) $ then the above lower bound is $ \Omega\left(\frac{1}{\alpha \sqrt{n}}\right) $, which suffices to prove the theorem. We give an explicit construction of $ p_0 $ and $ p_1 $ in the Appendix. Briefly, $ p_0 $ places probability mass $ \frac{1}{2} $ on one element and uniformly distributes the remaining mass on the other $ k - 1 $ elements, while $ p_1 $ is nearly the same as $ p_0 $ except for a $ \Theta\left(\frac{1}{\alpha \sqrt{n}}\right) $ perturbation applied to each probability. 4.2 EFFICIENT IMPLEMENTATION In Mechanism T, the server computes the all-pairs hash collision frequency per group. If each group contains $ m $ samples, a naive implementation would require $ \Omega(m^2) $ time per group. The next theorem shows how this can be reduced to $ O(m) $ time per group by computing the histogram of hash values. Theorem 4. For any values $ v_1, \ldots, v_m $ if $ \bar{c} = \frac{2}{m(m-1)} \sum_{i<j} 1\{v_i = v_j\} $ is the all-pairs collision frequency and $ \hat{n}_v = \sum_i 1\{v_i = v\} $ is the multiplicity of value $ v $ then \[ \bar{c} = \frac{1}{m(m-1)} \sum_v \hat{n}_v^2 - \frac{1}{m-1}. \] 4.3 COMPARISON TO PREVIOUS WORK Butucea & Issartel (2021) gave a non-interactive $ (\alpha, 0) $-locally differentially private mechanism for estimating collision probability with sample complexity $ \tilde{O}\left(\frac{(\log k)^2}{\varepsilon^2 \alpha^2}\right) $ and communication complexity $ O(k) $. Bravo-Hermsdorff et al. (2022) gave a non-interactive mechanism with the same privacy guarantee, sample complexity $\tilde{O}\left(\frac{1}{\alpha^2 \varepsilon^2}\right)$, and communication complexity $O(1)$. Thus the latter mechanism is better suited to distributions with very large support sizes, but is a worse choice when the privacy parameter $\alpha$ is very small. Our mechanism combines the advantages of these mechanisms, at the expense of a slightly weaker privacy guarantee and an additional $\tilde{O}(\log \frac{1}{\delta})$ samples. Notably, the earlier mechanism due to Bravo-Hermsdorff et al. (2022) is also based on counting collisions among salted hash values. But there are key differences between the mechanisms which lead to our improved sample complexity. In their mechanism, the server assigns salts to the users, each user adds noise to their hash value, and the server counts hash collisions among $\frac{n}{2}$ disjoint user pairs. In our mechanism, the salts are chosen privately, no additional noise is added to the hash values, and the server counts hash collisions among all $\binom{n}{2} = O(n^2)$ user pairs. Using all available pairs to count collisions is a more efficient use of data (although it significantly complicates the analysis, as the pairs are not all independent), and choosing the salts privately eliminates the need for additional randomness, which improves the accuracy of the estimate. 5 Sequential Testing In this section we describe an algorithm for sequentially testing whether $C(p) = c_0$ (the null hypothesis) or $\|C(p) - c_0\| \geq \varepsilon > 0$ (the alternative hypothesis), where $c_0$ is given but $\varepsilon$ is unknown. Algorithm 2 below draws samples from the distribution $p$ one at a time. Whenever the algorithm observes a sample $x_i$, it updates a running estimate of $\|C(p) - c_0\|$ based on the all-pairs collision frequency observed so far. The algorithm compares this estimate to a threshold that shrinks like $\Theta(\sqrt{\frac{i-1}{i} \log \log i})$ and rejects the null hypothesis as soon as the threshold is exceeded. Although our algorithm is simple to describe, its proof of correctness is non-trivial, as it involves showing that a sequence of dependent random variables (the running estimates) become concentrated. Our proof uses a novel decoupling technique to construct martingales based on the running estimates. Algorithm 2 Sequential testing of collision probability Given: Null hypothesis value $c_0$, confidence level $\delta \in [0, 1]$. 1: for $i = 1, 2, 3, \ldots$ do 2: Draw sample $x_i$ from distribution $p$. 3: Let $T_i = \sum_{j=1}^{i-1} 1\{x_i = x_j\} - 2(i-1)c_0$. 4: if $\left\|\frac{2}{i(i-1)} \sum_{j=1}^{i-1} T_j\right\| > 3.2 \sqrt{\frac{\log \log i + 0.72 \log (20.8/\delta)}{i}}$ then 5: Reject the null hypothesis. 6: end if 7: end for The next theorem provides a guarantee about the accuracy of Algorithm 2. Theorem 5. If $C(p) = c_0$ then Algorithm 2 does not reject the null hypothesis with probability $1 - \delta$. If $\|C(p) - c_0\| \geq \varepsilon$ then Algorithm 2 rejects the null hypothesis after observing $N$ samples, where $$N \in O\left(\frac{1}{\varepsilon^2} \log \log \frac{1}{\varepsilon} \log \frac{1}{\delta}\right)$$ with probability $1 - \delta$. The $\log \log \frac{1}{\varepsilon}$ factor in Theorem 5 results from our application of a confidence interval due to Howard et al. (2021) that shrinks like $\Theta(\sqrt{\frac{i-1}{i} \log \log i})$. Note that $\log \log \frac{1}{\varepsilon} < 4$ if $\varepsilon \geq 10^{-10}$, so this factor is negligible for nearly all problem instances of practical interest. Note that Bravo-Hermsdorff et al.'s original NeurIPS paper claimed $\tilde{O}\left(\frac{1}{\alpha^2 \varepsilon^2}\right)$ sample complexity, but a more recent version on Arxiv claims $\tilde{O}\left(\frac{1}{\alpha^2 \varepsilon^2}\right)$ sample complexity and explains that the original version contained mistakes. See References for a link to the Arxiv version. Proof sketch of Theorem 5. First note that $ T_1, T_2, \ldots $ which are used in Line 3 of Algorithm 2 are dependent sequences, so $ T_i $ depends on all $ x_1, \ldots, x_i $, which prevent us from computing a concentration bound for it. Therefore we shall apply a decoupling technique to derive a martingale sequence. Let us define $ \tilde{U}_m := U(X_1, \ldots, X_m) = \sum_{i<j} g(X_i, X_j) $ with \[ g(X_i, X_j) = 1 \{ X_i = X_j \} - E[1 \{ X_i = X_j \} \| X_i] - E[1 \{ X_i = X_j \} \| X_j] + E[1 \{ X_i = X_j \}] = 1 \{ X_i = X_j \} - Pr(X_i = X_j \| X_i) - Pr(X_i = X_j \| X_j) + c_0 . \] This decoupling technique is motivated by Theorem 8.1.1 of Tsypkov (2008) since the kernel function $ g $ has become centered and degenerated, i.e. $ E[g(X_i, X_j) \| X_j] = E[g(X_i, X_j) \| X_i] = 0 $ which implies that $ \tilde{U}_n $ is a zero-mean martingale with $ m \geq 2 $. The empirical sequence is $ \tilde{u}_m = \sum_{i=1}^m y_m $ with \[ y_j = \sum_{i=1}^{m-1} 1 \{ x_i = x_j \} - \sum_{i=1}^{m-1} p_{x_i} - (m-1)p_{x_j} + (m-1)c_0 \] which has bounded differences such that $ \|\tilde{U}_k - \tilde{U}_{k-1}\| = \|Y_k\| \leq 4m $ and $ y_1 = 0 $. However we cannot compute this empirical sequence, since the parameters of distribution are not known. As a remedy, we further decompose $ \tilde{U}_n $ as the sum of two sequences based on the observation that \[ E[p_{X_i}] = \sum_i p_{x_i}^2 = c_0 \] which implies that $ \sum_{i=1}^m (p_{X_i} - c_0) $ is again a zero-mean martingale sequence with the same filtration $ F_m $ such that the difference $ \|p_{X_i} - c_0\| < 1 $ for all $ i $. This motivates the following decomposition of $ \tilde{U}_n $ as \[ Y_j = \sum_{i=1}^{j-1} 1 \{ X_i = X_j \} - 2(j-1)c_0 + 2(j-1)c_0 - \sum_{i=1}^{j-1} p_{X_i} - (j-1)p_{X_j} \] Note that $ T_m $, used in Algorithm 2, can be computed and it is a zero-mean martingale sequence up to an error term $ E_n $ which cannot be computed, since the parameters of the underlying distribution $ p $ is not available. However $ E_n $ can be again decomposed into sequence of sums of zero mean-mean terms which we can upper bound with high probability. Important to note that if $ H_0 : c_0 = 1/K $, the error term is equal to zero in any time step, i.e. $ E_n = 0, \forall n \in [1, 2, \ldots] $, therefore $ T_m $ is a zero-mean martingale itself. Finally, we rely on the work of Howard et al. (2021) in which a sequence of confidence intervals is introduced for martingales that hold uniformly for each time step, even with random stopping time. We remark that our proof technique bears some superficial resemblance to the approach used in recent work by Oufkir et al. (2021). They make use of the fact that for any random variable $ T $ taking values from $ \mathbb{N} $ and for all $ T \in \mathbb{N}_+ $, it holds that $ E[T] \leq N + \sum_{t>N} P(T \geq t) $. Then with a carefully selected $ N $ and Chernoff bounds with infinite many applications of union bound implies upper bound on the expected sample complexity. By contrast, we construct a test martingale that is specific to collision probability and apply an anytime or time-uniform concentration bound to the martingale introduced by Waudby-Smith & Ramdas (2020). 5.1 Lower bound The next theorem proves that sample complexity bound in Theorem 5 is tight up to log-log factors. Theorem 6. Let $ N $ be the number of samples observed by a sequential testing algorithm for collision probability. For all $ \varepsilon, \delta \in [0, 1] $ there exists a distribution $ p $ and $ c_0 \in [0, 1] $ such that $ \|C(p) - c_0\| \geq \varepsilon $ and if the algorithm rejects the null hypothesis with probability $ 1 - \delta $ then \[ E[N] \geq \Omega \left( \frac{\log(1/\delta)}{\varepsilon^2} \right). \] Proof sketch. Our proof is based on a reduction to the problem of identity testing and a lower bound for that problem due to Oufkir et al. (2021). In an identity testing problem we are given a distribution $ p_0 $ and sample access to a distribution $ p_1 $ and the goal is to decide whether $ p_0 = p_1 $. or $\\|p_0 - p_1\\|_1 \geq \varepsilon > 0$. Oufkir et al. (2021) proved that if $\\|p_0 - p_1\\|_1 \geq \varepsilon$ then the number of samples $N$ needed to make a correct decision must satisfy $E[N] \geq \frac{\log(1/(3\varepsilon))}{D_{KL}(p_0\|\|p_1)}$. We complete the proof by showing that there exist distributions $p_0$ and $p_1$ such that $\\|p_0 - p_1\\|_1 \geq \Omega(\varepsilon)$, $\|C(p_0) - C(p_1)\| \geq \Omega(\varepsilon)$ and $D_{KL}(p_0\|\|p_1) \leq O(\varepsilon^2)$. An explicit construction of $p_0$ and $p_1$ is in the Appendix, and they are the same distributions as in the proof of Theorem 3. 6 EXPERIMENTS We compare our mechanism for private collision probability estimation (Mechanism 1) to the recently proposed mechanism from Bravo-Hernsdorff et al. (2022). As discussed in Section 4.3 we expect Mechanism 1 to outperform their mechanism when the support size of the distribution is large and the privacy requirement is strict. We also compare to an indirect method: Privately estimate the distribution itself, and then compute the collision probability of the estimated distribution. In our experiments we use an open-source implementation of a private heavy hitters algorithm due to Cormode et al. (2021). In Figure 1, we use each mechanism to privately estimate the collision probability of two distributions supported on 1000 elements: the uniform distribution ($p_i = 1/k$) and the power law distribution ($p_i \propto 1/i$). Our simulations show that Mechanism 1 has significantly lower error for small values of the privacy parameters $\alpha$ and $\beta$. ![Figure 1](https://github.com/Samuel-Maddock/pure-LDP) Figure 1: Sample complexity of private collision probability estimation mechanisms for $\alpha = 0.25$. Both mechanisms use the MD5 hash function and confidence level $\delta = 0.1$. For Mechanism 1 we let $\beta = 10^{-5}$. Error bars are one standard error. We next evaluate our sequential testing algorithm (Algorithm 2). Since we are not aware of any existing algorithm for sequential testing of collision probability, we compare Algorithm 2 to two batch testing algorithms, both of which are described in a survey by Canonne (2022a): - Plug-in: Form empirical distribution $\hat{p}$ from samples $x_1, \ldots, x_n$, and let $\hat{C} = C(\hat{p})$. - U-statistics: Let $\hat{C} = \frac{2}{n(n-1)} \sum_{i<j} 1 \{x_i = x_j\}$ be the all-pairs collision frequency. Each batch testing algorithm takes as input both the null hypothesis value $c_0$ and a tolerance parameter $\varepsilon$, and compares $\|\hat{C} - c_0\|$ to $\varepsilon$ to decide whether to reject the null hypothesis $C(p) = c_0$. The sample complexity of a batch testing algorithm is determined via worst-case theoretical analysis in terms of $\varepsilon$ (see Appendix). On the other hand, sequential testing algorithms automatically adapt their sample complexity to the difference $\|C(p) - c_0\|$. In Figure 2, we evaluate batch and sequential testing algorithms on both the uniform distribution and power law distributions. We use 20 different support sizes for each distribution, evenly spaced on a log scale between 10 and $10^6$ inclusively. Varying the support size also varies $\|C(p) - c_0\|$. As expected, when $\|C(p) - c_0\|$ is large, our sequential testing algorithm requires many fewer samples than the batch algorithm to reject the null hypothesis, and as $\|C(p) - c_0\|$ shrinks the number of samples required sharply increases (see grey areas in Figure 2). In all cases our sequential testing algorithm is never outperformed by the batch testing algorithms. ![Figure 2](image) Figure 2: Sample complexity of the sequential tester compared to the sample complexity of the batch testers. For the batch testers, the tolerance parameter $\epsilon$ is set to 0.01. Note that in Figure 3, the plug-in tester has a worse sample complexity than the U-statistics tester. Since these sample complexities are determined by theoretical analysis, we experimentally confirmed that this discrepancy is not simply an artifact of the analysis. In Figure 3, we run simulations comparing the algorithms in terms of their error $\|\hat{C} - C(p)\|$, and find that the plug-in tester is also empirically worse than the U-statistics tester. ![Figure 3](image) Figure 3: Empirical absolute error of plug-in and U-statistic estimators when the data is generated from uniform distribution and power law with domain size 1000. ### 7 CONCLUSIONS AND FUTURE WORK We introduced a locally differentially private estimator for collision probability that is near-optimal in a minimax sense and empirically superior to the state-of-the-art method introduced by Bravo-Hermsdorff et al. (2022). Our method is based on directly estimating the collision probability using all pairs of observed samples, unlike in previous work. 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KbDzdqevfV	Bellman equations to characterise safety and equivalently reachability properties have been already developed in the literature, and also in the stochastic setting [1,2]. The one proposed by the authors is an extension of those ones, with the added constraint that only actions that have probability 1 of being safe are admitted. This should be clarified and discussed in the paper.	CORRECT-BY-DESIGN SAFETY CRITICS USING NON-CONTRACTIVE BINARY BELLMAN OPERATORS Anonymous authors Paper under double-blind review ABSTRACT The inability to naturally enforce safety in Reinforcement Learning (RL), with limited failures, is a core challenge impeding its use in real-world applications. One notion of safety of vast practical relevance is the ability to avoid (unsafe) regions of the state space. Though such a safety goal can be captured by means of an action-value-like function, a.k.a. safety critics, the associated operator lacks the desired contraction and uniqueness properties that the classical Bellman operator enjoys. In this work, we overcome the non-contractiveness of safety critic operators by leveraging the fact that safety is a binary property. To that end, we study the properties of the binary safety critic associated with a deterministic dynamical system that seeks to avoid reaching an unsafe region. We formulate the corresponding binary Bellman equation (B2E) for safety and study its properties. While the resulting operator is still non-contractive, we provide a full characterization of the fixed points representing—except for a spurious solution—maximal persistently safe regions of the state space that can always avoid failure. Interestingly, while maximality is often a desired notion for performance, in the context of safety, it means that the learned classification boundary is dangerously close and often crosses the region where failure is unavoidable. We thus further propose a one-sided version of the B2E that allows for more robust fixed points that are non-maximal. Finally, we provide an algorithm that, by design, leverages axiomatic knowledge of safe data points to avoid spurious fixed points. We provide initial empirical validation of our theory, showing how the proposed safety critic outperforms existing solutions, particularly regarding the number of samples (and failures) needed to secure safe policies. 1 INTRODUCTION The last decade has witnessed a resurgence of Reinforcement Learning (RL) as a core enabler of Artificial Intelligence (AI). Today, RL algorithms can provide astonishing demonstrations of super-human performance in multiple settings, such as Atari (Mnih et al., 2015), Go (Silver et al., 2016), StarCraft II (Vinyals et al., 2017), and even poker (Nichols et al., 2019). However, this super-human success in RL is overwhelmingly limited to virtual domains (particularly games), where not only one has a vast amount of data and computational power, but also there is little consequence to failure in achieving a task. Unfortunately, physical domain applications (autonomous driving, robotics, personalized medicine) lack most of these qualities and are particularly sensitive to scenarios where the consequences of poor decision-making are catastrophic (Yu et al., 2021; Brunke et al., 2022). Guaranteeing safety in an RL setting is a challenging task, as agents often lack a priori knowledge of the safety of states and actions (Gu et al., 2022). Inspired by these challenges, numerous methods have been proposed to imbue RL methods with safety constraints, including expectation constraints (Paternain et al., 2022; Castellano et al., 2023), probabilistic/conditional value at risk constraints (Chow et al., 2017; Chen et al., 2023), and stability constraints (Li & Bastani, 2020; Taylor et al., 2020), among others. Such methods naturally lead to different safety guarantees, some of which can be theoretically characterized (Robey et al., 2020; Castellano et al., 2022). However, the majority of these methods fail to capture the safety-critical nature of some types of events that must be avoided at all costs, i.e., with probability one. One type of safety constraint of practical relevance in safety-critical applications is reachability constraints (e.g., Bertsekas (1972); Sontag (2013), Ch. 3; Bansal et al., 2017), wherein one seeks to avoid regions of the state space that are associated with failure events by computing sets that are either, persistently safe (a.k.a. control invariant safe sets (Gurriet et al., 2018)), i.e., regions of the state that can avoid failure regions for all times by proper choice of actions, or unsafe regions (a.k.a. as backward reachable tubes [Mitchell] (2007)) where failure is unavoidable irrespectively of the actions taken. Recent research efforts incorporating such constraints in RL problems have shaped the notion of safety critics ([Fisac et al.] (2019), [Srinivasan et al.] (2020), [Thananjeyan et al.] (2021)), which aim to compute action value-like functions that, based on information about either the (signed) distance to failure or a logical fail/not fail feedback, predict whether certain state-action pair is safe to take or is likely to lead to catastrophic failures. Unfortunately, the computation and learning of safety critics is a challenging task since their corresponding Bellman-like equations (and associated operators) lack typical uniqueness (resp. contraction) properties that guarantee the validity of the solution (and convergence of RL algorithms). As a result, most works seek to compute approximate safety critics by introducing an artificial discount factor ([Fisac et al.] (2019), [Hsu et al.] (2021)). This approximation, however, can have drastic effects on the accuracy of the critic, as approximately safe sets are not, by design, safe. Contributions of our work In this work, we seek to overcome the difficulties in computing accurate safety critics by developing supporting theory and algorithms that allow us to learn accurate and more robust safety critics directly from the original non-contractive safety critic operator. Precisely, we consider a setting with deterministic, continuous state dynamics that are driven by stochastic policies on discrete action spaces, and model safety as a binary (safe/unsafe) quantity. Building on the literature of risk-based safety critics, we develop a deeper theoretical understanding of the properties of the corresponding binary safety (action-)value function and how to exploit them to learn accurate safety critics. In doing so, we make the following contributions. • Characterization of solutions to the binary Bellman equations for safety We study the properties of the action-value function associated with the binary safe/unsafe feedback and formulate a binary Bellman equation (B2E) that such function must satisfy. This binary Bellman equation is undiscounted and has a non-contractive operator with multiple fixed points. Nevertheless, we show (Theorem 1) that all (but one) of the possibly infinite solutions to the B2E represent regions of the state space that are: (i) persistently safe regions that can avoid failure for all future times and (ii) maximal, in the sense that no state that is declared to be unsafe can reach the declared safe region. • One-sided binary Bellman equations to compute non-maximal, persistently safe regions While non-spurious solutions to the B2E represent valid, persistently safe regions, the maximality property ensures that the classification boundary often lies exactly at the edge of the unsafe region, making such a solution non-robust. This motivates the introduction of a one-sided B2E (O-B2E) that only requires solutions to satisfy the persistently safe property (and not maximality) (Theorem 2). The novel O-B2E induces a set-valued operator that drastically increases the number of fixed points and allows for solutions whose classification boundary has a larger margin from the truly unsafe boundary. • Algorithm for learning fixed points of a non-contractive set-valued operator Finally, we provide an algorithm that is able to find a fixed point of the novel set-valued operator despite the lack of contraction. Our algorithm has two distinctive features that make this possible. First, it uses axiomatic data points, i.e., points of the state space that are a priori known to be safe. Secondly, it uses a classification loss that enforces self-consistency of the one-sided Bellman equation across samples. Preliminary numerical evaluations indicate that our proposed methodology outperforms a well-known safety critic ([Fisac et al.] (2019)) in a simple setup, and show good performance in a 32-dimensional environment. 2 Problem Formulation Environment We consider a Markov Decision Process $(S, A, F, G, i, \rho)$ where the state space $S$ is continuous and compact, the action space $A$ is discrete and finite, the map $F : S \times A \rightarrow S$ is a deterministic transition function. The set $G$ represents a set of “failure” states to be avoided. At each time step the agent receives as feedback the insecurity of state $s_t$, that is $i(s_t) = I\{s_t \in G\} \in \{0, 1\}$. Episodes start at a state $s_0 \sim \rho$ and run indefinitely or end when the system enters $G$. Policies We consider stochastic, stationary policies $\pi : S \rightarrow \Delta_A$, and denote $\pi(a\|s)$ the probability of picking $a \in A$ when at state $s \in S$. Since $A$ is discrete and finite, and the transition dynamics are deterministic, the set of reachable states starting from any $s$ is finite as well, as defined next. Figure 1: The optimal $b^$ describes different regions of the state space. The set $\mathcal{G}$ (solid, dark red) is the one to be avoided at all times. Due to the system dynamics, there is a region of the state space $\mathcal{R}(\mathcal{G})$ (shaded red) such that any trajectory starting there (e.g., from $s_0$) will inevitably enter $\mathcal{G}$. For any point in its complement $\mathcal{S}_{\text{safe}}$ (e.g., $s_1$), the optimal policy avoids $\mathcal{G}$ at all times. Definition 1* ($t$-step reachable sets) For any policy $\pi$ and any state $s \in S$, the $t$-step reachable set from $s$ under $\pi$ is $\mathcal{F}_t^\pi(s) \triangleq \{ s' \in S : \Pr^\pi(s_t = s' \| s_0 = s) > 0 \}$. Similarly, for any $a \in A$ we define $\mathcal{F}_t^\pi(s, a) \triangleq \{ s' \in S : \Pr^\pi(s_t = s' \| s_0 = s, a_0 = a) > 0 \}$. Given these notions of reachable sets, we can define the binary safety value functions for any policy. Definition 2 (Binary safety value functions) The binary safety (action-)value function of policy $\pi$ at state $s$ (and action $a$) is: $$v^\pi(s) \triangleq \sup_{t \geq 0} \max_{s_t \in \mathcal{F}_t^\pi(s)} i(s_t), \quad b^\pi(s, a) \triangleq \sup_{t \geq 0} \max_{s_t \in \mathcal{F}_t^\pi(s, a)} i(s_t).$$ We choose the notation $b(\cdot, \cdot)$ instead of the usual $Q$ to emphasize that it is a binary action-value function. Note that $b^\pi(s, a) = 1$ if and only if starting from $(s, a)$ and following $\pi$, there is positive probability of entering $\mathcal{G}$. The optimal (action-)value functions are then defined. Definition 3 (Optimal binary value functions) For all $s \in S$ and $a \in A$, the optimal value and action-value functions are $v^(s) \triangleq \min_\pi v^\pi(s)$ and $b^(s, a) \triangleq \min_\pi b^\pi(s, a)$. Relationship between safety and the optimal binary functions These optimal value functions fully characterize the logical safe/unsafe nature of each state or state-action pair, and have nice interpretations in terms of how they partition the state-space, as illustrated in Fig. 1. Recall that the safety goal is to avoid $\mathcal{G}$. However, due to the MDP dynamics, this might not be possible for every state outside $\mathcal{G}$.\footnote{A car heading to a wall ($\mathcal{G}$) one meter away at 100mph will hit it, regardless of the actions taken.} A state $s$ is persistently safe if trajectories from $s$ can avoid $\mathcal{G}$ at all times—in other words, if $\exists a \in A : b^(s, a) = 0$. Conversely, a state $s$ is doomed to fail if $b^(s, a) = 1 \forall a \in A$. We use $\mathcal{R}(\mathcal{G})$ to denote this set of “unsafe states” that are doomed to fail. The complement of this set is the set of persistently safe states, and the “safe” actions for each state are given by: $$\mathcal{S}_{\text{safe}} = \{ s \in S : \min_{a \in A} b^(s, a) = 0 \}, \quad \mathcal{A}_{\text{safe}}(s) = \{ a \in A : b^(s, a) = 0 \}. \quad (2)$$ Just like in the standard RL setup, each (action-)value function has associated Bellman equations. Proposition 1 (Binary Bellman Equations) For any policy $\pi$, the following set of Bellman equations hold for all $s \in S$, for all $a \in A$: $b^\pi(s, a) = i(s) + (1 - i(s)) v^\pi(s')$, where $b^\pi(s, a) = i(s) + (1 - i(s)) v^\pi(s')$. In particular, any optimal policy satisfies: $$b^(s, a) = i(s) + (1 - i(s)) \min_{a' \in A} b^(s', a'). \quad (3)$$ Proof: See Appendix A.2 Unsafety as a logical OR The Bellman equation for the optimal $b^$ can be understood as: “an $(s, a)$ pair is unsafe ($b^(s, a) = 1$) if either: the current state is unsafe ($i(s) = 1$), OR it leads to an unsafe state later in the future ($\min_{a'} b^(s', a') = 1$).” Non-contractive Bellman operator The optimal binary function of equation (3) has an associated operator, acting on the space of functions $ B = \{ b : S \times A \rightarrow \{0,1\} \} $: \[ T : B \rightarrow B : (Tb)(s,a) = i(s) + (1 - i(s)) \min_{a' \in A} b(s',a') \quad \forall (s,a) \in S \times A \] One of the key features in the standard (discounted) Bellman equations for infinite-horizon problems is that it has an associated operator that is contractive (Bertsekas [2015] p.45), and as such, it admits a unique fixed point (the optimal value function). This is crucial for the application of value iteration procedures or for methods reliant on temporal differences (Schwartz [1993]). Surprisingly, the operator defined in equation (4) is non-contractive, and as such, it admits more fixed points than the optimal $ b^ $. In particular, there are even fixed points of equation (4) that have no physical meaning. We will soon see, however, that all—except for one—of them do have a physical interpretation. 2.1 Closely Related Work Control-theoretic approaches for computing $ S_{safe} $ Standard tools from Control Theory exist to approximate the safe regions corresponding to $ b^(\cdot,\cdot) $, both for linear (Grard et al. [2006]) and nonlinear dynamics (Mitchell & Templeton [2005]). The latter requires knowledge of the transition map $ F(\cdot,\cdot) $ along with the signed distance to the unsafe region (Mitchell et al. [2005]). This amounts to solving partial differential equations (PDEs) of the Hamilton-Jacobi-Isaacs (HJI) type (Bansal et al. [2017]), and yields value functions whose zero super-level sets correspond to $ S_{safe} $. Risk-based vs Reachability-based safety critics The binary action-value function $ b^ $ defined in equation (5) is closely related to recent work on Risk-based safety critics (Srinivasan et al. [2020]; Thananjeyan et al. [2021]), which use binary information to indicate the risk of unsafe events. However, unlike risk-based critics, which seek to measure a cumulative expected risk $ b^_{risk}(s,a) = \max_\pi E_\pi \sum_{k=1}^\infty \gamma^{k-t}i(s_k) \| s_t = s, a_t = a \in [0,1] $, our binary critic only takes values $ b^(s,a) \in \{0,1\} $, and outputting 1 whenever unavoidable failure has positive probability. Reachability based safety critics, build on the literature of HJI equations and seek to measure the largest (signed) distance $ h(s_t) $ that one can sustain from the failure set $ G $, i.e., $ b^_{reach}(s,a) = \sup_\pi \inf_{t \geq 0} h(s_t) $ almost surely (Fisac et al. [2019]). Our binary critic $ b^ $ is indeed related to $ b^_{reach} $ when the signed distance $ h(s) $ is replaced with the binary signal $ -i(s) $. We will soon show that this particular choice of safety measure allows for a precise characterization of the fixed points of equation (4). To contract or not to contract Despite the diversity of safety critics present in the literature, a common practice in both risk-based critics (Srinivasan et al. [2020]; Thananjeyan et al. [2021]) and reachability-based critics (Fisac et al. [2019]; Chen et al. [2021]) is the introduction of a discount factor $ \gamma < 1 $. While this leads to desired uniqueness and contraction properties for the operator, it comes with trade-offs: it degrades the accuracy, requiring the introduction of conservative thresholds (Srinivasan et al. [2020]; Chen et al. [2021]), which further limits exploration. Notably, such an approach is particularly worrisome when seeking to guarantee persistent safety (the ability to avoid failure for all future times), as such property is not preserved for finite accuracy approximation, even for thresholded ones. In this work, we overcome this limitation by seeking to learn directly using the non-contractive operator, thus guaranteeing, by design, the correctness of the solution. 3 Binary Characterization of Safety The fixed points $ b^ $ of the binary Bellman operator have a meaningful interpretation in terms of the topology of the state-space, and can be used to derive persistently safe policies. This connection will be better understood once we define the notion of control invariant safe sets. Definition 4 (Control invariant safe (CIS) set) A set $ C \subset S $ is a control invariant safe (CIS) set if there exists a policy $ \pi $ such that: i) (Control invariance): $ \forall s_0 \in C, \forall t \geq 0, \quad F^\pi_t(s_0) \subset C $ ii) (Safety): $ \forall s_0 \in C, \forall t \geq 0, \quad F^\pi_t(s_0) \cap G = \emptyset $. In essence, (i) means that there exists a controller that guarantees that trajectories starting in $ C $ can be made to remain in $ C $ forever, which is a standard notion in control theory (Bertsekas [1972]; Blanchini [1999]). Property (ii) means this can be done while also avoiding the unsafe region $ G $. Figure 2: An illustration of Thm. 1 and Thm. 2. Left: a valid fixed point $ \tilde{b} $ of $ T $ and its corresponding safe control invariant set. Trajectories starting in $ C $ can be made to remain in $ C $. Middle: a function $ \tilde{b} $ that is not a fixed point. A state $ s_{\text{int}} $ in the intersection will inevitably lead to the unsafe region $ G $, so $ \tilde{b}(s, a) $ should be 1 for all states in the trajectory (which would mean $ s_{\text{int}} \notin C $). Similarly, a state $ s_{\text{out}} $ outside $ C $ cannot reach inside. If it could, $ \tilde{b}(s_{\text{out}}, a) = 1 $ for some $ a \in A $, but it would transition to a state where $ \min_a \tilde{b}(s', a') = 0 $, violating equation 3. Right: a valid fixed point of the one-sided operator $ O $ of Thm. 2. States starting in $ C $ can be made to remain there; there is no guarantee that a state in $ C^O $ cannot enter $ C $. This set is CIS, and a subset from the fixed point of $ T $. With these definitions in place, we are ready for the main result of this section. Theorem 1 (Fixed points and control invariant safe sets) Let $ \tilde{b}: S \times A \rightarrow \{0, 1\} $ be a fixed point of equation 4. Then either $ \tilde{b}(s, a) = 1 \forall (s, a) $ (spurious fixed point), or: i) $ C \triangleq \{ s \in S : \min_a \tilde{b}(s, a) = 0 \} $ is control invariant safe (CIS). ii) $ C $ is unreachable from outside, i.e., $ F_t^\pi(s_0) \cap C = \emptyset \quad \forall s_0 \in S \setminus C, \forall \pi, \forall t \geq 0 $. iii) Any policy $ \pi $ that satisfies equation 5 renders $ C $ CIS. \[ \tilde{b}(s, a) = 1 \Rightarrow \pi(a\|s) = 0, \quad \forall s \in C. \] (5) Proof: The proof is in Appendix A.3 The first statement proclaims that, starting in $ C $, the system can be made to remain in $ C $ forever (thus ensuring safety). The contrapositive of property (ii) sheds light on a notion of maximality of $ C $: Corollary 1 (Maximality of the CIS set) Let $ X $ be a strict subset of $ C $. If $ X $ is reachable$^2$ from $ C \setminus X $, then $ X $ cannot be associated$^3$ with any fixed point of equation 4. We refer the reader to Fig. 2 for an illustration of valid and nonvalid fixed points. By means of Theorem 1 and Corollary 1, we achieve our goal of identifying the fixed points of the binary Bellman operator to maximal persistently safe states. In the next section, we relax the binary Bellman operator to increase the number of fixed points and safe regions associated with them, making it simpler to find safe policies. ### 4 Safety through a One-Sided Operator Theorem 1 states that any non-spurious fixed point of equation 4 yields a CIS set, along with a policy that guarantees said invariance. This set is maximal (in the sense of Corollary 1), and cannot be reached from outside. While maximality is a desired property, trying to learn the boundary of such maximal CIS sets $ C $ under limited data and with high fidelity is challenging. Moreover, overestimating $ C $ with an approximate set $ \hat{C} $ would make the unsafe region $ U = \hat{C} \setminus C $ to be declared safe. To prevent this problem, we will avoid the boundary, aiming for inner safe sets included in $ C $. Thus, we will sacrifice the maximality given by property (ii) in Theorem 1 and focus on the safety property (i). --- $^2$i.e. if $ \exists \pi, \exists t \geq 0, \exists s_0 \in C \setminus X : F_t^\pi(s_0) \cap X \neq \emptyset $ $^3$that is to say: $ \forall \tilde{b} : \tilde{b} = T \tilde{b}, X \neq \{ s \in S : \min_a \tilde{b}(s, a) = 0 \} $ In this direction, we relax the binary Bellman equations equation (3) to yield fixed points that only certify property (i). As such, for any $(s, a) \in S \times A$ and $s' = F(s, a)$, we want a function satisfying: \[ b(s, a) \geq i(s) + (1 - i(s)) \min_{a' \in A} b(s', a'), \] (6) This inequality has an associated set-valued operator mapping functions into sets of functions. Definition 5 (One-sided operator) Let $\mathcal{N}(B)$ denote the class of non-empty subsets of $B = \{b : S \times A \to \{0, 1\}\}$. We define the set-valued, one-sided operator $\mathcal{O} : B \to \mathcal{N}(B)$ as: \[ (\mathcal{O}b) = \left\{ b' \in B : b'(s, a) - i(s) - (1 - i(s)) \min_{a' \in A} b(s', a') \geq 0 \quad \forall (s, a) \in S \times A \right\} \] (7) A binary function $\tilde{b}$ is a fixed point of equation (7) iff $\tilde{b} \in (\mathcal{O}\tilde{b})$. Given a fixed point $\tilde{b}$ of the one-sided operator $\mathcal{O}$, the pair $(s, a)$ could be declared unsafe by $\tilde{b}$ even if $s$ is safe $(i(s) = 0)$ and the next state $s'$ can be driven to safety as well, i.e., $\tilde{b}$ can be potentially conservative in describing the persistently safe region. As the next theorem shows, the fixed points of this operator have, indeed, the desired CIS property. Moreover, as shown in Fig. 2 and stated next, the set may no longer be maximal as it could potentially be reached from outside. Theorem 2 (Fixed points of the one-sided operator) Let $\tilde{b} : S \times A \to \{0, 1\}$ be a fixed point of equation (7). Then either $\tilde{b}(s, a) = 1 \forall (s, a)$ (spurious fixed point), or: i) $C \triangleq \{ s \in S : \min_a \tilde{b}(s, a) = 0 \}$ is control invariant safe (CIS). ii) Any policy $\pi$ that satisfies equation (5) renders $C$ CIS. Proof: The proof is in Appendix A.4 Theorem 2 proves that the fixed points of equation (7) and their associated sets retain the desired CIS property. In addition, the one-sided operator can accommodate more fixed points, allowing for inner approximations of the maximal sets whose classification boundaries have a larger margin from the truly unsafe boundary. In the following section, we leverage these results using the one-sided operator to build an algorithm that aims to find fixed points of $\mathcal{O}$. Algorithm 1: Pseudocode for learning the binary value function Input: Safe dataset $D_{safe}$; Output: Binary value function $b^\theta(\cdot, \cdot)$; 1. Initialize $b^\theta(\cdot, \cdot)$ using $D_{safe}$ and $M = []$; ▷ Transition buffer. 2. repeat 3. for $i=0,\ldots,\text{NUM\_EPISODES}-1$ do 4. Run episodes, store $(s_k, a_k, i(s_k), s'_k)_{k=1}^K$ transitions in $M$; 5. end 6. $D_{unsafe} \leftarrow \text{build\_unsafe\_dataset}(b^\theta, M)$; ▷ Use $b^\theta$ to compute labels. 7. Build $D = D_{safe} \cup D_{unsafe}$; ▷ Complete dataset. 8. repeat 9. Run gradient steps on $L_{train}$; ▷ Update $b^\theta$ 10. until $\text{Accuracy}(b^\theta, D) = 1$; 11. $D_{unsafe} \leftarrow \text{build\_unsafe\_dataset}(b_i, M)$; ▷ $b^\theta$ has changed w.r.t. 6 12. Build $D = D_{safe} \cup D_{unsafe}$; ▷ New dataset 13. if $\text{Accuracy}(b^\theta, D) \neq 1$; ▷ Check consistency of B2E 14. go to Line 8; ▷ Not self-consistent ⇒ Re-train the network 15. end 16. until termination; 4.1 ALGORITHM We propose learning fixed points of $ O $ in equation [7] by training a neural network classifier. We will denote the learned function by $ b^\theta(\cdot, \cdot) $ where $ \theta $ collects the parameters of the network. The network takes as input each state and outputs the value $ b^\theta(s, a) $ for each of the possible actions. The last layer is a point-wise sigmoid activation function ensuring $ b^\theta(s, a) $ lies in the unit interval. We use $ \hat{b}^\theta(s, a) \triangleq \text{Round}(b^\theta(s, a)) $ to denote the predicted label. Note that our threshold (at 1/2) will be fixed during training and testing. The pseudocode for the main algorithm can be found in Alg. 1. We provide a comprehensive breakdown of its main components next. Dataset: The dataset $ D $ consists of $(s, a, y)$ tuples, where $ y $ is a $\{0, 1\}$ label, and has two components. A prescribed safe set $ D_{\text{safe}} $ (for which $ y = 0 $) and a dynamically updated $ D_{\text{unsafe}} $ of unsafe transitions detected during data collection. We have observed empirically that the addition of $ D_{\text{safe}} $ helps prevent the collapse to the trivial fixed point described in Theorem 1. The algorithm iterates over the following three loops: Environment interaction: Episodes start from a state $ s_0 $ sampled from the initial distribution $ \rho $. To collect $(s, a, s', i(s))$ transitions and store them in a memory buffer $ M $ we run episodes by following a policy that satisfies equation [5] for example the uniform safe policy, which takes actions uniformly over the presumed-safe ones: \[ \pi^\theta(a\|s) = \begin{cases} 0 & \text{if } \hat{b}^\theta(s, a) = 1 \\ 1/\sum_{a' \in A} 1\{\hat{b}^\theta(s, a') = 0\} & \text{if } \hat{b}^\theta(s, a) = 0 \end{cases} \] Building the dataset: After collecting transitions, the binary value function is used to compute labels via the right hand side of equation [3] that is, $ y_k^\theta = i(s) + (1 - i(s)) \min_{a'} b^\theta(s'_k, a') $ for all $(s_k, a_k, i(s_k), s'_k) \in M $. Note that these are “soft” labels $ y_k^\theta \in [0, 1] $. Those that satisfy $ y_k^\theta \geq \frac{1}{2} $ are added to $ D_{\text{unsafe}} $. This procedure is dubbed `build_unsafe_dataset(b, M)` in Algorithm 1. Training the network: The network is trained by running mini-batch gradient descent on the binary cross-entropy loss until it can correctly predict all the labels in $ D := D_{\text{safe}} \cup D_{\text{unsafe}} $. Once that is achieved, the labels in $ D_{\text{unsafe}} $ are re-computed (some might have changed since $ b^\theta $ was updated during this process), and the program checks whether it can correctly predict the labels again. It repeats this process until all labels are predicted correctly, yielding a binary function that is self-consistent across the whole dataset. 5 NUMERICAL EXPERIMENTS We present numerical validations of our algorithm on two different environments. We contrast our method first against SBE (Fisac et al., 2019), a well-known safety-critic, and against PPO (Schulman et al., 2017), a state-of-the-art RL algorithm. 5.1 INVERTED PENDULUM We begin by showcasing our algorithm on a modified version of the inverted pendulum problem (Towers et al., 2023). We choose this environment because it allows easy visualization of the learned control invariant safe sets, and because these can be compared against numerically obtained “ground truth” references. Environment The state of the system $ s = [\theta, \omega]^T $ is the angular position and angular velocity of the pendulum with respect to the vertical. The action $ a \in [-a_{\text{max}}, a_{\text{max}}] $ is the torque applied on the axis, which we discretize in 5 equally spaced values. The goal in this task is to avoid falling past the horizontal, i.e. $ G = \{(\theta, \omega) : \|\theta\| \geq \frac{\pi}{2} \} $. $^4$e.g. $(s, a)$ pairs close to the equilibrium of the system, or sampled trajectories from a known, safe policy. Figure 3: Learned safe regions for the inverted pendulum problem during early (left, middle) and latter (right) stages of training. The white area corresponds to states classified as safe. The solid maroon lines show the boundary of the unsafe region $G$ (falling past the horizontal). The green region shows the set of states that can avoid $G$ at all times, and the purple region shows the set of safe states reachable from $D_{safe}$. These two sets were computed numerically using an optimal control toolbox (Mitchell & Templeton [2005]). As learning progresses, the classifier learns a control invariant safe set inside the green region. Animations at https://tinyurl.com/6u8fvaux. Figure 4: Left: cumulative failures during training of our algorithm (red) and SBE (blue) for the inverted pendulum. Solid lines represent the means across 5 seeds, shaded areas are 95% confidence intervals. Our algorithm is 5 times safer. Right: safety rate (fraction of safe episodes) and entropy of each learned model. Our algorithm (shaded lines) always uses the uniform safe policy. SBE is tested for different threshold values $\eta$. Our policy is 100% safe and is exploratory (high entropy). Only the most conservative SBE policies (large $\eta$) are 100% safe, but have low entropy (limited exploration). Training protocol We take $D_{safe}$ to be a small grid of $(s,a)$-pairs near the unstable equilibrium. Episodes are started from $D_{safe}$ and end whenever the pendulum reaches the unsafe region, or after 200 steps. The behavioral policy is the “uniform safe” as defined in equation 8. We alternate between collecting data for 10 episodes, building the dataset and training the network as explained in Sec. 4.1. Details on network architecture and hyperparameters are relegated to the Appendix A.5. Ground truth We compare the safe region learned by our algorithm against ground truths computed numerically with optimal control tools (Mitchell & Templeton [2005]). Fig. 3 shows in green (resp. light gray) the maximum CIS set in the entire state (resp. the maximum CIS for trajectories that start in the support of $\rho$). The learned safe region (in white) at different stages of training is also shown. At the beginning, the network is only fit to $D_{safe}$. As episodes run and it collects unsafe transitions, it effectively learns a CIS set included in the true safe region for the problem. Training performance We benchmark our proposed methodology against the Safety Bellman Equation (SBE) of Fisac et al. (2019). This algorithm learns a safety-critic $q(s,a)$ and considers “safe” those actions that have $q(s,a) \geq \eta$, for a threshold $\eta$. Hyperparameters for that algorithm are taken from Hsu et al. (2021) and can be seen in Appendix A.5. Fig. 4 (left) shows the cumulative failures during training; (a failure is an episode that touched the unsafe region $G$). Our algorithm is clearly safer during training. Post-training evaluation We evaluate the performance of each model after training and show it in Fig. 4 (right). We test the uniform safe policy of our model against the safety critic for SBE. In the latter, we consider—for varying threshold $\eta$—the safe policy that maximizes exploration, i.e., the uniform policy taking actions $a$ such that $q(s, a) \geq \eta$. We illustrate the safety rate, defined as the proportion of safe episodes, and the average entropy of each policy $\tilde{H}_\pi \triangleq \mathbb{E}_{a \sim \mathcal{R}_A} [H(\pi(\cdot \| s))]$, where $\mathcal{R}_A$ is the set of safe states reachable from the origin (see ‘reach-avoid’ set in Fig. 3). Our algorithm obtains perfect safety rate, while SBE only achieves it for safer policies (large enough $\eta$). These latter policies, though safe, are less exploratory—i.e. smaller entropy—than ours. In summary, our achievements are twofold: we learn a persistently safe family of policies that is more exploratory than the SBE counterpart. As argued in Section 2.1, for traditional safety critics, there is no straightforward connection between the threshold $\eta$ and discount factor $\gamma < 1$ needed to achieve safe policies, and safety comes at the expense of less exploration, which is undesired and difficult to balance. The solution found with our algorithm strikes a good balance between safety and the richness of the class of policies guaranteed to be safe. 5.2 Autonomous driving We finish the experiment section by showing the applicability of our method in a high-dimensional, autonomous driving environment (Leurent, 2018), comparing against PPO (Schulman et al., 2017). Environment The observation space is 25 dimensional, corresponding to the position and relative velocities of vehicles on the highway. The goal is to drive the car while avoiding crashes with other vehicles (see Fig. 5 left). Further details of the environment in A.5. Performance comparison We set the horizon of this environment to 100, more than doubling its default value (Leurent, 2018). In this context, a safer policy is one that runs for longer without crashing. Fig. 5 on the right shows the episode length as a function of environment steps for our algorithm and PPO. Results are averaged over five runs. After 700,000 steps, our algorithm slightly outperforms PPO in terms of safety. This warrants special merit, since our algorithm learns a family of safe policies, while PPO only learns one. 6 Conclusion In this work we proposed a framework for obtaining correct-by-design safety critics in RL, under the goal of always avoiding a region of the state space. Our framework exploits the logical safe/unsafe nature of the problem and yields binary Bellman equations with multiple fixed points. We argue that all these fixed points are meaningful, by characterizing their structure in terms of guaranteeing safety and maximality. 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ORUiqcLpV6	The proposed idea which pays attention to not just the target object, has been proposed before in “PhraseRefer”. They use human annotators to obtain fine grained annotation. Which share the same underlying idea with this paper.	CoT3DRef: Chain-of-Thoughts Data-Efficient 3D Visual Grounding Eslam Mohamed Bakr, Mohamed Ayman, Mahmoud Ahmed, Habib Slim, Mohamed Elhoseiny King Abdullah University of Science and Technology (KAUST) {eslam.abdelrahman, mohamed.mohamed.2, mahmoud.ahmed, habib.slim, mohamed.elhoseiny}@kaust.edu.sa Abstract 3D visual grounding is the ability to localize objects in 3D scenes conditioned by utterances. Most existing methods devote the referring head to localize the referred object directly, causing failure in complex scenarios. In addition, it does not illustrate how and why the network reaches the final decision. In this paper, we address this question “Can we design an interpretable 3D visual grounding framework that has the potential to mimic the human perception system?”. To this end, we formulate the 3D visual grounding problem as a sequence-to-sequence (Seq2Seq) task by first predicting a chain of anchors and then the final target. Interpretability not only improves the overall performance but also helps us identify failure cases. Following the chain of thoughts approach enables us to decompose the referring task into interpretable intermediate steps, boosting the performance and making our framework extremely data-efficient. Moreover, our proposed framework can be easily integrated into any existing architecture. We validate our approach through comprehensive experiments on the Nr3D, Sr3D, and Scanrefer benchmarks and show consistent performance gains compared to existing methods without requiring manually annotated data. Furthermore, our proposed framework, dubbed CoT3DRef, is significantly data-efficient, whereas on the Sr3D dataset, when trained only on 10% of the data, we match the SOTA performance that trained on the entire data. The code is available at github.com/eslambakr/CoT3DVG 1 Introduction The 3D visual grounding task involves identifying and localizing objects in a 3D scene based on a natural language description or query. This task is crucial for many applications, such as robotics (Nguyen et al., 2019; Karnan et al., 2022; Wijmans et al., 2019), virtual reality (Puig et al., 2018; Ghasemi et al., 2022; Park et al., 2020; Osborne-Crowley, 2020; Liu et al., 2019), and autonomous driving (Qian et al., 2022; Cui et al., 2021; Jang et al., 2017; Deng et al., 2021). The goal is to enable machines to understand natural language and interpret it in the context of a 3D environment. Although 3D visual grounding has significantly advanced, current solutions cannot imitate the human perception system nor be interpretable. To address this gap, we propose a Chain-of-Thoughts 3D visual grounding framework, termed CoT3DRef. One of the biggest challenges in machine learning is understanding how the model arrives at its decisions. Thus, the concept of Chain-of-Thoughts (CoT) comes in. Although CoT is widely applied in Natural Language Processing (NLP) applications (Wei et al., 2022; Chowdhery et al., 2022; Lyu et al., 2023; Wang et al., 2022; Zhang et al., 2023; Madaan & Yazdanbakhsh, 2022), it is less explored in vision applications. Understanding the CoT is crucial for several reasons. Firstly, it helps explain how the model arrived at its decision, which is essential for transparency and interpretability. Secondly, it helps identify potential biases or errors in the model. Figure 1: Overview of our approach, where we first predict a chain of anchors in a logical order. In this example, to reach the chair target, we first have to localize the white and red boxes, then the bookshelf. which can be addressed to improve its accuracy and reliability. Third, it is a critical step toward intelligent systems that mimic human perception. Similar to machine learning models, our perception system can be thought of as a Chain-of-Thoughts (McVay & Kane [2009], Chen et al. [2017]) - a series of intermediate steps that enable us to arrive at our final perception of the world. In this paper, we mainly answer the following question: Can we design an interpretable 3D visual grounding framework that has the potential to mimic the human perception system? To this end, we formulate the 3D visual grounding problem as a sequence-to-sequence (Seq2seq) task. The input sequence combines 3D objects from the input scene and an input utterance describing a specific object. On the output side, in contrast to the existing 3D visual grounding architectures, we predict the target object and a chain of anchors in a causal manner. This chain of anchors is based on the logical sequence of steps a human will follow to reach the target. For instance, in Figure 1, to reach the chair target, we first have to localize the white and red boxes, then the bookshelf. By imitating the human learning process, we can devise a transparent and interpretable 3D framework that details the model’s steps until localizing the target. To show that our proposed framework can be easily integrated into any existing architecture, we incorporated it into four different baselines: LAR (Bakr et al. [2022]), SAT (Yang et al. [2021]), MVT (Huang et al. [2022]), and ViL (Chen et al. [2022]). CoT3DRef achieves state-of-the-art results on Sr3D, Nr3D (Achlioptas et al. [2020]), and ScanRefer (Chen et al. [2020]) without requiring additional manual annotations by devising an efficient pseudo-label generator to provide inexpensive guidance to improve learning efficiency. Whereas it boosts the performance by 3.6%, 4%, 5%, 0.5% on Nr3D and 10%, 11%, 9%, 1% on Sr3D, respectively. Proper design of such an approach is pivotal in attaining a significant performance gain while circumventing the need for intensive human annotations. A pertinent example can be drawn from the labeling procedure employed in PhraseRefer (Yuan et al. [2022]), which demanded a cumulative workforce commitment of 3664 hours, roughly equivalent to an extensive five-month timespan. Moreover, using additional manual annotations on the Nr3D dataset led to a noteworthy enhancement in referring accuracy, boosting the performance by 9% compared to the baselines: LAR, SAT, and MVT, respectively. Consequently, using ScanRefer, our approach surpasses SAT and MVT by 6.5% and 6.8%, respectively. In addition, as depicted in Figure 2, CoT3DRef shows a remarkable capability of learning from limited data, where training on only 10% of the data is enough to beat all the baselines, which are trained on the entire data. Our contributions are summarized as follows: - We propose a 3D data-efficient Chain-of-Thoughts based framework, CoT3DRef, that generates an interpretable chain of predictions till localizing the target. - We devise an efficient pseudo-label generator to provide inexpensive guidance to improve learning efficiency. - Our proposed framework achieves state-of-the-art performance on Nr3D, Sr3D, and ScanRefer benchmarks without requiring manually annotated data. - Using 10% of the data, our framework surpasses the existing state-of-the-art methods. Figure 3: An overview of our Chain-of-Thoughts Data-Efficient 3D visual grounding framework (CoT3DRef). First, we predict the anchors $ O^T $ from the input utterance, then sort the anchors in a logical order $ O^P $ using the Pathway module. Then, we feed the multi-modal features $ F $, the parallel localized objects $ R^T $, and the logical path $ O^P $ to our Chain-of-Thoughts decoder to localize the referred object and the anchors in a logical order $ R^P $. 2 RELATED WORK 3D visual grounding. Significant progress has been made in 3D visual grounding thanks to advancements in deep learning and computer vision, as well as the availability of grounded datasets (Achlioptas et al., 2020; Chen et al., 2020; Abdelreheem et al., 2022a; Yuan et al., 2022). One approach involves using graph-based models (Achlioptas et al., 2020; Feng et al., 2021; Yuan et al., 2021; Huang et al., 2021) to represent the scene as nodes and edges, while attention mechanisms help to focus on relevant parts of the scene. Another approach (Roh et al., 2022) is to convert the visual input into language tokens using a classification head. These tokens can then be combined with the input utterance and fed into a transformer architecture to learn the relationships between input sequence elements. Moreover, recent work (Bakr et al., 2022; Yang et al., 2021) explores distilling knowledge from 2D to 3D in a multi-view setup. However, none of the existing works models the explicit reasoning process behind the prediction of the target object. Chain of thoughts. The chain-of-thought concept has been used in many different machine learning applications, including natural language processing (Wei et al., 2022; Chowdhery et al., 2022; Lyu et al., 2023; Wang et al., 2022; Zhang et al., 2023; Madaan & Yazdanbakhsh, 2022), and robotics (Jia et al., 2023; Yang et al., 2022). In the context of 3D Visual grounding, developing a chain-of-thought approach provides a natural way to explicitly model the grounding reasoning process, which to the best of our knowledge has not been explored. An extended version is discussed in the Appendix A.9. 3 CoT3DRef In this section, we propose a simple yet effective approach to decompose the referring task into multiple interpretable steps by modeling the problem as Seq2Seq, as shown in Figure 3. First, we briefly cover the general 3D visual grounding models’ skeleton and its main components (Sec. 3.1). Then, we present our method in detail (Sec. 3.2). Finally, we detail our loss function (Sec. 3.4). 3.1 PRELIMINARIES We built our framework in a generic way where it could be integrated into any state-of-the-art 3D visual grounding model. An arbitrary 3D visual grounding model mainly consists of three essential blocks; a Visual Encoder, a Language Encoder, and a Multi-Modal Fusion module. Visual Encoder. An arbitrary 3D scene $ S \in \mathbb{R}^{N \times 6} $ is represented by $ N $ points with spatial and color information, i.e., XYZ, and RGB, respectively. Using one of the off-the-shelf 3D object detectors or manual annotations, we have access to the object proposals $ P = \{P_k\}_{k=1}^L $, where $ P_k \in \mathbb{R}^{N' \times 6} $, $ N' $ represents the number of the object’s points and $ L $ is the number of proposals in the scene. Then, the visual encoder encodes the proposals into lower-resolution feature maps $ V = \{V_k\}_{k=1}^L $, where $ V_k \in \mathbb{R}^{1 \times d} $ and $ d $ is the number of hidden dimensions. Language Encoder. Simultaneously, given an input utterance that describes a particular object in a scene, called target, a pre-trained BERT model (Devlin et al., 2018) encodes the input sentence into $ T = \{T_j\}_{j=1}^W $, where $ W $ is the maximum sentence length. Then, a language classification head is utilized to predict only the referred object. We notice that limiting the language encoder to only predict the target object restricts its capability of learning representative features. Multi-Modal Fusion. After encoding the object proposals and the utterance, a multi-modal fusion block is exploited to refine the visual features $ V $ based on the language embeddings $ T $ generating fused features $ F = \{F_k\}_{k=1}^L $, where $ F_k \in \mathbb{R}^{1 \times d} $. Both graphs [Achlioptas et al., 2020] and transformers [Huang et al., 2022; Bakr et al., 2022; Yang et al., 2021] are explored to capture the correlation between the two different modalities. However, our CoT framework can be integrated easily into any existing 3D visual grounding architecture, regardless of how the multi-modal features $ F $ are obtained. ### 3.2 Chain-of-Thoughts We decompose the referring task into multiple interpretable steps, whereas to reach the final target the model must first predict the anchors one by one, in a logical order called Chain-of-Thoughts. To this end, we first have to predict the anchors from the input utterance, then sort the anchors in a logical order using our Pathway module. Then, we replace the naive referring decoder with our Chain-of-Thoughts decoder. We formulate the referring task as a Seq2Seq problem by localizing the anchors as an intermediate step. Instead of anticipating the target directly, we first predict the chain of anchors sequentially, then utilize them to predict the target. Pathway generation. First, we extend the language head, i.e., Target Anchors Head, to extract both the target and the anchors $ O_T = \{O_i^T\}_{i=1}^M $, where $ M $ is the maximum number of objects in the sentence, depicted in the lower red part of Figure 5. We add a “no_obj” class to pad the output to the maximum length $ M $. However, the predicted anchors are unsorted or sorted based on the occurrence order in the sentence, which cannot be fit in our Chain-of-Thoughts framework. Accordingly, we introduce a “Pathway Head” which takes the encoded sentence $ T $ and the predicted objects of the utterance $ O_T $ to produce logically ordered objects $ O_P = \{O_i^P\}_{i=1}^M $. One possible solution is to exploit an MLP head to predict the logical order for each object. However, for better performance, we use a single transformer encoder layer to capture the correlation between different objects. Sequence-to-Sequence. Similar to the language stream, we first employ a parallel referring head to localize the referred object alongside the anchors, ignoring their logical order. The parallel referring head only takes the multi-modal features $ F $ as input and localizes the target and the anchors $ R_F = \{R_i^F\}_{i=1}^M $. We experiment both with localizing the target and the anchors in parallel, and localizing them one by one in a sequential logical order using previously localized objects as prior information. In other words, we formulate the 3D referring task as a Seq2Seq task, where the input is two sequences; 1) a set of object proposals $ P $, 2) a sequence of words (utterance). The output is a sequence of locations for the anchors and the target. Specifically, we add positioning awareness to the plain localized objects $ R_F $ using the predicted ordered objects $ O_P $, which act as a positional encoding. These positions indicate a logical order that mimics human perception [McVay & Kane, 2009; Chen et al., 2017]. Furthermore, we employ a single transformer decoder layer, depicted in Figure 3, to localize the objects in sequence w.r.t the predicted logical order $ O_P $. In this decoder layer, the queries are $ R_F + O_P $ and the values and keys are $ F $. Accordingly, the attention maps denoted as $ A $ follow Eq. 1 \[ A = \sigma((R_F + O_P)F_T / \sqrt{d}), \] where $ \sigma $ is the Softmax function and $ d $ is the embedding dimensions. We use a masked self-attention layer to enforce the Chain-of-Thoughts, where while predicting the next object’s location, we only attend to the previously located objects. This could be interpreted as a one-direction CoT. We also experiment with another variant where no masking is applied so that we can attend to any object in the chain, as shown in Appendix A.6. ### 3.3 Pseudo Labels During the training phase, our proposed framework requires more information than the standard available GT in existing datasets [Achlioptas et al., 2020; Chen et al., 2020]. These datasets only annotate the referred object and the target. However, our framework requires anchor annotations. Three types of extra annotations are needed: 1) Given an input utterance, we need to identify the mentioned objects other than the target; the anchors. 2) Once we extract the anchors from the utterance, we need to know their logical order to create a chain of thoughts. 3) Finally, we need the localization information for each anchor, i.e., to assign a bounding box to every anchor. To make our framework self-contained and scalable, we do not require any manual effort. Instead, we collect pseudo-labels automatically without any human intervention. Anchors parser. We extract the textual information from the utterance using rule-based heuristics and a scene graph parser (Schuster et al., 2015; Wu et al., 2022). First, we extract the whole mentioned objects and their relations from the utterance using the scene graph parser. Then, we match the objects to their closest matching class from the ScanNet labels (Dai et al., 2017a) using SBERT (Reimers & Gurevych, 2019). Due to the free-from nature of Nr3D, the anchors mentioned in the GT descriptions sometimes do not precisely match the ScanNet class labels. For instance, the GT description is “The plant at the far right-hand side of the bookcase tucked in the furthest corner of the desk.” However, there is no “bookcase” class in ScanNet. Therefore, we need to match it to the nearest ScanNet class labels, which in this case will be “bookshelf.” Anchors pathway. We utilize GPT-3.5 (Ouyang et al., 2022) to extract the logical order of objects given an input utterance, using in-context learning (Brown et al., 2020). The full prompt used for pathway extraction is provided in Appendix A.12. Anchors localization. The anchors localization module employs object proposals $ P $, extracted relations $ R $ and utterance objects $ \mathcal{O}^T $ to establish associations between anchors and object bounding boxes within an input scene. Our method involves iterating over all anchors extracted from the utterance and searching for candidate objects in $ P $ that share the same class. When the anchor class is represented singularly in the scene, we return the matched object. However, in scenarios where disambiguation is required due to multiple objects belonging to the anchor class, we leverage the parsed spatial relations and the localized objects within the scene to identify the intended anchor accurately, termed FIND in Algorithm 1. However, it is not guaranteed that the FIND function will be able to localize the remaining unlocalized anchors accurately. Thus, in this case, as shown in the last step in Algorithm 1, we randomly sample an object of the same class. We summarize our localization method in Algorithm 1. Algorithm 1 Localizing objects mentioned in an input utterance Input: - $ \mathcal{O}^T = \{ o_i \}_{i=1}^{K} $: unlocalized objects mentioned in the utterance, extracted using the syntactic parser. - $ P $: object proposals for the scene $ P = (B, C) $, with: - $ B = \{ b_j \}_{j=1}^{L} $: bounding boxes $ b_j \in \mathbb{R}^6 $ - $ C = \{ c_i \}_{i=1}^{L} $: classes of the $ L $ object proposals for the scene. - $ \text{RELATE}: \mathcal{O}^T \rightarrow R \times \mathcal{O}^T $: map objects mentioned in the input utterance to a (spatial relation, object) pair. - $ \text{FIND}: P \times R \times P^m \rightarrow P $: map a localized object, a relation and a set of $ m $ candidate objects to an output localized object. Output: - $ A $: localized anchors used in the utterance, where $ A \subseteq P $. ``` function LOCALIZE(o, P, A) K ← {P_j \| o ∼ c_j, c_j ∈ C} if \|K\| = 1 then return K else for (r_m, o_k) ∈ RELATE(o) do if o_k ∈ A then return FIND(P_k, r_m, K) else p ← LOCALIZE(o_k, P, A) if p ≠ ∅ then return {p} ∪ FIND(p, r_m, K) return {p} ∪ {p_i \| p_i = (b_i, c_i), o ∼ c_i, p_i ∈ P} end end ``` 3.4 LOSSES Mainly, three losses exist in most existing 3D visual grounding architectures. Two of them are considered auxiliary losses, i.e., 3D classification loss $ L_{cls}^V $ and language classification loss $ L_{cls}^T $. Hence, the third one is the primary loss, i.e., referring loss $ L_{ref} $. First, we extend the language classification loss $ L_{cls}^T $ by recognizing the referred object class and the anchors based on the input utterance. Similarly, the referring loss is extended to localize both the target and the anchors, termed as parallel referring loss $ L_{ref}^P $, as we localize the target and anchors in one step. Furthermore, we add another referring loss after the transformer decoder, termed as CoT referring loss $ L_{ref}^{COT} $. Finally, an auxiliary distractor binary classification loss $ L_{\text{dist}} $ is introduced to distinguish between the target and the distractors, i.e., objects with the same class as the target. Grouping all these losses, we optimize the whole model in an end-to-end manner with the following loss function: \[ L = (\lambda_V \cdot L_{\text{cls}}^V) + (\lambda_T \cdot L_{\text{cls}}^T) + \lambda_{\text{ref}} \cdot (L_{\text{ref}}^P + L_{\text{ref}}^{\text{CoT}}) + \lambda_{\text{dist}} \cdot L_{\text{dist}}, \] where $ \lambda $ is the corresponding loss weight for each term. ### 4 EXPERIMENTAL RESULTS #### Datasets. To probe the effectiveness of our proposed framework, CoT3DRef, we conduct evaluations on three 3D visual-grounding benchmarks, namely Nr3D, Sr3D (Achlioptas et al., 2020) and ScanRefer (Chen et al., 2020). Nr3D contains 41.5K natural, free-form utterances gathered from humans through a referring game, while Sr3D consists of 83.5K synthetic utterances. Consequently, ScanRefer provides 51.5K utterances of 11K objects for 800 3D indoor scenes. #### Network Configuration. We model the Pathway module using only one transformer encoder layer, and the CoT decoder using a single transformer decoder layer. The number of heads used are 7 and 16 for the Pathway module and CoT decoder, respectively. The number of proposals $ L $ and the maximum sentence length $ W $ are 52 and 24, respectively. $ L $ and $ W $ define the sizes of the input sequences to our CoT decoder. The maximum number $ M $ of objects in the sentence, the output sequence length for our CoT decoder, is 8 and 3 for Nr3D and Sr3D, respectively. Following previous works (AbdelReheem et al., 2022b; Achlioptas et al., 2020; He et al., 2021; Roh et al., 2022; Jain et al., 2021; Yang et al., 2021; Qi et al., 2017), we randomly sample 1024 points for each proposal, set the hidden dimensions $ d $ to 768, and train the model for 100 epochs from scratch using the weight initialization strategy described in (He et al., 2015). The initial learning rate is set to $ 10^{-4} $ and decreases by 0.65 every ten epochs. The Adam optimizer (Kingma & Ba, 2014) and a mini-batch size of 24 per GPU are used for training all the models. We set the losses weights as follows: $ \lambda_V = 5 $, $ \lambda_T = 0.5 $, $ L_{\text{ref}} = 5 $, and $ \lambda_{\text{dist}} = 1 $. We used the PyTorch framework and a single NVIDIA A6000 GPU for training. #### 4.1 Ablation Studies We conducted several ablation studies to validate each module in our framework, termed CoT3DRef. CoT vs. Parallel. To assess our CoT3DRef framework, we have to disentangle the CoT from the pseudo label generation module. In other words, it could be thought that the achieved gain caused by just accessing additional supervision signal, i.e., anchors’ annotations. To this end, we implement a parallel approach, that has access to the anchors’ labels, however, localizes the targets and anchors in one shot without any interaction between them. In contrast, our CoT3DRef framework leverages the causality between the anchors and the target through our chain-of-thoughts decoder. As shown in Table 1 on the challenging setup, where we assume access for only 10% of the training data while testing on the entire testing dataset, the parallel variant boosts the performance by 4% and 6.5% over the vanilla MVT using Nr3D and Sr3D, respectively (row b). On the other hand, our CoT3DRef framework surpasses the vanilla MVT by 10% and 16.4% using Nr3D and Sr3D, respectively (row c). Consequently, using the entire data, our CoT surpasses both the parallel and the vanilla MVT approaches by 3% and 5% on Nr3D, and by 1% and 6.7% on Sr3D, respectively. \| Data Percentage \| +Distractor Loss \| +Parallel \| +CoT \| Nr3D ↑ \| Sr3D ↑ \| \|-----------------\|------------------\|-----------\|------\|--------\|--------\| \| (a) 10% \| - \| - \| 27.6 \| 48.8 \| \| (b) 10% \| - \| ✓ \| 31.7 \| 55.3 \| \| (c) 10% \| ✓ \| - \| 37.5 \| 65.2 \| \| (d) 10% \| ✓ \| ✓ \| 38.2 \| 66.4 \| \| (e) 100% \| - \| - \| 55.2 \| 66.0 \| \| (f) 100% \| ✓ \| - \| 57.0 \| 71.5 \| \| (g) 100% \| - \| ✓ \| 60.0 \| 72.7 \| \| (h) 100% \| ✓ \| ✓ \| 60.4 \| 73.2 \| Table 1: Ablation study for different components of our CoT3DRef framework. First, we compare the baseline, i.e., MVT (Huang et al., 2022), against the parallel and the chain-of-thoughts approaches. Then, we show the effect of adding distractor loss. All the experiments are conducted using the Nr3D dataset (Achlioptas et al., 2020). \| # Transformer Blocks \| Nr3D ↑ \| Sr3D ↑ \| \|----------------------\|--------\|--------\| \| 1 \| 60.4 \| 73.2 \| \| 2 \| 60.4 \| 73.3 \| \| 4 \| 60.1 \| 72.9 \| Table 2: Ablation study for different numbers of the transformer blocks used in our CoT decoder, based on MVT. \| Method \| Anchors \| Target \| \|---------------------------------\|---------\|--------\| \| Baseline \| N/A \| 55.1 \| \| +CoT + Zeroing Anchor Loss \| 4.5 \| 55.0 \| \| +CoT + Pseudo Labels \| 6.0 \| 60.1 \| \| +CoT + Human Labels \| 73.6 \| 64.4 \| Table 3: Ablation study to highlight the effect of the anchors’ quality on the final target referring accuracy, based on Nr3D dataset and MVT as a baseline. Table 4: Benchmarking results on Nr3D and Sr3D datasets (Achlioptas et al., 2020). We emphasize that we did not require any additional GT annotations, in contrast to SAT (Yang et al., 2021) which requires access to real 2D images, and PhraseRefer (Yuan et al., 2022) and ScanEnts (Abdelreheem et al., 2022a) require manual annotations for the whole anchors in the data. We report the standard-deviation $\sigma$ in green. \| Method \| GT Anchors \| Nr3D ↑ \| Sr3D ↑ \| \|-----------------\|------------\|--------\|--------\| \| ReferIt3D \| N/A \| 35.6 \| 43.6 \| \| Text-Guided \| N/A \| 37.2 \| 44.2 \| \| Instance-Ref \| N/A \| 38.8 \| 46.0 \| \| 3DRefTrans \| N/A \| 39.0 \| 46.4 \| \| 3DVG-Trans \| N/A \| 40.8 \| 48.5 \| \| FFL-3DDO \| N/A \| 41.7 \| 48.2 \| \| TransRef \| N/A \| 42.4 \| 48.5 \| \| LanguageRef \| N/A \| 43.9 \| 51.0 \| \| 3D-SPS \| N/A \| 51.5 \| 58.1 \| \| TAK \| N/A \| 56.0 \| 63.6 \| \| +CoT3DRef (Ours)\| ✓ \| 52.5±0.1\| 65.1±0.2\| \| SAT \| N/A \| 49.2 \| 56.3 \| \| +PhraseRefer \| ✓ \| 54.4 \| 62.1 \| \| +ScanEnts \| ✓ \| 52.5 \| 59.8 \| \| +CoT3DRef \| ✓ \| 53.1±0.2\| 60.8±0.3\| \| +CoT3DRef (Ours)\| ✓ \| 58.1±0.4\| 65.1±0.6\| \| MVT \| N/A \| 55.1 \| 61.3 \| \| +SceneRef \| ✓ \| 59.3 \| 65.4 \| \| +PhraseRefer \| ✓ \| 59.0 \| - \| \| +CoT3DRef (Ours)\| ✓ \| 60.4±0.2\| 66.2±0.4\| \| +CoT3DRef (Ours)\| ✓ \| 64.4±0.2\| 70.0±0.2\| \| VLT3DRef \| N/A \| 63.6 \| 70.0 \| \| +CoT3DRef (Ours)\| ✓ \| 64.1±0.1\| 70.4±0.2\| \| +CoT3DRef (Ours)\| ✓ \| 64.0±0.2\| 70.4±0.2\| Table 5: Benchmarking results while training jointly on Nr3D and Sr3D datasets (Achlioptas et al., 2020). \| Method \| Overall(σ) \| Easy \| Hard \| View-dep. \| View-indep. \| \|-----------------\|------------\|------\|------\|-----------\|-------------\| \| ReferIt3D \| 37.2 ± 0.3 \| 44.0 ± 0.6 \| 30.6 ± 0.3 \| 33.3 ± 0.6 \| 39.1 ± 0.2 \| \| +CoT3DRef (Ours)\| 43.9 ± 0.3 \| - \| - \| - \| - \| \| SAT \| 53.9 ± 0.2 \| 55.4 ± 0.5 \| 39.1 ± 0.5 \| 40.3 ± 0.4 \| 50.6 ± 0.2 \| \| MVT \| 58.5 ± 0.2 \| 65.6 ± 0.2 \| 51.6 ± 0.3 \| 56.6 ± 0.3 \| 59.4 ± 0.2 \| \| MVT+CoT3DRef (Ours)\| 62.5±0.4\| 70.0±0.5\| 53.0±0.2\| 58.3±0.1\| 63.9±0.3 \| Table 6: Benchmarking results on ScanRefer dataset (Chen et al., 2020). \| Method \| Data Percentage \| \|-----------------\|-----------------\| \| MVT \| 36.4 51.5 54.5 57.4 \| \| +PhraseRefer \| 48.6 60.1 62.5 64.2 \| \| SAT \| 54.9 49.5 50.5 53.8 \| \| +PhraseRefer \| 57.5 \| \| +CoT3DRef (Ours)\| 49.1 57.7 58.9 60.3 \| Distractor Loss. We add an auxiliary distractor binary classification loss $L_{dist}$ to disambiguate the target and the distractors, as discussed in Sec. 3.4. As shown in Table 1 rows d and h, incorporating it boosts the performance by 0.5-1%. Anchors Quality Effect. To establish the affirmative role of anchors in refining target localization accuracy without inducing detrimental effects, we conducted a worst-case simulation. In this simulation, all anchors were falsely detected. In other words, we deliberately zero the localization loss associated with anchors during training, causing the decoder to randomly and inaccurately predict anchor locations. Remarkably, the target accuracy remained unaltered, as shown in Table 3, reflecting the robustness of the approach. The accuracy experienced a decrement of approximately 5%, dropping from 60.4% to 55%. Encouragingly, even in this scenario, the accuracy mirrored the baseline performance, steadfast at 55.1%. This substantiates that while anchor detection may exhibit inaccuracies, the broader framework’s efficacy in target localization remains largely unaffected. In contrast, we replaced the pseudo labels with manual annotations (Abdelreheem et al., 2022a). This substitution serves as an upper-bound reference point for evaluation. As shown in Table 3, the exchange of noisy pseudo labels with precise manual annotations led to a noteworthy 4% enhancement in referring accuracy for Nr3D, elevating it from 60.4% to 64.4%. Number of Transformer Blocks. As shown in Table 2, we have explored using 1, 2, and 4 transformer blocks in our CoT referring decoder. However, we didn’t notice a significant gain in performance; therefore, we preferred to use a single transformer block. ### 4.2 Comparison to State-of-the-Art We verify the effectiveness of our proposed framework, termed CoT3DRef on three well-known 3D visual grounding benchmarks, i.e., Nr3D, Sr3D (Achlioptas et al., 2020) and ScanRefer (Chen et al., 2020). By effectively localizing a chain of anchors before the final target, we achieve state-of-the-art results without requiring any additional manual annotations. As shown in Table 4, when we integrate our module into four baselines; LAR (Bakr et al., 2022), SAT (Yang et al., 2021), MVT (Huang et al., and ViL (Chen et al., 2022), it boosts the accuracy by 3.6%, 4%, 5%, 0.5% on Nr3D and 10%, 11%, 9%, 1% on Sr3D, respectively. The disparity between the gain achieved on Nr3D and Sr3D is due to our pseudo label module that hinders achieving more gain on Nr3D. Exchanging our noisy pseudo labels with precise manual annotations led to a noteworthy enhancement in referring accuracy, where our module boosts the performance by 9% compared to the baselines; LAR, SAT and MVT, respectively. This outcome underscores our model’s ability to yield enhanced performance not only for simpler descriptions (Sr3D) but also in the context of more intricate, free-form descriptions (Nr3D). We have a limited gain on only ViL. A detailed analysis that justifies this behaviour is mentioned in the Appendix A.4. Nr3D+Sr3D. In addition, we jointly train on Nr3D and Sr3D. Whereas, we augment the Nr3D training data with Sr3D, while testing on the same original Nr3D test-set. Consistently with the previous results mentioned in Table 1 and 4, we surpass all the existing work by 3.5%. As shown in Table 5, we achieve 62.5% grounding accuracy, while MVT only achieves 58.5% accuracy. ScanRefer. To further show the effectiveness of our proposed method, we have conducted several experiments on the ScanRefer dataset across different data percentages on MVT and SAT baselines. As shown in Table 6, we outperform both MVT and SAT by a significant margin across the entire data percentages, i.e., 10%, 40%, 70%, and 100%. More specifically, integrating our CoT framework into MVT has boosted the performance by 12.2%, 8.6%, 8%, and 6.8%, respectively. In addition to MVT, we have integrated our CoT framework into the SAT baseline, where the same performance gain has been achieved, probing our method’s effectiveness across a comprehensive range of baseline models, datasets, and different available data percentages. Data Efficiency. To further validate the effectiveness of our framework, we assess it on a more challenging setup, where we assume access to only limited data. Four percentage of data is tested, i.e., 10%, 40%, 70%, and 100%. As shown in Figure 2, on the Sr3D dataset, using only 10% of the data, we match the same performance of MVT and SAT that are trained on 100% of the data. This result highlights the data efficiency of our method. Furthermore, when trained on 10% of the data on Nr3D with noisy pseudo labels (Sec. 3.3), we still surpass all the baselines with considerable margins. Qualitative results. As shown in Figure 4, the first three examples show that our model successfully localizes the referred objects by leveraging the mentioned anchors, such as “the table with 5 chairs around”. However, in the ambiguous description shown in the fourth example: “2nd stool from the left”, the model incorrectly predicts the stool, as it is view-dependent. In other words, if you look at the stools from the other side, our predicted box will be correct. Additionally, the last example shows a challenging scenario where a negation in the description is not properly captured by our model. 5 DISCUSSION AND LIMITATIONS Comparison with PhraseRefer (Yuan et al., 2022) and ScanEnts (Abdelreheem et al., 2022a). We acknowledge the great work proposed by PhraseRefer (Yuan et al., 2022) and ScanEnts (Abdelreheem et al., 2022a), which paved the way for the importance of paying attention not just to the target object but also to the anchors. Thus, underlying approaches have some similarities in terms of demonstrating the importance of including anchors in the pipeline. However, there are a lot of significant differences: 1) Design a CoT framework for 3D visual grounding that explicitly models causal reasoning while interpreting the instruction inspired by the human perception system. 2) Show that we can design an efficient framework that achieves state-of-the-art results on three challenging benchmarks, i.e., Nr3D, Sr3D (Achlioptas et al., 2020), and ScanRefer (Chen et al., 2020), without requiring human labels. Proper design of such an approach is pivotal in attaining a good performance while circumventing the need for intensive human annotations. A pertinent example can be drawn from the labeling procedure employed in PhraseRefer (Yuan et al., 2022), which demanded a cumulative workforce commitment of 3664 hours, roughly equivalent to an extensive five-month timespan. Pseudo labels accuracy. The accuracy of the pseudo-labels plays a vital role in the overall performance. To evaluate its performance, we manually collect ground-truth labels for 1) the predicted orderings of the anchors in the utterance and 2) the final localization of the anchors as predicted by the geometry module, based on 10% of the Nr3D. To evaluate orderings predicted by in-context learning, we use the normalized Levenshtein edit distance between two sequences, where a length of 1 means that every object in the sequence is incorrect. We achieve an average distance of 0.18 between predicted and ground-truth orderings. We consider anchor-wise localization accuracy to evaluate the geometry module’s accuracy, where the anchors considered for each sequence are those mentioned in the input utterance. We achieve a 77% accuracy for this task compared to human annotators. Overall, a significant accuracy gap is measured between automatically collected pseudo-labels and ground-truth data, contributing to the performance loss observed on the Nr3D dataset. Pseudo module limitations. Despite designing a data efficient framework that could be integrated into any existing 3D visual grounding architecture, and achieve SOTA results without requiring any manual annotation efforts, our pseudo label module hinders achieving more gain on Nr3D. Accordingly, we encourage the future efforts to try to enhance the pseudo module performance. In addition, the anchor localization block in our pseudo module is tailored on ScanNet dataset (Dai et al., 2017a), and will thus need some adaptations to be usable on other 3D scene datasets. Pathway module limitations. Our Pathway module is responsible of generating a chain of logical order for the extracted objects from the input utterance. However, it does not handle the multi-path scenario, where multiple paths are valid. For instance, given this utterance “It is the chair besides the desk, which has a book and a lamp above it”, we have two possible starting points, i.e., locating the lamp first or the book. Thus, for simplicity, we start by the last mentioned object in the utterance, in the multiple paths scenario. Nevertheless, one possible solution to handle this limitation implicitly through building a graph that reasons the different possibilities (Salzmann et al., 2020). 6 CONCLUSION We propose CoT3DRef: a novel and interpretable framework for 3D visual grounding. By formulating the problem of 3D visual grounding from a natural language instruction as a sequence-to-sequence task, our approach predicts a chain of anchor objects that are subsequently utilized to localize the final target object. This sequential approach enhances interpretability and improves overall performance and data efficiency. Our framework is data-efficient and outperforms existing methods on the Nr3D and Sr3D datasets when trained on a limited amount of data. Furthermore, our proposed chain-of-thoughts module can easily be integrated into other architectures. Through extensive experiments, we demonstrate consistent performance gains over previous state-of-the-art methods operating on Referit3D. Importantly, our approach does not rely on any additional manual annotations. Instead, we leverage automatic rule-based methods, syntactic parsing, and in-context learning to collect pseudo-labels for the anchor objects, thereby eliminating the laborious and time-consuming process of manually annotating anchors. Overall, our work advances 3D visual grounding by making a step towards bridging the gap between machine perception and human-like understanding of 3D scenes. REFERENCES Ahmed Abdelreheem, Kyle Olszewski, Hsin-Ying Lee, Peter Wonka, and Panos Achlioptas. Scannents3d: Exploiting phrase-to-3d-object correspondences for improved visio-linguistic models in 3d scenes. arXiv preprint arXiv:2212.06250, 2022a. 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0IaTFNJner	In section 3, the paper proposes Information Abundance to measure the degree of collapse of embedding matrices. As the paper focuses on the scaling law of embedding layers, the paper should discuss whether Information Abundance is a fair metric when comparing embedding matrices of different dimension sizes.	On the Embedding Collapse When Scaling up Recommendation Models Anonymous authors Paper under double-blind review Abstract Recent advances in deep foundation models have led to a promising trend of developing large recommendation models to leverage vast amounts of available data. However, we experiment to scale up existing recommendation models and observe that the enlarged models do not improve satisfactorily. In this context, we investigate the embedding layers of enlarged models and identify a phenomenon of embedding collapse, which ultimately hinders scalability, wherein the embedding matrix tends to reside in a low-dimensional subspace. Through empirical and theoretical analysis, we demonstrate that the feature interaction module specific to recommendation models has a two-sided effect. On the one hand, the interaction restricts embedding learning when interacting with collapsed embeddings, exacerbating the collapse issue. On the other hand, feature interaction is crucial in mitigating the fitting of spurious features, thereby improving scalability. Based on this analysis, we propose a simple yet effective multi-embedding design incorporating embedding-set-specific interaction modules to capture diverse patterns and reduce collapse. Extensive experiments demonstrate that this proposed design provides consistent scalability for various recommendation models. 1 Introduction Recommender systems are significant machine learning scenarios that predict users’ actions on items based on multi-field categorical data (Zhang et al., 2016), which play an indispensable role in our daily lives to help people discover information about their interests and have been adopted in a wide range of online applications, such as E-commerce, social media, news feeds, and music streaming. Recently, researchers have developed deep-learning-based recommendation models to dig feature representations flexibly. These models have been successfully deployed across a multitude of application scenarios, thereby demonstrating their widespread adoption and effectiveness. In recommender systems, there is a tremendous amount of Internet data, while mainstream models typically tuned with embedding size 10 (Zhu et al., 2022) do not adequately capture the magnitude of the available data. Motivated by the advancement of large foundation models (Kirillov et al., 2023; OpenAI, 2023; Radford et al., 2021; Rombach et al., 2022), which benefit from increasing parameters, it would be a promising trend to scale up the recommendation model size. However, when scaling up the embedding size, the bottleneck of mainstream recommendation models (Qu et al., 2016; Lian et al., 2018; Wang et al., 2021), we find an unsatisfactory improvement or even performance drop as shown in Figure 1a. This suggests a deficiency in the scalability of existing architecture designs, constraining the maximum potential for recommender systems. We take a spectral analysis on the learned embedding matrices based on singular value decomposition and exhibit the normalized singular values in Figure 1b. Surprisingly, most singular values are significantly small, i.e., the learned embedding matrices are nearly low-rank, which we refer to as the embedding collapse phenomenon. With the enlarged model size, the model does not learn to capture a larger dimension of information, implying a learning process with ineffective parameter utilization, which restricts the scalability. In this work, we study the mechanism behind the embedding collapse phenomenon through empirical and theoretical analysis. We shed light on the two-sided effect of the feature interaction module, the characteristic of recommendation models to model higher-order correlations, on model scalability. On the one hand, interaction with collapsed embeddings will constrain the embedding learning... Figure 1: Unsatisfactory scalability of existing recommendation models. (a): Increasing the embedding size does not improve remarkably or even hurts the model performance. (b): Most embedding matrices do not learn large singular values and tend to be low-rank. and, thus, in turn, aggravate the collapse issue. On the other hand, the feature interaction also plays a vital role in reducing overfitting when scaling up models. Based on our analysis, we conclude the principle to mitigate collapse without suppressing feature interaction about how to design scalable models. We propose multi-embedding as a simple yet efficient design for model scaling. Multi-embedding scales the number of independent embedding sets and incorporates embedding-set-specific interaction modules to jointly capture diverse patterns. Our experimental results demonstrate that multi-embedding provides scalability for extensive mainstream models, pointing to a methodology of breaking through the size limit of recommender systems. Our contributions can be summarized as: • To the best of our knowledge, we are the first to point out the non-scalability issue for recommendation models and discover the embedding collapse phenomenon, which is an urgent problem to address for model scalability. • We shed light on the two-sided effect of the feature interaction process on scalability based on the collapse phenomenon using empirical and theoretical analysis. Specifically, feature interaction leads to collapse while providing essential overfitting reduction. • Following our concluded principle to mitigate collapse without suppressing feature interaction, we propose multi-embedding as a simple unified design, which consistently improves scalability for extensive state-of-the-art recommendation models. 2 PRELIMINARIES Recommendation models aim to predict an action based on features from various fields. Throughout this paper, we consider the fundamental scenario of recommender systems, in which categorial features and binary outputs are involved. Formally, suppose there are $N$ fields, with the $i$-th field denoted as $\mathcal{X}_i = \{1, 2, ..., D_i\}$ where $D_i$ denotes the field cardinality. The value of $D_i$ may vary in a wide range, adding difficulty to recommender systems. Let $$\mathcal{X} = \mathcal{X}_1 \times \mathcal{X}_2 \times ... \times \mathcal{X}_N$$ and $\mathcal{Y} = \{0, 1\}$, then recommendation models aim to learn a mapping from $\mathcal{X}$ to $\mathcal{Y}$. In addition to considering individual features from diverse fields, there have been numerous studies (Koren et al., 2009; Rendle, 2010; Juan et al., 2016; Guo et al., 2017; Lian et al., 2018; Pan et al., 2018; Sun et al., 2021; Wang et al., 2021) within the area of recommender systems to model combined features using feature interaction modules. In this work, we investigate the following widely adopted architecture for mainstream models. A model comprises: (1) embedding layers $E_i \in \mathbb{R}^{D_i \times K}$ for each field, with embedding size $K$; (2) an interaction module $I$ responsible for integrating all embeddings into a combined feature scalar or vector; and (3) a subsequent postprocessing module $F$ used for prediction purposes, such as MLP and MoE. The forward pass of such a model is formalized as $$e_i = E_i^\top 1_{x_i}, \forall i \in \{1, 2, ..., N\},$$ $$h = I(e_1, e_2, ..., e_n),$$ $$\hat{y} = F(h),$$ where $1_{x_i}$ indicates the one-hot encoding of $x_i \in X_i$, in other words, $e_i$ refers to (transposed) $x_i$-th row of the embedding table $E_i$. 3 Embedding Collapse Singular value decomposition has been widely used to measure the collapse phenomenon [Jing et al., 2021]. In Figure 1b, we have shown that the learned embedding matrices of recommendation models are approximately low-rank with some extremely small singular values. To determine the degree of collapse for such matrices with low-rank tendencies, we propose information abundance as a generalized quantification. Definition 1 (Information Abundance) Consider a matrix $E \in \mathbb{R}^{D \times K}$ and its singular value decomposition $E = U\Sigma V = \sum_{k=1}^{K} \sigma_k u_k v_k^T$, then the information abundance of $E$ is defined as \[ IA(E) = \frac{\\|\sigma\\|_1}{\\|\sigma\\|_\infty}, \] i.e., the sum of all singular values normalized by the maximum singular value. Intuitively, a matrix with high information abundance demonstrates a balanced distribution in vector space since it has similar singular values. In contrast, a matrix with low information abundance suggests that the components corresponding to smaller singular values can be compressed without significantly impacting the result. Compared with matrix rank, information abundance can be regarded as a simple extension by noticing that $\text{rank}(E) = \\|\sigma\\|_0$, yet it is applicable for non-strictly low-rank matrices, especially for fields with $D_i \gg K$ which is possibly of rank $K$. We calculate the information abundance of embedding matrices for the enlarged DCNv2 [Wang et al., 2021] and compare it with that of randomly initialized matrices, shown in Figure 2. It is observed that the information abundance of learned embedding matrices is extremely low, indicating the embedding collapse phenomenon. 4 Feature Interaction Revisited In this section, we delve deeper into the embedding collapse phenomenon for recommendation models. Our investigation revolves around two questions: (1) How is embedding collapse caused? (2) How to properly mitigate embedding collapse for scalability? Through empirical and theoretical studies, we shed light on the two-sided effect of the commonly employed feature interaction module on model scalability. 4.1 Interaction-Collapse Theory To determine how feature interaction leads to embedding collapse, it is inadequate to directly analyze the raw embedding matrices since the learned embedding matrix results from interactions with all other fields, making it difficult to isolate the impact of field-pair-level interaction on embedding learning. Under this obstacle, we propose empirical evidences on models with sub-embeddings and theoretical analysis on general models, and conclude that feature interaction causes embedding collapse, named the interaction-collapse theory. Evidence I: Experiments on FFM. Field-aware factorization machines (FFM) [Juan et al., 2016] split an embedding matrix of field $i$ into multiple sub-embeddings with \[ E_i = \begin{bmatrix} E_i^{i+1}, E_i^{i+2}, \ldots, E_i^{(i-1)}, E_i^{(i+1)}, \ldots, E_i^{N} \end{bmatrix}, \] where sub-embedding $E_i^{j} \in \mathbb{R}^{D_i \times K/(N-1)}$ is only used when interacting field $i$ with field $j$ for $j \neq i$. To determine the collapse of sub-embedding matrices, we calculate $IA(E_i^{j})$ for all $i, j$ and show them in Figure 3a. For convenience, we pre-sort the field indices by the ascending order. Figure 3: Visualization of information abundance of sub-embedding matrices for FFM (left) and DCNv2 (right), with field indices sorted by information abundance of corresponding raw embedding matrices. Higher or warmer indicates larger. It is observed that IA($E_i^{j}$) are co-influenced by both IA($E_i$) and IA($E_j$). of information abundance, i.e., $i$ is ordered according to IA($E_i$), similar to $j$. We can observe that IA($E_i^{j}$) is approximately increasing along $i$, which is trivial since $E_i^{j}$ is simply a split of $E_i$. Interestingly, another correlation can be observed that the information abundance of sub-embeddings is co-influenced by the fields it interacts with, reflected by the increasing trend along $j$, especially with larger $i$. This is amazing in the sense that even using independent embeddings to represent the same field features, these embeddings get different information abundance after learning. For instance, we calculate the summation of IA($E_i^{j}$) over $j$ or $i$ to study the effect of the other single variable, shown in Figure 3b and Figure 3c. Both of them show an increasing trend, confirming the co-influence of $i$ and $j$. Evidence II: Experiments on DCNv2. An improved deep & cross network (DCNv2) (Wang et al., 2021) incorporates a crossing network which is parameterized with transformation matrices $W_{i \rightarrow j}$ (Sun et al., 2021) over each field pair to project an embedding vector from field $i$ before interaction with field $j$. By collecting all projected embedding vectors, DCNv2 can be regarded to implicitly generate field-aware sub-embeddings $E_i^{1}, E_i^{2}, ..., E_i^{N}$ to interact with all fields from embedding matrix $E_i$, with $$E_i^{j} = E_i W_{i \rightarrow j}^T.$$ DCNv2 consists of multiple stacked cross layers, and for simplification, we only discuss the first layer throughout this paper. Similar to Evidence I, we calculate IA($E_i^{j}$) together with the axis-wise summations and show them in the right part of Figure 3. Consistent with previous observation as FFM, the information abundance of sub-embedding matrices approximately increases along $j$ with the same $i$, even though they are projected from the same embedding matrix $E_i$. Theoretical analysis: Collapse on non-sub-embedding-based models. We now present how collapse is caused by feature interaction in non-sub-embedding-based recommendation models from a theoretical view. For simplicity, we consider an FM-style (Rendle, 2010) feature interaction. Formally, the interaction process is defined by $$h = \sum_{i=1}^{N} \sum_{j=1}^{i-1} e_i^\top e_j = \sum_{i=1}^{N} \sum_{j=1}^{i-1} 1_x^\top E_i E_j^\top 1_x,$$ where $h$ is the combined feature as mentioned before. Without loss of generality, we discuss one specific row $e_1$ of $E_1$ and keep other embedding matrices fixed. Consider a minibatch with batch size $B$. Denote $\sigma_{i,k}$ as the $k$-th singular value of $E_k$, similar for $u_{i,k}$, $v_{i,k}$. We have \[ \frac{\partial \mathcal{L}}{\partial e_1} = \frac{1}{B} \sum_{b=1}^{B} \frac{\partial \ell(b)}{\partial h(b)} \cdot \frac{\partial h(b)}{\partial e_1} = \frac{1}{B} \sum_{b=1}^{B} \frac{\partial \ell(b)}{\partial h(b)} \cdot \sum_{i=2}^{N} E_i^\top x_i^{(b)} \] \[= \frac{1}{B} \sum_{b=1}^{B} \frac{\partial \ell(b)}{\partial h(b)} \cdot \sum_{i=2}^{N} \sum_{k=1}^{K} \sigma_{i,k} v_{i,k} u_{i,k}^\top 1_{x_i^{(b)}}\] \[= \sum_{i=2}^{N} \sum_{k=1}^{K} \left( \frac{1}{B} \sum_{b=1}^{B} \frac{\partial \ell(b)}{\partial h(b)} u_{i,k}^\top 1_{x_i^{(b)}} \right) \sigma_{i,k} v_{i,k}\] \[= \sum_{i=2}^{N} \sum_{k=1}^{K} \alpha_{i,k} \sigma_{i,k} v_{i,k} = \sum_{i=2}^{N} \theta_i\] The equation means that the gradient can be decomposed into field-specific terms. We analyze the component $\theta_i$ for a certain field $i$, which is further decomposed into spectral for the corresponding embedding matrix $E_i$. From the form of $\theta_i$, it is observed that $\{\alpha_{i,k}\}$ are $\sigma_i$-agnostic scalars determined by the training data and objective function. Thus, the variety of $\sigma_i$ significantly influences the composition of $\theta_i$. For those larger $\sigma_{i,k}$, the gradient component $\theta_i$ will be weighted more heavily along the corresponding spectral $v_{i,k}$. When $E_i$ is low-information-abundance, the components of $\theta_i$ weigh imbalancedly, resulting in the degeneration of $e_1$. Since different $e_1$ affects only $\alpha_{i,k}$ instead of $\sigma_{i,k}$ and $v_{i,k}$, all rows of $E_1$ degenerates in similar manners and finally form a collapsed matrix. To further illustrate, we conduct a toy experiment over synthetic data. Suppose there are $N = 3$ fields, and we set $D_3$ to different values with $D_3 < K$ and $D_3 \gg K$ to simulate low-information-abundance and high-information-abundance cases, which matches the diverse range of the field cardinality in real-world scenarios. We train $E_1$ while keeping $E_2, E_3$ fixed. Details of experiment setups are discussed in Appendix A. We show the information abundance of $E_1$ along the training process for the two cases in Figure 4. It is observed that interacting with a low-information-abundance matrix will result in a collapsed embedding matrix. Summary: How is collapse caused in recommendation models? Evidence I&II highlight that interacting with a field with a low-information-abundance embedding matrix will result in a more collapsed sub-embedding. By further considering the fact that sub-embeddings reflect the effect when fields interact since it originates from raw embeddings, we recognize the inherent mechanism of feature interaction to cause collapse, which is further confirmed by our theoretical analysis. We conclude the interaction-collapse theory: Finding 1 (Interaction-Collapse Theory). In feature interaction of recommendation models, fields with low-information-abundance embeddings constrain the information abundance of other fields, resulting in collapsed embedding matrices. The interaction-collapse theory generally suggests that feature interaction is the primary catalyst for collapse, thereby imposing constraints on the ideal scalability. ### 4.2 IS IT SUFFICIENT TO AVOID COLLAPSE FOR SCALABILITY? Following our discussion above, we have shown that the feature interaction process of recommendation models leads to collapse and thus limits the model scalability. We now discuss its negative proposition, i.e., whether suppressing the feature interaction to mitigate collapse leads to model scalability. To answer this question, we design the following two experiments to compare standard models and models with feature interaction suppressed. Evidence III: Regularization on DCNv2 to mitigate collapse. Evidence II shows that a projection $W_{i \rightarrow j}$ is learned to adjust information abundance for sub-embeddings and thus lead to collapse.\footnote{Further explanation is referred to Appendix F} We now investigate how suppressing such effect would result in scalability by introducing the following regularization with learnable parameter $\lambda_{ij}$ $$\ell_{reg} = \sum_{i=1}^{N} \sum_{j=1}^{N} \\| W_{i \rightarrow j}^T W_{i \rightarrow j} - \lambda_{ij} I \\|_F^2$$ to regularize the projection matrix to be a multiplication of an unitary matrix. This way, $W_{i \rightarrow j}$ will preserve all normalized singular values and maintain the information abundance after projection. We experiment with various embedding sizes and compare the changes in performance, the information abundances, and the optimization dynamics for standard and regularized models. Results are shown in Figure 5. As anticipated, regularization in DCNv2 helps learn embeddings with higher information abundance. Nevertheless, from the performance perspective, the model presents unexpected results whereby the scalability does not improve or worsen as the collapse is alleviated. We further find that such a model overfits during the learning process, with the training loss consistently decreasing and the validation AUC dropping. ![Figure 5](image) (a) IA w/ 10x model size. (b) Test AUC w.r.t. model size. (c) Training vs. validation. Figure 5: Experimental results of Evidence III. Restricting DCNv2 leads to higher information abundance, yet the model suffers from over-fitting, thus resulting in non-scalability. Evidence IV: Scaling up DCNv2 and DNN. We now discuss DNN, which consists of a plain interaction module by concatenating all feature vectors from different fields and processing with an MLP, formulized by $$h = G([e_1, e_2, ..., e_N]).$$ Since DNN does not conduct explicit 2-order feature interaction (Rendle et al., 2020), following our previous interaction-collapse theory, it would suffer less from collapse. We compare the learned embeddings of DCNv2 and DNN and their performance with the growth of embedding size. Considering that different architectures or objectives may differ in modeling, we mainly discuss the performance trend as a fair comparison. Results are shown in Figure 6. DNN learns less-collapsed embedding matrices, reflected by higher information abundance than DCNv2. Yet, perversely, the AUC of DNN drops when increasing the embedding size, while DCNv2 sustains the performance. Such observations show that DNN falls into the issue of overfitting and lacks scalability, even though it suffers less from collapse. ![Figure 6](image) (a) IA w/ 10x model size. (b) Test AUC w.r.t. model size. Figure 6: Experimental results of Evidence IV. Despite higher information abundance, the performance of DNN drops w.r.t. model size. Summary: Does suppressing collapse definitely improve scalability? Regularized DCNv2 and DNN are both models with feature interaction suppressed, and they learn less-collapsed embedding matrices than DCNv2, as expected. Yet observations in evidence III&IV demonstrate that regularized DCNv2 and DNN are both non-scalable with the growth of model size and suffer from serious overfitting. We conclude the following finding: Finding 2. A less-collapsed model with feature interaction suppressed is insufficient for scalability due to overfitting concern. Such a finding is plausible, considering that feature interaction brings domain knowledge of higher-order correlations in recommender systems and helps form generalizable representations. When feature interaction is suppressed, models tend to fit noise as the embedding size increases, resulting in reduced generalization. 5 MULTI-EMBEDDING DESIGN In this section, we present a simple design of multi-embedding, which serves as an effective scaling design applicable to a wide range of model architecture designs. We introduce the overall architecture, present experimental results, and analyze how multi-embedding works. 5.1 MULTI-EMBEDDING FOR BETTER SCALABILITY The two-sided effect of feature interaction for scalability implies a principle for model design. That is, a scalable model should be capable of less-collapsed embeddings within the existing feature interaction framework instead of removing interaction. Based on this principle, we propose multi-embedding or ME as a simple yet efficient design to improve scalability. Specifically, we scale up the number of independent and complete embedding sets instead of the embedding size, and incorporate embedding-set-specific feature interaction layers. Similar to previous works such as group convolution (Krizhevsky et al., 2012), multi-head attention (Vaswani et al., 2017), and other decoupling-based works in recommender systems (Liu et al., 2022; 2019; Weston et al., 2013), such design allows the model to learn different interaction patterns jointly, while a single-embedding model would be limited to the only interaction pattern that causes severe collapse. This way, the model is capable of learning diverse embedding vectors to mitigate collapse while keeping the original interaction modules. Formally, a model with $M$ sets of embeddings is defined as $$e_i^{(m)} = \left(E_i^{(m)}\right)^T 1_{x_i}, \forall i \in \{1, 2, ..., N\},$$ $$h^{(m)} = I^{(m)}(e_1^{(m)}, e_2^{(m)}, ..., e_N^{(m)}),$$ $$h = \frac{1}{M} \sum_{m=1}^{M} h^{(m)}, \quad \hat{y} = F(h),$$ where $m$ stands for the index of embedding set. One requirement of multi-embedding is that there should be non-linearities such as ReLU in interaction $I$; otherwise, the model is equivalent to single-embedding and hence does not capture different patterns (see Appendix B). As a solution, we add a non-linear projection after interaction for the model with linear interaction layers and reduce one MLP layer for $F$ to achieve a fair comparison. An overall architecture comparison of single-embedding and mult-embedding models with $N = 2$ and $M = 2$ is shown in Figure 7. Figure 7: Architectures of single-embedding (left) and multi-embedding (right) models with $N = 2$ and $M = 2$. Figure 8: Scalability of multi-embedding on Criteo dataset. 5.2 Experiments Setup. We conduct our experiments on two datasets for recommender systems: Criteo (Jean-Baptiste Tien, 2014) and Avazu (Steve Wang, 2014), which are large and hard benchmark datasets widely used in recommender systems. We experiment on baseline models including DNN, IPNN (Qu et al., 2016), NFwFM (Pan et al., 2018), xDeepFM (Lian et al., 2018), DCNv2 (Wang et al., 2021), FinalMLP (Mao et al., 2023) and their corresponding multi-embedding variants with 2x, 3x, 4x and 10x model size. Here NFwFM is a variant of NFM (He & Chua, 2017) by replacing FM with FwFM. All experiments are performed with 8/1/1 training/validation/test splits, and we apply early stopping based on validation AUC. More details are shown in Appendix C.2. Results. We repeat each experiment 3 times and report the average test AUC with different scaling factors of the model size. Results are shown in Table 1. For the experiments with single-embedding, we observe that all the models demonstrate poor scalability. Only DCNv2 and NFwFM show slight improvements with increasing embedding sizes, with gains of 0.00036 on Criteo and 0.00090 on Avazu, respectively. For DNN, xDeepFM, and FinalMLP, which rely highly on non-explicit interaction, the performance even drops (0.00136 on Criteo and 0.00118 on Avazu) when scaled up to 10x, as discussed in Section 4.2. In contrast to single-embedding, our multi-embedding shows consistent and remarkable improvement with the growth of the embedding size, and the highest performance is always achieved with the largest 10x size. For DCNv2 and NFwFM, multi-embedding gains 0.00099 on Critio and 0.00202 on Avazu by scaling up to 10x, which is never obtained by single-embedding. Overall models and datasets, compared with baselines, the largest models averagely achieve 0.00106 improvement on the test AUC. Multi-embedding provides a methodology to break through the non-scalability limit of existing models. We visualize the scalability of multi-embedding on Criteo dataset in Figure 8. The standard deviation and detailed scalability comparison are shown in Appendix C.3. Table 1: Test AUC for different models. Higher indicates better. Underlined and bolded values refer to the best performance with single-embedding (SE) and multi-embedding (ME), respectively. \| Model \| Criteo \| Avazu \| \|-----------\|-----------------\|----------------\| \| \| base 2x 3x 4x 10x \| base 2x 3x 4x 10x \| \| DNN \| SE 0.81228 0.81207 0.81213 0.81142 \| ME 0.78744 0.78759 0.78752 0.78728 0.78648 \| \| \| SE 0.81261 0.81288 0.81289 0.81287 \| ME 0.78805 0.78826 0.78862 0.78844 \| \| IPNN \| SE 0.81272 0.81272 0.81271 0.81262 \| ME 0.78732 0.78741 0.78738 0.78750 0.78745 \| \| \| SE 0.81268 0.81270 0.81273 0.81311 \| ME 0.78806 0.78868 0.78902 0.78894 \| \| NFwFM \| SE 0.81059 0.81087 0.81100 0.81112 \| ME 0.78684 0.78757 0.78783 0.78794 – \| \| \| SE 0.81128 0.81153 0.81171 0.81210 \| ME 0.78868 0.78901 0.78932 – \| \| xDeepFM \| SE 0.81217 0.81180 0.81167 0.81116 \| ME 0.78743 0.78750 0.78714 0.78735 0.78693 \| \| \| SE 0.81236 0.81239 0.81255 0.81299 \| ME 0.78848 0.78886 0.78894 0.78927 \| \| DCNv2 \| SE 0.81341 0.81345 0.81346 0.81357 \| ME 0.78786 0.78835 0.78854 0.78852 0.78856 \| \| \| SE 0.81348 0.81361 0.81382 0.81385 \| ME 0.78862 0.78882 0.78907 0.78942 \| \| FinalMLP \| SE 0.81259 0.81248 0.81240 0.81175 \| ME 0.78751 0.78797 0.78795 0.78742 0.78662 \| \| \| SE 0.81290 0.81302 0.81303 0.81303 \| ME 0.78821 0.78831 0.78836 0.78830 \| 5.3 Analysis Information abundance. Multi-embedding models achieve remarkable scalability compared with single-embedding. We verify that such scalability originates from the mitigation of collapse. We compare the information abundance of single-embedding and multi-embedding DCNv2 with the 10x embedding size. As shown in Figure 9a, multi-embedding offers higher information abundance and indicates less collapsed embedding matrices. Variations of embeddings. Multi-embedding utilizes embedding-set-specific interactions to enrich embedding learning. We analyze the information abundance for each embedding set as shown --- 2The embedding of NFwFM with 10x size on Avazu costs nearly 37.6GB memory, which exceeds our GPU memory limit. Therefore, we do not conduct 10x NFwFM on Avazu. On the other hand, the existing experiment with 4x is already sufficient for NFwFM on Avazu. 3A slightly higher AUC at 0.001-level is regarded significant (Cheng et al., 2016; Guo et al., 2017; Song et al., 2019; Tian et al., 2023). in Figure 9b. It is observed that the embedding matrices of different sets vary in information abundance. Different interaction patterns. To justify that the scalability of multi-embedding originates from different interaction patterns, we visualize $\\|W_{i \rightarrow j}^{(m)}\\|_F$ as the interaction pattern (Wang et al., 2021) for a multi-embedding DCNv2 model in Figure 9c. It is shown that the interaction layers learn various patterns. To further illustrate, we conduct an ablation study by restricting the divergence of $\\|W_{i \rightarrow j}^{(m)}\\|_F$ across all embedding sets. From results in Figure 9d, it is observed that the divergence-restricted multi-embedding model does not show similar scalability as standard multi-embedding models, indicating multi-embedding works from the diversity of interaction layers. Ablation study on sharing one interaction layer across all embedding sets are provided in Appendix H. ![Figure 9](image) (a) IA($E_i$). (b) IA($E_i^{(m)}$). (c) $\\|W_{i \rightarrow j}^{(m)}\\|_F$. (d) Restricting diversity. Figure 9: Analysis of multi-embedding. (a): Multi-embedding learns higher information abundance. (b): Each embedding set learns diverse embeddings, reflected by varying information abundance. (c): Embedding-set-specific feature interaction layers capture different interaction patterns. (d): Restricting diversity of $\\|W_{i \rightarrow j}^{(m)}\\|_F$ across all embedding sets leads to non-scalability. 6 RELATED WORKS Modules in recommender systems. Plenty of existing works investigate the module design for recommender systems. A line of studies focuses on feature interaction process (Koren et al., 2009; Rendle, 2010; Juan et al., 2016; Qu et al., 2016; He & Chua, 2017; Guo et al., 2017; Pan et al., 2018; Lian et al., 2018; Song et al., 2019; Cheng et al., 2020; Sun et al., 2021; Wang et al., 2021; Mao et al., 2023; Tian et al., 2023), which is specific for recommender systems. These works are built up to fuse domain-specific knowledge of recommender systems. In contrast to proposing new modules, our work starts from a view of machine learning and analyzes the existing models for scalability. Collapse phenomenon. Neural collapse or representation collapse describes the degeneration of representation vectors with restricted variation. This phenomenon is widely studied in supervised learning (Papyan et al., 2020; Zhu et al., 2021; Tirer & Bruna, 2022), unsupervised contrastive learning (Hua et al., 2021; Jing et al., 2021; Gupta et al., 2022), transfer learning (Aghajanyan et al., 2020; Kumar et al., 2022), and generative models (Mao et al., 2017; Miyato et al., 2018). Chi et al. (2022) discuss the representation collapse in sparse MoEs. Inspired by these works, we realize the embedding collapse of recommendation models when regarding embedding vectors as representations by their definition, yet we are facing the setting of field-level interaction, which has not previously been well studied. Intrinsic dimensions and compression theories. To describe the complexity of data, existing works include intrinsic-dimension-based quantification (Levina & Bicke, 2004; Ansuini et al., 2019; Pope et al., 2020) and pruning-based analysis (Wen et al., 2017; Alvarez & Salzmann, 2017; Sun et al., 2021). Our SVD-based concept of information abundance is related to these works. 7 CONCLUSION In this paper, we highlight the non-scalability issue of existing recommendation models and identify the embedding collapse phenomenon that hinders scalability. From empirical and theoretical analysis around embedding collapse, we conclude the two-sided effect of feature interaction on scalability, i.e., feature interaction causes collapse while reducing overfitting. We propose a unified design of multi-embedding to mitigate collapse without suppressing feature interaction. Experiments on benchmark datasets demonstrate that multi-embedding consistently improves model scalability. REPRODUCIBILITY STATEMENT For toy experiments, we show the detailed settings in Appendix A. For experiments on benchmark datasets, we follow the default data pre-processing according to the repository of pytorch-fm[^1]. We present the general model architecture in Section 5.1, and demonstrate the specific design and all hyperparameters in Appendix C.2. We show the confidence of results with empirical standard deviations in Appendix C.3. We will release our code in case our paper is accepted. REFERENCES Armen Aghajanyan, Akshat Shrivastava, Anchit Gupta, Naman Goyal, Luke Zettlemoyer, and Sonal Gupta. Better fine-tuning by reducing representational collapse. In ICLR, 2020. Jose M Alvarez and Mathieu Salzmann. Compression-aware training of deep networks. In NeurIPS, 2017. Alessio Ansuini, Alessandro Laio, Jakob H Macke, and Davide Zoccolan. Intrinsic dimension of data representations in deep neural networks. In NeurIPS, 2019. Heng-Tze Cheng, Levent Koc, Jeremiah Harmsen, Tal Shaked, Tushar Chandra, Hrishi Aradhye, Glen Anderson, Greg Corrado, Wei Chai, Mustafa Ispir, et al. Wide & deep learning for recommender systems. In DLRS, 2016. Weiyu Cheng, Yanyan Shen, and Linpeng Huang. Adaptive factorization network: Learning adaptive-order feature interactions. In AAAI, 2020. Zewen Chi, Li Dong, Shaohan Huang, Damai Dai, Shuming Ma, Barun Patra, Saksham Singhal, Payal Bajaj, Xia Song, Xian-Ling Mao, et al. On the representation collapse of sparse mixture of experts. In NeurIPS, 2022. Huifeng Guo, Ruiming Tang, Yuning Ye, Zhenguo Li, and Xiuqiang He. Deepfm: a factorization-machine based neural network for ctr prediction. In IJCAI, 2017. Kartik Gupta, Thalaiyasingam Ajanthan, Anton van den Hengel, and Stephen Gould. Understanding and improving the role of projection head in self-supervised learning. In NeurIPS, 2022. Xiangnan He and Tat-Seng Chua. Neural factorization machines for sparse predictive analytics. In SIGIR, 2017. Tianyu Hua, Wenxiao Wang, Zihui Xue, Sucheng Ren, Yue Wang, and Hang Zhao. On feature decorrelation in self-supervised learning. In ICCV, 2021. Olivier Chapelle Jean-Baptiste Tien, joycenv. Display advertising challenge, 2014. URL https://kaggle.com/competitions/criteo-display-ad-challenge. Li Jing, Pascal Vincent, Yann LeCun, and Yuandong Tian. Understanding dimensional collapse in contrastive self-supervised learning. In ICLR, 2021. Yuchin Juan, Yong Zhuang, Wei-Sheng Chin, and Chih-Jen Lin. Field-aware Factorization Machines for CTR Prediction. In RecSys, 2016. Alexander Kirillov, Eric Mintun, Nikhila Ravi, Hanzi Mao, Chloe Rolland, Laura Gustafson, Tete Xiao, Spencer Whitehead, Alexander C Berg, Wan-Yen Lo, et al. Segment anything. In arXiv preprint arXiv:2304.02643, 2023. Yehuda Koren, Robert Bell, and Chris Volinsky. Matrix factorization techniques for recommender systems. In Computer, 2009. Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton. Imagenet classification with deep convolutional neural networks. In NeurIPS, 2012. [^1]: https://github.com/rixwew/pytorch-fm
sxGugrYhP9	While BatteryML puts a lot of effort into standardizing publicly available datasets and making them accessible to work with, for the wider community, I'm curious how much of the detailed battery science gets communicated to the end-user, or whether the layer of abstraction necessary to standardize the framework obfuscates the nuances to a large degree. The reason this is important is to enable the development of models that actively support/engage with physical phenomena in the correct fashion (physically plausible explanations).	BatteryML: An Open-source Platform for Machine Learning on Battery Degradation Han Zhang\textsuperscript{1,}, Xiaofan Gui\textsuperscript{2}, Shun Zheng\textsuperscript{2}, Ziheng Lu\textsuperscript{2}, Yuqi Li\textsuperscript{3}\textsuperscript{}, Jiang Bian\textsuperscript{2} \textsuperscript{1}Institute for Interdisciplinary Information Sciences, Tsinghua University \textsuperscript{2}Microsoft Research \textsuperscript{3}Department of Materials Science and Engineering, Stanford University han-zhan17@mails.tsinghua.edu.cn, {xiaofangu, shun.zheng, zihenglu, jiang.bian}@microsoft.com, yuqili@stanford.edu Abstract Battery degradation remains a pivotal concern in the energy storage domain, with machine learning emerging as a potent tool to drive forward insights and solutions. However, this intersection of electrochemical science and machine learning poses complex challenges. Machine learning experts often grapple with the intricacies of battery science, while battery researchers face hurdles in adapting intricate models tailored to specific datasets. Beyond this, a cohesive standard for battery degradation modeling, inclusive of data formats and evaluative benchmarks, is conspicuously absent. Recognizing these impediments, we present BatteryML\textsuperscript{1}, a one-step, all-encompass, and open-source platform that integrates data preprocessing, feature extraction, and the implementation of both conventional and state-of-the-art models. This streamlined approach promises to enhance the practicality and efficiency of research applications. BatteryML seeks to fill this void, fostering a collaborative platform where experts from diverse specializations can contribute, thereby accelerating collective progress in battery research. 1 Introduction Lithium-ion batteries, characterized by their high energy density and prolonged cycle life, have revolutionized energy storage across sectors like electric vehicles, consumer electronics, and renewable energy solutions. However, the ubiquitous adoption of these batteries comes with inherent challenges surrounding their capacity degradation and performance stability (Edge et al., 2021). Continuous cycling tends to diminish their charging and discharging capacities, posing dire implications for real-world applications. For instance, “range anxiety” becomes prevalent among electric vehicle owners, and reliability concerns arise for energy storage systems. Beyond the user experience, rapid degradation introduces broader issues, such as escalated maintenance costs, heightened resource usage, environmental strain, and potential economic inefficiencies. As such, decoding and forecasting battery performance degradation has ascended as a pivotal topic in industrial artificial intelligence. Peeling back the layers of lithium-ion batteries reveals their intricate, non-linear electrochemical dynamics (Hu et al., 2020). Degradation, observed as diminishing performance with increased charge-discharge iterations, branches mainly into losses in lithium-ion inventory (e.g., solid electrolyte interphase film formation and electrolyte decomposition) and active material losses (e.g., graphite delamination and binder decomposition) (Pop et al., 2007; Dubarry et al., 2012; Sarasketa-Zabala et al., 2015). Moreover, the internal resistance and excessive electrolyte losses further contribute to the battery’s declining health. Such losses in electrolytes, in particular, can precipitate a stark capacity plunge towards a battery’s lifecycle end (Edge et al., 2021). Confronting this degradation complexity, reliably predicting a battery’s remaining useful life (RUL), state of health (SOH), and state of charge (SOC) becomes a herculean endeavor (Lipu et al., 2018). Han Zhang and Yuqi Li worked on this project during their internship at Microsoft Research. \textsuperscript{1}Project repository: \url{https://github.com/microsoft/BatteryML} The significance of RUL, especially in battery management, second-hand vehicle evaluation, and more, has spurred extensive research. For instance, integrating techniques like electrochemical impedance spectroscopy with machine learning has been demonstrated to hold promise (Zhang et al., 2020; Severson et al., 2019; Attia et al., 2021, 2020). Similarly, SOH and SOC estimation has seen advances through capacity-based, Coulomb counting, impedance methods, and model-based techniques, with machine learning bringing innovative dimensions (Wang et al., 2011; Plett, 2004; Barsoukov et al., 2005; Doyle et al., 1993; He et al., 2011b). Yet, a glaring gap persists in the domain. While individual studies have made strides in understanding battery degradation, their focal points often remain narrowly defined by specific use scenarios or charge-discharge strategies. Existing research predominantly uses particular battery types and operation paradigms, making findings less generalizable. The disparities across datasets — in terms of battery forms, chemistries, operational profiles, or environmental conditions — render a universal approach elusive. Consequently, the absence of a consistent standard in battery research underscores the need for a comprehensive and unified methodology. Challenges* In the realm of battery research and modeling, diverse challenges often impede the streamlined application and integration of machine learning techniques. - Data heterogeneity. Battery data exhibits considerable heterogeneity in terms of both data format and data patterns. The output format of various battery testing systems differs with respect to the recorded fields, time granularity, file types, etc. Sometimes, severe conceptual confusion may arise due to differences in terminology conventions. For example, the capacity of the battery may be reported through the areal specific capacity, the total capacity, or even normalized capacity. Subsequent data processing further adds to the data heterogeneity. Even for the same data format, different cathode material compositions such as LiCoO$_2$ (LCO), LiFePO$_4$ (LFP), and LiNiMnCoO$_2$ (NMC) lead to diverse degradation patterns. - Domain knowledge. On the one hand, machine learning professionals struggle to craft effective feature spaces due to the high-dimensional and heterogeneous characteristics of battery data. This intricacy presents a significant challenge in applying advanced machine learning techniques to battery performance modeling. - Model development. On the other hand, battery experts, while proficient in understanding degradation mechanisms, often face challenges in building robust machine learning models due to the nuanced data cleaning, feature engineering, and model fine-tuning processes. Existing tools and models that are crafted for specific data structures might not seamlessly adapt to other scenarios. Contributions BatteryML addresses the above challenges in a holistic manner. As an inclusive open-source platform, BatteryML simplifies every stage of battery modeling, from data preprocessing and feature construction to model training and inference. - Unified data representation. Recognizing the challenges of diverse battery data, BatteryML introduces a standardized data representation method. It provides comprehensive processing tools to collate and harmonize virtually all public battery datasets. With this consistent data representation, a uniform evaluation criterion for assessing battery degradation becomes feasible, promoting robust comparisons and insights across diverse battery contexts. - Comprehensive open-source platform. BatteryML covers essential battery research tasks like State of Charge (SOC), State of Health (SOH), and Remaining Useful Life (RUL). It offers a holistic suite of tools encompassing data preprocessing, feature and target extraction, model training, prediction, and visualization. This integrative design allows experts from varied fields to contribute, nurturing ongoing innovation in battery research. - State-of-the-art model integration. BatteryML seamlessly integrates a wide array of models, spanning both traditional and cutting-edge techniques. The platform’s modular design ensures clear demarcation between models and data processing stages, facilitating effortless integration and refinement by machine learning experts. With a unified data representation, researchers can leverage multiple datasets simultaneously, unlocking techniques... like transfer learning. This fluidity not only accelerates research but also sets the stage for the integration of more sophisticated models in the coming times. 2 RELATED WORK Battery modeling tasks. Lithium-ion battery lifetime modeling has been the subject of numerous studies. A vast number of researchers have proposed both physical and semi-empirical models to capture various mechanisms, including the growth of the solid-electrolyte interphase, lithium plating, active material loss, and impedance increase (Das et al., 2019; Palacín, 2018; Woosung et al., 2020). Predictive state estimation for remaining useful life in battery management systems often hinges upon these mechanistic and semi-empirical models. Specialized diagnostic measurements, such as coulombic efficiency and impedance spectroscopy, further assist in lifetime estimation (Burns et al., 2013; Chen et al., 2001; Tröltzsch et al., 2006; Love et al., 2014). Despite their success, these chemistry or mechanism-specific models often struggle to accurately characterize the battery degradation on the cell level due to the intricate interactions between multiple degradation modes and the thermal and mechanical variances within a cell (Waldmann et al., 2014, 2015; Bach et al., 2016; Jain et al., 2013; Aykol et al., 2016). While semi-empirical approaches require in-depth battery and chemistry domain knowledge to model various intricate degradation mechanisms, the rapid evolution of machine learning provides a fully data-driven methodology, using linear models, support vector machines, and neural networks for accurate battery degradation modeling (Severson et al., 2019; Segler et al., 2018; Ng et al., 2020; Lu et al., 2023; Chemali et al., 2018; Zhang et al., 2018; Li et al., 2020; Atta et al., 2021; Ma et al., 2022; Ren et al., 2018; Khumprom & Yodo, 2019; Sahinoglu et al., 2018; Jiménez-Bermejo et al., 2018; Wu et al., 2018; Zhang et al., 2019). Such data-driven approaches allow seamlessly integration of different degradation factors such as electrical signals, temperature and electrochemical impedance spectroscopy data, for flexible battery modeling (Han et al., 2019; Li et al., 2019; Meng & Li, 2019; Liu et al., 2020; Hossain Lipu et al., 2021; Ayob et al., 2022; Rauf et al., 2022). On one hand, the flourishing of these data-driven approaches continues to propel advancements in battery data acquisition and the performance of machine learning models. On the other hand, due to the absence of a unified modeling framework, the problem settings and data representations vary across different models, making it challenging to achieve stable replication and comparison. This accentuates the urgent need for a consolidated platform that standardizes battery degradation research, further propelling the field’s progression. Battery Early Prediction Framework. The Battery Evaluation and Early Prediction Software Package (BEEP) offers an open-source, Python-centric framework designed for the efficient handling and processing of extensive battery cycling data streams (Herring et al., 2020). Notable features of BEEP encompass file-system-oriented organization of raw cycling data, validation procedures for data authenticity, linear model learning for anomaly detection and cycle life early-prediction. While BEEP positions itself as a tool designed to assist battery experts with coding skills to more efficiently conduct battery life predictions to validate design ideas, BatteryML aims to bridge the gap between the battery and machine learning communities. Through its modular design, BatteryML decouples the knowledge dependencies of the two communities, enabling battery experts to utilize the most advanced machine learning models, while also allowing machine learning professionals to more effectively optimize models for battery data. BatteryML’s data representation naturally supports the transformation of battery data into multidimensional tensors, thereby facilitating seamless integration for battery experts into existing deep learning frameworks. Moreover, BatteryML empowers machine learning professionals to explore advanced learning paradigms, such as transfer learning, on a variety of battery data. 3 BatteryML PLATFORM Pipeline Overview As depicted in Figure 7, the BatteryML pipeline comprises an organized sequence of functional modules, guiding users through the process of model creation and application. The initial step involves converting all incoming data into a consistent format. Following this, a configuration file is crafted to specify data locations, partitioning strategies, feature and label generation methods, as well as the associated model parameters. An elaborate sample of these settings is presented in Code 1. Once configured, the comprehensive pipeline process allows end-to-end battery degradation modeling. Components integrated in this process encompass the train-test data split module, label extractor module, feature extractor module, data preprocessing module, and the model module. Detailed examples of the pipeline’s core functions, including pipeline initialization, model training, and result extraction, is available in Appendix E.1. - Train-test split module. This module determines the split strategy of the learning process. BatteryML allows users to randomly allocate battery cells into training and test subsets with respect to a given split proportion. Alternatively, BatteryML also provide standard data splits for popular datasets such as MATR (Severson et al., 2019) and HUST (Ma et al., 2022) to enable reproducible and comparable experiments. The module also offers high flexibility for custom data partitioning. - Label extractor module. This module automatically annotates the prediction target for major battery modeling tasks such as RUL and SOH prediction. - Feature extractor module. This module implements the popular features designed by domain experts such as the cycle-difference features for battery life prediction (Severson et al., 2019). This module also enables extraction of raw electric signals and conversion to tensors for training neural networks. This module is highly extensible to support custom feature design. - Data preprocessing module. This module supports flexible transformation and refinement independently for both features and labels, including data normalization and augmentation techniques to boost model performance. - Model module. This module specifies the model structure and learning parameters. BatteryML currently supports a broad spectrum of machine learning models including linear models, tree-based models, and neural network-based models. Users can effortlessly manipulate the model behavior without the concerning the intricacies of the specialized domain knowledge. ### 3.1 BatteryData: A Unified Battery Data Representation The multifaceted landscape of battery data stems from the diverse data collection apparatuses and preprocessing methodologies employed by manufacturers. This diversity results in varied data formats, terminological conventions, data fields, and signal recording strategies. Recognizing the challenge this poses, we introduced a unified data representation `BatteryData` that encompasses both meta information and charge/discharge cycles. Figure 2: MATR1 degradation curve showcasing the train and test dataset discharge capacity - Meta information specifications. Battery attributes such as the anode, cathode, and electrolyte materials are recorded as meta information fields. Parameters such as nominal capacity, depth of charge/discharge, and the operating thresholds for voltage and current are also summarized as cell-level attributes. - Charge/Discharge cycles. BatteryData organizes the time series records as a list of cycles, detailing charge/discharge protocols and capacity, voltage, current, time, temperature, and internal resistance records. A comprehensive outline of this BatteryData is presented in Appendix E.2. Our commitment to this unified data representation not only facilitates efficient data management from diverse sources but also supports insightful comparisons of battery performance and highlights degradation nuances as battery ages. It creates a conducive environment for deploying machine learning and sophisticated data analysis strategies, propelling advancements in battery data operation and maintenance. For users, BatteryData allows flexible visualization for battery data. As evident in Figure 2, one can glean insights into capacity degradation trends by aggregating all the cycles of the battery cell. Additionally, users can delve into cycle-level signals such as the trajectory of voltage curves of successive cycles or the evolution of Coulombic efficiency for further degradation analysis, as highlighted in Figure 3. Such graphical interpretations provide users with clearer and more intuitive understanding of battery performance and degradation pathways, which in turn benefits the feature and model design. BatteryML seamlessly supports converting output formats from various battery cycler systems into BatteryData out of the box. Moreover, we offer a suite of automated processing tools to transform existing datasets into the BatteryData format. These dual capabilities ensure BatteryML’s excellent compatibility with a wide range of battery data sources. ### 3.2 Feature Engineering BatteryML boasts an array of degradation features, enabling users to flexibly tailor the data based on distinct experimental needs. These features bifurcate into two categories: within-cycle and between-cycle features. Within-cycle features. This category encompasses characteristics observed within individual charge/discharge cycles. Examples include - QdLinear, which is derived by linear interpolation of the capacity-voltage curve in discharge cycles (Attia et al., 2021). - Coulombic efficiency, which is the fraction of charge and discharge capacity within a single cycle, an important indicator of how efficiently a battery can release its stored energy. - Internal resistance, which can be calculated by the voltage and current signals to measure the opposition to the flow of electric current within the battery. Between-cycle features. These features capture the degradation patterns of any battery cell on a higher level, usually across multiple cycles. Examples include - Variance of the difference of QdLinear curves, an intuitive yet effective feature that indicates battery degradation speed (Severson et al., 2019). - Capacity decay dynamics, the slope of capacity decay curve fitted in early cycles. - Average charging time, which reflects the irreversible structural changes within the battery, such as lithium plating and the growth of the SEI layer. - Temperature dynamics, which indicates the intensity of the electrochemical reactions occurring within the battery. - Minimal internal resistance, reflecting the upper bound of battery health state. Both the within- and between-cycle features are represented as tensors that are compatible with modern machine learning frameworks. This allows a flexible manipulation of the feature space and a natural combination with deep learning models. We provide a detailed introduction of BatteryML feature module in Appendix E.3. Through these feature extraction methodologies, BatteryML empowers users to freely design and combine features, significantly simplifying the process of reproducing existing work and testing new features. ### 3.3 Automatic Label Annotation BatteryML supports automatic label annotation for supervised battery degradation modeling. For different degradation modeling tasks, BatteryML calculates the label as tensors according to the definition. Here we briefly introduce the most important degradation modeling tasks. As batteries undergo continuous cycles, their capacity inevitably declines due to aging, resulting in slight variances in performance with each use. From the moment of manufacture until a battery is fully aged and retired, a direct and crucial question is to measure the loss of the battery’s maximum available capacity in any given cycle relative to its capacity when new, namely, the cycle’s State of Health (SOH). Specifically, if we denote the battery’s nominal capacity as $C_{\text{nom}}$ and the discharge capacity in the current cycle as $C_{\text{full}}$, the SOH is defined as the percentage of these two quantities $$\text{SOH} = \frac{C_{\text{full}}}{C_{\text{nom}}} \times 100\%. \quad (1)$$ Here the $C_{\text{full}}$ in a strict definition of SOH refers to the capacity measured under the same discharge protocol as the nominal capacity. Note that in practice, this requires additional cycles under constrained temperature and current conditions and is usually infeasible due to significant human labor costs. This requires accurately estimating the ratio of the capacity already discharged in the current cycle to the original total capacity, that is, the State of Charge (SOC). Essentially, when the depth of discharge is 100%, and the Battery Management System (BMS) has recorded the discharged capacity, SOC is equivalent to SOH. Estimating SOC becomes more challenging when the depth of discharge is less than 100% (for example, when discharging begins again after a brief charge). Specifically, denoting the remaining capacity of the battery as $C_{\text{curr}}$, the definition of SOC is $$\text{SOC} = \frac{C_{\text{curr}}}{C_{\text{full}}} \times 100\%. \quad (2)$$ SOH requires a prediction for each cycle, whereas SOC demands a prediction at every moment during charging and discharging phases. Both tasks require online prediction, with SOC estimation imposing higher demands on the real-time performance of the model. Another offline task, early battery life prediction, involves conducting a limited number of charge-discharge tests to predict the full lifespan of the battery—typically until its capacity falls below 80% of the nominal capacity (Li et al., 2019)—before significant degradation occurs. This task can greatly shorten the cycling test duration for batteries, playing a crucial role in downstream battery optimization tasks such as the selection of battery materials, optimization of charging strategies, and control of operating temperatures. BatteryML automates the task of label annotation through sequential traversal of each cycle’s charge/discharge stages, significantly reducing the reliance of model design on battery domain knowledge and allowing machine learning experts to focus on developing superior models. ### 3.4 Model Development Following (Attia et al., 2021), BatteryML incorporates multiple off-the-shelf baselines for battery lifetime predictions, including linear models, tree-based models and neural networks. Domain-Enhanced methods like the ‘Variance’, ‘Discharge’, and ‘Full’ models from the (Severson et al., 2019) are implemented using handcrafted features and linear models. BatteryML also includes statistical models such as Ridge regression (Hoerl & Kennard, 2000), Principal Component Regression(PCR) (Tipping & Bishop, 1999), Partial Least Squares Regression(PLSR) (Geladi & Kowalski, 1986), Gaussian process (Williams & Rasmussen, 2006), XGBoost (Chen & Guestrin, 2016), Random forest (Breiman, 2001) models. For high-performance needs, we offer neural network models like Multi-Layer Perceptron (MLP) (Haykin, 1994), Convolutional Neural Networks (CNN) (Krizhevsky et al., 2012), Long Short-Term Memory networks (LSTM) (Hochreiter & Schmidhuber, 1997). Additionally, we introduce Transformer (Vaswani et al., 2017), a ground-breaking architecture in language and vision domains (Brown et al., 2020; Dosovitskiy et al., 2021), as a new neural baseline. These implementations utilize scikit-learn (Pedregosa et al., 2011) for all statistical models, barring XGBoost, and PyTorch (Paszke et al., 2019) for neural networks. We re-train each model with 10 random seeds and report averaged results to eliminate the effect of random initialization. To cater to the diversity of experimental contexts, users have the liberty to tweak and customize these models as per their needs. A deeper dive into model intricacies can be found in Appendix B. BatteryML’s versatility lies in offering a spectrum of models, ensuring users can cherry-pick and tailor the most fitting analytical approach aligned with their research objectives. Anchored on BatteryData, BatteryML paves the way for integrating cutting-edge machine learning paradigms like transfer learning and multi-task learning into battery modeling. Moreover, as we sail through an era where large-scale model architectures are blossoming, BatteryML lays a robust foundation to harness the power of these expansive models for future battery research. ### 4 Evaluation In this section, we provide an in-depth evaluation of model performance across various datasets to inform model selection. Through a comprehensive analysis, our intent is to offer a holistic perspective on the efficacy of each model, empowering researchers and practitioners to make informed decisions tailored to their specific goals. #### 4.1 Data We based our evaluation on several publicly accessible battery datasets: CALCE (Xing et al., 2013; He et al., 2011a), HNEI (Devie et al., 2018), HUST (Ma et al., 2022), MATR (Severson et al., 2019; Hong et al., 2020), RWTH (Li et al., 2021), SNL (Preger et al., 2020), and UL_PUR (Juarez-Robles et al., 2020, 2021). These datasets encompass LFP, LCO, NMC, NCA and NMC_LCO battery types. Further details are outlined in Table I. Certain datasets were excluded due to their unsuitability for tasks such as RUL estimation. The datasets differ in terms of materials, capacities, voltages, and RUL ranges. For RUL tasks, we also created combined datasets from the public sources to assess training efficacy when various battery data are combined. Notably, CRUH combines CALCE, RWTH, UL_PUR, and HNEI datasets; CRUSH merges CALCE, RWTH, UL_PUR, SNL, and HNET datasets; and MIX incorporates all datasets used in our study. For more detailed information on the data, please refer to the Appendix A. #### 4.2 Battery Degradation Modeling BatteryML currently supports battery degradation tasks, including RUL prediction, SOH estimation and SOC estimation. Here we report the main benchmark results, and leave the detailed analysis and further ablation studies in the appendix. Table 1: Specifications of data sources. \| Data source \| Electrode chemistry \| Nominal capacity \| Voltage range (V) \| RUL dist. \| SOC dist. (%) \| SOH dist. (%) \| Cell count \| \|-------------\|---------------------\|------------------\|-------------------\|-----------\|---------------\|---------------\|------------\| \| CALCE \| LCO/graphite \| 1.1 \| 2.7-4.2 \| 566±106 \| 77±17 \| 48±30 \| 13 \| \| MATR \| LFP/graphite \| 1.1 \| 2.0-3.6 \| 823±368 \| 93±7 \| 36±36 \| 180 \| \| HUST \| LFP/graphite \| 1.1 \| 2.0-3.6 \| 1899±389 \| 100±10 \| 43±28 \| 77 \| \| HNEI \| NMC_LCO/graphite \| 2.8 \| 3.0-4.3 \| 248±15 \| 64±17 \| 49±28 \| 14 \| \| RWTH \| NMC/carbon \| 1.11 \| 3.5-3.9 \| 658±64 \| 60±24 \| 46±22 \| 48 \| \| SNL \| NCA,NMC,LFP/graphite\| 1.1 \| 2.0-3.6 \| 1256±1321 \| 86±7 \| 45±27 \| 61 \| \| UL_PUR \| NCA/graphite \| 3.4 \| 2.7-4.2 \| 209±50 \| 89±6 \| 41±33 \| 10 \| Table 2: Benchmark results for remaining useful life prediction. The comparison methods are split into four types, including 1) dummy regressor, a trivial baseline that uses the mean of training label as predictions; 2) linear models with features designed by domain experts; 3) traditional statistical models with QdlLinear feature; 4) deep models with QdlLinear feature. For models sensitive to initialization, we present the error mean across ten seeds and attach the standard deviation as subscript. \| Models \| MATR1 \| MATR2 \| HUST \| SNL \| CLO \| CRUH \| CRUSH \| MIX \| \|-------------------------\|-------\|-------\|------\|-----\|-----\|------\|-------\|-----\| \| Dummy regressor \| 398 \| 510 \| 419 \| 466 \| 331 \| 239 \| 576 \| 573 \| \| "Variance" model \| 136 \| 211 \| 398 \| 360 \| 179 \| 118 \| 506 \| 521 \| \| "Discharge" model \| 329 \| 149 \| 322 \| 267 \| 143 \| 76 \| >1000 \| >1000\| \| "Full" model \| 167 \| >1000 \| 335 \| 433 \| 138 \| 93 \| >1000 \| 331 \| \| Ridge regression \| 116 \| 184 \| >1000\| 242 \| 169 \| 65 \| >1000 \| 372 \| \| PCR \| 90 \| 187 \| 435 \| 200 \| 197 \| 68 \| 560 \| 376 \| \| PLSR \| 104 \| 181 \| 431 \| 242 \| 176 \| 60 \| 535 \| 383 \| \| Gaussian process \| 154 \| 224 \| >1000\| 251 \| 204 \| 115 \| >1000 \| 573 \| \| XGBoost \| 334 \| 799 \| 395 \| 547 \| 215 \| 119 \| 330 \| 205 \| \| Random forest \| 168<sub>9</sub> \| 233<sub>7</sub> \| 368<sub>7</sub> \| 532<sub>25</sub> \| 192<sub>3</sub> \| 81<sub>1</sub> \| 416<sub>5</sub> \| 197<sub>6</sub> \| \| MLP \| 149<sub>3</sub> \| 275<sub>27</sub> \| 459<sub>9</sub> \| 370<sub>81</sub> \| 146<sub>5</sub> \| 103<sub>4</sub> \| 565<sub>9</sub> \| 451<sub>42</sub> \| \| CNN \| 102<sub>94</sub> \| 228<sub>104</sub> \| 465<sub>75</sub> \| 924<sub>267</sub> \| >1000 \| 174<sub>92</sub> \| 545<sub>11</sub> \| 272<sub>101</sub> \| \| LSTM \| 119<sub>11</sub> \| 219<sub>33</sub> \| 443<sub>29</sub> \| 539<sub>40</sub> \| 222<sub>12</sub> \| 105<sub>10</sub> \| 519<sub>39</sub> \| 268<sub>9</sub> \| \| Transformer \| 135<sub>13</sub> \| 364<sub>25</sub> \| 391<sub>11</sub> \| 424<sub>23</sub> \| 187<sub>14</sub> \| 81<sub>8</sub> \| 550<sub>21</sub> \| 271<sub>16</sub> \| Remaining useful life prediction. In the task of RUL prediction, BatteryML models predict the number of cycles until a battery’s SOH falls below a certain threshold, e.g. 80%, in comparison with the nominal capacity. The performance metrics of various methods are presented in Table 2. Linear models using handcrafted features, such as the "Discharge" and "Full" model, offer relatively accurate predictions for LFP battery datasets. However, their performance diminishes on the CRUSH and MIX dataset, which features diverse aging conditions, due to the limited feature set and model capacity. Traditional statistical models, capable of discerning non-linear patterns from low-level features such as Q<sub>d</sub>(V<sub>t</sub>) curves, employ specific modeling mechanisms such as the decision tree ensemble approach in Random Forests and XGBoost. Despite robust performance on CRUH, CRUSH, and MIX, their efficacy decreases on datasets such as MATR2 and SNL, where the number of training samples are limited. This finding indicates that these statistical models require a larger volume of training data to effectively learn and represent meaningful insights in RUL task. Neural network models, through automatic representation learning on low-level features, offer advancements, but face significant performance variations due to different random parameter initializations. For instance, our observations of CNN reveal its ability to make accurate predictions with many random seeds (as exemplified by the results on MATR1, see Table C.1). However, certain seeds can lead to a surprising increase in error, causing significant regression error variations. This underscores both the potential benefits and challenges of applying neural networks to RUL prediction. tasks. The observed disparities in performance across various network architectures also highlight the absence of a universally optimal architecture for battery modeling. From the feature space perspective, linear models, utilizing handcrafted features, have demonstrated satisfactory performance on datasets such as MATR2, HUST, and CLO, which solely consist of one battery type, LiFePO4 (LFP). This finding validates the efficacy of domain knowledge. However, these models appear to be less successful when applied to datasets that encompass a wider range of battery types and aging conditions, such as CRUSH and MIX. In these instances, models that are directly fitted on the Qd(Vd) curve have proven to be more effective than those using manually crafted features. This highlights a deficiency in domain-specific feature design and underscores the necessity for more versatile, generalizable features, emphasizing the potential advantages of automated representation learning. Please refer to the appendix C.1 for more detailed comparison analysis. We also provided an in-depth exploration of the impact of features and model hyperparameters in appendix D. State of Health estimation. SOH estimation task requires model to predict the ratio of the current discharge capacity in reference performance test (RPT) to the nominal capacity. Since the RPT results are not always available in the public datasets, we turn to predict the ratio of observed discharge capacity to nominal capacity in this study. We directly employ cells from the data sources in Table 1 for training and evaluation. Table C.2 showcases the comparison results. The effectiveness of methods in SOH prediction varies across datasets. Linear models are generally effective but face challenges with the MATR cells due to variable charging strategies. Tree-based models show consistent, robust performance across datasets, establishing a strong baseline in SOH estimation. Deep learning models, however, haven’t consistently outperformed traditional methods, indicating potential areas for improvement. We provided detailed analysis in appendix C.2. State of Charge estimation. Similar to SOH estimation, the exact SOC value is unattainable in practice by definition. Given the fact that RPT results are also not available in most public datasets, in this study we predict the SOC derived from the observed discharge capacity. Table C.2 demonstrates the benchmark results. LightGBM consistently surpassed other methodologies in most tasks, thereby establishing tree-based models as the current state-of-the-art in SOC prediction. Moreover, linear models continue to excel over deep learning models, highlighting the need for further research to unlock the full potential of neural networks in battery modeling. For detailed insights, please see the appendix C.3. 5 CONCLUSION At the core of BatteryML is a commitment to fostering collaboration and bridging divides. As a comprehensive open-source platform, it effectively bridges the knowledge chasm between battery researchers and AI experts, streamlining data preprocessing, feature extraction, and model application, both traditional and advanced. This synthesis not only elevates battery modeling endeavors but also catalyzes a two-way exchange, i.e., empowering battery scientists to harness AI-driven tools for research and equipping AI experts with insights to tackle intricacies specific to the battery sector. Furthermore, BatteryML serves as an anchor in standardizing practices within the battery research realm. By pioneering a unified data format and integrating advanced models into the baseline, BatteryML promotes consistency and rigour, thereby catalyzing a harmonious evolution of the industry. It is our aspiration that through BatteryML, the pace of research in battery degradation is accelerated, fostering seamless integration across industry, academia, and research spheres. 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Q1vkAhdI6j	The mechanics of the “clicking annotation” remain somewhat elusive. How were the annotators guided in executing this task? Was there a specific strategy adopted for different instance types? Was this manual labeling extended to all point clouds during training? If not, how were the crucial clicking points discerned? This reviewer believes a more thorough exposition on this subject would be beneficial in the rebuttal phase.	MixSup: Mixed-Grained Supervision for Label-Efficient LiDAR-Based 3D Object Detection Yuxue Yang1,2,3,5 Lue Fan2,3,5† Zhaoxiang Zhang1,2,3,4,5† 1School of Artificial Intelligence, UCAS 2University of Chinese Academy of Sciences (UCAS) 3Institute of Automation, Chinese Academy of Sciences (CASIA) 4Centre for Artificial Intelligence and Robotics (HKISI_CAS) 5State Key Laboratory of Multimodal Artificial Intelligence Systems (MAIS) {yangyuxue2023, fanlue2019, zhaoxiang.zhang}@ia.ac.cn Abstract Label-efficient LiDAR-based 3D object detection is currently dominated by weakly/semi-supervised methods. Instead of exclusively following one of them, we propose MixSup, a more practical paradigm simultaneously utilizing massive cheap coarse labels and a limited number of accurate labels for Mixed-grained Supervision. We start by observing that point clouds are usually textureless, making it hard to learn semantics. However, point clouds are geometrically rich and scale-invariant to the distances from sensors, making it relatively easy to learn the geometry of objects, such as poses and shapes. Thus, MixSup leverages massive coarse cluster-level labels to learn semantics and a few expensive box-level labels to learn accurate poses and shapes. We redesign the label assignment in mainstream detectors, which allows them seamlessly integrated into MixSup, enabling practicality and universality. We validate its effectiveness in nuScenes, Waymo Open Dataset, and KITTI, employing various detectors. MixSup achieves up to 97.31% of fully supervised performance, using cheap cluster annotations and only 10% box annotations. Furthermore, we propose PointSAM based on the Segment Anything Model for automated coarse labeling, further reducing the annotation burden. The code is available at https://github.com/BraveGroup/PointSAM-for-MixSup. Figure 1: Illustration of distinct properties of point clouds compared to images. They make semantic learning from points difficult but ease the estimation of geometry, which is the initial motivation of MixSup. 1 Introduction LiDAR-based 3D perception is an indispensable functionality for autonomous driving. However, the laborious labeling procedure impedes its development in academia and industry. Therefore, many label-efficient learning approaches have emerged for LiDAR-based 3D object detection, such as semi-supervised learning (Zhao et al., 2020; Wang et al., 2021; Yin et al., 2022a; Liu et al., 2023a) and weakly supervised learning (Qin et al., 2020; Meng et al., 2020; 2021; Zhang et al., 2023b; Xia et al., 2023). In this paper, we propose a more practical label-efficient learning paradigm for LiDAR-based 3D object detection. Particularly, we leverage massive cheap coarse labels and a limited number of †Corresponding authors accurate labels for mixed-grained supervision (MixSup), instead of exclusively following one of the previous label-efficient learning paradigms. MixSup stems from our following observations of point clouds. (1) Texture absence: 3D point cloud lacks distinctive textures and appearances. (2) Scale invariance: point clouds in the 3D physical world are scale-invariant to the distance from sensors since there is no perspective projection like 2D imaging. (3) Geometric richness: consisting of raw Euclidean coordinates, the 3D point cloud naturally contains rich geometric information. We summarize these distinct properties in Fig. 1. These properties cut both ways. On the one hand, the lack of textures and appearances makes it challenging to learn the categories of point clouds and identify the approximate regions where objects are located, which are collectively referred to as semantics. On the other hand, scale invariance and geometric richness potentially make it relatively easy to estimate geometric attributes of objects, such as accurate poses and shapes. Therefore, we derive the motivation of MixSup: A good detector needs massive semantic labels for difficult semantic learning but only a few accurate labels for geometry estimation. Fortunately, object semantic labels can be coarse and are much cheaper than geometric labels since the former do not necessitate accurate poses and shapes. So, in particular, we opt for semantic point clusters as coarse labels and propose MixSup aiming to simultaneously utilize cheap cluster-level labels and accurate box-level labels. Technically, we redesign the center-based and box-based assignment in popular detectors to ensure compatibility with cluster-level labels. In this way, almost any detector can be integrated into MixSup. To further reduce annotation cost, we utilize the emerging Segment Anything Model (Kirillov et al., 2023) and propose PointSAM for coarse cluster label generation, enjoying the “freebie” from the advances of image recognition. Our contributions are listed as follows. 1. Based on the observations of point cloud properties, we propose and verify a finding that a good detector needs massive coarse semantic labels for difficult semantic learning but only a few accurate geometric labels for geometry estimation. 2. We propose to adopt semantic point clusters as coarse labels and build a practical and general paradigm MixSup to utilize massive cheap cluster labels and a few accurate box labels for label-efficient LiDAR-based 3D object detection. 3. We leverage the Segment Anything Model and develop PointSAM for instance segmentation, achieving automated coarse labeling to further reduce the cost of cluster labels. 4. Extensive experiments on three benchmarks and various detectors demonstrate MixSup achieves up to 97.31% performance of the fully-supervised counterpart with 10% box annotations and cheap cluster annotations. 2 RELATED WORK LiDAR-based 3D Object Detection The mainstream LiDAR-based 3D detection can be roughly categorized into point-based methods and voxel-based methods. Point-based detectors (Shi et al., 2019; Yang et al., 2020; Shi et al., 2020b; Li et al., 2021) generally employ PointNet series (Qi et al., 2017a;b) as the point feature extractor, following diverse architectures to predict 3D bounding boxes. Voxel-based approaches (Zhou & Tuzel, 2018; Yan et al., 2018; Yin et al., 2021; Fan et al., 2022a;b; Chen et al., 2023b; Wang et al., 2023a;b; Liu et al., 2023d) transform raw points into 3D voxels, which facilitates 3D sparse convolution or transformer regimes. Besides, hybrid methods (Yang et al., 2020; Shi et al., 2020a; 2023) are utilized to harness the benefits from both sides. Semi-supervised Learning in 3D Semi-supervised learning aims to reduce the annotation burden by training models with a small amount of labeled data and a large amount of unlabeled data. Inspired by the achievement in 2D, semi-supervised learning has been propagated into 3D domain. SESS (Zhao et al., 2020) inherits the Mean Teacher (Tarvainen & Valpola, 2017) paradigm and encourages consensus between the teacher model and the student model. 3DIoUMatch (Wang et al., 2021) focuses on improving the quality of pseudo labels with a series of handcrafted designs. Different from 3DIoUMatch, Proficient Teacher (Yin et al., 2022a) leverages the spatial-temporal ensemble module and clustering-based box voting module to enhance the teacher model and obtain the accurate pseudo labels, removing the deliberately selected thresholds. Considering the weak augmentation in the teacher-student framework, HSSDA (Liu et al., 2023a) proposes shuffle data augmentation to strengthen the training of the student model. Weakly Supervised Learning Weakly supervised learning employs inexpensive weak labels to mitigate the burden of annotation costs. Especially for outdoor scenes, the emerged methodologies mainly leverage weak annotations including click-level (Meng et al., 2020; 2021; Liu et al., 2022; 2023b; Zhang et al., 2023b), scribble-level (Unal et al., 2022) and image-level (Qin et al., 2020). Albeit these works achieve promising performance, they inevitably involve intricate training regimes or elaborate network architecture. In this paper, we find utilizing a few accurate labels can estimate good geometry. So it might be more practical to introduce some accurate labels instead of following a purely weakly-supervised setting. 3 Pilot Study: What Really Matters for Label Efficiency In Sec. 1, we argue that a good detector needs massive coarse labels for semantic learning but only a few accurate labels for geometry estimation. Here we conduct a pilot study to confirm the validity of our claim. We utilize predictions from a pre-trained detector (Fan et al., 2022b) to crop point cloud regions. Thus these regions are well-classified and we only need to focus on the objects’ geometry estimation in the cropped regions. Before cropping, we introduce strong noise to the proposals to avoid geometry information leakage. In particular, we expand the proposals by 2 meters in all three dimensions, randomly shift them $0.2 \sim 0.5$ meters, and rotate them $-45^\circ \sim 45^\circ$. In this way, we build a well-classified dataset that comprises the cropped noisy regions. Finally, we train a sparse convolution-based detector with different portions of the well-classified dataset. The illustration of the pilot study is shown in Fig. 2. ![Figure 2: Illustration of the pilot study. We develop a well-classified dataset to factor out the classification and only focus on the influence of varying data amounts on geometry estimation.](image) Table 1: Performances with varying data amounts on the well-classified dataset. \| Data amount \| Vehicle 3D L2 AP / APH \| Pedestrian 3D L2 AP / APH \| Cyclist 3D L2 AP / APH \| \|-------------\|------------------------\|--------------------------\|-----------------------\| \| \| IoU = 0.7 \| IoU = 0.5 \| IoU = 0.5 \| \| 100% \| 64.19 / 63.74 \| 81.70 / 80.86 \| 65.23 / 58.02 \| \| 20% \| 64.02 / 63.54 \| 81.60 / 80.74 \| 65.00 / 58.04 \| \| 10% \| 63.37 / 62.89 \| 81.50 / 80.60 \| 64.78 / 57.96 \| \| 5% \| 63.38 / 62.90 \| 81.45 / 80.54 \| 64.11 / 56.73 \| \| 1% \| 56.40 / 55.75 \| 79.01 / 77.51 \| 57.92 / 50.26 \| The results in Table 1 show that performance with data amounts from 5% to 100% are quite similar. This phenomenon suggests that LiDAR-based detectors indeed only need a very limited number of accurate labels for geometry estimation. Additionally, we explore the impact of varying data amounts on the 3D detector’s semantic learning in Appendix A.2, supporting our claim that massive data is only necessary for semantic learning. Fortunately, semantic annotations are relatively cheap and do not necessitate accurate geometry. So in the rest of this paper, we delve into the utilization of massive cheap coarse labels for semantic learning and limited accurate labels for geometric estimation. 4 Method In this section, we first propose utilizing cluster-level labels and compare them with prior coarse center-level labels (Sec. 4.1) and how to integrate the coarse labels into MixSup for general use (Sec. 4.2). Then, we elaborate on how to obtain the coarse labels with PointSAM to further release the annotation burden (Sec. 4.3). Figure 3: Overview of MixSup. The massive cluster-level labels serve for semantic learning and a few box labels are used to learn geometry attributes. We redesign the label assignment to integrate various detectors into MixSup. 4.1 Cluster-level Coarse Label Obtaining precise 3D bounding boxes is a demanding and time-consuming undertaking, necessitating meticulous fine-tuning to meet the need for high-level accuracy. A line of work has emerged to acquire cheaper coarse labels, such as center-level labels (Meng et al., 2020; 2021). They click the center of objects on the Bird’s Eye View to obtain center-level labels. Although straightforward, a single center point provides very limited information about an object and makes it inconvenient to adopt various types of detectors. In addition, it is also non-trivial for annotators to make an accurate center click. Henceforth, we introduce clusters serving as better coarse labels. The acquisition of cluster labels is quite simple. Basically, annotators could follow this protocol: Making three coarse clicks around an object in Birds Eye’s View. Then the three click points form a parallelogram serving as three corners of the parallelogram. The points inside the parallelogram form a coarse cluster. We emphasize the labeling of clusters is very efficient since it only needs three coarse clicks around the object corners instead of an accurate click in an exact object center. In Sec. 5.3, we empirically find the average labeling cost of a cluster is only around 14% of an accurate box. We provide a simple illustration of the labeling protocol in Appendix D.1. 4.2 Coarse Label Assignment In this subsection, we demonstrate how to integrate coarse cluster-level labels and box labels into different types of detectors for mixed-grained supervision, as illustrated in Fig. 3. The most relevant part to the labels in a detector is the label assignment module, responsible for properly assigning labels to the detector to provide classification and regression supervision. Thus, MixSup only needs to redesign the label assignments for cluster-level labels to ensure the generality. We categorize these assignments into two types: center-based assignment and box-based assignment. Center-based Assignment and Inconsistency Removal The center-based assignment is widely adopted in numerous detectors. For them, we substitute the original object centers with the cluster centers $\bar{c}$, which is defined in Eq. 1. The substitution inevitably leads to the inconsistency between the true object center (of accurate boxes) and the cluster center. To resolve the inconsistency, for box labels, we also use its inside cluster center as the classification supervision. As for regression supervision, it is only attained from a few box labels. $$\bar{c} = \left\{ \frac{\min x + \max x}{2}, \frac{\min y + \max y}{2}, \frac{\min z + \max z}{2} \right\},$$ where $x, y, z$ indicate the coordinate set of points in a cluster. Box-based Assignment Box-based assignment is the procedure of assigning labels to pre-defined anchors or proposals. For example, anchor-based methods consider anchors that have a high intersection over union (IoU) with box labels as positive. Similarly, two-stage methods select proposals having proper IoU with box labels for refinement and confidence learning. Below we only focus on assigning cluster-level labels to proposals, as the design for anchors is the same. Figure 4: Illustration of Box-cluster IoU. In order to implement box-based assignment, we first define box-cluster IoU, which is defined as the point-level IoU between point clusters in proposals and cluster-level labels. As depicted in Fig. 4, the gray dots represent the point cluster in the box, while the dots outlined in green denote the cluster-level label. The box-cluster IoU is computed as the ratio of the gray dots with green outlines to all the dots in the figure. With box-cluster IoU, we can assign cluster-level labels to proposals to train any anchor-based detectors and two-stage detectors. Ambiguity of Box-based Assignment It is worthwhile to note that the box-cluster IoU is essentially ambiguous. In particular, slight perturbations of bounding boxes can result in significant changes in the ordinary box IoU. However, slight perturbations on bounding boxes usually do not change the internal cluster, so box-cluster IoU may remain unchanged. Fortunately, we only rely on box-cluster IoU for semantic assignment instead of the geometric label assignment, and the former does not necessitate accurate IoU. In Sec. 5.5, we quantitatively demonstrate the adverse effect of the ambiguity is negligible. 4.3 PointSAM for Coarse Label Generation The utilization of cluster-level labels has greatly decreased the demand for human annotation. To further reduce the annotation burden of coarse labels, we propose PointSAM for automated coarse labeling, resorting to the mighty SAM (Kirillov et al., 2023) to generate coarse cluster-level labels. PointSAM is illustrated in Fig. 5, which comprises two modules: (1) SAM-based 3D Instance Segmentation: We use SAM to infer over-segmented masks and map them to 3D point clouds. (2) Separability-Aware Refinement: Since SAM’s over-segmentation and imprecise point-pixel projection, we propose SAR to mitigate the issues to enhance the quality of segmentation. SAM-assisted 3D Instance Segmentation We first utilize a pre-trained semantic segmentation model to generate 2D semantic mask. We then project 3D points into the 2D semantic mask. The points mapped into 2D foreground semantic masks serve as prompts for SAM to generate 2D over-segment masks, which significantly improves inference speed. For each mask generated by SAM, the semantic label is assigned based on the category with the highest pixel count within the mask. By the 3D-2D projection, we obtain initial 3D instance masks. Separability-Aware Refinement (SAR) Nonetheless, the over-segmentation of SAM and projection errors lead to mediocre segmentation quality. For example, there might be some points belonging to the same objects are assigned with different mask IDs or two far apart clusters in the same direction may be assigned the same mask ID. Fortunately, these issues can be alleviated by leveraging the spatial separability property inherent to point clouds. Specifically, we employ connected components labeling (CCL) on the foreground points. After performing CCL, we obtain multiple components. We split those masks which are across multiple components and then merge those masks belonging to a single component. A simple illustration of SAR can be found in Appendix C.2. We explore the resistance of SAR to inaccurate calibration in Appendix C.3. The comparison between PointSAM and other SAM-based methods for 3D tasks is presented in Appendix C.4. 4.4 Training Loss During training stage, coarse cluster labels only contribute to classification (or confidence) $ L_{cls} $ and accurate box labels only contribute to regression $ L_{reg} $. Based on label assignment, we denote positive samples assigned with accurate labels as $ S_a $, positive samples assigned with coarse labels as $ S_c $, and negative samples as $ S_n $. The loss function for MixSup can be formulated as Eq. 2. \[ L = \frac{1}{\|S_a \cup S_c \cup S_n\|} \sum_{s_a \cup s_c \cup s_n} L_{cls} + \frac{1}{\|S_a\|} \sum_{s_a} L_{reg}. \] (2) 4.5 Discussion: Comparing MixSup with Other Label-Efficient Methods MixSup and other label-efficient learning settings such as semi/weakly/self-supervised frameworks serve the same purpose of improving label efficiency. However, they are quite different in terms of design philosophy. For example, weakly supervised methods focus on how to utilize a certain type of weak labels. Popular semi-supervised methods design training schemes such as self-training to generate high-quality pseudo labels. MixSup follows a more practical philosophy to utilize different types of supervision and tries to integrate them into popular detectors for generality. Thanks to such essential differences, MixSup can seamlessly collaborate with other settings for better performance. To demonstrate the potential, in Sec. 5.5, we establish a simple baseline to utilize the self-training technique brought from semi-supervised learning. We will pursue a more effective combination of MixSup and other label-efficient methods in future work. 5 Experiments 5.1 Dataset nuScenes nuScenes (Caesar et al., 2020) is a popular dataset for autonomous driving research. It requires 10 object classes, so it is an ideal testbed to evaluate semantic learning with massive coarse labels. Since nuScenes also contains a panoptic segmentation benchmark (Fong et al., 2022), we use it to validate the effectiveness of PointSAM and evaluate the quality. Waymo Open Dataset (WOD) Waymo Open Dataset (Sun et al., 2020) is a widely recognized dataset utilized for 3D object detection. The evaluation metric for WOD is 3D IoU-based mean Average Precision. We set the IoU thresholds of 0.7 for vehicles and 0.5 for pedestrians and cyclists, following official guidelines. Such strict metric makes it a challenging benchmark for MixSup, since it only relies on a limited number of accurate box-level labels for geometry estimation. KITTI KITTI (Geiger et al., 2012) is one of the earliest datasets for 3D detection evaluation. Due to the occlusion and truncation levels of objects, the evaluation is reported with three difficulty levels: easy, moderate, and hard. Here we present the results in terms of the mean Average Precision (mAP) with 11 recall positions under moderate difficulty. IoU thresholds for Car, Pedestrian, and Cyclist are set as 0.7, 0.5, and 0.5, respectively. 5.2 Implementation Details To demonstrate the versatility of our method, we integrate four prominent detectors into MixSup. These include an anchor-based detector SECOND (Yan et al., 2018), an anchor-free detector CenterPoint (Yin et al., 2021), a two-stage detector PV-RCNN (Shi et al., 2020a), and an emerging fully sparse detector FSD (Fan et al., 2022b; 2023b). Notably, SECOND, PV-RCNN, and FSD leverage box-based assignment, while CenterPoint adopts center-based assignment. We randomly choose 10% and 1% of ground truth boxes to serve as box-level labels. However, in Waymo Open Dataset, since the Cyclist class is very rare, we give it more possibility to be selected. This is essentially another superiority of MixSup: we can flexibly adjust the budget as needed instead of following frame-by-frame selection in conventional methods. These selected labels also function as the database for CopyPaste augmentation, as opposed to the default database copied from fully labeled frames. The implementation of MixSup is based popular codebases MMDetection3D (Contributors, 2020) and OpenPCDet (Team, 2020). The training schedule and hyperparameters are all the same as the fully-supervised training, and all experiments are conducted in 8 RTX 3090 GPUs. In PointSAM, we solely employ the semantic segmentation head of HTC (Chen et al., 2019), pre-trained on nuImages, to obtain semantics. Notably, due to the negligible overlap in image data between nuImages and nuScenes, there is no data leakage during the process of PointSAM. 5.3 Labeling Protocol and Cost Analysis We ask experienced annotators to label 100 frames from different sequences of nuScenes. They follow this protocol: Making three coarse clicks around and object in Bird Eye’s View, and the three click points are regarded as three corners of a parallelogram. The points inside the parallelogram form a coarse cluster. We provide a simple illustration in Appendix D.1. The annotators time the whole process. The average time cost of a coarse cluster label is only 14% of an accurate box Table 2: Performances on WOD and nuScenes validation split. †: Using coarse cluster labels and 10% accurate box labels. The percentage in parentheses indicates the performance ratio to the fully supervised counterpart. \| Detector \| Mean \| Vehicle \| Pedestrian \| Cyclist \| mAP \| NDS \| \|---------------------------\|------\|---------\|------------\|---------\|-------\|-------\| \| CenterPoint (100% frames) \| 64.66\| 65.04 \| 61.20 \| 67.73 \| 62.41 \| 68.20 \| \| CenterPoint (10% frames) \| 51.64\| 55.39 \| 49.07 \| 50.46 \| 42.19 \| 55.38 \| \| CenterPoint (MixSup) † \| 62.34 (96.41%) \| 61.83 (95.06%) \| 57.72 (94.31%) \| 67.46 (99.60%) \| 60.73 (97.31%) \| 66.46 (97.45%) \| In experiments, we obtain clusters by using noisy GT boxes to crop the inside points. The GT boxes are randomly expanded 0% to 10% in each dimension to mimic the potential noise in coarse labeling. To clearly demonstrate the cost of annotation requirement of MixSup, we unify the annotation costs of box-level and cluster-level labels as Eq. 3. \[ \text{cost} = \frac{(N_b + 0.14N_c)}{N_t}, \] where $N_b, N_c$ represent the number of box-level labels and cluster-level labels, and $N_t$ denotes the total number of labels in training set. 5.4 Main results Performance on mainstream datasets. We first showcase the main results of MixSup in Table 2 (WOD and nuScenes), and Table 3 (KITTI). In particular, SECOND and PV-RCNN exhibit performance up to 95.20% of fully-supervised methods, demonstrating the effectiveness of box-based assignment for cluster-level labels. CenterPoint achieves performance levels between 94.31% and 99.60% of fully supervised performance, validating the feasibility of center-based assignment. Comparison with other label-efficient frameworks. Besides MixSup, there are several other label-efficient frameworks such as semi-supervised learning and self-supervised learning. Due to their different settings, an absolute fair comparison cannot be established among these methods. However, in order to gain an intuitive understanding of performance, we list their performances in Table 4, 5. The results suggest that MixSup is an effective paradigm, achieving better or on-par performance, compared with semi/self-supervised settings. We emphasize that MixSup and these methods are complementary and compatible with each other, which will be briefly demonstrated in Sec. 5.5. Table 3: Performances on KITTI validation split with moderate difficulty. Notations have the same meanings as those in Table 2. \| Detector \| Car \| Pedestrian \| Cyclist \| \|---------------------------\|-------\|------------\|---------\| \| SECOND (100% frames) \| 78.62 \| 52.98 \| 67.15 \| \| SECOND (MixSup) † \| 74.85 (95.20%) \| 50.18 (94.71%) \| 61.46 (94.53%) \| \| PV-RCNN (100% frames) \| 83.61 \| 57.90 \| 70.47 \| \| PV-RCNN (MixSup) † \| 76.09 (91.01%) \| 54.33 (93.83%) \| 65.67 (93.19%) \| Table 4: Comparison with other weakly-supervised learning on KITTI validation split (Car). \| Label-efficient method \| Annotation \| Easy \| Moderate \| Hard \| \|------------------------\|---------------------\|------\|----------\|------\| \| WS3D (Meng et al., 2020)\| 534 boxes + weak labels \| 84.04 \| 75.10 \| 73.29 \| \| WS3D (Meng et al., 2021)\| 534 boxes + weak labels \| 85.04 \| 75.94 \| 74.38 \| \| MixSup (ours) \| 534 boxes + weak labels \| 86.37 \| 76.20 \| 72.36 \| 5.5 Performance Analysis Comparison with handcrafted box fitting. Fitting pseudo box-level labels from cluster-level labels, such as L-shape fitting (Zhang et al., 2017), presents a trivial option for incorporating coarse labels into training. However, these methods cannot distinguish length and width, and are also confusing with a certain heading $\theta$ and a heading $\theta + \pi$. Therefore, we ignore the shape and heading supervision during the training. The results in Table 6 manifest box fitting is sub-optimal, particularly for large objects like Car and Truck. This is due to the fact that the point clusters of these large objects are more prone to displaying incomplete object parts. Consequently, the pseudo boxes derived from these clusters exhibit unreliable sizes. Integration with simple self-training. As we discussed in Sec. 4.5, MixSup can collaborate with semi-supervised methods. Deliberately designing the semi-supervised training scheme is out of Table 5: Comparison with other label-efficient detectors on Waymo Open Dataset validation split (L2 mAPH). †: The annotation cost contains both box labels and cluster labels, defined by Eq. 3. : From ProficientTeacher (Yin et al., 2022a). §: From MV-JAR (Xu et al., 2023). ˆ: From HS-DDA (Liu et al., 2023a). \| Detector \| Label-efficient method \| Annotation \| Mean \| Vehicle \| Pedestrian \| Cyclist \| \|----------\|------------------------\|------------\|------\|---------\|-----------\|---------\| \| SECOND \| - \| all frames \| 57.23\| 63.33 \| 51.31 \| 57.05 \| \| SECOND \| - \| 10% frames \| 49.11\| 56.81 \| 41.91 \| 48.62 \| \| SECOND* \| FixMatch (Sohn et al., 2020) \| 10% frames \| 51.45\| 58.37 \| 44.23 \| 51.75 \| \| SECOND* \| ProficientTeacher (Yin et al., 2022a) \| 10% frames \| 54.16\| 59.36 \| 46.97 \| 56.15 \| \| SECOND \| MixSup (ours) \| 10% annotation cost † \| 54.23\| 55.02 \| 49.61 \| 58.06 \| \| SST \| - \| all frames \| 65.54\| 64.56 \| 64.89 \| 67.17 \| \| SST† \| - \| 10% frames \| 50.46\| 54.37 \| 50.71 \| 46.29 \| \| SST† \| PointContrast (Xie et al., 2020) \| 10% frames \| 49.94\| 54.30 \| 50.12 \| 45.39 \| \| SST† \| ProposalContrast (Yin et al., 2022b) \| 10% frames \| 50.13\| 54.71 \| 50.39 \| 45.28 \| \| SST† \| MV-JAR (Xu et al., 2023) \| 10% frames \| 54.06\| 58.00 \| 54.66 \| 49.52 \| \| SST \| MixSup (ours) \| 10% annotation cost † \| 60.74\| 59.10 \| 60.00 \| 63.13 \| \| PV-RCNN \| - \| all frames \| 67.06\| 68.98 \| 64.42 \| 67.79 \| \| PV-RCNN* \| - \| 1% frames \| 20.90\| 43.30 \| 15.90 \| 2.90 \| \| PV-RCNN§ \| HSDDA (Liu et al., 2023a) \| 1% frames \| 28.27\| 47.30 \| 17.50 \| 20.00 \| \| PV-RCNN \| MixSup (ours) \| 1% annotation cost † \| 56.58\| 55.46 \| 52.02 \| 62.25 \| Table 6: Comparison with handcrafted box fitting on nuScenes. We adopt CenterPoint as the base detector, conducting training for 10 epochs. †: Ignore the shape and heading supervision for fitted pseudo boxes. ˆ: Ignore the heading supervision for pseudo boxes. \| Label Format \| mAP \| NDS \| Car \| Truck \| C.V. \| Bus \| Trailer \| Bar. \| Mot. \| Byc. \| Ped. \| T.C. \| \|-------------------------------\|-----\|-----\|------\|-------\|------\|------\|---------\|------\|------\|------\|------\|------\| \| MixSup (cluster-level) \| 59.48\| 64.97\| 82.35\| 53.65 \| 19.17\| 67.25\| 36.87 \| 66.48\| 64.79\| 53.45\| 83.46\| 67.29\| \| MixSup (fitted pseudo boxes†)\| 55.75\| 62.33\| 63.72\| 45.04 \| 19.51\| 65.80\| 21.42 \| 67.57\| 65.09\| 56.87\| 84.24\| 68.25\| \| MixSup (fitted pseudo boxes‡)\| 56.22\| 60.54\| 65.39\| 44.85 \| 20.74\| 67.23\| 25.45 \| 63.91\| 66.47\| 56.90\| 83.70\| 67.54\| the scope of this paper. For simplicity and generality, we establish a simple self-training baseline to verify our claim, which is one of the most common techniques in semi-supervised learning. In particular, we first use a trained MixSup detector to generate pseudo boxes in the training set, and pseudo boxes with scores higher than 0.7 are utilized to replace corresponding coarse cluster labels. Then the updated label set is adopted to train a new detector. As shown in Table 7, the simple self-training strategy consistently improves the performance, indicating MixSup is compatible with semi-supervised training schemes. We will delve into the combination of MixSup and semi-supervised framework in future work. Roadmap from coarse clusters to accurate boxes. To better understand the gap and differences between MixSup and fully supervised detectors. We incrementally incorporate additional supervisory information for cluster-level labels. Specifically, we sequentially augment the 90% original cluster-level labels with objects’ center coordinates, shape dimensions, and heading step by step. We employ CenterPoint as the fundamental detector and conduct experiments on nuScenens for 10 epochs. The noteworthy enhancements, as detailed in Table 8, are primarily observed in large objects, like Car and Truck. This can be attributed to the fact that the centers of these large-size cluster labels exhibit a more significant deviation from their true box centers. The ambiguity of box-cluster IoU. As mentioned in Sec. 4.2, the proposed box-based assignment relies on box-cluster IoU, which is inherently more ambiguous compared to the IoU between the proposal and the box-level labels. To demystify the effect of such ambiguity, we establish the following oracle experiment: Based on FSD, a state-of-the-art two-stage detector, we adopt standard box-to-box IoU for the matching between proposal and GTs during the label assignment of the second stage. The learning scheme after the matching is the same as MixSup, where only 10% of proposals are supervised by accurate poses and shapes. Table 7: Integration with simple self-training on KITTI validation split with moderate difficulty. \| Detector \| Car \| Pedestrian \| Cyclist \| \|---------------------------\|------\|------------\|---------\| \| SECOND (100% frames) \| 78.62\| 52.98 \| 67.15 \| \| SECOND (MixSup) \| 74.85\| 50.18 \| 61.46 \| \| Above + self-training \| 77.46\| 56.89 \| 64.40 \| \| PV-RCNN (100% frames) \| 83.61\| 57.90 \| 70.47 \| \| PV-RCNN (MixSup) \| 76.09\| 54.33 \| 65.67 \| \| Above + self-training \| 78.87\| 61.03 \| 70.91 \| Table 8: Roadmap from coarse cluster labels to accurate box labels on nuScenes. We adopt CenterPoint as the base detector, conducting training for 10 epochs. †: This setting is equivalent to the fully supervised baseline, while its performance is slightly worse due to the shorter training schedule. \| Supervision \| mAP \| NDS \| Car \| Truck \| C.V. \| Bus \| Trailer \| Bar. \| Mot. \| Byc. \| Ped. \| T.C. \| \|------------------------------------\|-------\|-------\|--------\|--------\|--------\|--------\|---------\|--------\|-------\|-------\|-------\|-------\| \| MixSup \| 59.48 \| 64.97 \| 82.35 \| 53.65 \| 19.17 \| 67.25 \| 36.87 \| 66.48 \| 64.79 \| 53.45 \| 83.46 \| 67.29 \| \| MixSup + Center \| 60.49 \| 65.89 \| 83.85 \| 57.00 \| 19.66 \| 69.09 \| 37.98 \| 66.30 \| 65.29 \| 53.86 \| 83.84 \| 68.00 \| \| MixSup + Center + Shape \| 60.79 \| 66.27 \| 83.80 \| 56.93 \| 21.30 \| 70.01 \| 37.44 \| 67.35 \| 64.93 \| 54.11 \| 83.82 \| 68.17 \| \| MixSup + Center + Shape + Heading †\| 60.95 \| 66.31 \| 83.79 \| 57.20 \| 21.75 \| 69.09 \| 36.99 \| 66.90 \| 66.96 \| 54.24 \| 84.12 \| 68.44 \| Table 11: Performances with generated labels by PointSAM on nuScenes validation split. : Using labels from PointSAM. †: Removing false positive clusters. ‡: Adding false negatives based on †. \| Detector \| mAP \| NDS \| Car \| Truck \| C.V. \| Bus \| Trailer \| Bar. \| Mot. \| Byc. \| Ped. \| T.C. \| \|-----------------------------------\|-------\|-------\|--------\|--------\|--------\|--------\|---------\|--------\|-------\|-------\|-------\|-------\| \| CenterPoint (10% frames) \| 42.19 \| 55.38 \| 77.18 \| 38.18 \| 3.60 \| 42.17 \| 9.12 \| 59.29 \| 36.31 \| 20.54 \| 78.97 \| 56.57 \| \| CenterPoint (MixSup) \| 49.49 \| 58.65 \| 64.63 \| 41.71 \| 15.61 \| 57.57 \| 28.19 \| 43.56 \| 62.28 \| 51.42 \| 75.07 \| 54.87 \| \| CenterPoint (MixSup†) \| 53.09 \| 60.93 \| 70.81 \| 43.66 \| 15.66 \| 62.05 \| 30.40 \| 59.92 \| 63.37 \| 48.80 \| 77.27 \| 59.00 \| \| CenterPoint (MixSup‡) \| 58.30 \| 64.21 \| 80.33 \| 50.74 \| 20.59 \| 65.38 \| 36.11 \| 65.52 \| 62.45 \| 51.92 \| 82.06 \| 67.89 \| As can be seen from Table 9, there is no significant performance boost in the oracle experiments, demonstrating that MixSup does not necessitate precise IoU measurements. The performance is especially robust for small objects like Pedestrian and Cyclist, indicating the box-cluster IoU is sufficient in semantics learning even if it is a little ambiguous. 5.6 Analysis of PointSAM Quantitative Analysis We perform PointSAM for automated coarse labeling on nuScenes and compare the labels with prior arts on LiDAR-based panoptic segmentation benchmark (Fong et al., 2022). As PointSAM disregards background, we only report the performance for foreground thing classes, in Table 10. Thanks to the mighty SAM, PointSAM is on par with the recently fully supervised panoptic segmentation models without any 3D annotations. Human Rectification Although SAM usually generates high-quality clusters, there are inevitable false-positive clusters and false negatives due to the errors of 3D-2D projection in nuScenes. These errors cannot be completely fixed due to imprecise calibration of sensors. We provide the analysis of these bad cases in Appendix C.1. Thus, we manually correct the false positive labels and false negatives, according to the labeling protocol in Sec. 4.1. The human rectification leads to significant results in Table 11, at a cost of 50% annotation burden of all coarse labels. 6 Conclusion and Future Work Based on the unique properties of point clouds, we first verify that a good LiDAR-based detector needs massive coarse labels for semantic learning but only a few accurate labels for geometry estimation. We then propose a general label-efficient LiDAR-based framework MixSup to utilize massive cheap cluster labels and a few accurate box labels. In addition, we develop PointSAM to further reduce the annotation burden. The effectiveness is validated in three mainstream benchmarks. MixSup has great potential to collaborate with well-studied semi-supervised methods. We have shown the potential with a simple attempt and will delve into the relevant investigation in the future. Moreover, the emerging auto-labeling methods, such as (Yang et al., 2021; Qi et al., 2021; Fan et al., 2023a; Ma et al., 2023), present a compelling way to generate massive coarse labels. These automatic labelers can be utilized to further improve the performance of MixSup. ACKNOWLEDGMENTS This work was supported in part by the National Key R&D Program of China (No.2022ZD0116500), the National Natural Science Foundation of China (No.U21B2042, No.62320106010, No.62072457), and in part by the 2035 Innovation Program of CAS. REFERENCES Holger Caesar, Varun Bankiti, Alex H Lang, Sourabh Vora, Venice Erin Liong, Qiang Xu, Anush Krishnan, Yu Pan, Giancarlo Baldan, and Oscar Beijbom. nuscenes: A multimodal dataset for autonomous driving. In CVPR, pp. 11621–11631, 2020. Kai Chen, Jiangmiao Pang, Jiaqi Wang, Yu Xiong, Xiaoxiao Li, Shuyang Sun, Wansen Feng, Ziwei Liu, Jianping Shi, Wanli Ouyang, et al. Hybrid task cascade for instance segmentation. In CVPR, pp. 4974–4983, 2019. 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ikwEDva1JZ	Why are the constructive proofs important for understanding in-context learning in transformers? I am aware that there are prior works that design transformers that are capable of in-context learning. However, I am not convinced of the importance and significance of these results. Couldn't we also find weights for other architectures (like large MLPs or LSTMs) and argue that they are capable of in-context learning. Is the existence of these model weights informative of what is learnt in practice?	How Do Transformers Learn In-Context Beyond Simple Functions? A Case Study on Learning with Representations Tianyu Guo¹ Wei Hu² Song Mei¹ Huan Wang³ Caiming Xiong³ Silvio Savarese³ Yu Bai³ ¹UC Berkeley ²University of Michigan ³Salesforce AI Research tianyu_guo@berkeley.edu Abstract While large language models based on the transformer architecture have demonstrated remarkable in-context learning (ICL) capabilities, understandings of such capabilities are still in an early stage, where existing theory and mechanistic understanding focus mostly on simple scenarios such as learning simple function classes. This paper takes initial steps on understanding ICL in more complex scenarios, by studying learning with representations. Concretely, we construct synthetic in-context learning problems with a compositional structure, where the label depends on the input through a possibly complex but fixed representation function (which we instantiate as multi-layer MLPs), composed with a linear function that differs in each instance. By construction, the optimal ICL algorithm first transforms the inputs by the representation function, and then performs linear ICL on top of the transformed dataset. We show theoretically the existence of transformers that approximately implement such algorithms with mild depth and size. Empirically, we find trained transformers consistently achieve near-optimal ICL performance in this setting, and exhibit the desired dissection where lower layers transform the dataset and upper layers perform linear ICL. Through extensive probing and a new pasting experiment, we further reveal several mechanisms within the trained transformers, such as concrete copying behaviors on both the inputs and the representations, linear ICL capability of the upper layers alone, and a post-ICL representation selection mechanism in a harder mixture setting. These observed mechanisms align well with our theory and may shed light on how transformers perform ICL in more realistic scenarios. 1 Introduction Large language models based on the transformer architecture have demonstrated remarkable in-context learning (ICL) capabilities (Brown et al., 2020), where they can solve newly encountered tasks when prompted with only a few training examples, without any parameter update to the model. Recent state-of-the-art models further achieve impressive performance in context on sophisticated real-world tasks (OpenAI, 2023; Bubeck et al., 2023; Touvron et al., 2023). Such remarkable capabilities call for better understandings, which recent work tackles from various angles (Xie et al., 2021; Chan et al., 2022; Razeghi et al., 2022; Min et al., 2022; Olsson et al., 2022; Wei et al., 2023). A recent surge of work investigates ICL in a theoretically amenable setting where the context consists of real-valued (input, label) pairs generated from a certain function class. They find that transformers can learn many function classes in context, such as linear functions, shallow neural networks, and decision trees (Garg et al., 2022; Akyürek et al., 2022; Li et al., 2023a), and further studies provide theoretical justification on how transformers can implement and learn various learning algorithms in-context such as ridge regression (Akyürek et al., 2022), gradient descent (von Oswald et al., 2022; Dai et al., 2022; Zhang et al., 2023a; Ahn et al., 2023), algorithm selection (Bai et al., 2023), and Bayes model averaging (Zhang et al., 2023b), to name a few. Despite the progress, an insufficiency of this line is that the settings and results may not actually resemble ICL in real-world scenarios—for example, ICL in linear function classes are well understood in theory with efficient transformer constructions (Bai et al., 2023), and transformers indeed learn them well empirically (Garg et al., 2022); however, such linear functions in the raw input may fail to capture real-world scenarios where prior knowledge can often aid learning. This paper takes initial steps towards addressing this by studying ICL in the setting of learning with representations, a more complex and perhaps more realistic setting than existing ones. We construct synthetic ICL tasks where labels depend on inputs through a fixed representation function composed with a varying linear function. We instantiate the representation as shallow neural networks (MLPs), and consider both a supervised learning setting (with input-label pairs) and a dynamical systems setting (with inputs only) for the in-context data. Our contributions can be summarized as follows. - Theoretically, we construct transformers that implement in-context ridge regression on the representations (which includes the Bayes-optimal algorithm) for both learning settings (Section 4). Our transformer constructions admit mild sizes, and can predict at every token using a decoder architecture, (non-trivially) generalizing existing efficient constructions that predict at the last token only using an encoder architecture. - Empirically, using $L$-layer MLPs as representations, we find that trained small transformers consistently achieve near-optimal ICL risk in both learning settings (Section 5 & Figure 1b). - Using linear probing techniques, we identify evidence for various mechanisms in the trained transformers. Our high-level finding is that the lower layers transforms the data by the representation and prepares it into a certain format, and the upper layers perform linear ICL on top of the transformed data (Figure 1c), with often a clear dissection between these two modules, consistent with our theory. See Figure 1a for a pictorial illustration. - We further observe several lower-level behaviors using linear probes that align well with our (and existing) theoretical constructions, such as copying (of both the input and the representations) where which tokens are being copied are precisely identifiable (Section 5.2), and a post-ICL representation selection mechanism in a harder setting (Section 5.1.1 & Appendix E). - We perform a new pasting experiment and find that the upper layers within the trained transformer can perform nearly-optimal linear ICL in (nearly-)isolation (Section 5.1), which provides stronger evidence that the upper module alone can be a strong linear ICL learner. 2 RELATED WORK In-context learning The in-context learning (ICL) capabilities of pretrained transformers have gained significant attention since first demonstrated with GPT-3 (Brown et al., 2020). Subsequent empirical studies have investigated the capabilities and limitations of ICL in large language models (Liu et al., 2021; Min et al., 2021a;b; Lu et al., 2021; Zhao et al., 2021; Rubin et al., 2021; Razeghi et al., 2022; Elhage et al., 2021; Kirsch et al., 2022; Wei et al., 2023). A line of recent work investigates why and how pretrained transformers perform ICL from a theoretical perspective (Garg et al., 2022; Li et al., 2023a; von Oswald et al., 2022; Akyürek et al., 2022; Xie et al., 2021; Bai et al., 2023; Zhang et al., 2023a;b; Ahn et al., 2023; Raventós et al., 2023). In particular, Xie et al. (2021) proposed a Bayesian inference framework explaining ICL. Garg et al. (2022) showed transformers could be trained from scratch for ICL of simple function classes. Other studies found transformers can implement ICL through in-context gradient descent. (von Oswald et al., 2022; Akyürek et al., 2022) and in-context algorithm selection (Bai et al., 2023). Zhang et al. (2023a) studied the training dynamics of a single attention layer on linear ICL tasks. Li et al. (2023b) used the ICL framework to explain chain-of-thought reasoning (Wei et al., 2022). Our work builds on and extends the work of (Garg et al., 2022; Akyürek et al., 2022; von Oswald et al., 2022; Bai et al., 2023), where we study the more challenging setting of ICL with a representation function, and also provide new efficient ICL constructions for predicting at every token using a decoder transformer, as opposed to predicting only at the last token in most of these work. In-weights learning versus in-context learning Recent work has investigated when transformers learn a fixed input-label mapping versus when they perform ICL (Chan et al., 2022; Wei et al., 2023; Bietti et al., 2023). Chan et al. (2022) refer to learning a fixed input-label mapping from the pre-training data as “in-weights learning” (IWL), in contrast with ICL. Our problem setting assumes the pre-training data admits a fixed representation function, which should be learned by IWL. In this perspective, unlike these existing works where IWL and ICL are typically treated as competing mechanisms, we study a model in which IWL (computing the fixed representation by transformer weights) and ICL (learning the changing linear function in context) occur simultaneously. Mechanistic understanding and probing techniques A line of work focuses on developing techniques for understanding the mechanisms of neural networks, in particular transformers (Alain & Bengio, 2016; Geiger et al., 2021; Meng et al., 2022; von Oswald et al., 2022; Akyürek et al., 2022; Wang et al., 2022; Räuker et al., 2023). We adopted the linear probing technique of (Alain & Bengio, 2016) in a token-wise fashion for interpreting the ICL mechanisms of transformers. Beyond probing, more convincing mechanistic interpretations may require advanced approaches such as causal intervention (Geiger et al., 2021; Vig et al., 2020; Wang et al., 2022); Our pasting experiment has a similar interventional flavor in that we feed input sequences (ICL instances) from another distribution directly (through a trainable embedding layer) to the upper module of a transformer. ### 3 Preliminaries Transformers We consider sequence-to-sequence functions applied to $N$ input vectors $\{h_i\}_{i=1}^N \subset \mathbb{R}^{D_{\text{hid}}}$ in $D_{\text{hid}}$ dimensions, which we write compactly as an input matrix $H = [h_1, \ldots, h_N] \in \mathbb{R}^{D_{\text{hid}} \times N}$, where each $h_i$ is a column of $H$ (also a token). We use a standard $L$-layer decoder-only (autoregressive) transformer, which consists of $L$ consecutive blocks each with a masked self-attention layer (henceforth “attention layer”) followed by an MLP layer. Each attention layer computes $$\text{Attn}_\theta(H) := H + \sum_{m=1}^M (V_m H) \times \overline{\sigma}(\text{MSK} \odot ((Q_m H)^T (K_m H))) \in \mathbb{R}^{D \times N},$$ where $\theta = \{(Q_m, K_m, V_m) \subset \mathbb{R}^{D_{\text{hid}} \times D_{\text{hid}}}\}_{m \in [M]}$ are the (query, key, value) matrices, $M$ is the number of heads, $\text{MSK} \in \mathbb{R}^{N \times N}$ is the decoder mask matrix with $\text{MSK}_{ij} = 1\{i \leq j\}$, and $\overline{\sigma}$ is the activation function which is typically chosen as the (column-wise) softmax: $[\overline{\sigma}(A)]_{j} = \text{softmax}(a_j) \in \mathbb{R}^N$ for $A = [a_1, \ldots, a_N] \in \mathbb{R}^{N \times N}$. Each MLP layer computes $$\text{MLP}_{W_1, W_2}(H) := H + W_2 \sigma(W_1 H),$$ where $W_{\{1,2\}} \in \mathbb{R}^{D_{\text{hid}} \times D_{\text{hid}}}$ are the weight matrices, and $\sigma(t) = \max\{t, 0\}$ is the ReLU activation. We use $\text{TF}$ to denote a transformer, and typically use $\tilde{H} = \text{TF}(H)$ to denote its output on $H$. In-context learning We consider in-context learning (ICL) on regression problems, where each ICL instance is specified by a dataset $D = \{(x_i, y_i)\}_{i \in [N]} \overset{\text{iid}}{\sim} P$, with $(x_i, y_i) \in \mathbb{R}^d \times \mathbb{R}$, and the model is required to accurately predict $y_i$ given all past observations $D_{i-1} := \{(x_j, y_j)\}_{j \leq i-1}$ and the test input $x_i$. Each instance $D = D^{(j)}$ is drawn from a different data distribution $P = P^{(j)}$. Accurate prediction requires learning $P$ in-context from the past observations $D_{i-1}$ (i.e. the context); merely memorizing any fixed $P^{(j)}$ is not enough. This is a main challenge of in-context learning. We consider using transformers to do ICL, where we feed a sequence of length $2N$ into the transformer TF using the following input format: $$H = [h_1, \ldots, h_{2N}] = \begin{bmatrix} x_1 & 0 & \cdots & x_N & 0 \\ 0 & y_1 & \cdots & 0 & y_N \\ p^x_1 & p^y_1 & \cdots & p^x_N & p^y_N \end{bmatrix} \in \mathbb{R}^{D_{\text{hid}} \times 2N},$$ where $p^x_i, p^y_i \in \mathbb{R}^{D_{\text{hid}} - d - 1}$ are fixed positional encoding vectors consisting of zero paddings, followed by non-zero entries containing information about the position index $i$ and indicator of being an $x$-token (1 in $p^x_i$, and 0 in $p^y_i$); see (12) for our concrete choice. We refer to each odd token $h_{2i-1}$ as an $x$-token (also the $x_i$-token), and each even token $h_{2i}$ as a $y$-token (also the $y_i$-token). After obtaining the transformer output $\tilde{H} = \text{TF}(H)$, for every index $i \in [N]$, we extract the prediction $\hat{y}_i$ from the output token at position $x_i$: $\hat{y}_i := (\tilde{H}_i)_d + 1$. Feeding input (1) into the transformer simultaneously computes $\hat{y}_i \leftarrow \text{TF}(x_1, y_1, \ldots, x_{i-1}, y_{i-1}, x_i)$ for all $i \in [N]$. Denote the parameters of transformers as $\theta$. In addition to the above setting, we also consider a dynamical system setting with $D = \{x_i\}_{i \in [N]}$ where the transformer predicts $\hat{x}_i$ from the preceding inputs $x_{<i}$. See Section 4.2 for details. ## 4 IN-CONTEXT LEARNING WITH REPRESENTATIONS ### 4.1 Supervised learning with representation We begin by considering ICL on regression problems with representation, where labels depend on the input through linear functions of a fixed representation function. Formally, let $\Phi^* : \mathbb{R}^d \to \mathbb{R}^D$ be a fixed representation function. We generate each in-context data distribution $P = P_w$ by sampling a linear function $w \sim N(0, \tau^2 I_D)$ from a Gaussian prior, and then generate the ICL instance $D = \{(x_i, y_i)\}_{i \in [N]} \sim P_w$ by a linear model on $\Phi^$ with coefficient $w$ and noise level $\sigma > 0$: $$y_i = \langle w, \Phi^(x_i) \rangle + \sigma z_i, \quad x_i \overset{\text{iid}}{\sim} P_x, \quad z_i \overset{\text{iid}}{\sim} N(0, 1), \quad i \in [N].$$ Note that all $D$'s share the same representation $\Phi^$, but each admits a unique linear function $w$. The representation function $\Phi^$ can in principle be chosen arbitrarily. As a canonical and flexible choice for both our theory and experiments, we choose $\Phi^$ to be a standard $L$-layer MLP: $$\Phi^(x) = \sigma^(B^_L \sigma^(B^_{L-1} \cdots \sigma^(B^_1 x) \cdots)), \quad B^_1 \in \mathbb{R}^{D \times d}, (B^_\ell)_{\ell=2}^L \subset \mathbb{R}^{D \times D}$$ where $D$ is the hidden and output dimension, and $\sigma^$ is the activation function (applied entry-wise) which we choose to be the leaky ReLU $\sigma^(t) = \sigma_\rho(t) := \max\{t, \rho t\}$ with slope $\rho \in (0, 1)$. Theory As $\Phi^$ is fixed and the $w$ is changing in model (2), by construction, a good ICL algorithm should compute the representations $\{\Phi^(x_i)\}_i$ and perform linear ICL on the transformed dataset $\{\Phi^(x_i), y_i\}_i$ to learn $w$. We consider the following class of $\Phi^$-ridge estimators: $$\hat{w}^{\Phi^, \lambda}_i := \arg\min_{w \in \mathbb{R}^d} \frac{1}{2(\tau^2 - 1)} \sum_{j=1}^{i-1} (\langle w, \Phi^(x_j) \rangle - y_j)^2 + \frac{\lambda}{2} \\|w\\|_2^2,$$ and we understand $\hat{w}^{\Phi^, \lambda}_1 := 0$. In words, $\hat{w}^{\Phi^, \lambda}_i$ performs ridge regression on the transformed dataset $\{\Phi(x_j), y_j\}_{j<i-1}$ for all $i \in [N]$. By standard calculations, the Bayes-optimal predictor for $y_i$ given $(D_{i-1}, x_i)$ is exactly the ridge predictor $\hat{y}^{\Phi^, \lambda}_i := \langle \hat{w}^{\Phi^, \lambda}_i, \Phi^(x_i) \rangle$ at $\lambda = \sigma^2/\tau^2$. We show that there exists a transformer that can approximately implement ($\Phi^$-Ridge) in-context at every token $i \in [N]$. The proof can be found in Appendix B. Theorem 1 (Transformer can implement $\Phi^$-Ridge). For any representation function $\Phi^$ of form (3), any $\lambda > 0$, $B_\Phi, B_w, B_y > 0$, $\varepsilon < B_\Phi B_w/2$, letting $\kappa := 1 + B_\Phi^2/\lambda$, there exists a transformer TF with $L + O(\kappa \log(B_\Phi B_w/\varepsilon))$ layers, 5 heads, $D_{\text{hid}} = 2D + d + 10$ such that the following holds. For any dataset $D$ such that $\\|\Phi^(x_i)\\|_2 \leq B_\Phi$, $\|y_i\| \leq B_y$ and the corresponding input $H \in \mathbb{R}^{D_{\text{hid}} \times 2N}$ of format (1), we have 1. There is no information leakage, as the “prefix” property of decoder transformers $\tilde{H}_i = \tilde{H}_{2i-1} = [\text{TF}(H_{1:(2i-1)})]_{2i-1}$ ensures that $\tilde{H}_i$ (and thus $\hat{y}_i$) only depends on $(D_{i-1}, x_i)$. 2. The predictor $\hat{y}_i = \hat{y}_i(D_{i-1}, x_i)$ that minimizes the posterior square loss $\mathbb{E}[\frac{1}{2}(\hat{y}_i - y_i)^2 \| D_{i-1}, x_i]$. (a) The first $(L + 2)$ layers of TF transforms $x_i$ to the representation $\Phi^(x_i)$ at each $x$ token, and copies them into the succeeding $y$ token: \[ \text{TF}^{(1:L+2)}(H) = \begin{bmatrix} \Phi^(x_1) & \Phi^(x_1) & \cdots & \Phi^(x_N) & \Phi^(x_N) \\ 0 & y_1 & \cdots & 0 & y_N \\ \tilde{p}_1^x & \tilde{p}_1^y & \cdots & \tilde{p}_N^x & \tilde{p}_N^y \end{bmatrix}, \] where $\tilde{p}_i^x, \tilde{p}_i^y$ only differ from $p_i^x, p_i^y$ in the dimension of the zero paddings. (b) For every index $i \in [N]$, the transformer output $\tilde{H} = \text{TF}(H)$ contains prediction $\hat{y}_i := [\tilde{h}_{2i-1}]_{D+1}$ that is close to the $(\Phi^* \text{-Ridge})$ predictor: $\|\hat{y}_i - (\Phi^(x_i), \hat{w}_i^\Phi, \lambda)\| \leq \varepsilon$. The transformer construction in Theorem 1 consists of two “modules”: The lower layers compute the representations and prepares the transformed dataset $\{\Phi^(x_i), y_i\}$ into form (4). In particular, each $\Phi^(x_i)$ appears both in the $i$-th $x$-token and is also copied into the succeeding $y$ token. The upper layers perform linear ICL (ridge regression) on top of the transformed dataset. We will test whether such mechanisms align with trained transformers in reality in our experiments (Section 5.1). Proof techniques The proof of Theorem 1 builds upon (1) implementing the MLP $\Phi^$ by transformers (Lemma B.3), and (2) an efficient construction of in-context ridge regression (Theorem B.5), which to our knowledge is the first efficient construction for predicting at every token using decoder transformers. The latter requires several new construction techniques such as a copying layer (Lemma B.1), and an efficient implementation of $N$ parallel in-context gradient descent algorithms at all tokens simultaneously using a decoder transformer (Proposition B.4). These extend the related constructions of von Oswald et al. (2022); Bai et al. (2023) who only consider predicting at the last token using encoder transformer, and could be of independent interest. In addition, the bounds on the number of layers, heads, and $D_{\text{hid}}$ in Theorem 1 can imply a sample complexity guarantee for (pre-)training: A transformer with $\varepsilon$-excess risk (on the same ICL instance distribution) over the one constructed in Theorem 1 can be found in $\tilde{O}((L + \kappa)^2(D + d)^2\varepsilon^{-2})$ training instances, by the generalization analysis of Bai et al. (2023, Theorem 20). We remark that the constructions in Theorem 1 & 2 choose $\sigma$ as the normalized ReLU instead of softmax, following (Bai et al., 2023) and in resonance with recent empirical studies (Wortsman et al., 2023). 4.2 Dynamical system with representation As a variant of model (2), we additionally consider a (nonlinear) dynamical system setting with data $D = (x_1, \ldots, x_N)$, where each $x_{i+1}$ depends on the $k$ preceding inputs $[x_{i-k+1}; \ldots; x_i]$ for some $k \geq 1$ through a linear function on top of a fixed representation function $\Phi^$. Compared to the supervised learning setting in Section 4.1, this setting better resembles some aspects of natural language, where the next token in general depends on several preceding tokens. Formally, let $k \geq 1$ denote the number of input tokens that the next token depends on, and $\Phi^ : \mathbb{R}^{kd} \to \mathbb{R}^D$ denotes a representation function. Each ICL instance $D = \{x_i\}_{i \in [N]}$ is generated as follows: First sample $P = P_W$ where $W \in \mathbb{R}^{D \times d}$ is sampled from a Gaussian prior: $W_{ij} \overset{\text{iid}}{\sim} N(0, \tau^2)$. Then sample the initial input $x_1 \sim P_x$ and let \[ x_{i+1} = W^\top \Phi^([x_{i-k+1}; \ldots; x_i]) + \sigma z_i, \quad z_i \overset{\text{iid}}{\sim} N(0, I_d), \quad i \in [N - 1], \] where we understand $x_j := 0_d$ for $j \leq 0$. We choose $\Phi^$ to be the same $L$-layer MLP as in (3), except that the first weight matrix has size $B_1^* \in \mathbb{R}^{D \times kd}$ to be consistent with the dimension of the augmented input $\bar{x}_i := [x_{i-k+1}; \ldots; x_i]$. We remark that (5) substantially generalizes the setting of Li et al. (2023a) which only considers linear dynamical systems (equivalent to $\Phi^* \equiv \text{id}$), a task arguably much easier for transformers to learn in context. As $x_i$ acts as both inputs and labels in model (5), we use the following input format for transformers: \[ H := \begin{bmatrix} x_1 & \cdots & x_N \\ p_1 & \cdots & p_N \end{bmatrix} \in \mathbb{R}^{D_{\text{hid}} \times N}, \] where $p_i := [0_{D_{\text{hid}}-d-4}; 1; i; i^2; i^3]$, and we extract prediction $\hat{x}_{i+1}$ from the $i$-th output token. Theory Similar as above, we consider the ridge predictor for the dynamical system setting \[ \hat{W}_{i}^{\Phi^, \lambda} := \arg\min_{W \in \mathbb{R}^{D \times d}} \frac{1}{2(t-1)} \sum_{j=1}^{i-1} \\| W^\top \Phi^(\bar{x}_j) - x_{j+1} \\|_2^2 + \frac{\lambda}{2} \\| W \\|_{F,r}^2. \quad (\Phi^-\text{Ridge-Dyn}) \] We understand $ \hat{W}_{0}^{\Phi^, \lambda} := 0_{D \times d} $, and let $ \\| W \\|_{2,\infty} := \max_{j \in [d]} \\| W_{:,j} \\|_2 $ for any $ W \in \mathbb{R}^{D \times d} $. Again, $ (\Phi^-\text{Ridge-Dyn}) $ gives the Bayes-optimal predictor $ (\hat{W}_{i}^{\Phi^, \lambda})^\top \Phi^(\bar{x}_t) $ at $ \lambda = \sigma^2/\tau^2 $. The following result shows that $ (\Phi^-\text{Ridge-Dyn}) $ can also be implemented efficiently by a transformer. The proof can be found in Appendix C.2. Theorem 2 (Transformer can implement $ \Phi^* $-Ridge for dynamical system). For the dynamical system setting where the $ L $-layer representation function $ \Phi^* : \mathbb{R}^{kd} \to \mathbb{R}^D $ takes form (3), but otherwise same settings as Theorem 1, there exists a transformer TF with $ L + 2 + O(\kappa \log(B_\Phi B_w/\varepsilon)) $ layers, max $ \{3d, 5\} $ heads, and $ D_{\text{hid}} = \max \{2(k+1), D\}d + 3(D+d) + 5 $ such that the following holds. For any dataset $ \mathcal{D} $ such that $ \\| \Phi^(\bar{x}_i) \\|_2 \leq B_\Phi $, $ \\| x_i \\|_\infty \leq B_y $, and $ \\| \hat{W}_{i}^{\Phi^, \lambda} \\|_{2,\infty} \leq B_w/2 $ (cf. $ (\Phi^-\text{Ridge-Dyn}) $) for all $ i \in [N] $, and corresponding input $ H \in \mathbb{R}^{D_{\text{hid}} \times N} $ of format (6), we have (a) The first transformer layer copies the $ k $ previous inputs into the current token, and computes the first layer $ \{\sigma_\rho(B_i^1 \bar{x}_i)\}_{i \in [N]} $ within $ \Phi^ $: \[ \text{Attn}^{(1)}(H) = \begin{bmatrix} \bar{x}_1 & \cdots & \bar{x}_N \\ \bar{p}_1 & \cdots & \bar{p}_N \end{bmatrix} = \begin{bmatrix} x_{1-k+1} & \cdots & x_{N-k+1} \\ x_1 & \cdots & x_N \\ \bar{p}_1 & \cdots & \bar{p}_N \end{bmatrix}; \] \[ \text{TF}^{(1)}(H) = \text{MLP}^{(1)}\left(\text{Attn}^{(1)}(H)\right) = \begin{bmatrix} \sigma_\rho(B_i^1 \bar{x}_1) & \cdots & \sigma_\rho(B_i^1 \bar{x}_N) \\ x_1 & \cdots & x_N \\ \bar{p}_1 & \cdots & \bar{p}_N \end{bmatrix}. \] (b) The first $ (L + 1) $ layers of TF transforms each $ x_i $ to $ \Phi^(\bar{x}_i) $, and copies the preceding representation $ \Phi^(\bar{x}_{i-1}) $ onto the same token to form the (input, label) pair $ (\Phi^(\bar{x}_{i-1}), x_i) $: \[ \text{TF}^{(1:L+1)}(H) = \begin{bmatrix} \Phi^(\bar{x}_1) & \Phi^(\bar{x}_2) & \cdots & \Phi^(\bar{x}_N) \\ 0_d & 0_d & \cdots & 0_d \\ 0_D & \Phi^(\bar{x}_1) & \cdots & \Phi^(\bar{x}_{N-1}) \\ x_1 & x_2 & \cdots & x_N \\ \bar{p}_1 & \bar{p}_2 & \cdots & \bar{p}_N \end{bmatrix}. \] Above, $ \bar{p}_i, \bar{p}_i', \bar{p}_i'' $ only differs from $ p_i $ in the dimension of the zero paddings. (c) For every index $ i \in [N] $, the transformer output $ \tilde{H} = \text{TF}(H) $ contains prediction $ \hat{x}_{i+1} := [\tilde{h}_i]_{1:d} $ that is close to the $ (\Phi^-\text{Ridge-Dyn}) $ predictor: $ \\| \hat{x}_{i+1} - (\hat{W}_{i}^{\Phi^, \lambda})^\top \Phi^(\bar{x}_t) \\|_\infty \leq \varepsilon $. To our best knowledge, Theorem 2 provides the first transformer construction for learning nonlinear dynamical systems in context. Similar as for Theorem 1, the bounds on the transformer size here imply guarantees $ \varepsilon $ excess risk within $ \tilde{O}((L + \kappa)^2((k + D)d)^2\varepsilon^{-2}) $ (pre-)training instances. In terms of the mechanisms, compared with Theorem 1, the main differences in Theorem 2 are (1) the additional copying step (7) within the first layer, where the previous $ (k - 1) $ tokens $ [x_{i-k+1}; \cdots; x_{i-1}] $ are copied onto the $ x_i $ token, to prepare for computing of $ \Phi^(\bar{x}_i) $; (2) the intermediate output (9), where relevant information (for preparing for linear ICL) has form $ [\Phi^(\bar{x}_{i-1}); x_i; \Phi^(\bar{x}_i)] $ and is gathered in the $ x $-tokens, different from (4) where the relevant information is $ [\Phi^(x_i); y_i] $, gathered in the $ y $-token. We will test these in our experiments (Section 5.2). 5 EXPERIMENTS We now empirically investigate trained transformers under the two settings considered in Section 4.1 & 4.2. In both cases, we choose the representation function $ \Phi^ $ to be a normalized version of the $ L $-layer MLP (3): $ \Phi^(x) := \tilde{\Phi}^(x)/\\|\tilde{\Phi}^(x)\\|_2 $, where $ \tilde{\Phi}^ $ takes form (3), with weight matrices $ (B_i^)_{i \in [L]} $ sampled as random (column/row)-orthogonal matrices and held fixed in each experiment, and slope $\rho = 0.01$. We test $L \in \{1, 2, 3, 4\}$, hidden dimension $D \in \{5, 20, 80\}$, and noise level $\sigma \in \{0, 0.1, 0.5\}$. All experiments use $P_x = N(0, I_d)$, $\tau^2 = 1$, $d = 20$, and $N = 41$. We use a small architecture within the GPT-2 family with 12 layers, 8 heads, and $D_{\text{hid}} = 256$, following (Garg et al., 2022; Li et al., 2023a; Bai et al., 2023). The (pre)-training objective for the transformer (for the supervised learning setting) is the average prediction risk at all tokens: $$\min_\theta \mathbb{E}_{w,D \sim P_w} \left[ \frac{1}{2N} \sum_{i=1}^{N} (\hat{y}_{\theta,i}(D_{i-1}, x_i) - y_i)^2 \right],$$ where $\hat{y}_{\theta,i}$ is extracted from the $(2i - 1)$-th output token of $\text{TF}_\theta(H)$ (cf. Section 3). The objective for the dynamical system setting is defined similarly. Additional experimental details can be found in Appendix D, and ablation studies (e.g. along the training trajectory; cf. Figure 9) in Appendix F. ### 5.1 Supervised Learning with Representation We first test ICL with supervised learning data as in Section 4.1, where for each configuration of $(L, D, \sigma)$ (which induces a $\Phi^$) we train a transformer on ICL data distribution (2) and evaluate ICL on the same distribution. Note that Figure 1c & 1b plots the results for $(L, D, \sigma) = (2, 20, 0.1)$. ICL performance Figure 2 reports the test risk across various settings, where we observe that trained transformers can consistently match the Bayes-optimal ridge predictor. This extends existing results which show that linear functions (without a representation) can be learned near-optimally in-context by transformers (Garg et al., 2022; Akyürek et al., 2022), adding our model (2) to this list of (empirically) nearly-optimally learnable function classes. Among the complexity measures $(L, D, \sigma)$, observe that the noise level $\sigma$ and hidden dimension $D$ of the representation (Figure 2a & 2b) appears to have a larger effect on the (nearly Bayes-optimal) risk than the depth $L$ (Figure 2c). Mechanisms via linear probing We conduct probing experiments to further understand the mechanisms of the trained transformers. In accordance with the theoretical construction in Theorem 1, our main question here is: Does the trained transformer perform the following in order: 1. Computes $\Phi^(x_i)$ at $x_i$ tokens; 2. Copies them onto the following $y_i$ token and obtains dataset $\{\Phi^(x_i), y_i\}_i$ in the form of (4); 3. Performs linear ICL on top of $\{\Phi^(x_i), y_i\}_i$? Figure 4: (a) Illustration of our pasting experiment, which examines the linear ICL capability of the upper module of a trained transformer. (b) Pasting results for the upper module of a trained transformer in setting $(L, D, \sigma) = (3, 20, 0.1)$. “TF_upper+...” correspond to feeding the upper module of trained transformer with different embeddings. It achieves nearly optimal linear ICL risk (in 20 dimension with noise 0.1), using a 1-layer transformer embedding, and also non-trivial performance using linear and linear copy embeddings. While such internal mechanisms are in general difficult to quantify exactly, we adapt the linear probing (Alain & Bengio, 2016) technique to the transformer setting to identify evidence. Linear probing allows us to test whether intermediate layer outputs (tokens) $\{h_{x_i}^{\ell}\}_{\ell \in [12]}$ ($\ell$ denotes the layer) and $\{h_{y_i}^{\ell}\}_{\ell \in [12]}$ “contains” various quantities of interest, by linearly regressing these quantities (as the $y$) on the intermediate tokens (as the $x$), pooled over the token index $i \in [N]$. For example, regressing $\Phi^(x_i)$ on $h_{x_i}^{\ell}$ tests whether the $x_i$ token after the $\ell$-th layer “contains” $\Phi^(x_i)$, where a smaller error indicates a better containment. See Appendix D.1 for further setups of linear probing. Figure 3 reports the errors of three linear probes across all 12 layers: The representation $\Phi^(x_i)$ in the $x_i$ tokens and $y_i$ tokens, and the optimal ridge prediction $y_i^* - \Phi^\lambda$ in the $x_i$ tokens. Observe that the probing errors for the representation decrease through lower layers and then increase through upper layers (Figure 3a & 3b), whereas probing errors for the ridge prediction monotonically decrease through the layers (Figure 3c), aligning with our construction that the transformer first computes the representations and then performs ICL on top of the representation. Also note that deeper representations take more layers to compute (Figure 3a). Further, the representation shows up later in the $y$-tokens (layers 5-6) than in the $x$-tokens (layers 1,3,4,5), consistent with the copying mechanism, albeit the copying appears to be lossy (probe errors are higher at $y$-tokens). Finally, observe that the separation between the lower and upper modules seems to be strong in certain runs—for example, the red transformer $(L = 4, \sigma = 0.1)$ computes the representation at layer 5, copies them onto $y$-tokens at layer 6, and starts to perform iterative ICL from layer 7, which aligns fairly well with our theoretical constructions at a high level. Investigating upper module via pasting To further investigate upper module, we test whether it is indeed a strong ICL learner on its own without relying on the lower module, which would provide stronger evidence that the upper module performs linear ICL. However, a key challenge here is that it is unclear how to feed raw inputs directly into the upper module, as they supposedly only admit input formats emitted from the lower module—the part we wanted to exclude in the first place. We address this by conducting a pasting experiment, where we feed $D$-dimensional linear ICL problems $(y_i' = \langle w', x_i'\rangle$ without a representation) with input format (1) directly to the upper module of the transformer trained on representation $\Phi^$, by adding a trainable embedding layer in between; see Figure 4a for an illustration of the pasting approach. This trainable embedding layer itself needs to be shallow without much ICL power—we test the following three choices: (1) Linear embedding: $h_{x_i} = W[x_i; 0]$ and $h_{y_i} = W[0; y_i]$; (2) Linear-copy embedding, where the $y$ tokens are instead $h_{y_i} = W[x_i; y_i]$, motivated by the format (4); (3) One-layer transformer embedding $TF$, which computes $H = TF(H)$. See Appendix D.2 for further setups of pasting. Figure 4b shows the pasting results on a trained transformer on $(L, D, \sigma) = (3, 20, 0.1)$ (an ablation in Figure 10b), where we dissect the lower and upper modules at layer 4 as suggested by the probing curve (Figure 3a green). Perhaps surprisingly, the upper module of the transformer can indeed perform nearly optimal linear ICL without representation when we use the one-layer transformer. embedding. Note that a (freshly trained) single-layer transformer itself performs badly, achieving about the trivial test risk 1.01, which is expected due to our specific input format$^3$ (1). This suggests that the majority of the ICL is indeed carried by the upper module, with the one-layer transformer embedding not doing much ICL itself. Also note that the linear-copy and linear embeddings also yield reasonable (though suboptimal) performance, with linear-copy performing slightly better. ### 5.1.1 Extension: Mixture of Multiple Representations We additionally investigate an harder scenario in which there exists multiple possible representation functions $(\Phi^_j)_{j \in [K]}$, and the ICL data distribution is a mixture of the $K$ distributions of form (2) each induced by $\Phi^_j$ (equivalent to using the concatenated representation $\Phi^* = [\Phi^_1; \ldots; \Phi^_K]$ with a group 1-sparse prior on $w \in \mathbb{R}^{KD}$). We find that transformers still approach Bayes-optimal risks, though less so compared with the single-representation setting. Using linear probes, we find that transformers sometimes implement the post-ICL algorithm selection mechanism identified in Bai et al. (2023), depending on the setting. Details are deferred to Appendix E due to the space limit. ### 5.2 Dynamical Systems We now study the dynamical systems setting in Section 4.2 using the same approaches as in Section 5.1. Figure 5a shows that transformers can still consistently achieve nearly Bayes-optimal ICL risk. An ablation of the risks and probing errors in alternative settings can be found in Appendix F.2. #### Probing copying mechanisms The main mechanistic question we ask here is about the data preparation phase, where the transformer construction in Theorem 2 performs copying twice: i) A copying of $[x_{i-k+1}; \ldots; x_{i-1}]$ onto the $x_i$ token as in (7), to prepare for the computation of $\Phi^(\bar{x}_i)$; As copying may not be distinguishable from the consequent matrix multiplication step $[x_{i-k+1}; \ldots; x_{i-1}; x_i] \mapsto B_1^[x_{i-k+1}; \ldots; x_{i-1}; x_i]$, we probe instead the result $B^_{1,-j} x_{i-j}$ after matrix multiplication, where $B^_{1,-j} \in \mathbb{R}^{D \times d}$ denotes the block within $B_1^$ hitting $x_{i-j}$. ii) A second copying of $\Phi^(\bar{x}_{i-1})$ onto the $x_i$ token to obtain (9), after $\{\Phi^(\bar{x}_i)\}_i$ are computed. We probe one transformer trained on the dynamical systems problem with $k = 3$ (so that the useful preceding inputs are $x_{i-1}$ and $x_{i-2}$), and find that the transformer indeed performs the two conjectured copyings. Figure 5b demonstrates copying i) onto the current token, where the copying of $x_{i-1}$ happens earlier (at layer 3) and is slightly more accurate than that of $x_{i-2}$ (at layer 4), as expected. Further observe that layer 4 (which we recall contains an attention layer and an MLP layer) have seemingly also implemented the (unnormalized) MLP representation $\tilde{\Phi}^(\bar{x}_i) = \sigma_p(B^_{2} \sigma_p(B_1^ \bar{x}_i))$, though the probing error for the actual representation $\Phi^(\bar{x}_i) = \tilde{\Phi}^(\bar{x}_i)/\\|\tilde{\Phi}^(\bar{x}_i)\\|_2$ continues to drop in layer 4-6 (Figure 5c). Figure 5c further demonstrates copying ii), where $\Phi^(\bar{x}_{i-1})$ are indeed copied to the $i$-th token, whereas by sharp contrast $\Phi^(\bar{x}_{i-k})$ for $k > 2$ are not* copied at all into the $x_i$ token, aligning with our conjectured intermediate output format (9). --- $^3$A one-layer transformer does not have much ICL power using input format (1)—$x_i$ and $y_i$ are stored in separate tokens there, which makes “one-layer” mechanisms such as gradient descent (von Oswald et al., 2022; Akyürek et al., 2022; Bai et al., 2023) unlikely to be implementable; see Appendix D.3 for a discussion. 6 CONCLUSION This paper presents theoretical and mechanistic studies on the in-context learning ability of transformers on learning tasks involving representation functions, where we give efficient transformer constructions for linear ICL on top of representations for the supervised learning and dynamical system setting, and empirically confirm the existence of various high-level mechanisms in trained transformers. We believe our work opens up the investigation of ICL beyond simple function classes, and suggests open questions such as further investigations of the mechanisms of the linear ICL modules, and theory for ICL in more complex function classes. One limitation of our work is that the setting still consists of synthetic data with idealistic representation functions; performing similar studies on more real-world data would be an important direction for future work. ACKNOWLEDGMENT WH acknowledges support from the Google Research Scholar program. S. Mei is supported by NSF DMS-2210827, CCF-2315725, NSF CAREER DMS-2339904, and an Amazon Research Award. REFERENCES Kwangjun Ahn, Xiang Cheng, Hadi Daneshmand, and Suvrit Sra. Transformers learn to implement preconditioned gradient descent for in-context learning. arXiv preprint arXiv:2306.00297, 2023. Ekin Akyürek, Dale Schuurmans, Jacob Andreas, Tengyu Ma, and Denny Zhou. 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Transformer Circuits Thread, 2021. Shivam Garg, Dimitris Tsipras, Percy S Liang, and Gregory Valiant. What can transformers learn in-context? a case study of simple function classes. Advances in Neural Information Processing Systems, 35:30583–30598, 2022.
zhZXk5Ctz2	The widely used dual-domain loss in models, such as MIMOUNet (Cho et al, ICCV'21) and SFNet (Cui et al, ICLR'23), can introduce global information refinement. How does aRGB compare to this loss function? This function does not lead to much computation overhead.	Rethinking RGB Color Representation for Image Restoration Models Anonymous authors Paper under double-blind review Abstract The per-pixel distance loss defined in the RGB color domain has been almost a compulsory choice for training image restoration models, despite its well-known tendency to guide the model to produce blurry, unrealistic textures. To enhance the visual plausibility of restored images, recent methods employ auxiliary objectives such as perceptual or adversarial losses. Nevertheless, they still do not eliminate the reliance on the per-pixel distance in the RGB domain. In this work, we try to redefine the very representation space over which the per-pixel distance is measured. Our augmented RGB (aRGB) space is the latent space of an autoencoder that comprises a single affine decoder and a nonlinear encoder, trained to preserve color information while capturing low-level image structures. As a direct consequence, per-pixel distance metrics, e.g., $L_1$, $L_2$, and smooth $L_1$ losses, can also be defined over our aRGB space in the same way as for the RGB space. We then replace the per-pixel losses in the RGB space with their counterparts in training various image restoration models such as deblurring, denoising, and perceptual super-resolution. By simply redirecting the loss function to act upon the proposed aRGB space, we demonstrate boosted performance without any modification to model architectures or other hyperparameters. Our results imply that the RGB color is not the optimal representation for image restoration tasks. 1 Introduction Since SRCNN (Dong et al., 2016) reinterpreted image restoration pipeline as a cascade of deep neural networks, the field of image restoration has undergone unprecedented improvements, most of which are attributed to the advancements in model architectures (Kim et al., 2016b; Lim et al., 2017; Nah et al., 2017; Tong et al., 2017; Wang et al., 2018b; Zhang et al., 2018b; Waqas Zamir et al., 2021; Liang et al., 2021; Chen et al., 2022). On the contrary, shifting our interest to the very objectives the models are optimized for, we see only a few variations: the per-pixel $L_1$ or $L_2$ distances are used almost unanimously. This particular fondness for the distance metrics in the RGB color space stems from the characteristics of the image restoration problem itself, where a low-quality input, the model’s reconstruction, and the corresponding ground truth images have extremely dense, pixel-grained correlations in between. Unfortunately, it is widely known that those per-pixel losses are the main cause of the blurriness easily found in the restored images (Ledig et al., 2017). Each spatial feature in the RGB color space is only responsible for the three-dimensional color information at that specific locus; it does not carry any information directly pertaining to local structures. In other words, the models do not learn structural information from the loss function. Instead, they only learn it implicitly from its architectural prior. The conventions to remedy the problem are to introduce auxiliary objectives such as perceptual loss (Johnson et al., 2016) or adversarial loss (Ledig et al., 2017; Kupyn et al., 2018; Wang et al., 2018b). Nonetheless, they cannot be used by themselves when accurate reconstruction is required. In particular, a perceptual loss (Johnson et al., 2016) is a distance metric defined over the range of another network, typically a pre-trained classifier (Simonyan and Zisserman, 2015). Those classifiers, despite being favorable latent encoders for the perceptual losses, are originally designed to prefer coarse semantic structures over high-frequency textual variations in order to achieve robust classification accuracy. To this end, a classifier typically downscales inputs (Krizhevsky et al., 2012), normalizes internal feature distributions (Ioffe and Szegedy, 2015; Ba et al., 2016), and filters out insignificant patterns using noninvertible rectifiers (von der Malsburg, 1973; Hendrycks and Gimpel, Such process can be advantageous in maintaining semantic information; however, the resulting embeddings inevitably lose information about pixel-grained alignments and colors, which is crucial when we want to reconstruct high-fidelity images that correctly match the given inputs. Adversarial losses (Goodfellow et al., 2014; Ledig et al., 2017; Kupyn et al., 2018; Wang et al., 2018b) cannot be used alone for restoration either, as they prioritize realism over pixel-level accuracy and content preservation. As a consequence, the per-pixel distance metrics have been regarded almost necessary evils in training a restoration network, despite their notoriety of producing blurry outputs. In summary, yet the per-pixel distances defined over the RGB color representation does provide fine-grained supervision for the paired data, it fails to convey information regarding local structures within an image. On the other hand, despite their structural awareness, existing solutions such as perceptual or adversarial losses cannot change the way of using the per-pixel distances. Because these loss functions do not preserve the exact fine-grained information, the per-pixel distances are still required to assist their supervision. We believe that, however, the lack of structural information within the guidance of per-pixel distances is not attributed to the metrics themselves but rather, to the very space those metrics are defined over, i.e., the RGB color domain. What we need is a representation space where each pixel captures its neighboring structure while not losing its original color value so as to provide better supervision with a per-pixel distance. For this goal, we design an encoder that augments images into latent features that satisfy this condition. Our encoder is trained with a linear decoder in an autoencoder fashion to ensure those latent features to be decoded back to the original images almost losslessly (> 60 dB PSNR). We refer to this latent feature space as the augmented RGB (aRGB) space. Replacing the RGB representation with our aRGB space in calculation of per-pixel distances enjoys several benefits: Versatility. Directly altering the underlying representation space allows us an additional degree of freedom in choosing the loss function. Among various high-performing image restoration models, we choose frameworks employing different per-pixel and auxiliary losses for demonstration, namely: MPRNet (Waqas Zamir et al., 2021), NAFNet (Chen et al., 2022), and ESRGAN (Wang et al., 2018b). Performance improvement. Replacing per-pixel RGB losses with our aRGB space-based ones improves not only in perceptual super-resolution tasks but, to our surprise, in the image denoising and deblurring tasks in terms of PSNR and SSIM. Better PSNR metrics could be achieved without using the per-pixel RGB distances, despite their mathematical equivalence. Interpretability. In Section 4, we provide comprehensive analysis on our aRGB space. Thanks to the linear decoder, we can separate the information added to the augmented space from the existing RGB color information. We investigate further into the topology of the aRGB space and the characteristics of the gradients from the aRGB distances using various visualization techniques. 2 LIFTING THE RGB COLOR SPACE 2.1 THE aRGB AUTOENCODER Our primary goal is to design a representation space for low-level vision tasks in order to facilitate training of image restoration networks. Designing a representation space is achieved by defining the encoder and the decoder to translate images back and forth between the RGB space and the target space. Building upon the discussion from Section 1, we can split our goal into two parts: (1) the feature at each pixel in our space is required to encode its neighboring structure, and (2) the integrity of the color information should be preserved. To fulfill the first requirement, our encoder is a size-preserving ConvNet with nonlinearities to capture the structure among adjacent pixels. For the latter, we employ a per-pixel linear decoder, i.e., a $1 \times 1$ convolution, to strongly constrain the embedding of a pixel to include its RGB color information. We start from an RGB image $x \in \mathbb{R}^{3 \times H \times W}$. Our convolutional encoder $f$ transforms image $x$ into a feature $\xi \in \mathbb{R}^{C \times H \times W}$ of a new representation space. Unlike typical undercomplete autoencoders, which removes information from its inputs, we aim to add more information regarding local structures for each pixel $[\xi]_{ij}$ at coordinate $(i, j)$. Therefore, $C$ must be greater than 3, and the receptive field size $R$ should be greater than unity. Our decoder $g : \xi \mapsto x$ is effectively a single $1 \times 1$ convolution. That is, we can express $g([\xi]_{ij})$ as a per-pixel linear operation: $g([\xi]_{ij}) = A[\xi]_{ij} + b$, where $A \in \mathbb{R}^{3 \times C}$ and $b \in \mathbb{R}^3$. This ensures that each feature $[\xi]_{ij}$ in our representation space extends the color information presented in $[x]_{ij}$, hence the name of our new representation, augmented RGB. Additionally, using a linear decoder $g$ offers an interpretability: we can regard the nullspace of $A$, i.e., the set of undecoded information, as a reservoir of any extra information captured by the encoder $f$ other than local colors. What is crucial at this juncture is to define our aRGB space to effectively capture the highly varying, complex mixture of information from the color and the neighboring structure at each pixel. To this end, we employ a mixture-of-experts (MoE) architecture (Jacobs et al., 1991; Shazeer et al., 2017; Fedus et al., 2022) within our encoder. We choose this design based on our conjecture that the topology of the space of image patches is disconnected, and therefore can be more efficiently modeled with a MoE architecture than a single ConvNet. For the set of the smallest images, i.e., a set of pixels, we can argue that their domain is a connected set under absence of quantization, since a pixel can take arbitrary color value. This does not hold in general if the size of the patches become large enough to contain semantic structures. In fact, we cannot interpolate between two images of semantically distinct objects in the natural image domain, e.g., there is no such thing as a half-cat half-airplane object in nature. This implies that topological disconnectedness emerge from the domain of patches as the size of its patches increases. Since a single-module encoder is a continuous function, learning a mapping over a disconnected set may require deeper architecture with a lot of parameters. An MoE encoder, per contra, can model a discontinuous map more effectively through its discrete routing strategy between small, specialized experts. We will revisit our conjecture in Section 4. In practice, an RGB image $x \in \mathbb{R}^{3 \times H \times W}$ is fed into the router $f_r$ as well as $K$ encoders $f_1, \ldots, f_K$. The router $f_r$ is a five-layer ConvNet classifier with a softmax at the end. The output of the router $y = f_r(x) \in [0, 1]^{K \times H \times W}$ partitions each pixel of $x$ into $K$ different bins with top-1 policy. This is equivalent to generating mutually exclusive and jointly exhaustive $K$ masks $m_1, \ldots, m_K$ of size $H \times W$. Finally, the features $\xi_1 = f_1(x), \ldots, \xi_K = f_K(\xi)$ are aggregated into a single feature $\xi$, i.e., $$\xi = f(x) = \sum_{k=1}^{K} m_k \odot f_k(x) = \sum_{k=1}^{K} \mathbb{1}_{\arg\max_{l'}[f_l(x)]_{l'} = k} \odot f_k(x) \in \mathbb{R}^{C \times H \times W},$$ where $\odot$ is an element-wise multiplication and $\mathbb{1}$ is the indicator function. We ensure that $(g \circ f)(x) = x'$ by training $f$ and $g$ jointly in an autoencoder scheme. After the training, the decoder $g$ is discarded and the encoder $f$ is used to generate aRGB representations from RGB images. 2.2 TRAINING THE AUTOENCODER Our objective is to ensure that the aRGB encoder $f$ effectively learns accurate low-level features from clean (or sharp) and natural images. To achieve this goal, we make use of a dataset $D$, consisting of clean image patches. With this dataset, the aRGB autoencoder is trained to minimize the $L_1$ distance between a patch $x \in D$ and its reconstruction $(g \circ f)(x)$. In addition, likewise in Switch Transformer (Fedus et al., 2022), a load-balancing loss $ L_{\text{balance}} $ is applied to encourage the router $ f_r $ to distribute pixels evenly across the $ K $ experts during training: \[ L_{\text{balance}} = K^2 \sum_{i=1}^{H} \sum_{j=1}^{W} \left[ \max_k [f_r(x)]_k \right]_{ij}, \] which is minimized when the distribution is uniform with the value of unity. Furthermore, to increase the sensitivity of the encoder $ f $, we simply add an isotropic Gaussian noise at the output of the encoder only during the training of the $ a $RGB autoencoder. That is, we have the reconstruction loss: \[ L_{\text{recon}} = \\| g(f(x) + z) - x \\|_1, \] where $ z \sim \mathcal{N}(0, I) $. Although the decoder is only informed with three color channels of each pixel during the training, we observe that the latent space does not degenerate into trivial solutions. See Appendix A for more information. Overall, the training loss for the $ a $RGB autoencoder is: \[ L_{\text{AE}} = L_{\text{recon}} + \lambda L_{\text{balance}}. \] In practice, we choose $ \lambda = 0.01 $. The final autoencoder achieves 67.21 dB in reconstruction of the Set5 benchmark (Bevilacqua et al., 2012). In other words, the average RGB color difference is below tenth of the quantization step. Henceforth, we will consider our $ a $RGB autoencoder lossless in the analysis in Section 4. More implementation details are provided in Appendix B. 3 TRAINING IMAGE RESTORATION MODELS IN $ a $RGB SPACE 3.1 INTEGRATION INTO EXISTING RESTORATION FRAMEWORKS Training image restoration models with respect to the $ a $RGB space only requires a few lines of code modified. An image restoration model is typically trained to minimize a per-pixel distance $ L_{\text{pixel}} $, optionally with some auxiliary losses $ L_{\text{aux}} $ for perceptual quality, such as a perceptual loss (Johnson et al., 2016) or an adversarial loss (Ledig et al., 2017). The overall loss can be represented as: \[ L_{\text{total}}(x_H, \hat{x}_H) = L_{\text{pixel}}(x_H, \hat{x}_H) + L_{\text{aux}}(x_H, \hat{x}_H), \] where $ x_H $ is the ground-truth image and $ \hat{x}_H $ is the restoration result. To train the model in the $ a $RGB space, we are only required to modify the the input to the per-pixel loss $ L_{\text{pixel}} $. That is, the per-pixel distances are now computed between the images in the $ a $RGB space, namely, $ f(x_H) $ and $ f(\hat{x}_H) $. \[ L_{\text{total}, a \text{RGB}}(x_H, \hat{x}_H) = L_{\text{pixel}}(f(x_H), f(\hat{x}_H)) + L_{\text{aux}}(x_H, \hat{x}_H). \] Since what we present is not a specific loss function but the underlying space itself, our method can be seamlessly integrated with any existing restoration framework regardless of the type of per-pixel loss it uses. Typical per-pixel losses used for these tasks can be grouped into three categories: an $ L_1 $ loss; an $ L_2 $ loss and its equivalents; and a group of smooth $ L_1 $ losses that interpolate between the former two. To demonstrate the versatility of our solution, we choose a high-performing image restoration model trained by a loss from each of the group to solve different type of tasks. In specific, a perceptual image super-resolution model trained for an $ L_1 $ loss, a real image denoising model trained for a PSNR loss, an equivalent to the $ L_2 $ loss, and finally a motion blur deblurring model trained for a Charbonnier loss, a type of smooth $ L_1 $ loss, are chosen. A notable feature of our method is that the trained image restoration models with respect to our $ a $RGB representation space are generally better at reconstructing the underlying edge structures. This offers visual artifact reduction for perceptual image super-resolution in Section 3.2, sharper edges and enhanced alignments for image denoising and deblurring in Section 3.3 and 3.4. More visual comparisons are provided in Appendix D. 3.2 PERCEPTUAL IMAGE SUPER-RESOLUTION WITH $ L_1 $ LOSS Our initial hypothesis revolved around the potential of our $ a $RGB encoder $ f $ to enrich the supervision of the per-pixel loss with structural information. Perceptual super-resolution should be a natural starting point to search for the evidence, since in the task, the supervision from the original per-pixel loss is heavily interfered by structure-aware auxiliary losses, i.e., the VGG perceptual loss (Simonyan and Zisserman, 2015; Johnson et al., 2016) and the adversarial loss (Ledig et al., 2017). We trained ESRGAN (Wang et al., 2018b) models and summarized the results in Table 1. Fine-tuned over the Table 1: Quantitative results on training $4\times$ super-resolution ESRGAN in the $a$RGB space. In our methods using $a$RGB representation, we modify only the $L_1$ loss by exchanging it with the $L_{1,a}$RGB loss. All the other training hyperparameters are left untouched. Better scores in each block are shown in boldface text. \| Objective \| DIV2K-Val \| Urban100 \| \|-----------\|-----------\|----------\| \| \| PSNR↑ \| SSIM↑ \| LPIPS↓ \| NIQE↓ \| FID↓ \| PSNR↑ \| SSIM↑ \| LPIPS↓ \| NIQE↓ \| FID↓ \| \| Pre-trained RRDBNet† \| 29.466 \| 0.8306 \| 0.2537 \| 5.4860 \| 15.910 \| 25.496 \| 0.7951 \| 0.1963 \| 5.6236 \| 23.729 \| \| $0.01L_1 + 0.005L_{Adv}$ \| 27.102 \| 0.7687 \| 0.1282 \| 3.0419 \| 13.593 \| 23.535 \| 0.7373 \| 0.1322 \| 3.9479 \| 18.428 \| \| $0.01L_{1,a}$RGB \| 27.218 \| 0.7622 \| 0.1235 \| 3.0896 \| 12.936 \| 23.348 \| 0.7204 \| 0.1289 \| 3.8524 \| 18.015 \| \| $0.01L_1 + L_{VGG} + 0.005L_{Adv}$ \| 26.627 \| 0.7033 \| 0.1154 \| 3.0913 \| 13.557 \| 22.776 \| 0.7033 \| 0.1232 \| 4.2067 \| 20.616 \| \| $0.01L_{1,a}$RGB + $L_{VGG} + 0.005L_{Adv}$ \| 26.845 \| 0.7500 \| 0.1110 \| 2.9615 \| 12.799 \| 23.270 \| 0.7196 \| 0.1183 \| 3.8982 \| 17.739 \| † The official ESRGAN model (Wang et al., 2018b). Figure 2: Qualitative comparison of ESRGAN models trained with different loss functions. Each column corresponds to each row in Table 1. The loss weights are omitted for brevity, ESRGAN corresponds to the $0.01L_1 + L_{VGG} + 0.005L_{Adv}$ in Table 1. same PSNR-oriented pre-trained RRDBNet, various combinations for the adversarial training are examined. Here, our method simply modifies the $L_1$ loss to act within the $a$RGB space. First, as Table 1 indicates, the modified $L_1$ metric, $L_{1,a}$RGB, provides sufficient constraints for stabilizing the adversarial training of a super-resolution model. Remarkably, even in the absence of the perceptual loss, our $L_{1,a}$RGB loss generally improves perceptual scores over the original $L_1$ loss while maintaining similar PSNR scores during adversarial training. This implies that our $a$RGB representation provides complementary information that the conventional per-pixel $L_1$ distances does not provide. Furthermore, the last two rows of Table 1 demonstrate that the benefit of training in our $a$RGB space is maximized in the presence of the perceptual loss. This implies that the local structural information captured within our $a$RGB representation is also complementary to the supervision from a pre-trained classifier. As a result, this leads to superior performance in every distortion-based and perceptual metric compared to the original ESRGAN. In particular, the improvements in the PSNR and SSIM scores aligns with our design philosophy that the RGB colors are included as a subspace in our $a$RGB representation; in other words, the effect of minimizing the $L_1$ loss can also be achieved by minimizing the $L_{1,a}$RGB loss. From visual results in Figure 2 and Appendix D, we can observe how artifacts are suppressed using our $L_{1,a}$RGB loss, successfully guiding the adversarial training towards visually pleasing restoration. More quantitative results are provided in Appendix C. 3.3 Real noise denoising with $L_2$ loss To demonstrate the effect of $a$RGB representation with $L_2$ loss, we choose NAFNet (Chen et al., 2022), which employs a per-pixel PSNR loss $L_{PSNR}$, a mathematically equivalent form of the $L_2$ loss. We first train a NAFNet-width32 on the SIDD Medium sRGB dataset (Abdelhamed et al., 2018) with our new PSNR loss $L_{PSNR,a}$RGB, the same metric but defined within the $a$RGB space. To our surprise, Table 2 and Figure 3 reveal that our $a$RGB representation provides better PSNR and SSIM scores than the original model directly trained using the PSNR metric $L_{PSNR}$. The results imply that our $a$RGB representation not only maintains most of original RGB information but also incorporates additional local structural information that leads to better supervision in the denoising task. Additional experiments using different metrics for the same task reveal another noteworthy characteristics of... changing the representation space. As elaborated in Section 4.3, changing the underlying space can profoundly alter the scale and the shape of a metric and its gradients, resulting in different training dynamics. A direct consequence is that the optimal hyperparameters and their resulting performance may change for restoration frameworks in use. Better performance obtained with NAFNets trained for the $L_1$ metric in our $a$RGB space in the last rows of Table 3 clearly demonstrates this issue, revealing a potential unexpected benefit from changing the underlying representation. ### 3.4 Motion Blur Deblurring with Smooth $L_1$ Loss A Charbonnier loss (Bruhn et al., 2005) is a type of smooth $L_1$ loss defined as $L_{\text{Char}}(\hat{x}_H, x_H) = (\\|\hat{x}_H - x_H\\|_2^2 + \epsilon^2)^{1/2}$, where $\epsilon$ is a small constant. To show the effectiveness of our $a$RGB representation with this type of loss, we train an MPRNet (Waqas Zamir et al., 2021) for motion blur deblurring task using GoPro dataset (Nah et al., 2017). The MPRNet is originally trained with a Charbonnier loss with $\epsilon = 10^{-3}$ together with an edge loss, an auxiliary loss defined as another Charbonnier loss calculated between the Laplacians of two images. We leave the edge loss and its weight untouched and change only the Charbonnier loss to act upon our $a$RGB space, i.e., $L_{\text{MPRNet}, aRGB} = L_{\text{Char}}(f(\hat{x}_H), f(x_H)) + 0.05L_{\text{Char}}(\Delta \hat{x}_H, \Delta x_H)$. We observe clear improvements in Table 3 and Figure 4. As shown, the performance gain was orthogonal to existing enhancement techniques, e.g., test-time local converter (TLC) (Chu et al., 2022). From the experiments, we conclude that our $a$RGB representation indeed helps training image restoration models better than the RGB color representation in a variety of tasks, architectures, loss functions, and lead to synergic effect with a variety of other enhancement techniques, such as perceptual loss, adversarial training, edge loss, and test-time local converter. ### 4 Discussion In order to understand the representation learned by the $a$RGB autoencoder, we first explore the consequence of our two key design choices: the linear decoder and the mixture-of-experts encoder. Table 2: Results on real image denoising using NAFNet. \| Model \| Objective \| PSNR↑ \| SSIM↑ \| \|---------------------\|-----------------\|-------\|-------\| \| NAFNet-width32 \| $L_{PSNR}$ \| 39.9672 \| 0.9599 \| \| NAFNet-width32 \| $L_{PSNR,RGB}$ \| 39.9864 \| 0.9601 \| \| NAFNet-width32 \| $L_1,RGB$ \| 40.0106 \| 0.9602 \| \| NAFNet-width64 \| $L_{PSNR}$ \| 40.3045 \| 0.9614 \| \| NAFNet-width64 \| $L_1,RGB$ \| 40.3364 \| 0.9620 \| (a) Inverting orthogonal mixture of two aRGB embeddings. (b) Expert selection map of the MoE router $f_t$. (c) t-SNE plot of the aRGB embedding $\xi$ of pixels in image 5b. (d) Change of $L_2$ metrics in the aRGB space relative to the $L_2$ metrics in the RGB space. Figure 5: Understanding the learned aRGB representation. Figure 5a show a visual example of aRGB embedding inversion. Figure 5b and 5c reveal clear evidence that the experts of our aRGB encoder $f$ are specialized for a particular type of input structures, and that even the embedding vectors within a single patch are clustered in a complicated manner, justifying our usage of MoE architecture. Figure 5d shows how the distance metric changes in the aRGB space relative to the distance in the RGB space. Mean distances and their standard deviations are measured by MSE losses between an image and the same image with 100 AWGNs with the same standard deviation. Note that the aRGB space slightly exaggerates the distance more outside natural image domain, e.g., Gaussian noise, and the metric’s variance is negligibly small. Then, we quantify the effect of changing the representation space on the scale of metrics defined over the space and their gradients. We conclude our discussion with ablation studies. 4.1 NULLSPACE OF THE DECODER In addition to the design simplicity, our pixel-wise linear decoder enjoys an additional benefit: decomposability. Since our autoencoder is almost lossless as demonstrated in Table 8, we will consider that the RGB $x \in \mathbb{R}^3$ and the aRGB $\xi = f(x) \in \mathbb{R}^C$ representations of any given image equivalent. That is, $x' = g(\xi) = A\xi + b = x$. As a result of the linearity of our decoder $g$, the aRGB representation $\xi$ can be decomposed into the sum of two orthogonal components: $$\xi = \xi_\parallel + \xi_\perp, \quad \text{s.t.} \quad \xi_\parallel = A^\dagger A \xi =: f_\parallel(x) \quad \text{and} \quad \xi_\perp = (I - A^\dagger A)\xi =: f_\perp(x),$$ where $A^\dagger$ is the Moore-Penrose pseudoinverse of $A$. The parallel component $\xi_\parallel$ of the aRGB representation lies in the three-dimensional subspace of $\mathbb{R}^C$ that is projected onto the RGB colors by the decoder $g$, i.e., $A\xi_\parallel = AA^\dagger A \xi = A \xi$. The remaining perpendicular part $\xi_\perp$ can be regarded as the information the aRGB space encodes in addition to the RGB colors. The contribution of the two components can be visualized by inverting the encoder $f$ with respect to a mixed embedding: $$f^{-1}(\xi_{mix}) = \arg\min_z \\|f(z) - \xi_{mix}\\|^2_2, \quad \text{s.t.} \quad \xi_{mix} = \xi_\parallel + \xi_\perp = A^\dagger A f(x_1) + (I - A^\dagger A)f(x_2).$$ We use a SGD optimizer with a learning rate of 0.1 for 50 iterations. As shown in Figure 5a and Appendix E, the inversion of the mixed embedding inherits color information from the parallel embedding $\xi_\parallel$, while the perpendicular part $\xi_\perp$ contributes to the high-frequency edge information. 4.2 SPECIALIZATION OF THE EXPERTS AND LEARNED STRUCTURES Figure 5b visualizes how individual pixels of a natural image are distributed into $K = 20$ experts. Unlike in semantic segmentation, where segmentation maps are chunked into large blocks of semantically correlated pixels, our pixel-wise router $f_t$ generates fine-grained distributions of pixels. That is, multiple experts jointly involve in encoding the same texture such as the blue sky and the leafy trees. Another salient feature we can observe in the figure is that edges of different orientations are dealt with different experts, implying their specialization. Visualizing the aRGB embedding space using t-SNE (van der Maaten and Hinton, 2008) provides us with additional insights on the topology of Table 3: Results on motion blur deblurring using MPRNet. \| Model \| Objective \| PSNR↑ \| SSIM↑ \| PSNR↑ \| SSIM↑ \| \|----------------\|-----------------\|-------\|-------\|-------\|-------\| \| MPRNet \| $L_{Char} + 0.05L_{Edge}$ \| 32.6581 \| 0.9589 \| 30.9622 \| 0.9394 \| \| MPRNet \| $L_{Char,RGB} + 0.05L_{Edge}$ \| 32.7118 \| 0.9594 \| 31.0248 \| 0.9398 \| \| MPRNet-TLC \| $L_{Char} + 0.05L_{Edge}$ \| 33.3137 \| 0.9637 \| 31.1868 \| 0.9418 \| \| MPRNet-TLC \| $L_{Char,RGB} + 0.05L_{Edge}$ \| 33.3886 \| 0.9642 \| 31.2082 \| 0.9421 \| the space. Figure 5c reveals that the aRGB embeddings cluster into multiple disconnected groups in two different types: common groups where multiple experts are involved in encoding process and specialized groups where a single expert is exclusively allocated for the embeddings. These observations align well with our initial design principles in Section 2.1, where the feature embeddings occupy highly complicated, disconnected set, and an MoE architecture effectively deals with this structure by specializing each expert to a subset of the embedding space. ### 4.3 aRGB Metric Space and Produced Gradients The main purpose of our the aRGB space is to provide alternative supervision to the existing image restoration framework. This supervision is realized with a metric defined over the space and its gradients generated from pairs of images. To this end, we first visualize the correlation between $L_2$ distances defined in the RGB and our aRGB spaces in Figure 5d. We plotted additional figure with title $X - X_0X_1$ to show the deviation of the graph over the straight line, showing clear convexity of the graph. This implies that the metrics within aRGB spaces are inflated when the given two images are similar. Figure 6 shows the gradients from two per-pixel $L_1$ losses between a restored image and its high-quality counterpart defined over both spaces. Unlike RGB $L_1$ loss which exhibits a highly off-centered, discrete distribution, the $L_{1,aRGB}$ loss shows smooth and centered distribution of gradients. We believe that this allows for the stable training of the image restoration models despite its huge scale of the generated gradients from the $L_{1,aRGB}$ loss, which is more than a hundredfold as shown in the x axis of Figure 6b. In the RGB domain, the same scale of gradient is achievable only through increasing the learning rate, which leads to destabilization of the training. Overall, the analyses show how our aRGB encoder helps the training of image restoration models. ### 4.4 Ablation Study Lastly, we provide ablation studies to determine the best hyperparameters for our aRGB autoencoder. We compare the models by the results of training an RRDBNet (Wang et al., 2018b) only on DIV2K dataset. The results are summarized in Table 4. More information is elaborated in Appendix B. #### Number of experts. The first four rows of Table 4 show the effect of the number of experts of the aRGB encoder $f$ on its supervision quality. From the results, we choose to fix the number of experts to 20 throughout our experiments. #### Dataset dependence. As the second part of Table 4 presents, the training data for the aRGB autoencoder decides the quality of supervision the model gives. This implies that our aRGB autoencoder utilizes structural priors of its training data. Appendix 7 provides additional theoretical and empirical evidence that our aRGB autoencoder learns image structures to reconstruct given images. #### Regularizers. In the last row of Table 4, we observe that the regularizing noise $z$ added at the end of the encoder during training helps the aRGB encoder to produce stronger supervision for image restoration models. In practice, we observe more than tenfold reduction in the scale of produced gradients when the aRGB autoencoder trained without the regularizing noise is applied. This correlates to our discussion in Section 4.3, that our aRGB encoder helps training image restoration models by stably increasing the scale of gradients. --- Table 4: Ablation studies on the aRGB autoencoder. RRDBNets (Wang et al., 2018b) are trained with DIV2K (Agustsson and Timofte, 2017) for 300k iterations for $4 \times$ SISR tasks with only the $L_1$ loss between the aRGB embeddings. \| # experts \| Routing \| aRGB train set \| Reg. noise \| Set14 PSNR \| SSIM \| Urban100 PSNR \| SSIM \| DIV2K-Val PSNR \| SSIM \| \|-----------\|---------\|----------------\|------------\|-------------\|------\|---------------\|------\|----------------\|------\| \| 1 \| MoE \| DIV2K \| ✓ \| 26.87 \| 0.7467 \| 24.75 \| 0.7735 \| 29.08 \| 0.8222 \| \| 5 \| MoE \| DIV2K \| ✓ \| 26.87 \| 0.7477 \| 24.83 \| 0.7745 \| 29.12 \| 0.8231 \| \| 10 \| MoE \| DIV2K \| ✓ \| 26.89 \| 0.7474 \| 24.84 \| 0.7750 \| 29.11 \| 0.8231 \| \| 20 \| MoE \| DIV2K \| ✓ \| 26.91 \| 0.7471 \| 24.87 \| 0.7745 \| 29.14 \| 0.8227 \| \| 30 \| MoE \| DIV2K \| ✓ \| 26.89 \| 0.7476 \| 24.84 \| 0.7750 \| 29.11 \| 0.8231 \| \| 20 \| MoE \| GoPro \| ✓ \| 26.89 \| 0.7459 \| 24.83 \| 0.7728 \| 29.12 \| 0.8220 \| \| 20 \| MoE \| SIDD \| ✓ \| 26.86 \| 0.7420 \| 24.80 \| 0.7690 \| 29.06 \| 0.8186 \| \| 20 \| MoE \| None \| ✓ \| 26.63 \| 0.7441 \| 24.66 \| 0.7722 \| 28.86 \| 0.8212 \| \| 20 \| MoE \| DIV2K \| ✓ \| 26.91 \| 0.7469 \| 24.85 \| 0.7722 \| 29.13 \| 0.8223 \| 5 RELATED WORK Pairwise loss in image restoration. Training a deep neural network that translates low-quality images into high-quality estimates has undoubtedly become the standard way of solving image restoration. While most of the advancements have been made in the network architecture (Kim et al., 2016b; Lim et al., 2017; Nah et al., 2017; Tong et al., 2017; Wang et al., 2018b; Zhang et al., 2018b; Waqas Zamir et al., 2021; Liang et al., 2021; Waqas Zamir et al., 2022; Chen et al., 2022), the importance of loss functions is also widely acknowledged. Since SRCNN (Dong et al., 2016), the first pioneer, employed the MSE loss, the first image restoration models had been trained with the MSE loss (Kim et al., 2016a;b; Nah et al., 2017; Zhang et al., 2017). However, after EDSR (Lim et al., 2017) reported that better convergence can be achieved with $L_1$ loss, various pairwise loss functions are explored. LapSRN (Lai et al., 2017) rediscovers Charbonnier loss (Bruhn et al., 2005), a type of smooth $L_1$ loss, for image super-resolution, which is also employed in image deraining (Jiang et al., 2020) with a new edge loss, defined as a Charbonnier loss between Laplacians, which is then employed in general restoration by MPRNet (Waqas Zamir et al., 2021). NAFNet (Chen et al., 2022), on the other hand, uses the PSNR score directly as a loss function. In accordance with these approaches, we attempt a more general approach to design a representation space, over which those loss functions can be redefined. Structural prior of natural images. It is generally recognized that a convolutional neural network, either trained (Simonyan and Zisserman, 2015) or even untrained (Ulyanov et al., 2018), contains structural prior that resonates with the internal structure of natural images. This prior information permeates through the network into its output space. Attempts to exploit this information include the perceptual loss (Johnson et al., 2016) and various perceptual metrics (Zhang et al., 2018a; Ding et al., 2020). Those are pairwise distance metrics defined over the range space of pre-trained classifier networks (Krizhevsky et al., 2012; Simonyan and Zisserman, 2015). However, as mentioned in Section 1, such losses cannot be used alone when it is required to respect the strong correspondence between the generated and the desired images. Different from the strategies sought for the perceptual metrics, our aRGB encoder is designed to preserve the full information of its inputs by a scale-preserving architecture and a linear decoder to strictly constrain the representation. Mixture of Experts. Instead of relying on a single model to handle complex large-scaled data, a more effective approach is to distribute the workload among multiple workers. To achieve this, a routing strategy (Shazeer et al., 2017) can be employed to divide information between different models, each of which processes a subset of the training data. These individual models, referred to as experts, collectively form a Mixture of Experts (MoE) (Jacobs et al., 1991). Recent studies (Zhou et al., 2022; Fedus et al., 2022) have shown the advantages of MoE in deep learning. However, there are two main challenges when working with multiple experts: limited computational resources and training stability. The conventional routing strategy can lead to unstable training of the MoE unless appropriate regularization methods are applied. Moreover, without advanced techniques (Fedus et al., 2021; He et al., 2021), MoE experience longer processing times as the number of experts increases. In response to these challenges, we employ a balancing loss (Fedus et al., 2022) to ensure the stable training of expert networks and incorporate MoE exclusively during the training phase, leaving the testing phase unaffected. 6 CONCLUSION It is a well-known phenomenon (Ledig et al., 2017) that per-pixel pairwise loss functions, such as $L_1$ or $L_2$ distances, defined in the RGB color space have a strong tendency to guide the trained image restoration model to produce blurry, unrealistic textures. We hypothesize that such problem can be alleviated if we have a representation space that contains accurate color information as well as the local structural information of an image. Our augmented RGB (aRGB) representation is designed with a nonlinear mixture-of-experts encoder and a linear decoder to meet the requirements. From diversified set of experiments, we demonstrate the improved performance across a variety of image restoration tasks such as perceptual super-resolution, denoising, and deblurring could be achieved by only changing the representation space to our aRGB space. Given our results suggesting that the RGB color space may not be the optimal representation space for low-level computer vision tasks, we hope our work spurs more interests and exploration in this research direction. REFERENCES Abdelrahman Abdelhamed, Stephen Lin, and Michael S. Brown. A high-quality denoising dataset for smartphone cameras. In CVPR, 2018. Eirikur Agustsson and Radu Timofte. NTIRE 2017 challenge on single image super-resolution: Dataset and study. In CVPR Workshop, 2017. Jimmy Lei Ba, Jamie Ryan Kiros, and Geoffrey E. Hinton. Layer normalization. arXiv preprints, 2016. Marco Bevilacqua, Roumy Aline, Guillemot Christine, and Morel Marie line Alberi. 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DluJpvRF69	In equation(3), the activation of s^t_m includes phi^k_m and phi^t_m, it looks like a recursive definition, then the activation of s^k_m should include phi from its nearest model and the phi for itself, and so on following the recursive rules. Why all the historical nearest models are omitted in equation (3)?	StyleCL: Latent Dictionary Learning for StyleGAN without Forgetting Anonymous authors Paper under double-blind review Abstract StyleGAN is one of the most versatile generative models that have emerged in recent times. However, when it is trained continually on a stream of data (potentially previously unseen distributions), it tends to forget the distribution it has learned, as is the case with any other generative model, due to catastrophic forgetting. Recent studies have shown that the latent space of StyleGAN is very versatile, as data from a variety of distributions can be inverted onto it. In this paper, we propose to leverage this property to facilitate lifelong learning of StyleGAN without forgetting. Specifically, given a StyleGAN trained on a certain task (dataset), we propose to learn a set of dictionary vectors in its latent space, one for each novel, unseen task (or dataset). Additionally, we also learn a relatively small set of shared parameters (feature adaptors) in the weight space to complement the dictionary learning in the latent space. During inference, given a dataset/task, our method invokes the corresponding learned latent dictionary and the shared parameters for that particular task. Our method avoids catastrophic forgetting because the set of dictionary and the feature adaptor parameters are unique for each task. However, the generator for each task shares all of the parameters except for the newly added parameters of the feature adaptor. We demonstrate that our method, StyleCL, achieves better generation quality on multiple datasets. Additionally, our method requires significantly fewer additional parameters per task compared to previous methods. This is a consequence of learning task-specific dictionaries in the latent space, which has a much lower dimensionality compared to the weight space. We also demonstrate that our method, StyleCL, offers the capability for positive forward transfer for semantically similar tasks. 1 Introduction Continual learning (CL) is a fundamental machine learning paradigm that focuses on the model’s ability to learn and adapt to new tasks or evolving data streams over time while ensuring that previously acquired knowledge remains intact. Extensive research has explored continual learning within the context of discriminative models De Lange et al. (2022), but relatively less attention has been devoted to the application of this paradigm in the realm of generative models. However, recent progress in the field of generative modelling has brought them to the forefront of application domains. Specifically, models such as Generative Adversarial Networks (GANs) Goodfellow et al. (2014) and denoising diffusion models Ho et al. (2020) have found their utility in a wide variety of tasks such as semantic editing Ling et al. (2021), image in-painting, Yu et al. (2018) etc. Thus, it is imperative to consider the problem of continual learning in the context of generative models Lesort et al. (2019). In particular, we direct our attention to continual learning in StyleGAN Karras et al. (2020b) which is one of the most popular variants of GANs. We hypothesize, that StyleGAN is suited for such cases because of its versatility, in that, a large variety of datasets can be inverted onto its extended latent space ($W^+$) as observed in Abdal et al. (2019). Motivated by these observations, we investigate whether the latent space of StyleGAN can be exploited to generate data from a stream of datasets without forgetting. Towards that end, we propose a method to learn a per-task, style-wise dictionary of vectors that define a subspace in the latent space of StyleGAN. In addition to latent dictionary learning, we also learn a set of shared parameters in the weight space, to accommodate a richer knowledge in tandem with the learned latent subspace. Knowledge transfer, a cornerstone of continual learning, assumes a central role in StyleCL. StyleCL utilizes the latent space to identify the most similar task unlike GAN Memory Cong et al. (2020) and CAM-GAN Varshney et al. (2021) where the most similar task is characterized using the most recent task or the task with high Fisher information respectively. We also determine the nature of forward knowledge transfer (positive or negative) by measuring the cosine similarity of dictionary vectors to its projection onto the latent subspace of the most similar task which is then used to prevent negative forward transfer. Moreover, we expand the scope of generative continual learning to encompass real-world scenarios where data from multiple tasks arrive simultaneously without task identification (task ID). Notably, StyleCL adeptly extends its applicability to such settings with minimal adjustments to its training strategy. StyleCL accomplishes this by segregating distinct tasks into distinct regions in the latent space. Even in these scenarios where supervision on task ID is not available, StyleCL consistently delivers high-quality generation capabilities. The following is a summary of our contributions: - Latent subspace learning for StyleGAN: We propose a latent subspace learning approach that enables learning without forgetting for StyleGAN. - Improved generation quality: By harnessing the versatility of StyleGAN’s latent space, our method outperforms contemporary approaches like CAM-GAN and GAN Memory in terms of generation quality, all while employing fewer parameters (28.95% reduction) and FLOPs (11.6% reduction). - Prevention of negative forward transfer: We further propose a simple way to identify the most similar previous task and also characterize the nature of forward transfer between any two tasks to prevent negative forward transfer. - Extension to task ID free setting: We extend StyleCL to scenarios where task ID is not available wherein StyleCL discovers different data distributions automatically. ## 2 RELATED WORK Generative Continual Learning: Continual Learning methods are broadly categorized into three categories: replay-based, regularization-based and parameter isolation-based methods. These categorizations are defined for a discriminative continual setting but they can be applied to generative continual learning as well. Chenshen et al. (2018) introduces MerGAN, a replay-based GAN that combines generated samples from previous tasks with new task data to form an extended training dataset. They also introduce a replay-alignment loss to ensure consistent generation for previous tasks as the number of tasks increases. Zhai et al. (2019) presents Lifelong GAN for continual image-conditioned image generation, employing knowledge distillation and auxiliary data generation by creating patch montages from training batches to mitigate catastrophic forgetting. However, replay-based approaches face scalability issues due to cumulative inaccuracies when a single generator is incrementally updated. Parameter isolation techniques like PiggybackGAN Zhai et al. (2020) freeze old task parameters and introduce smaller new parameters for learning without forgetting. GAN memory Cong et al. (2020) employs normalization parameters to adapt the generator’s weights to incoming data streams. CAM-GAN Varshney et al. (2021) introduces adaptation modules via group-wise convolutions at the output of each convolution layer in the base network. In contrast, StyleCL takes a different approach by learning a latent subspace alongside shared weight space parameters, facilitating continual learning. Even though few regularisation-based approaches like Liang et al. (2018) and Seff et al. (2017) use regularisations to enable continual learning, their generation quality still degrades over time and thus parameter isolation methods appear to be a better choice and have been receiving more attention. Knowledge transfer in continual learning: Knowledge transfer is a crucial aspect of continual learning, predicated on the notion that similar tasks inherently possess shared knowledge that can be effectively transferred between them. However, previous approaches, like MerGAN Chenshen et al. (2018), Lifelong GAN Zhai et al. (2019), and Piggyback GAN Zhai et al. (2020), often lack explicit mechanisms to facilitate this positive knowledge transfer. While GAN Memory Cong et al. (2020) demonstrates promise in enabling knowledge transfer, it relies on the assumption that the most recent task is invariably the most similar, a notion that does not consistently hold. In contrast, CAM-GAN Varshney et al. (2021) quantifies task similarity by approximating the Fischer information matrix (FIM) and posits that initializing the current task with parameters from the most similar task would consistently yield positive forward transfer which may not always hold true. StyleCL distinguishes itself by characterizing both the most similar task and the nature of forward transfer using the latent space, thus effectively capturing the state of the generator while identifying the most similar task and elucidating the nature of the forward knowledge transfer. Continual Learning beyond GANs: Continual learning is a dynamic field that extends beyond GANs. While Variational Autoencoders (VAEs) have been considered in the past, their subpar generation quality has led to a recent decline in attention. In contrast, the exploration of continual learning in Diffusion models represents an emerging paradigm. Recent studies, such as those by Gao & Liu (2023) and Chen et al. (2023), have delved into the utility of diffusion models for replaying previous data in the context of discriminative continual learning. It is essential to note that Diffusion models offer remarkable generation quality enhancements, albeit with a trade-off of increased inference time. On the other hand, GANs excel in efficiency, requiring only a single forward pass. Furthermore, the introduction of GigaGAN, Kang et al. (2023) and StyleGAN-T Sauer et al. (2023) have illustrated their ability to provide competitive generation quality while maintaining faster inference speeds. Given these advantages, we turn our attention to continual learning in GANs, leveraging their rapid inference capabilities while upholding competitive performance compared to other generative models. Additionally, it is worth highlighting that many recent state-of-the-art GANs for various tasks, as seen in works like Kang et al. (2023), Sauer et al. (2023), and Fu et al. (2022), employ StyleGAN-based architectures. This inspires our investigation into StyleGAN-based architectures for continual learning. 3 Proposed Method: StyleCL 3.1 Problem Setting and Method Overview Our setting is that a stream of datasets (or tasks) arrive sequentially with a unique task ID. We assume that at any given time, only one dataset is available for training. Formally, let $ \{X^t\}_{t=1}^{T} $ denote the sequential stream of datasets where $ X^t = \{x^t_j\}_{j=1}^{N} $, $ X^t \sim p_t(x^t) $, with $ x^t_j $ denoting the $ j^{th} $ instance from the $ t^{th} $ task/dataset. The objective is to train a GAN that can sample from the current dataset without forgetting to sample from all the previously seen $ t - 1 $ distributions. Our method starts by training a GAN as in Karras et al. (2020a) on the first (or base) dataset (task), denoted by $ G^1 $. The parameters of $ G^1 $ are denoted by $ \phi^1 $ and are shared by all the subsequent tasks. For each dataset $ X^t $, our method first selects the most similar previous task and the corresponding generator $ G^k $ and learn the following components to obtain $ G^t $: (i) A set of dictionary vectors $ U^t $ on the latent space of $ G^k $ and (ii) a set of feature adaptor blocks $ \phi^t $ ($ 1 \times 1 $ convolutions) on the weight space of $ G^k $. To maintain simplicity, we make the assumption throughout this paper that the feature adaptor $ \phi^k $ of generator $ G^k $ encompasses $ \phi^1 $, the feature adaptor of task $ k $, and the feature adaptors of the most similar previous tasks of task $ k $. It is noteworthy that in our method, the dictionary vectors are unique for each task, whereas the feature adaptors are shared and added recursively based on the similarity of tasks. Specifically, every $ G^t $ comprises $ \phi^1 $, the feature adaptor block of its own as well as the feature adaptor block of the most similar previous tasks. Fig. 1 presents an overview of our method, which we name ‘StyleCL’. 3.2 Latent Dictionary Learning We employ StyleGAN2 Karras et al. (2020a) architecture for the generators $ G $ that contains $ M $ style blocks and for simplicity of discussion, we assume each of these style blocks comprises of just 1 layer. The first stage of our method is to learn a set of dictionaries on the extended latent space ($ W^+ $ space) of the StyleGAN. Given a dataset $ X^t $, a dictionary $ U^t_m = \{u^t_{m1}, \ldots, u^t_{mK}, \ldots, u^t_{mK}\}, u^t_{mi} \in \mathbb{R}^d $, containing $ K $ vectors are learned for each of the $ m = 1, 2, \ldots, M $ style blocks of the generator. Figure 1: Overview of StyleCL: Given a dataset $X^t$ at time $t$, the most similar previous generator $G^k$ is first selected as the base generator. A set of $K$ dictionary vectors, each for a style block $m$, is learned for $X^t$. Further, a feature adaptor block $\phi_m^t$ is added to the existing shared feature adaptor block $\phi_m^k$ in $G^k$. During inference, sampling from $X^t$ is done by giving stochastic combinations of elements of the corresponding dictionary vectors as input to $G^t$. During the training, the parameters of $U_m^t$ are initialized randomly. First, a batch of vectors is stochastically sampled from each dictionary $U_m^t$ as follows: $$w_m^t = z_{m1} u_{m1}^t + z_{m2} u_{m2}^t \ldots + z_{mK} u_{mK}^t + b_m^t$$ where $z_m = [z_{m1}, \ldots, z_{mK}] \sim N(0, I)$ and $b_m^t$ is the bias term. Further, each $w_m^t$ corresponding to every style block is concatenated to form $w^t$ as: $$w^t = [w_1^t, \ldots, w_M^t], w^t \in W^+$$ Finally, $w^t$ is passed as the input to the fixed generator $G^k$ obtained from the most similar task to generate images from $p_t(x^t)$. 3.3 Feature Adaptors in the Weight Space We observed empirically that the ability of the latent dictionary to capture a distribution $p_t(x^t)$, depends on $G^k$. Therefore, learning $U_m^t$ alone may not be fully sufficient to model $p_t(x^t)$. Therefore we also introduce additional feature adaptor blocks on the weight space of the generator of the most similar task $G^k$ to obtain $G^t$. Since the latent subspace would have already captured some characteristics of the datasets, the number of feature adaptor parameters to be learned would be lesser (Tab. 1). Let $S_m^t$ denote the $m^{th}$ style block within the generator $G^t$ for task $t$. Initially, we identify the most similar task to $t^{th}$ task which is denoted by $k$ and the corresponding generator $G^k$ is selected as the base generator. We introduce a trainable feature adaptor block $\phi_m^t$ ($1 \times 1$ Convolution layer) to the existing shared feature adaptor block $\phi_m^k$ in $G^k$ to obtain $S_m^t$ of $G^t$. When the $t^{th}$ task emerges, we learn $\phi_m^t$ and compute the new activation map of $S_m^t$ as follows: $$f_m^t = \alpha_m^k \times \phi_m^k(f_m^{t-1}) + \alpha_m^t \times \phi_m^t(f_m^{t-1})$$ The feature adaptor block $\phi_m^t$ is intended to learn additional information that is absent in $G^k$. Here both $\phi_m^t$ and scaling coefficient $\alpha_m^t$ are learnable and jointly learned with the latent dictionary. It is important to note that $\phi^k_m$ is shared between $G^k$ and $G^t$, whereas $\phi^t_m$ is exclusively for $G^t$. We follow the training paradigm in Karras et al. (2020a), which includes adversarial loss $L_1$ and to ensure smoothness and facilitate better convergence, we use the Perceptual Path Regularizer (PPL) Karras et al. (2020b) and $R_1$ regularization Mescheder et al. (2018) as in Karras et al. (2020b). Algorithm 1 StyleCL : Training Input: $\{X^t\}_{t=1}^{T}$; Sequential Data stream, where $T$ is the total number of tasks Output: $\{U^t\}_{t=2}^{T}$; $\{b^t\}_{t=2}^{T}$; $G^1$; $\{\phi^t\}_{t=2}^{T}$ where $\phi^t = \{\phi^t_m\}_{m=1}^{M}$ 1: Train StyleGAN2 on $X^1$ to obtain $G^1$ 2: for $t = 2 \ldots T$ do 3: Initialize Discriminator parameters $\psi$, and set of dictionary vectors $U^t$ and $b^t$ 4: Find the most similar previous task $k$ using Eq. (6) to obtain $G^k$ 5: for each training iteration do 6: Obtain $w^t$ using Eq. (1) and Eq. (2) 7: Optimize parameters $U^t$, $b^t$ using Eq. (4) combined with PPL and $R_1$ regularization $$L_1 = \mathbb{E}_{x \sim p_t(x^t)}[\log D_\psi(x)] + \mathbb{E}_{z_1,\ldots,z_M \sim N(0,1)}[1 - \log D_\psi(G^k(w^t))]$$ 8: end for 9: Compute $sim(t,k)$ using Eq. (7) 10: Initialize parameters $\phi^t$, $\alpha_t$ 11: if $sim(t,k) > 0$ then 12: $G^t = G^k \cup \phi^t$ 13: else 14: $G^t = G^1 \cup \phi^t$ 15: end if 16: Optimize parameters $U^t$, $b^t$ and $\phi^t$ using Eq. (5) combined with PPL and $R_1$ regularization $$L_2 = \mathbb{E}_{x \sim p_t(x^t)}[\log D_\psi(x)] + \mathbb{E}_{z_1,\ldots,z_M \sim N(0,1)}[1 - \log D_\psi(G^t(w^t))]$$ 17: end for ### 3.4 Forward Transfer: Choosing the Most Similar Previous Task Given a task $t$, our method first chooses the generator $G^k$ of the most similar task and learns feature adaptors over it. This is akin to the idea of forward transfer in the continual learning literature Chen & Liu (2018). The dictionary vectors learned for each task allow easy characterisation of the task and we use these to find the most similar task. In order to find the task that is most similar to an incoming task, we need to characterize both the previous task as well as the current task in the latent space. We characterize the current task by learning the dictionary vectors alone using the base generator $G^1$. It is to be noted that the dictionary vectors are already learnt for previous tasks from 2 to $t-1$. Given any task $t$, we use the set of bias vectors learned as task embedding, $b^t = [b^t_1, \ldots, b^t_M]$, $b^t \in W^+$ since it captures the relative position of the learned latent subspace in the $W^+$ space. Given the task embedding for the current and previous tasks, we define the most similar task $k$ as the one whose embedding has the least Euclidean distance from the embedding of the current task as provided in Eq. (6). This is motivated by the fact that the latent vectors of similar tasks lie close together while being distant from dissimilar tasks as observed in Fig. 4. $$k = \arg\min_{r:t \in \{2,\ldots,t-1\}} \\|b^t - b^r\\|_2$$ Preventing negative forward transfer: Choosing the most similar task facilitates selecting a task with a similar set of features as that of the current task, however, it may lead to negative forward transfer. In order to alleviate this problem, we estimate the nature of forward transfer (positive or negative) by computing the cosine similarity of dictionary vectors of the current task to its projection onto the latent subspace of the most similar task $k$. Let $V^k$ correspond to the orthonormal vectors obtained using the Gram-Schmidt orthogonalisation procedure on the dictionary vectors $U^k$. The projection of $U^t$ onto the latent subspace characterized by the orthonormal vectors $V^k$ is then defined... by $V_i^k U_i^t V_i^k$. Subsequently, we define the nature of forward transfer as follows: $$\text{sim}(t, k) = \frac{1}{M} \sum_{m=1}^{M} \frac{U_m^t \cdot (V_m^k U_m^t V_m^k)}{\\|U_m^t\\| \\|V_m^k U_m^t V_m^k\\|}$$ (7) $\text{sim}(t, k) \in [-1, 1]$, and when $\text{sim}(t, k) \leq 0$, it signifies zero or negative forward transfer. In such cases, we avoid reusing the parameters of the most similar task, thus preventing negative forward transfer. We provide a comprehensive overview of the StyleCL training process in Algorithm 1. 3.5 Overcoming Task ID Constraints with StyleCL The previously defined problem setting assumes a sequential arrival of datasets (or tasks) with unique task IDs. However, in real-life scenarios, this assumption may not always hold. For instance, data collection from multiple sources can occur simultaneously, resulting in a set of data without task distinction. Specifically, we address scenarios where task $t$ comprises contributions from $Q$ datasets, and task ID is not available. We operate under the assumption that $Q$ is known apriori. We demonstrate that StyleCL can be seamlessly extended to accommodate these scenarios. We initialize $Q$ sets of dictionary vectors and feature adaptors. The intuition is that this modelling choice forces each of the $q$ latent adaptors to concentrate on latent vectors originating from separate regions in the latent space. As a result, the model would effectively allocate distinct data sources to distinct dictionaries. During training, we exclusively employ the $q^{th}$ feature adaptor set for latent vectors generated from $q^{th}$ dictionary where $q \in \{1, 2, \ldots, Q\}$. This ensures that $q^{th}$ feature adaptors capture task-specific knowledge exclusively for $q^{th}$ dataset. 4 Experiments and Results 4.1 Experimental settings We conducted three experiments to evaluate our method’s effectiveness. First, we tested its ability to generate from perceptually distinct datasets. Second, we assessed knowledge transfer across similar tasks, using six butterfly categories from ImageNet Russakovsky et al. (2015). Third, we demonstrated StyleCL’s effectiveness in scenarios without task ID information. 4.2 Baselines and Metrics We employ StyleGAN2 Karras et al. (2020a) as the base architecture for all experiments. StyleCL is compared to GAN Memory, CAM-GAN with task similarity learning, and MerGAN. Evaluation metrics include Fréchet inception distance (FID) Heusel et al. (2017), Density, and Coverage Naeem et al. (2020). We also consider computational and memory overhead during inference, measured in FLOPs and parameter count Dehghani et al. (2022), crucial factors for CL scalability. 4.3 Results for Perceptually Distant Tasks Following the experimental setup used in CAM-GAN and GAN-Memory, we begin by training a GAN on CelebA-HQ Karras et al. (2018) dataset and then consider a stream of six perceptually distinct datasets, namely Oxford 102 Flowers Nilsback & Zisserman (2008), LSUN Church Yu et al. (2015), LSUN Cats Yu et al. (2015), Brain MRI Cheng et al. (2016), Chest X-Ray Keromaly et al. (2018), and Anime Faces$^1$. Samples of generated data from all the methods considered can be found in Fig. 2, and it can be observed that StyleCL produces higher-quality generated images. Tab. 2 summarizes the quantitative results, which shows that StyleCL outperforms all other baselines for most cases in terms of FID, Density, and Coverage metrics. Furthermore, Tab. 1 shows the amount of parameter reduction and percentage increase in FLOPs. StyleCL has relatively lower per-task parameter requirements compared to other methods, even though it does not have efficient adaptation modules like CAM-GAN. This reduction in the number of parameters can be attributed to the fact that StyleCL achieves continual adaptation using a combination of feature transformation and latent space modulation. The latent space modulation requires a lesser number of parameters, while still allowing the generation of some features of the target manifold. While MerGAN has no increase in parameter or FLOP count, it comes at the expense of decreased generation quality on earlier tasks, which does not occur in our no-forgetting setting. \| Algorithm \| parameter increase per task ↓ \| FLOPs Increase(%) ↓ \| \|-----------------\|-------------------------------\|---------------------\| \| GAN Memory \| 4.21M \| 15.7 \| \| CAM-GAN \| 1.52M \| 23.32 \| \| StyleCL \| 1.08M \| 4.1 \| Table 1: Comparison of our approach against the baselines GAN Memory and CAM-GAN with respect to parameter increase per task and percentage increase in FLOPS. Note that both the baselines also store the parameters for each task. \| Method \| MerGAN \| GAN Memory \| CAM-GAN \| StyleCL \| \|-----------------\|--------\|------------\|---------\|---------\| \| Dataset/Metric \| FID \| D \| Cov \| FID \| D \| Cov \| FID \| D \| Cov \| \| Flowers \| 45.14 \| 0.6 \| 0.49 \| 23.97 \| 0.73 \| 0.71 \| 23.38 \| 0.89 \| 0.71 \| 18.48 \| 0.67 \| 0.77 \| \| LSUN Church \| 31.41 \| 0.56 \| 0.18 \| 37.9 \| 0.30 \| 0.11 \| 24.25 \| 0.20 \| 0.17 \| 17.36 \| 0.59 \| 0.41 \| \| LSUN Cat \| 53.52 \| 1.10 \| 0.20 \| 53.22 \| 0.86 \| 0.32 \| 52.59 \| 0.62 \| 0.22 \| 34.43 \| 1.15 \| 0.41 \| \| Brain MRI \| 78.80 \| 0.16 \| 0.29 \| 45.78 \| 0.32 \| 0.55 \| 31.26 \| 0.18 \| 0.77 \| 29.42 \| 0.38 \| 0.82 \| \| Chest X-Ray \| 58.51 \| 0.13 \| 0.11 \| 58.82 \| 0.23 \| 0.3 \| 24.81 \| 0.36 \| 0.73 \| 25.83 \| 0.55 \| 0.75 \| \| Anime \| 39.83 \| 0.35 \| 0.09 \| 16.20 \| 0.63 \| 0.38 \| 21.52 \| 0.50 \| 0.27 \| 12.38 \| 0.62 \| 0.39 \| Table 2: Comparison of the performance of StyleCL, CAM-GAN, GAN-Memory, and MerGAN on six tasks using FID (lower is better), Density (D) (higher is better), and Coverage (Cov) (higher is better). The tasks are listed along the rows and methods are listed in the columns. 4.4 Results on Perceptually Similar Tasks In order to evaluate the forward transfer capability of StyleCL, we consider six varieties of butterflies from ImageNet to create a sequence of perceptually similar generation tasks, $X^1$ to $X^6$. We consider 2 scenarios: (a) StyleCL that enables forward transfer by considering the generator of the most similar previous task, and (b) StyleCL with parameter sharing only with the base task $G^1$ (without forward transfer). Tab. 3 summarizes the results for both scenarios. We observe improved performance on most datasets for scenario (a) compared to scenario (b), confirming the benefit of forward transfer. $^1$https://github.com/jayleicn/animeGAN Figure 3: Qualitative illustration of forward transfer in StyleCL: Fig. 3a and Fig. 3c correspond to real and generated samples from current task $\mathcal{X}^5$. StyleCL employs feature adaptors from previous tasks (samples of which are shown in Fig. 3b) to generate shared features across tasks (Fig. 3d). Meanwhile, it utilizes newly added feature adaptors exclusively for the unique features of the current tasks (Fig. 3e). Inherent in our method, also the amount of knowledge that could be reused varies (positive or negative forward transfer) which leads to varying degrees of improvement. As observed from Tab. 3 in case of $\mathcal{X}^3$, $sim(t,k) < 0$ indicates potential negative transfer and hence when the model is forced to reuse the most similar task, it results in a performance drop. This empirically validates our characterization of the nature of forward transfer by using $sim(t,k)$. In such cases, we prevent negative forward transfer by avoiding parameter reuse from the most similar task. To qualitatively evaluate the forward transfer capability of our approach, StyleCL, we train it on dataset $\mathcal{X}^5$ shown in Fig. 3a using the generator of the most similar previous task, $\mathcal{X}^3$ whose samples are shown in Fig. 3b. The generated samples are illustrated in Fig. 3c. To analyze the individual contribution of current and previous feature adaptors in StyleCL, we separately disable their individual contribution by setting $\alpha_m^i$ and $\alpha_m^k$ to 0 in equation 3. The corresponding generated samples are illustrated in Fig. 3d and Fig. 3e. Our results show that $\phi_m^3$ is reused to capture shared characteristics of $\mathcal{X}^5$ and $\mathcal{X}^3$, such as shape and background (as seen in Fig. 3d), whereas newly introduced feature adaptors $\phi_m^5$ capture features unique to $\mathcal{X}^5$, such as the orange colour of the wings (as seen in Fig. 3e). These findings confirm that StyleCL enables forward transfer by reusing knowledge from previous tasks. ### 4.5 Overcoming task ID constraints with StyleCL To provide empirical evidence that StyleCL inherently segregates datasets within a task, we created two distinct tasks: one that combines Flowers and Brain MRI images and another that merges Anime and LSUN Church images. For each task, we randomly sampled from the individual datasets to simulate a balanced mixture. We initialized StyleCL with two sets of dictionary vectors and feature adaptors, one for each dataset in a task. After completing training by generating samples from each component distribution using the corresponding dictionary and \| Task \| Datasets \| FID \| \|-----------------------\|----------------\|-------\| \| Flowers & Brain-MRI \| Flowers \| 24.38 \| \| \| Brain-MRI \| 35.22 \| \| LSUN Church & Anime \| LSUN Church \| 23.48 \| \| \| Anime \| 14.63 \| Table 4: Performance of StyleCL on the task ID free setting on two data mixtures. feature adaptor pair. The results are presented in Tab. 4. As observed from Tab. 4, StyleCL maintains high generation quality even in the absence of task ID information. 5 ABLATIONS AND ANALYSIS 5.1 ANALYSIS OF LEARNED LATENT SUBSPACES The learned latent dictionary for a task characterizes its position within the latent space and plays a crucial role in identifying both the most similar tasks and in preventing negative forward transfer. To validate the effectiveness of these learned latent vectors in capturing the semantics of each task, we present t-SNE visualizations of the latent vectors for a selected set of tasks. In particular, we aim to demonstrate that latent vectors associated with similar tasks are clustered closely together while remaining distinct from those associated with dissimilar tasks. To illustrate this, we generate t-SNE visualizations for the latent vectors of two distinct Butterfly datasets Sec. 4.4 and a perceptually different task Brain-MRI Sec. 4.3. The resulting t-SNE visualization is presented in Fig. 4. As observed in Fig. 4, latent vectors of different tasks forms clusters in latent space with latent vectors of semantically similar task lying close together. ![Figure 4: t-SNE visualization of latent vectors of similar and dissimilar tasks.](image) 5.2 EFFECT OF GENERATOR INITIALIZATION The generator $G^1$ is obtained by training on $\mathcal{X}^1$ and shares parameters with all subsequent tasks. To analyze this initialization’s impact on StyleCL, we experiment by initializing the generator with weights trained on Brain MRI and ImageNet datasets. We evaluate StyleCL’s performance on a data stream consisting of CelebA-HQ, Flowers, LSUN Church, and Chest X-Ray, using these different initializations. The results in Tab. 5 demonstrate a significant performance boost when the generator is initialized with weights from a diverse dataset like ImageNet-1K, compared to a more domain-specific base task like Brain MRI. This suggests our method benefits from initial weights trained on a diverse dataset. \| \| CelebA-HQ \| Flowers \| LSUN Church \| Chest X-Ray \| \|------------------\|-----------\|---------\|-------------\|-------------\| \| Brain-MRI \| 22.82 \| 31.98 \| 55.45 \| 29.93 \| \| ImageNet \| 15.86 \| 14.25\| 11.71 \| 23.54 \| Table 5: This table presents the performance comparison of StyleCL (measured by FID) on four different datasets(along columns), using various generator initialization methods(along rows). 6 CONCLUSIONS AND FUTURE WORK We introduce StyleCL, a lightweight expansion-based approach for generative continual learning with StyleGAN. Unlike prior methods of transforming feature maps or weights, we harness StyleGAN’s latent space for continual learning. For each new task, we learn a latent subspace via dictionary learning in the $W^+$ space and a feature adaptor. 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zEkvV65Wi1	If setting 2 is adopted, the comparison between the proposed method and the baselines (e.g., KD with Mixup vs Mixup) is not fair, since the hyperparameters of baselines are not optimized in terms of the calibration error.	Understanding Calibration Transfer in Knowledge Distillation Anonymous authors Paper under double-blind review Abstract Modern deep neural networks are often miscalibrated, leading to overconfident mistakes that erode their reliability, and limit their use in critical applications. The existing confidence calibration techniques range from train-time modification of loss functions to post-hoc smoothing of the classifier’s predicted confidence vector. Despite the success of these approaches, it is relatively unclear if supervision from an already trained expert classifier can further enhance a given classifier’s confidence calibration. Knowledge distillation (KD) has been shown to help classifiers achieve better accuracy. However, little to no attention has been paid to a systematic understanding if the calibration can also be transferred via KD. In this work, we provide new insights into how and when expert supervision can produce well-calibrated classifiers, by studying a special class of linear teacher and student classifiers. Specifically, we provide theoretical insights into the working mechanisms of KD and show that calibrated teachers can distill calibrated students. We further show that unlike traditional KD where a smaller capacity classifier learns reliably from a larger capacity expert, transfer of calibration can be induced from lower capacity teachers to larger capacity students (aka reverse-KD). Furthermore, our findings indicate that not all training regimes are equally suitable and that a teacher classifier trained using dynamic label smoothing leads to the better calibration of student classifiers via KD. Moreover, the proposed KD based calibration leads to a state-of-the-art (SOTA) calibration framework surpassing all existing calibration techniques. Our claims are backed up by extensive experiments on standard computer vision classification tasks. On CIFAR100 using WRN-40-1 feature extractor, we report an ECE of 0.98 compared to 7.61 and 2.1 by the current SOTA calibration techniques Adafoal (Ghosh et al. (2022)) and CPC Cheng & Vasconcelos (2022) respectively, and 11.16 by the baseline NLL loss (lower ECE is better). The calibration improvement is achieved across various architectures. Using MobileNetv2 on CIFAR100 we report an ECE of 0.88/1.83/4.17/7.76 using Ours/Adafoal/CPC/NLL. 1 Introduction Calibration. Deep neural network (DNN) models have become increasingly prevalent in critical applications such as healthcare (Kononenko (2001); Miotto et al. (2018)), and autonomous driving (Bojarski et al. (2016)). In such applications, it is crucial for DNN predictions to not only be accurate but also reliable and trustworthy (Nixon et al. (2019); Dusenberry et al. (2020)). Yet, it has been shown that the softmax probabilities (referred to as predicted confidence in this paper) produced by DNNs come with no formal probabilistic guarantees (Guo et al. (2017)). The phenomenon known as Calibration, refers to the alignment between a DNN model’s predicted confidence and the actual frequency of the event it represents. Calibration indicates model’s ability to provide reliable uncertainty estimates, and most modern DNNs are shown to be highly miscalibrated. Reasons for Miscalibration and Our Investigation. Mukhoti et al. (2020) have shown that a DNN model overfitting on NLL loss is the main reason behind highly overconfident predictions, leading to miscalibration. This begs the question, if access to richer class structure and label uncertainties during training can prevent such overfitting and generate a calibrated classifier. Knowledge distillation (KD) has been used for transferring learnt representations from a (typically large) teacher model to a (usually smaller) student model in multitude of works. In this work we investigate if access to learnt class structure through a teacher model’s representation also helps in calibration of a student model. Background on Knowledge Distillation. Since its introduction in 2015 by Hinton et al. (2015), KD has become a go-to method for transferring information between two classifiers with different capacities or architectures. It has been shown repeatedly that student classifiers trained via expert supervision from a teacher classifier or ensemble of classifiers via soft-label training exhibit improved performance than when trained with hard label-based training (for eg., one hot encoding) via cross-entropy loss Sun et al. (2019); Mirzadeh et al. (2020); Gou et al. (2021). The improved performance is reflected both in terms of increased classification accuracy, as well as stable behavior during training, requiring fewer optimization tricks (Phuong & Lampert (2019)). In summary, student classifiers tend to inherit properties of the teacher classifiers through knowledge sharing between them via KD. Our Proposal: KD for Calibration. Unfortunately, existing explanations of the process of KD rarely go beyond simple qualitative statements attributing improved performance to learning from soft-labels of the expert classifiers. Phuong & Lampert (2019) provide a first theoretical insight into the working mechanism of KD, albeit from an optimization viewpoint. Allen-Zhu & Li (2023) elucidates the effectiveness of ensemble learning and KD in enhancing the test accuracy of classifiers, without placing particular emphasis on the transfer of calibration properties. In our work, we view the role of KD beyond its well-studied role of accuracy transfer and provide theoretical and empirical insights into the transfer of calibration to student classifiers. We show, arguably for the first time, that only calibrated teachers potentially distill the best-calibrated students, and thus, a recipe for producing accurate and calibrated classifiers must also involve KD through calibrated teacher classifiers. Departure from Current Belief: Does KD Conflict with Calibration. Interestingly, there is some precedence in investigating the role of calibrating teacher classifiers via label-smoothing (LS) Müller et al. (2019). However, LS was observed to impair KD, i.e., the accuracy of student classifiers degrades when teacher classifiers are calibrated with LS, which potentially points to pitfalls of KD Shen et al. (2021). In our work, we show that this impairment is not the artefact of KD but of the LS itself, which when used to calibrate teacher classifiers and distill their understanding to student classifiers at higher temperatures, ends up over-smoothing a student’s predictions, thereby significantly degrading its accuracy (Chandrasegaran et al. (2022)). We show that teachers trained via dynamic label-smoothing methods (E.g., Hebbalaguppe et al. (2022b)) consistently distil calibrated students across all temperatures. To this end, we highlight the role of KD in calibrating classifiers and argue strongly in favor of using knowledge sharing from calibrated experts to student classifiers as the most promising calibration technique. Mathematical Definition. Formally, given a data distribution $ D $ of $(x, y) \in X \times \{0, 1\} $ and let $ c $ be the predictive confidence, the predictor $ f : X \rightarrow [0, 1] $ is said to be calibrated (Dawid (1982)), if: \[ \mathbb{E}_{(x,y)\sim D} \left[ y \mid f(x) = c \right] = c, \quad \forall c \in (0, 1) \] Key Contributions. To achieve calibration of DNNs, we bring together two seemingly unrelated subfields: KD and confidence calibration. We make the following key contributions in this direction: 1. Understanding calibration transfer via distillation: We develop a theoretical framework to analyze KD and its ability to transfer learning of a teacher classifier to a student classifier, and show, arguably for the first time, that only calibrated teachers can distill calibrated students. We corroborate our theoretical results later through exhaustive experiments. 2. Achieving best student network calibration: Our experiments demonstrate that students trained via KD from teachers that are first calibrated using dynamic/adaptive label-smoothing, exhibit the best calibration compared to other train-time/post-hoc calibration techniques. (Sec. 5.1). Our framework is dubbed KD(C) (Knowledge Distillation from Calibrated teacher). 3. Not all calibration techniques are compatible with KD: It has been observed empirically that LS impairs KD (Müller et al. (2019)). This impairment is argued to be a high-temperature phenomenon (Chandrasegaran et al. (2022)). In our experiments, we, too, observe a similar behavior when teacher classifiers are trained via static calibration methods, such as LS. However, we show that when the teacher classifiers are calibrated using dynamic LS methods the distillation produces calibrated student classifiers consistently across wide temperature regimes. 4. Calibration distillation works both ways: Contrary to popular belief that only larger models can effectively distill their learning to smaller models, we show that smaller calibrated models can also yield better calibrated larger models. The observation is consistent with our key insight that the availability of label ambiguities through soft-labels during training is extremely useful for calibration. This setting is relevant when large calibrated models or large datasets for training such large models not readily available and significantly widens the applicability of our framework. 2 RELATED WORK Research pertaining to calibration of DNNs can be categorized into following approaches: (a) train-time calibration, (b) post-hoc calibration, (c) Bayesian inference, and (d) data augmentation. A train-time calibration integrates model calibration during the training phase through suitable modification of loss function. For instance, label smoothing (LS), originally introduced by Szegedy et al. (2015) to improve the classifier accuracy by computing cross-entropy with a weighted sum of one-hot vector and the uniform distribution, was adopted by Müller et al. (2019) for improving calibration. Most train-time methods for calibration improvement inherently look to smooth confidence scores in a sample-agnostic manner. Noteworthy works in this category include those by Moon et al. (2020); Kim et al. (2021); Liu et al. (2022); Hebbalaguppe et al. (2022b); Park et al. (2023). Conversely, post-hoc calibration focuses on optimizing calibration measures using a separate hold-out set post-training. Guo et al. (2017), demonstrated that temperature scaling (TS), a technique that smooths confidence scores by dividing the logits of a classifier by a scalar $T > 1$, enhances its calibration. Other notable contributions in this category also include the studies by Platt et al. (1999); Kull et al. (2017; 2019); Bohdal et al. (2021); Islam et al. (2021). However, despite its simplicity, it was observed in Hebbalaguppe et al. (2022b) that train-time approaches offer superior performance over post-hoc methods. Prominent examples of calibration methods relying on data augmentation encompass Thulasidasan et al. (2019) and Hebbalaguppe et al. (2022a). Meanwhile, Bayesian methodologies are exemplified by Gal & Ghahramani (2016); Lakshminarayanan et al. (2017); Ovadia et al. (2019) and Wenzel et al. (2020). However, in the context of our research, train-time and KD-based approaches are especially pertinent. KD Hinton et al. (2015) was originally proposed to enhance the accuracy of student classifiers by transferring knowledge from high-capacity teacher classifiers. However, recent empirical evidence points to regularization effects of KD over student classifiers akin to training classifiers separately via LS (see Tang et al. (2020)), which seems to suggest direct calibration benefits of KD. It was shown in Yuan et al. (2021) that when the temperature parameter during KD is set to unity and the probability distribution of teacher classifiers are assumed to be uniform, KD via teacher classifier and LS of student classifier exhibit identical behaviors in terms of gradient propagation. The observations prompted to explore the scenario where a teacher classifier itself is first calibrated via LS and then distills knowledge to a student classifier (i.e., distill knowledge from an LS calibrated teacher) with the hope of doubling the regularization benefits. However, it was observed in Müller et al. (2019) that LS representations impaired with those of KD, thus nullifying any regularization benefits. This viewpoint, nonetheless, was later shown to be incomplete by Shen et al. (2021), where the authors argued that such an impairment is only a high-temperature phenomenon. While this interplay between LS and KD provided some insights into the regularization benefits of KD, its role as a potential calibrator of smaller teacher classifiers has not been addressed in the literature. In our work, we look beyond just the vanilla LS of teacher classifiers and provide direct theoretical and empirical evidence towards the benefits of working with calibrated teachers and how they distill SOTA calibrated student classifiers via KD. We also systematically analyze various calibration techniques for the teacher classifiers so that the resulting student classifiers exhibit significantly improved calibration performance over directly calibrating them via train-time or post-hoc methods. We show that dynamic LS methods, such as the MDCA (Hebbalaguppe et al. (2022b)), consistently exhibit better accuracy and calibration trade-off across wider temperature ranges. 3 UNDERSTANDING CALIBRATION VIA KNOWLEDGE DISTILLATION We analyze the mechanics of obtaining calibrated models via KD from a theoretical standpoint, focusing on linear teacher and student networks in a binary classification problem. Such linear classifiers, which were initially explored in Phuong & Lampert (2019) to gain a general understanding of KD, have not been previously investigated for their potential to transfer learned representations, particularly calibration, to student networks. Furthermore, the authors in Phuong & Lampert (2019) utilized a simplified version of the KD loss function, which did not consider the significance of distillation weights and quadratic temperature scaling. These factors play a crucial role in showcasing the transfer of calibration to student models. To this end, we represent an $i\text{th}$ training instance by $x_i \in \mathbb{R}^d$. The set of all training examples is represented by $X \in \mathbb{R}^{d \times N}$. We use $z_{i,s}$ and $z_{i,t}$ to represent logits of the student and teacher networks for the $i\text{th}$ training instance, respectively. These logits can be converted into valid probability distributions $p_{i,s}$ and $p_{i,t}$, respectively, using Sigmoid activation function. In knowledge distillation, the output probabilities of the teacher network are softened using inverse temperature scaling of the logit, leading to prediction probabilities $\tilde{p}_{i,t}$. The true class labels are denoted by $\{y_i \in \{0, 1\}\}$. Since, the teacher and student networks are assumed to be linear networks, an arbitrary deep network can equivalently be represented using a single layer network. We use $W_s$ and $W_t$ to represent weight matrices of the student and teacher networks, respectively. Finally, we use $T \in \mathbb{R}_+$ to depict temperature parameter for temperature scaling, while $\alpha \in [0, 1)$ represents the relative importance of the student’s binary cross-entropy loss. Below we list the key assumptions before presenting the key theoretical results. Assumption 1. The feature dimension $d$ is larger than the number of training examples $N$. Assumption 2. The student and teacher networks are represented by linear networks. Remark 1. A direct consequence of Assumption 1 is that the data matrix $X$ is full column rank almost surely, since if one randomly samples $N$ training examples (with $N < d$), the probability that any two sample vectors are perfectly aligned is nearly zero. Consequently, the matrix $X^\top X$ is invertible. Assumption 2 ensures that both student and teacher networks can be compactly represented as single layer linear networks. Though, the assumption implicitly enforces student network to be of the same capacity as that of the teacher network, it makes it easier to better understand the mechanics of the KD, specifically when we later distill larger models from smaller models. Extending to nonlinear networks poses significant challenges, similar to the lack of a general theory for DNNs and non-convex optimization. However, we can still extract valuable insights from the theory of linear networks and utilize them to establish a framework applicable to general nonlinear networks. In KD, the student aims to minimize the weighted combination of the binary cross-entropy loss $L_{BCE}$, and the KD loss $L_{KD}$, given by: \[ L_{BCE} = -\sum_{i=1}^{N} [y_i \log p_{i,s} + (1 - y_i) \log (1 - p_{i,s})], \] \[ L_{KD} = -T^2 \sum_{i=1}^{N} [\tilde{p}_{i,t} \log \tilde{p}_{i,s} + (1 - \tilde{p}_{i,t}) \log (1 - \tilde{p}_{i,s})], \] \[ L_{tot} = (1 - \alpha)L_{BCE} + \alpha L_{KD}, \] where $p_{i,s} := \sigma(W_s^\top x_i)$, $\tilde{p}_{i,s} := \sigma\left(\frac{W_s^\top x_i}{T}\right)$, $\tilde{p}_{i,t} := \sigma\left(\frac{W_t^\top x_i}{T}\right)$ and $\sigma(\cdot)$ is the Sigmoid function. Below we provide our key theoretical results. Theorem 1. Let $X \in \mathbb{R}^{d \times N}$ be a data matrix satisfying Assumption 1, and $W_s$ and $W_t$ represent the parameters of the student and the teacher networks, respectively. Then, under Assumption 2 and using the gradient-descent algorithm, the parameters $W_s$ of the student network converge to: \[ W_s \approx \alpha W_t + 4(1 - \alpha)X(X^\top X)^{-1}Y_{1/2}, \] where $Y_{1/2} := [y_i - \frac{1}{2}]_{i=1}^{N}$ is an $N$-dimensional vector. Proof. Please refer to Section 2.1 in the supplementary material for the detailed proof. Remark 2. Theorem 1 shows that when $\alpha$ is close to unity, the weights of the student network are almost identical to those of the teacher network. Thus, properties of the teacher network transfer directly to the student. For $\alpha \neq 1$, the student also learns to update its weight from the labeled data. Calibrated Teachers produce Calibrated Students. A neural network classifier is said to be well calibrated if the predicted probability distribution is similar to the observed probability distribution (Naeini et al. (2015)). Mathematically speaking, if a teacher network with predicted probabilities $\{p_{i,t}\}$ is well calibrated, then the following holds: $$\sum_{i=1}^{N} p_{i,t} = \sum_{i=1}^{N} y_i.$$ (3) We now prove that well-calibrated teachers distill well-calibrated students. On the contrary, if the teacher classifier is not well-calibrated, it is impossible to distill well-calibrated student classifiers. The result extends our understanding of KD beyond accuracy transfer and formally characterizes the transfer of calibration from student-to-teacher networks. Theorem 2. Let Assumptions 1-2 hold. Let $t_c$ and $t_{uc}$ be two teacher classifiers with output probabilities $\{p_{i,t,c}\}$ and $\{p_{i,t,uc}\}$, respectively. Also, let $s_c$ and $s_{uc}$ depict two student classifiers trained independently from the corresponding teacher classifiers $t_c$ and $t_{uc}$ through KD, with output probabilities $\{p_{i,s,c}\}$ and $\{p_{i,s,uc}\}$, respectively. Furthermore, assume that the teacher classifier $t_c$ is well calibrated, then the student classifier $s_c$ is also well calibrated. Conversely, if the teacher classifier $t_{uc}$ is uncalibrated, the corresponding student classifier $s_{uc}$ mimics a similar behavior, i.e., $$\sum_{i=1}^{N} p_{i,s,c} = \sum_{i=1}^{N} y_i,$$ and $$\sum_{i=1}^{N} p_{i,s,uc} \neq \sum_{i=1}^{N} y_i.$$ Proof. Please refer to Section 2.2 in the supplementary material for detailed proof. Below we describe our framework for calibrating classifiers that leverage KD as a key tool. ### 3.1 Recipe for Joint Optimization of Calibration and Accuracy Our main goal is to create well-trained DNNs that demonstrate the highest possible accuracy and confidence calibration during inference. Traditional loss functions like negative log-likelihood (NLL) have been found to encourage over-confident models. To overcome the shortcomings of NLL-based training, one potential but simplistic approach involves calibrating classifiers pre-trained via KD, with the hope that these models will retain enhanced accuracy from the teacher model (refer to Theorem 1) and achieve confidence calibration through subsequent post-hoc calibration. However, our experiments illustrate that this two-step approach does not guarantee the best results. We hypothesize that the suboptimal performance of the student classifier primarily arises from the interference between the representations learned through KD and the post-hoc calibration technique. As shown in Theorem 2, calibrated teachers are guaranteed to distill calibrated students. Hence, we experiment with KD using calibrated teachers which can potentially distill accurate and calibrated students in one go. For calibrating teacher networks, we restrict our attention to train-time methods, such as MDCA (Hebbalaguppe et al. (2022b)) and LS (Szegedy et al. (2015)), since post-hoc methods \| Abbreviation \| Description Calibration Technique \| \|--------------\|----------------------------------\| \| NLL \| Negative Log Likelihood \| \| CE+TS (Gao et al. (2017)) \| Loss function with Temperature scaling on student model trained with cross-entropy loss \| \| MMDA (Sun et al. (2018)) \| Minimizing Maximum Mean Discrepancy Error \| \| MoUp (Thulasidasan et al. (2019)) \| Calibration through MoUp data augmentation \| \| CRD (Sun et al. (2019)) \| Correctness Ranking Loss \| \| PSKD (Kumar et al. (2021)) \| Posterior Sharpening with Self Knowledge Distillation \| \| MDCA (Hebbalaguppe et al. (2022b)) \| Multi-class difference in Confidence and Accuracy \| \| AdaLoss (Zhang et al. (2021)) \| AdaLoss \| \| CPC (Cheng & Vasconcelos (2022)) \| Calibration via Pairwise Constraints \| \| MLD (Hebbalaguppe et al. (2022a)) \| Margin based Label Smoothing \| \| KDUCY \| Knowledge Distillation from Uncalibrated teacher \| \| KD with TS (Ours) \| Temperature Scaling on distilled Student classifier \| \| KD with LS (Ours) \| KD framework with teacher trained on LS \| \| KD with MDCA (Ours) \| KD framework with teacher trained on MDCA \| \| KD with CRD (Ours) \| KD framework with teacher trained on CRD \| \| KD with CPC (Ours) \| KD framework with teacher trained on CPC \| \| KD with AdaLoss (Ours) \| KD framework with teacher trained on AdaLoss \| \| KD with PSKD (Ours) \| KD framework with teacher trained on PSKD \| \| KD with MBLS (Ours) \| KD framework with teacher trained on Margin based Label Smoothing \| \| KD with MixUp (Ours) \| KD framework with teacher trained on MixUp data augmentation \| Table 1: Nomenclature used for baseline methods and the corresponding description of calibration technique. Our proposed approach KD(C) variants are highlighted where ‘C’ is calibration type. \| Method \| Architecture \| WideResNet-40-1 (0.56M) \| MobileNetV2 (2.25M) \| \|-------------------------\|-----------------------\|-------------------------\|---------------------\| \| \| Top1 (%) ↑ \| ECE (%) ↓ \| SCE (%) ↓ \| ACE (%) ↓ \| Top1 (%) ↑ \| ECE (%) ↓ \| SCE (%) ↓ \| ACE (%) ↓ \| \| NLL \| 70.04 \| 11.16 \| 0.30 \| 11.19 \| 66.09 \| 7.76 \| 0.25 \| 7.80 \| \| LS Szeredy et al. (2015)\| 70.07 \| 11.30 \| 0.21 \| 1.49 \| 66.96 \| 4.24 \| 0.23 \| 4.18 \| \| CE+TS Gao et al. (2017) \| 70.04 \| 2.57 \| 0.19 \| 2.50 \| 66.09 \| 2.33 \| 0.19 \| 2.37 \| \| MMCE Kumar et al. (2018)\| 69.69 \| 2.34 \| 0.25 \| 2.31 \| 67.90 \| 2.21 \| 0.21 \| 2.15 \| \| MixUp Thulasidas et al. (2019) \| 72.04 \| 2.57 \| 0.21 \| 2.52 \| 67.53 \| 8.69 \| 0.28 \| 9.73 \| \| CRL Johnson et al. (2020)\| 65.80 \| 13.91 \| 0.37 \| 13.91 \| 67.05 \| 12.06 \| 0.33 \| 12.06 \| \| PSKD Kim et al. (2021) \| 72.56 \| 3.73 \| 0.20 \| 3.72 \| 69.09 \| 6.95 \| 0.23 \| 6.94 \| \| MDCA Hebbalaguppe et al. (2022b) \| 68.51 \| 1.35 \| 0.21 \| 1.34 \| 66.96 \| 1.61 \| 0.20 \| 1.92 \| \| AdaFocal Ghosh et al. (2022) \| 67.36 \| 2.10 \| 0.21 \| 1.97 \| 65.34 \| 1.83 \| 0.20 \| 1.53 \| \| CPC Cheng & Vasconcelos (2022) \| 69.99 \| 7.61 \| 0.23 \| 7.55 \| 67.30 \| 4.17 \| 0.22 \| 4.07 \| \| MbLS Liu et al. (2022) \| 69.97 \| 5.37 \| 0.22 \| 5.37 \| 67.22 \| 1.25 \| 0.20 \| 1.25 \| \| KD(UC) \| 69.60 \| 15.18 \| 0.37 \| 15.18 \| 66.82 \| 5.40 \| 0.22 \| 5.36 \| \| Ours (KD with MixUp) \| 72.48 \| 1.21 \| 0.20 \| 1.17 \| 69.92 \| 2.17 \| 0.24 \| 2.10 \| \| Ours (KD with AdaFocal) \| 71.70 \| 1.19 \| 0.19 \| 1.34 \| 66.64 \| 1.55 \| 0.20 \| 1.43 \| \| Ours (KD with CPC) \| 70.00 \| 3.02 \| 0.26 \| 9.01 \| 67.83 \| 0.88 \| 0.19 \| 0.95 \| \| Ours (KD with MDCA) \| 71.07 \| 0.98 \| 0.20 \| 1.10 \| 67.17 \| 1.10 \| 0.20 \| 1.17 \| Table 2: [Large-to-small] Comparison of calibration performance of small student models calibrated using KD(C) framework vs. SOTA calibration techniques, employing a relatively larger calibrated teacher on CIFAR100 dataset. WRN-40-2 (Zagoruyko & Komodakis (2016)) and ResNeXt-18x4 (Xie et al. (2017)) were used as teachers for WRN-40-1 (Zagoruyko & Komodakis (2016)) and MobileNetV2 (Sandler et al. (2018)) respectively as student models. For ECE/SCE computation, 15 bins were used in accordance with prior work. ACE uses an adaptive binning strategy. For full results refer to the supplementary. Numbers in bold: best performance; underlined: second best. KD(C) gives best all around performance, in one instance PSKD is slightly better than KD(C) in accuracy, but lags much behind in calibration metric (the focus of this paper). are known to result in relatively inferior calibration performance (Müller et al. (2019); Platt et al. (1999); Kull et al. (2019)). Supported by the theoretical insights of generic property transfer from a teacher to a student network, we describe our training regime in two steps: 1. Train-time calibration of teachers: We draw inspiration from train-time calibration techniques that have shown superior performance than post-hoc calibration, and have experimented with the following techniques: (Kumar et al. (2018); Müller et al. (2019); Hebbalaguppe et al. (2022b); Cheng & Vasconcelos (2022); Moon et al. (2020)) to name a few. A simple gradient analysis reveals that train-time calibration methods, such as MDCA (Hebbalaguppe et al. (2022b)) and ACLS (Park et al. (2023)), act as dynamic/adaptive label smoothing, which is arguably better than the traditional static label smoothing (Müller et al. (2019)). 2. Knowledge distillation from calibrated teacher: Once trained for calibration and accuracy, teacher classifiers distill their behavior to student classifiers through KD loss Eq. (2). As a result, the student classifiers are both accurate and confidence calibrated (see Theorem 2). Proposed comprehensive framework KD(C) encompasses the full spectrum, enabling models with varying capacity (smaller/larger) to distill student models with the least calibration error and better accuracy compared to the SOTA post-hoc/train-time calibration methods. Note: We do not advocate a specific train-time calibrator but rather a KD-style calibration where an expert model helps enhance the calibration performance of a student. Tab. 1 shows the KD(C) framework encompasses the variations highlighted in Cyan. With this, we study systematically the effect of various direct calibration methods. The best results among “KD with” methods are shown in each table. Please refer to the supplementary for a full version of these results. ### 4 EXPERIMENTS We now validate our theoretical claims on calibrated teachers distilling calibrated students (see Theorem 2 in supplementary material) through extensive experiments. Evaluation Metrics: We benchmark our framework KD(C) against other competing methods using (a) calibration error metrics (lower value is better), Expected calibration error (ECE) (Guo et al. Figure 1: Comparative study of accuracy vs. calibration trade-offs associated with existing calibration techniques and ours (Top-left is most preferred): The mean and one standard scatter error bars for Top1, ECE and SCE of WideResNet-40-1 trained on CIFAR100 using SOTA calibration techniques. WideResNet-40-2 was used as Teacher for KD(UC) and the proposed, KD(C) variants. Note: KD(C) variants (magenta, cyan, and green) achieve the best results in terms of ECE, ACE and SCE, along with slight boosts in Top1 (an inherent KD-property). Further, the lower variances emphasize the reliability of KD(C) variants. All plots were generated by training WideResNet-40-1 models through every calibration technique on 3 runs. Static Calibration Error (SCE) and Adaptive Calibration Error (ACE) (Nixon et al. (2019)), as well as (b) Top1 accuracy (higher value is better), indicative of generalization performance. Datasets and Baselines. We use widely accepted diverse datasets, CIFAR10 (Krizhevsky et al. (2009)), CIFAR100 (Krizhevsky et al. (2009)), and Tiny-ImageNet (Le & Yang (2015)) for benchmarking. To test robustness of our approach, we report additional results on CIFAR100-C (Hendrycks & Dietterich (2019)) dataset in the supplementary. We could not experiment on ImageNet due to limited computational constraints in our lab. We include models trained through standard NLL, as well as LS, MixUp, Adafoocal, MMCE, CRL, CPC, MDCA, and PSKD (please refer to Tab. 1 for citations on calibration techniques). Along with this, we include student KD(UC) distilled from an uncalibrated teacher obtained by training using NLL as one of the baselines. Training details. The architectures used in the experiments include ResNet (He et al. (2016)), MobileNetV2 (Sandler et al. (2018)), and ShuffleNetV2 (Ma et al. (2018)) DenseNet (Huang et al. (2018)), WideResNet (Zagoruyko & Komodakis (2016)) architectures. The exact details of training and model hyperparameters, along with the details on compute resources are included in the supplementary material. Source code /trained models for all benchmark methods will be made public upon acceptance for reproducibility. 5 RESULTS Large calibrated teacher models distilling into smaller models. We now present compelling evidence supporting the superiority of our proposed KD(C) method over the SOTA train-time and post-hoc techniques for calibrating smaller student classifiers. To this end, we leverage distillation to create a smaller model (e.g., WRN-40-1/MobileNetV2) from a well-calibrated teacher model (e.g., WRN-40-2) and compare its performance with models directly subjected to train-time calibration techniques, as well as the progressive-KD (PSKD) method introduced by Kim et al. (2021). We also report the impact of distillation from an uncalibrated teacher model, denoted as KD(UC), as a baseline. The summarized results are detailed in Tab. 2. Notably, KD(C) demonstrates significantly lower calibration errors (ECE/SCE/ACE) while simultaneously achieving higher accuracy compared to models calibrated directly using metrics such as NLL/MDCA/LS. Additionally, Fig. 1 provides a visual representation of our findings, illustrating the mean and standard deviations of accuracy and calibration errors over three random runs. Notably, KD(C) variants exhibit (a) the best balance between accuracy and calibration while (b) displaying higher reliability, as evidenced by their lower variance. Importantly, our results confirm our theoretical findings discussed in Sec. 3, establishing that calibrated teachers are capable of effectively distilling calibrated students. This underscores the successful transfer of learned representations, encompassing both accuracy and calibration aspects, from a calibrated teacher model to a smaller student. We give additional results in supplementary showing our approach consistently yields improved calibration across various model architectures. Reliability diagrams corresponding to Tab. 2 can also be found in the supplementary. Self-distillation. A significant question that arises pertains to the generalizability of insights gleaned from the prior set of experiments. Particularly whether these insights can be extended to produce accurately calibrated classifiers with identical architecture and capacity. Our research demonstrates that this process referred to as “self-distillation,” results in classifiers that exhibit superior calibration compared to their teachers. However, the increase in accuracy is only marginal, likely due to the absence of distillation from a teacher with greater capacity. Our findings on the CIFAR-10 dataset are succinctly presented in Tab. 3. It is worth noting that, unlike the PSKD approach proposed by Kim et al. (2021), which progressively distills knowledge by leveraging the previous epoch’s trained model, KD(C) employs self-distillation just once with a fixed teacher throughout the training process, following a methodology akin to Zhang & Sabuncu (2020). Nevertheless, KD(C) achieves remarkable improvements in terms of calibration errors, notably surpassing the performance of the baseline models by a significant margin. Small calibrated teacher models distilling into large models. In settings where large trained models are not available, it is desirable to be able to distill the knowledge from smaller models to larger models. Jiang & Deng (2023) have shown that smaller models can also be valid teachers for large students, however, it was observed that the gains in accuracy were not significant as compared to distilling from a large teacher comparatively. Our results for the configuration are summarized in Tab. 4, where smaller model MobileNetV2 is used as teacher network to calibrate DenseNet-121. We note a trade-off between accuracy and calibration performance, primarily arising from the inherent capacity differences between larger student models and smaller-capacity teachers. Larger student models possess greater capacity and can potentially match or surpass the performance of smaller-capacity teachers. Consequently, this limits the additional knowledge that can be effectively distilled from smaller-capacity teachers. Nevertheless, the process still yields improvements in calibration for larger models through KD. Iterative self-distillation. Taking inspiration from previous works such as Mirzadeh et al. (2020); Yalburgi et al. (2020); Kim et al. (2021), we investigate whether KD(C) can iteratively distill more accurate and calibrated models. Here both teacher and student have identical architectures, and a \| Calibration Method \| Top1 (%) \| ECE (%) \| SCE (%) \| AECE (%) \| \|--------------------\|----------\|---------\|---------\|----------\| \| NLL \| 89.87 \| 3.30 \| 0.75 \| 3.28 \| \| LS Szedegy et al. (2015) \| 89.60 \| 7.10 \| 1.78 \| 6.35 \| \| CE+TS Guo et al. (2017) \| 89.90 \| 0.96 \| 0.40 \| 0.77 \| \| MMCE Kumar et al. (2018) \| 89.38 \| 1.20 \| 0.51 \| 0.94 \| \| MixUp Thulasidasan et al. (2019) \| 89.57 \| 9.42 \| 2.07 \| 9.41 \| \| CRL Moon et al. (2020) \| 90.31 \| 2.48 \| 0.72 \| 2.81 \| \| PSKD Kim et al. (2021) \| 89.21 \| 5.27 \| 0.93 \| 3.25 \| \| MDCA Hebbalaguppe et al. (2022b) \| 88.74 \| 0.99 \| 0.46 \| 0.80 \| \| AdaFocal Ghosh et al. (2022) \| 88.98 \| 0.79 \| 0.44 \| 0.86 \| \| CPC Chung & Vasconcelos (2022) \| 89.26 \| 3.47 \| 0.79 \| 3.44 \| \| MblS Liu et al. (2022) \| 89.86 \| 0.83 \| 0.69 \| 0.78 \| \| KD(UC) \| 89.88 \| 0.99 \| 0.43 \| 0.82 \| \| Ours (KD with TS) \| 90.23 \| 0.51 \| 0.41 \| 0.59 \| \| Ours (KD with MixUp) \| 89.27 \| 9.16 \| 2.19 \| 9.05 \| \| Ours (KD with AdaFocal) \| 89.56 \| 0.63 \| 0.41 \| 0.65 \| \| Ours (KD with CPC) \| 89.92 \| 0.48 \| 0.48 \| 0.57 \| \| Ours (KD with MDCA) \| 88.79 \| 0.48 \| 0.48 \| 0.54 \| \| Ours (KD with MMCE) \| 89.97 \| 0.85 \| 0.54 \| 0.84 \| Table 3: [Self-distillation] using MobileNetV2 feature extractor on CIFAR10 dataset. Only top-3 performing KD(C) variants are reported, for full results refer to supplementary. \| Method \| Top1 Acc.(%) \| ECE(.) \| SCE(.) \| ACE(.) \| \|--------\|--------------\|--------\|--------\|--------\| \| NLL \| 94.81 \| 3.37 \| 0.72 \| 3.34 \| \| LS Szedegy et al. (2015) \| 93.85 \| 4.58 \| 0.85 \| 4.36 \| \| CE+TS Guo et al. (2017) \| 94.81 \| 0.97 \| 0.40 \| 0.23 \| \| MMCE Kumar et al. (2018) \| 93.08 \| 1.01 \| 0.39 \| 0.92 \| \| MixUp Thulasidasan et al. (2019) \| 95.18 \| 2.85 \| 0.67 \| 2.83 \| \| CRL Moon et al. (2020) \| 93.67 \| 1.36 \| 0.45 \| 1.25 \| \| PSKD Kim et al. (2021) \| 94.49 \| 1.81 \| 0.43 \| 2.02 \| \| MDCA Hebbalaguppe et al. (2022b) \| 92.69 \| 0.31 \| 0.35 \| 0.35 \| \| AdaFocal Ghosh et al. (2022) \| 93.48 \| 1.44 \| 0.37 \| 1.10 \| \| CPC Chung & Vasconcelos (2022) \| 94.39 \| 4.36 \| 0.91 \| 4.18 \| \| MblS Liu et al. (2022) \| 94.45 \| 3.42 \| 0.75 \| 3.43 \| \| KD(UC) \| 90.2 \| 2.17 \| 0.60 \| 2.13 \| \| Ours (KD with AdaFocal) \| 91.68 \| 0.54 \| 0.34 \| 0.61 \| \| Ours (KD with MDCA) \| 90.09 \| 0.53 \| 0.45 \| 0.51 \| \| Ours (KD with MblS) \| 93.1 \| 0.61 \| 0.38 \| 0.40 \| Table 4: [small-to-large] Calibration performance of a large student model (DenseNet-121) (6.95M) on CIFAR10 when distilled from small (un)calibrated teacher (MobileNetV2 (2.25M)). Top3 performing KD(C) variants have been shown. See to supplementary for additional results. Figure 2: Iterative self-KD on CIFAR100 using ResNet56. We use KD with MDCA for calibration. student in $t^{th}$ iteration (called generation hereon) becomes teacher for $(t+1)^{th}$ generation. We refer to this as iterative self-distillation. Fig. 2 shows the self-distillation process for six generations. As expected the gap between KD(UC) and KD(C) gradually diminishes with each generation (the only difference between the two is initialization: generation zero teacher is uncalibrated in KD(UC) but calibrated in KC(C)). Our observation aligns with findings from (Zhang & Sabuncu (2020); Kim et al. (2021)). Note: From Tables 2, 3, and 4, we observe no other method is as consistent in improving calibration performance as KD(C); notice that while direct calibration can be the best-performing in one of the instances, it fails miserably in other instances. The KD(C) framework variants are consistently the best-performing ones or the second-best. Other Experiments. We report (a) calibration performance under dataset drift on vision datasets; (b) ablation study on the effect of hyper-parameters like $T$ (temperature) and $\alpha$ (distillation weight) in the supplementary along with experiments involving other DNN architectures. 5.1 Discussion Our work offers promising results with a recipe to combine KD and calibration in one go. Unlike traditional LS, dynamic/adaptive smoothing based regularizers (Park et al. (2023); Hebbalaguppe et al. (2022b)) offer sample-specific dynamic label-smoothing. However, not many of these methods tend to capture inter-class semantics, which are inherently captured through the interplay of teacher-based knowledge transfer and learning directly from data. Inspired from Chandrasegaran et al. (2022), in the supplementary, we give additional rationale for the superior performance of the proposed KD(C) framework using penultimate layer visualizations. We also provide more justification on why teacher classifiers need to be calibrated at train-time in the supplemental material. 6 Conclusions Our primary contribution lies in providing a robust theoretical foundation, including formal proofs, for the transfer of calibration and accuracy between teacher and student DNN models. We have demonstrated consistent superior calibration achieved through our KD(C) method. Our experiments span diverse scenarios, encompassing large-to-small, small-to-large, and self-distillation settings, featuring a variety of architectures and datasets. The consistent success of KD(C) across these different configurations highlights its effectiveness and broad applicability. Additionally, we advocate for the adoption of dynamic calibration techniques (such as MDCA) on the teacher model before the distillation. Our findings are further enriched by the insights gained from penultimate layer visualizations, shedding light on the inner workings of calibration in DNNs. From an application perspective, as the utilization of KD continues to grow, particularly for obtaining lightweight models beneficial for edge computing, neural architecture search, model compression, and other domains, our work contributes by introducing a method to additionally enhance the trustworthiness of these neural networks through calibration. REFERENCES Zeyuan Allen-Zhu and Yuanzhi Li. Towards understanding ensemble, knowledge distillation and self-distillation in deep learning. In The Eleventh International Conference on Learning Representations, 2023. URL https://openreview.net/forum?id=Uuf2q9TfXGA. Ondrej Bohdal, Yongxin Yang, and Timothy Hospedales. Meta-calibration: Meta-learning of model calibration using differentiable expected calibration error. In ICML Uncertainty in Deep Learning Workshop, 2021. 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YLJs4mKJCF	One of the problem of introducing elastic penalty is how to choose properly the two penalty parameters $\lambda_{1}$ and $\lambda_2$. Though it can be chosen empirically, it can be dataset-dependent. Would it make significantly difference if we simply choose the L1 norm penalty instead?	Towards Poisoning Fair Representations Tianci Liu1, Haoyu Wang1, Feijie Wu1, Hengtong Zhang2, Pan Li3, Lu Su1, Jing Gao1 1Purdue University 2Tencent AI Lab 3Georgia Institute of Technology 1{liu3351,wang5346,wu1977,lusu,jinggao}@purdue.edu 2htzhang.work@gmail.com 3panli@gatech.edu Abstract Fair machine learning seeks to mitigate model prediction bias against certain demographic subgroups such as elder and female. Recently, fair representation learning (FRL) trained by deep neural networks has demonstrated superior performance, whereby representations containing no demographic information are inferred from the data and then used as the input to classification or other downstream tasks. Despite the development of FRL methods, their vulnerability under data poisoning attack, a popular protocol to benchmark model robustness under adversarial scenarios, is under-explored. Data poisoning attacks have been developed for classical fair machine learning methods which incorporate fairness constraints into shallow-model classifiers. Nonetheless, these attacks fall short in FRL due to notably different fairness goals and model architectures. This work proposes the first data poisoning framework attacking FRL. We induce the model to output unfair representations that contain as much demographic information as possible by injecting carefully crafted poisoning samples into the training data. This attack entails a prohibitive bilevel optimization, wherefore an effective approximated solution is proposed. A theoretical analysis on the needed number of poisoning samples is derived and sheds light on defending against the attack. Experiments on benchmark fairness datasets and state-of-the-art fair representation learning models demonstrate the superiority of our attack. 1 Introduction Machine learning algorithms have been thriving in high-stake applications such as credit risk analysis. Despite their impressive performance, many algorithms suffered from the so-called fairness issue, i.e., they were shown biased against under-represented demographic subgroups such as females in decision making (Barocas & Selbst, 2016; Buolamwini & Gebru, 2018). As remedies, fair machine learning has accumulated a vast literature that proposes various fairness notions and attempts to achieve them (Pedreshi et al., 2008; Dwork et al., 2011; Hardt et al., 2016). For shallow models such as logistic regression and support vector machine, fairness notions are often defined upon scalar model predictions and have been well-studied, see Kamishima et al. (2011); Zafar et al. (2017) and reference therein for instances. We refer to this family of work as classical fair machine learning. Recently, fair representation learning (FRL) with deep neural networks (DNNs) has attracted great attention (Xie et al., 2017; Madras et al., 2018; Creager et al., 2019; Sarhan et al., 2020). FRL learns high-dimensional representations for downstream tasks that contain minimal information of sensitive features (i.e., the memberships of demographic subgroups). These information-based fairness notions equip FRL with higher transferability than classical methods (Creager et al., 2019). Despite the success of fair machine learning methods, not much is known about their vulnerability under data poisoning attacks until very recent studies (Chang et al., 2020; Solans et al., 2021; Mehrabi et al., 2021). Data poisoning attacks aim to maliciously control a model’s behaviour to achieve some attack goal by injecting poisoning samples to its training data and are widely used to benchmark the robustness of machine learning models in adversarial scenarios (Bard & Falk, 1982; Biggio et al., 2012). Recently researchers successfully attacked classical fair machine learning methods such as fair logistic regression (Mehrabi et al., 2021) and exacerbated bias in model predictions, thereby hurting fairness. But it is still an open question whether FRL suffers from a similar threat$^1$. $^1$See App A for a more detailed discussion on attacking fair representations versus downstream classifiers. Notwithstanding, devising a poisoning attack to degrade fairness of representations proves to be a non-trivial task. The difficulty is from two aspects. First, high-dimensional representations are more complicated to evaluate fairness than scalar predictions in classical fair machine learning. This makes existing attack goals for fairness degradation against the latter fall short to apply. Secondly, fair representations implicitly depend on the training data, and non-convex DNNs make this dependency hard to control by previous optimization- or heuristic-based attacks on classical fair machine learning. Optimization-based attacks (Solans et al., 2021) need the victim model to be simple enough (e.g., convex) to analyze. Heuristics such as label flipping (Mehrabi et al., 2021) do not directly optimize the attack goal, thereby often requiring great effort to design good manipulations to success. We propose the first data poisoning attack that directly targets on fairness of high-dimensional representations as shown in Figure 1. Our attack is optimization-based with a new attack goal. Specifically, following the common principle behind FRL that ideal fair representations should contain no information of sensitive features (Moyer et al., 2018; Zhao et al., 2019; Creager et al., 2019), we devise our attack goal to maximize the mutual information (MI) between the learned representations and sensitive features. The attack goal entails a bilevel optimization, whose challenges lie in two-folds. Firstly, MI does not admit analytic form, making its maximization challenging. Instead, a variational lower bound of MI (Barber & Agakov, 2003) inspires us to use the negative loss of the optimal classifier that predicts sensitive features from representations as a proxy. To avoid the complexity of training such a classifier in the inner loop of the optimization, we further connect the classification loss with a principled notion of data separability. FLD scores give us an analytic measure of such and simplify the optimization substantially. Notably, FLD score as a valid data separability measure does not rely on the mixture-of-Gaussian assumption (Fisher, 1936), and our empirical results also show that this assumption is not essential for our purpose in practice. To our best knowledge, we propose the first attack goal based on MI explicitly towards fair machine learning that is highly principled and flexible. We also analyze its connection to demographic parity, one of the most popular fairness notions. This is one of the most significant contributions of our work. Secondly, representations are learned by DNNs, whose dependency on poisoning samples is impossible to track exactly. Consequently, one cannot identify training on what poisoning samples the victim will produce the most fairness-violating representations. We solve this problem approximately by matching the upper- and lower-level gradients following Geiping et al. (2020): The gradient of the victim from FLD score gives the desired update direction; when it matches the gradients of the victim on poisoning samples, training the victim on poisoning samples will solve the attack goal approximately. To improve stealthiness of the attack on fairness degradation in our scenarios, we design better constraints on how much poisoning samples can deviate from clean samples. In specific, we define a more proper valid range based on the data type, and an elastic-net penalty (Chen et al., 2018) to encourage such deviation to be sparse. Meanwhile, we derive the first theoretical minimal number of poisoning samples required by gradient-matching based attacks (Geiping et al., 2020) to succeed under regular conditions. This bound is crucial for practical attacks, as using more poisoning samples will increase the chance of being detected (Geiping et al., 2020; Koh et al., 2022). This theoretical analysis is another contribution of this paper and can be of independent interest to users of all gradient-matching based attacks. Extensive experimental results in Section 4 demonstrate high effectiveness of our attacks on four representative FRL methods using as few as 5% of training data for poisoning. In the remaining part of this paper, we review related works in Section 5, and conclude the work in Section 6. 2 PROPOSED METHOD We propose the first white-box clean-label data poisoning attack on deep learning-based fair representation learning (FRL) to degrade fairness. We follow previous attacks on classical fair machine learning (Chang et al., 2020; Mehrabi et al., 2021) and suppose a worst-case threat model, which has full knowledge and control of the victim model, as well as the access to part of the training samples. 2.1 PRELIMINARIES We denote a dataset consisting of $ N $ datapoints by $ D = \{x_i, a_i, y_i\}_{i=1}^N $, where $ x_i \in \mathbb{R}^M $ represents a multivariate nonsensitive feature, $ a_i \in \{0, 1\} $ is a binary sensitive feature, and $ y_i $ is a binary class label. As adopted in literature (e.g., Moyer et al. (2018); Reddy et al. (2021)), we assume $ x $ has continuous values that can be fed into neural networks. A FRL model parameterized by $ \theta $ has an encoder $ h $ to learn representation from the nonsensitive feature by $ z(\theta) = h(x; \theta) $ and is trained on $ D $ to minimize some fairness-aware loss $ L(D; \theta) $. A data poisoning attack aims to maliciously control the training of this victim model by perturbing a few training samples to minimize some attack loss $ U(\theta) $. The perturbed training samples are referred to as poisoning samples. Mutual information (MI)-based fairness aims to minimize $ I(a, z) $ between sensitive feature $ a $ and representation $ z $. It makes fair representations highly transferable on a wide range of downstream classification tasks and has become the de facto principle in FRL (Moyer et al., 2018; Creager et al., 2019; Zhao et al., 2019). Formally, the data processing inequality (Cover, 1999) states that $ I(a, z) \geq I(a, g(z)) \geq 0 $ holds for any classifier $ g $ acting on $ z $. In addition, $ I(a, g(z)) = 0 $ is equivalent to demographic parity (DP, Zemel et al. (2013)), one of the most popular group fairness notions. At a colloquial level, if representations from different demographic groups are similar to each other, then any classifier acting on the representations will be agnostics to the sensitive feature and fair thereof. This condition remains valid without access to $ y $ when DP cannot be evaluated. 2.2 POISONING FAIR REPRESENTATIONS FORMULATION Motivated by the importance of MI-based fairness in FRL, we attack the fairness on some target data $ D_{ta} $ by maximizing $ I(a, z) $. This involves a bilevel optimization problem. Given a victim $ \theta $ and its lower-level loss $ L(D; \theta) $, the training data $ D = D_{po} \cup D_{cl} $ consists of $ P $ poisoning samples $ D_{po} $ and clean samples $ D_{cl} $, the attacker wants to minimize $ -I(a, z) $ over target data $ D_{ta} $ by learning perturbations $ \Delta = \{\delta_p\}_{p=1}^P $ to add on $ D_{po} $ through solving \[ \min_{\Delta \in C} -I(a, z(\theta^(\Delta))), \quad \text{s.t.} \quad \theta^(\Delta) = \arg\min_\theta L(D_{po}(\Delta) \cup D_{cl}; \theta). \] (1) Our clean-label attack leaves original label $ y $ unpoisoned and only perturbs nonsensitive feature $ x $, i.e., $ D_{po}(\Delta) = \{x_p + \delta_p, a_p, y_p\}_{p=1}^P $, under constraint $ C $ which will be detailed shortly. Connection to attacking group fairness. Our attack is principled and jeopardizes DP. We use well-defined metric variation of information (VI, Kraskov et al. (2005)). For $ y $ and $ a $, their VI distance is $ VI(y, a) = H(y) + H(a) - 2I(y, a) $ where $ H(\cdot) $ is the entropy. Applying triangle inequality to $ g(z), y $, and $ a $ gives us $ I(g(z), a) \geq I(g(z), y) + I(a, y) - H(y) = I(g(z), y) - H(y \| a) $. By maximizing $ I(z, a) $ that upper bounds $ I(g(z), a) $, a successful attack diminishes the guarantee for the MI-based fairness. When $ H(y \| a) < I(g(z), y) $, fitting $ g $ to predict $ y $ from $ z $ that maximizes $ I(g(z), y) $ will force $ I(g(z), a) $ to increase, thereby exacerbating the DP violation. Notably, $ H(y \| a) $ depicts how dependent $ y $ and $ a $ are and is expected small when a fairness issue exists (Hardt et al., 2016). We provide empirical success of attacking DP with our attack goal in Appendix E.3. Unfortunately, Eq. (1) is intractable with difficulty lying in finding the most fairness-violating representations and the method for acquiring them. Mathematically, the first problem involves $ I(a, z) $ which lacks an analytic expression. This makes its computation non-trivial, let alone its maximization. The second entails a feasible set in the lower-level optimization that is NP-hard to identify due to the non-convexity of deep neural networks. We solve Eq. (1) approximately as follows. 2.3 UPPER-LEVEL APPROXIMATION: FISHER’S LINEAR DISCRIMINANT (FLD) The lack of analytic form for $ I(a, z) $ necessitates some approximations. Our first step is to lower bound MI by a negative binary-cross-entropy (BCE) loss which is easier to estimate. For any classi- fier $g$ that predicts $a$ from $z$, let $q(a \mid z)$ be the distribution learned by the optimal $g^$, we have $$I(a, z) = \mathbb{E}_{p(a,z)} \left[ \log \frac{p(a\|z)q(a\|z)}{p(a)q(a\|z)} \right] \overset{(a)}{\geq} \mathbb{E}_p \left[ \log \frac{q(a\|z)}{p(a)} \right] = \mathbb{E}_p [\log q(a\|z)] + \mathbb{E}_p [-\log p(a)],$$ where the inequality $(a)$ holds from omitting some non-negative KL terms and is tight when $g^$ recovers the true distribution $p(a \mid z)$ as shown in Barber & Agakov (2003). On the other hand, since $\mathbb{E}_p [-\log p(a)] = H(a) \geq 0$, the first term, negative BCE loss of $g^$, is a lower bound for $I(a, z)$, and is a measure of how fair the representations are (Feng et al., 2019; Gupta et al., 2021). We dub the BCE loss of $g^$ the optimal BCE loss. However, substituting the MI maximization from Eq. (1) with minimizing the optimal BCE loss does not make it solvable: $g^$ depends on $z$’s for $D_{ta}$ and how to update them to minimize the BCE loss of $g^$ is unknown. This requires differentiate through the whole optimization procedure. To walk around this challenge, we note that the optimal BCE loss of $g^$ measures how difficult to separate $z$ for $D_{ta}$ with $a = 0$ or $1$. While this difficulty is hard to tackle directly, it can be approximated by the data separability: If the two classes of data are more separable, one can expect the classification simpler and the optimal BCE loss lower. Motivated by this, we instead maximize Fisher’s linear discriminant (FLD) score, a closed-formed data separability measure. Specifically, suppose the two classes of representations have mean $\mu^0, \mu^1$ and covariance $S^0, S^1$ respectively. FLD maps them to a 1-dimensional space via linear transformation $v$ which induces separation $s_v = (\mu^0 - \mu^1)^T(S^0 + S^1)^{-1}(\mu^0 - \mu^1)$ when $v \propto (S^0 + S^1)^{-1}(\mu^0 - \mu^1)$. This equation allows us to compute its gradient with respect to $z$ for $D_{ta}$, which gives the direction to update these representations in order to make it less fair. For stability we regularize $s$ by $$s = (\mu^0 - \mu^1)^T(S^0 + S^1 + cI)^{-1}(\mu^0 - \mu^1),$$ and resort to solving the following bilevel optimization: $$\min_{\Delta \in C} -s(\theta^(\Delta)), \quad \text{s.t. } \theta^(\Delta) = \arg\min_\theta L(D_{po}(\Delta) \cup D_{cl}; \theta).$$ In Appendix B we extend our attack to multi-class sensitive feature scenarios. Remark 2.1. Maximizing $I(z, a)$ is a general framework to poison FRL and admits other proxies such as sliced mutual information (Chen et al., 2022), kernel canonical correlation analysis (Akaho, 2007), and non-parametric dependence measures (Székely et al., 2007). In this work, we use FLD because of its conceptual simplicity, interpretability, and good empirical performance. As one may recognize, when $p(z \mid a = 1)$ and $p(z \mid a = 0)$ are Gaussian with equal variance, FLD is optimal (Hamsici & Martinez, 2008) whose BCE loss attains the tight lower bound of $I(z, a)$ up to constant $H(a)$. In this case, our method provably optimize the lower bound of $I(z, a)$ whereas other proxies may not due to the lack of direct connections to mutual information. While the Gaussianity may not hold in general, FLD score as a measure of data separability is still valid (Fisher, 1936), and we verify its efficacy for our goal in Appendix E.2, where we show that FLD score is highly informative for the empirical minimal BCE loss of a logistic regression. 2.4 LOWER-LEVEL APPROXIMATION: ELASTIC-NET GRADMATCH (ENG) Bilevel optimization in Eq. (3) enjoys a tractable upper-level loss but its lower-level optimization remains challenging due to the use of deep neural networks in FRL and needs further approximation. To this end, we fix parameter $\theta$ in a pre-trained victim model and treat its lower-level gradient on poisoning samples $\nabla_\theta L(D_{po}(\Delta); \theta)$ as a function of $\Delta$. An attack is launched by aligning $\nabla_\theta L(D_{po}(\Delta); \theta)$ and $-\nabla_\theta s(\theta)$ by maximizing their cosine similarity $$B(\theta, \Delta) = -\langle \nabla_\theta s(\theta), \nabla_\theta L(D_{po}(\Delta); \theta) \rangle / \\|\nabla_\theta s(\theta)\\| \\|\nabla_\theta L(D_{po}(\Delta); \theta)\\|,$$ with respect to $\Delta$. When the two directions are matched, gradient descents on the lower-level loss $L$ from poisoning samples will decrease the upper-level loss $-s$ as well. The concept of gradient matching in poisoning attacks was initially introduced in GradMatch by Geiping et al. (2020) for image classification. To enhance the stealthiness of such attacks in fairness degradation scenarios, we impose two new constraints on the learning perturbations. First, a hard constraint $ C $ mandates that poisoning samples must reside within the clean data domain, i.e., \[ \min_{x_c \in D_c} x_{cm} \leq \delta_{pm} + x_{pm} \leq \max_{x_c \in D_c} x_{cm} \] for all feature dimensions $(1 \leq m \leq M)$ and poisoning samples $(1 \leq p \leq P)$, this allows using dimension-specific ranges. Second, the soft constraint employs the elastic-net penalty (Zou & Hastie, 2005; Chen et al., 2018) to promote sparsity in each $\delta_p$ (i.e., only a few dimensions are perturbed). Combining these constraints, we formulate below optimization problem aimed at poisoning fair representations: \[ \min_{\Delta \in C} -B(\theta, \Delta) + \sum_{p=1}^{P} (\lambda_1 \\|\delta_p\\|_1 + \lambda_2 \\|\delta_p\\|_2^2). \] In execution, we rescale the three terms before tuning $\lambda$'s to avoid bearing with the magnitude difference. Non-differentiable $L_1$ norms are tackled with the iterative shrinkage-thresholding algorithm (ISTA, Beck & Teboulle (2009)) $\delta_k^{p+1} = \text{Proj}_C \left( S_{\lambda_1} [\delta_k^p + \alpha \nabla B(\theta, \Delta) - \lambda_2 \\|\delta_p\\|_2^2] \right)$, where $S_{\lambda_1}$ is the element-wise projected shrinkage-thresholding function and $\text{Proj}_C$ is the projection onto $C$. We refer to our attack by Elastic-Net GradMatch (ENG) and summarize it in Algorithm 1. Computation complexity.* Denote the number and dimension of poisoning samples by $P$ and $M$, iteration numbers of running the attack by $T$, dimension of $\theta$ by $D$. The computation complexity of ENG and GradMatch (Geiping et al., 2020) are $O(TM + TDPM)$ and $O(TDPM)$, respectively. The additional computation cost $O(TM)$ due to the one-step ISTA in ENG is marginal. ### 3 ANALYSIS ON MINIMAL NUMBER OF POISONING SAMPLES It is non-trivial to minimize the number of poisoning samples that deteriorate the victim model’s performance. An insufficient amount may not impact lower-level optimization, while a large amount makes it easy to detect the attack, leading to a direct failure (Huang et al., 2020; Koh et al., 2022). Our analysis is built upon the convergence of upper-level loss $U(\theta) \equiv -s(\theta)$. Our conclusion relies on the following assumptions. Assumption 3.1 (smooth upper- and lower-level losses). There exists $C > 0$ such that upper-level loss $U(\theta)$ and lower-level loss $L(\theta)$ are $C$-smooth, i.e., their gradients are $C$-Lipschitz continuous. Assumption 3.2 (attack well-trained victims only). Before attack, the victim is well-trained so its gradient on each clean sample performs like a random noise with mean zero and finite norm $\sigma$. Assumption 3.3 (well-matched gradients). After being attacked, gradient $\nabla_\theta L(\theta)$ evaluated on any poisoning sample is an unbiased estimator of the gradients $\nabla_\theta U(\theta)$ with bounded error norm, i.e., for any poisoning sample $p$, $\nabla L_p(\theta) = \nabla_\theta U(\theta) + \epsilon_p$ where $E[\epsilon_p] = 0$ and $E[\\|\epsilon_p\\|] \leq \sigma$. Assumption 3.1 is a fairly weak condition that has been widely used (Colson et al., 2007; Mei & Zhu, 2015; Sinha et al., 2017), the rest two are introduced by us and are well-suited to the context of our proposed framework. Assumption 3.2 is valid because $\theta$ undergoes thorough training with the clean samples. Assumption 3.3 presumes that given constant gradient $\nabla_\theta U(\theta)$, we can construct an unbiased estimator with $P$ poisoning samples $E[\frac{1}{P} \sum_{p=1}^{P} \nabla_\theta L_p(\theta)] = \nabla_\theta U(\theta)$. With these assumptions, we obtain the following theorem. Theorem 3.4. Suppose that Assumption 3.1, 3.2, and 3.3 hold. Let $P$ and $N$ be the number of poisoning and total training samples, respectively. Set the learning rate to $\alpha$ and the batch size to $n$. Then, the ratio of poisoning data $P/N$ should satisfy \[ \frac{P}{N} \geq c + \frac{\alpha C \sigma^2}{2n \\|\nabla_\theta U(\theta)\\|^2} + \frac{\alpha C}{2}, \] such that the upper-level loss $U(\theta)$ is asymptotic to an optimal model. Here $c$ is a small constant (e.g., $10^{-4}$) for sufficient descent, and $\theta$ is a well-trained model on the clean data. Deferring the proof to Appendix C, we summarize the underlying idea. We assume the pretrained victim converged with respect to $\nabla_\theta L(\theta)$. Besides, after applying ENG, a poisoning sample can (in expectation) induce a gradient equal to $\nabla_\theta U(\theta)$ so training the victim model on it will optimize the upper-level loss $U(\theta)$. However, clean samples may obfuscate this goal as their lower-level gradients do not depend on the upper-level loss. To counteract with this effect, a minimal portion of poisoning... samples are needed to dominate the training. In practice, learning rate $\alpha$ is often small compared with $C$, batch size $n$ is large, and $\\|\nabla_\theta U(\theta)\\|$ is much greater than 0. Therefore, the minimal portion bound is expected to be smaller than 1. Practical Significance. Theorem 3.4 sheds light on the difficulty of ENG and other GradMatch-based attacks, whereon a defending strategy can be built. If batch size $n$ is large, the attack is simpler and needs fewer poisoning samples. So reducing $n$ should help defense and we evaluate its performance in Appendix E.7. In addition, term $\sigma^2$ is affected by the lower-level gradients, whose increase will require more poisoning samples. This helps explain why adding noises to gradients can defend against GradMatch as verified in Geiping et al. (2020). ### 4 EXPERIMENTS We evaluate the proposed attack on four FRL models trained on two fairness benchmark datasets and show its effectiveness through extensive experiments. Ablation studies and practical insights are also given to help understand how the attack succeeds in poisoning victims. #### 4.1 SETUP Attack Goals. We consider three variants of Eq. (2) to maximize: (a) FLD follows Eq. (2) with $c = 10^{-4}$ to stabilize the covariance estimation. (b) sFLD takes the same form as FLD but does not back-propagate through covariance terms when computing $-\nabla_\theta s(\theta)$. (c) EUC replaces the covariance terms with an identity matrix$^2$. Attacks. An attack to maximize score $X$ using ENG is referred to as ENG-X, for instance, ENG-FLD is to maximize FLD. Conceptually, ENG-EUC is suitable for small $D_{ta}$ as it omits covariance matrix, which can be unstable to estimate. ENG-FLD should be favored when $D_{ta}$ is large as it is based on the exact FLD score $s$ and should measure separability more accurately, which is expected to benefit solving Eq. (5) as well. ENG-sFLD strikes a balance in between by using only part of covariance information. FRL Victims. We select four representative FRL models as our victims. CFAIR and CFAIR-EO (Zhao et al., 2019) are based on adversarial learning and seek to achieve different fairness notions. Non-adversarial learning based ICVAE-S and ICVAE-US (Moyer et al., 2018) differ in whether $y$ is accessible. We follow the official codes to implement the victims as detailed in Appendix D. Datasets. We train victims on two benchmark datasets from UCI repository that are extensively studied in fair machine learning, which are pre-processed$^3$ following Zhao et al. (2019); Reddy et al. (2021). Adult (Kohavi, 1996) contains 48,842 samples of US census data with 112 features and the objective is to predict whether an individual’s yearly income is greater than $50K dollars or not. The sensitive feature is gender. German (Dua & Graff, 2017) consists of 1,000 samples of personal financial data with 62 features and the objective is to predict whether or not a client has a good credit score. The sensitive feature is binarized age as in Moyer et al. (2018) and we adjust the threshold to increase its correlation with label$^4$. In Appendix E.6 we study multi-class sensitive feature race. On both datasets, we leave 20% of total samples out as $D_{ta}$. More results on COMPAS (Dieterich et al., 2016) and Drug Consumption Datasets (Dua & Graff, 2017) are presented in Appendix E.8. --- $^2$This score is equivalent to the squared Euclidean distance between $\mu^0$ and $\mu^1$ and gets its name thereof. $^3$In Appendix E.1 we discuss the practicability of perturbing pre-processed versus raw data. $^4$We increase the lower threshold of defining advantaged group from 25 to 30. --- Algorithm 1 Craft Poisoning Samples with ENG Attack 1. Input: clean data $D_{cl}$, poisoning data $D_{po}$, target data $D_{ta}$; victim $\theta$ and its lower-level loss $L$; number of pre-training epochs $E$ and attack iterations $T$. 2. Fix $D_{po}$ unperturbed. Pretrain victim on $D_{po} \cup D_{cl}$ for $E$ epochs and obtain $\theta^E$. 3. Compute upper-level gradient $\nabla_\theta U(D_{ta}; \theta^E)$. 4. Randomly initialize $\Delta^0 = \{\delta_p, p = 1, \ldots, P\}$ in $C$. 5. for $t = 1, \ldots, T$ do 6. Compute lower-level gradient $\nabla_\theta L(D_{po}(\Delta^t); \theta^E)$ as a function of $\Delta^t$. 7. Compute ENG loss in (5) and update $\Delta^t$ with ISTA. 6. end for 7. return $D_{po}$ when using $\Delta^T$. Evaluation. We treat decrease in the BCE loss of a logistic regression predicting $a$ from $z$ as a measure of increase in $f(z, a)$. To verify how group fairness and representation utility can be affected, we present the exacerbation of DP violation and accuracy of predicting $y$ from $z$ in Appendix E.3 and Appendix E.4 respectively. Representations are extracted after training the victims on poisoned Adult and German dataset for 20 and 50 more epochs respectively considering their size difference. Baselines. We compare ENG-based attacks with four variants of anchor attack (AA), a recent heuristic generic poisoning attack on classical fair machine learning (Mehrabi et al., 2021). RAA-y and RAA-a randomly picks one training sample from the subgroup with $(y = 1, a = 0)$ and one with $(y = 0, a = 1)$ after each epoch, then makes copies of the two chosen samples with flipped $y$ or $a$ respectively. NRAA-y and NRAA-a replaces the random selection in RAA counterparts with picking from each subgroup the training sample that has the most neighbors within a pre-specified radius and has not been selected yet. (N)RAA-y were proposed in Mehrabi et al. (2021), and we implement (N)RAA-a following the same logic. Note that (N)RAA-y are not clean-label, (N)RAA-a are but they directly modify the sensitive feature. In contrast, our attacks only modify the nonsensitive feature. Moreover, all baselines are allowed to use any training sample to poison, while ours can only perturb a given randomly selected set of poisoning samples. These differences put our proposed attacks in an unfavorable situation under direct comparison. Nevertheless, ours can still outperform the four baselines by a large margin. 4.2 Comparison Between ENG and (N)RAA. We compare three ENG-based attacks with penalty coefficients $\lambda_1 = 0.0025$, $\lambda_2 = 0.005$ (Eq. (5)) against four AA attacks under different settings where 5% to 15% of training data are used for poisoning. Performance of an attack is measured by the decrease of BCE loss, higher is better, and corresponding DP violations are reported in Appendix E.3. Figure 2 shows results averaged over 5 replications. Three ENG-based attacks achieved notable performance (on both BCE loss and DP violations) in various settings. In contrast, AA encountered severe failures, for instance, when attacking CFAIR trained on German dataset, only RAA-a succeeded with all the three budgets. Such failures cannot be fully attributed to the budgets: (N)RAA-y succeeded with budget 10% but failed with budget 15%. (N)RAA-a occasionally achieved the best performance because of much stronger capacities. Nonetheless, the proposed ENG-based attacks beat AA baselines in terms of better and more reliable performance by a large margin in most case. When comparing the three proposed attacks with each other, their performance difference matched our previous analysis, e.g., ENG-FLD gave the best result on larger Adult dataset. These results clearly establish the efficacy of our attack. ![Figure 2](image-url) Figure 2: ENG-based attacks reduce BCE loss more than AA baselines with less portion of poisoning samples. Results are averaged over 5 independent replications and bands show standard errors. 4.3 Performance of Elastic-Net Penalty Next, we provide a sensitivity analysis of ENG-based attacks against the choices of $\lambda_1$ and $\lambda_2$. We set $\lambda_2 = 2\lambda_1$ and vary $\lambda_1$ following Chen et al. (2018). Results are again averaged over 5 replications. Figure 3a exhibits how elastic-net penalty affects the performance of ENG-FLD under different poisoning budgets where victims are trained on Adult dataset. More results are deferred to Appendix E.5 due to page length but conclusions here hold in general. All three attacks are relatively robust to small and intermediate level of elastic-net penalty. Moreover, improvement from applying the elastic-net penalty is observed (e.g., column 1 in Fig 3a). This implies that the penalty can actually help stabilize the optimization. We further study how elastic-net penalty regularizes the learned perturbations by computing $L_1$ and $L_2$ norms. We only present $L_1$ norms of perturbations using ENG-EUC attack on Adult dataset as an illustration and defer others to Appendix E.5 after observing similar trends. According to Figure 3b, $L_1$ norms were significantly shrank by elastic-net penalty with mild degradation on the attack performance. For instance, elastic-net penalty with $\lambda_1 = 0.0025$ effectively reduced the $L_1$ norm of perturbations by a third without hurting the performance of attacking CFAIR. When a more stealthy attack is wanted, $\lambda_1 = 0.01$ was able to launch a satisfactory attack with only one half budget of perturbation norm. These results clearly show the efficacy of our proposed ENG-based attacks. Given these merits of elastic-net penalty, one may ask if it can be used in other attacks such as AA. In Appendix E.5 we discuss the difficulty of doing so and highlight its affinity to our attack by nature. ![Figure 3](image) (a) Decrease of BCE loss is insensitive to under small and intermediate level of elastic-net penalty. (b) $L_1$ norm of learned perturbations is effectively restricted by elastic-net penalty. Figure 3: Decrease of BCE loss and $L_1$ norm of perturbations learned by ENG-FLD attack. Victims are trained on Adult dataset and results are averaged over 5 replications. 4.4 Robust Features of Victims ENG-based attacks can do feature selections, and we consider unselected features as robust ones for the victim, in the sense that they did not help in attacking. We end this section with a case study on this aspect. Table 1 shows the percentage of selected features and corresponding robust features when attacking ICVAE-S and ICVAE-US trained on Adult dataset with ENG-FLD. From the table, elastic-net penalty successfully reduced the perturbed features by 20%, and robust features for ICVAE-US and ICVAE-S are largely overlapped. These results helped understand how ENG-FLD was launched by walking around these robust features when attacking the victims. Note that this robust features identification is not applicable for AA-based attacks. 5 Related Work Fair representation learning (FRL) is a family of algorithms to learn fair representations from nonsensitive features such that any downstream tasks (e.g., classification) acting on these representations will be fair. Towards this goal, different approaches to removing sensitive information from learned representations have been proposed. To name a few, Zemel et al. (2013) set representations to be multinomial and used fairness violations based on the representations as penalties, and Madras et al. (2018) derived different bounds for these violations used for adversarial regularization. Mutual information and other measures between distributions have also been used as penalties to encourage independence between the representation and sensitive features, either in an adversarial (Xie et al., 2017; Creager et al., 2019) or non-adversarial way (Louizos et al., 2015; Moyer et al., 2018; Sarhan et al., 2020; Wang et al., 2021). Recently, Zhao et al. (2019) proposed to learn different encoders on different label groups with theoretical guarantees and achieved state-of-the-art performance in an adversarial manner Reddy et al. (2021). Moyer et al. (2018), on the other hand, provided a non-adversarial based solution and also got promising results. We select representative methods from adversarial and non-adversarial regimes to test our attack. Data poisoning attack aims to achieve some attack goal with a victim model by controlling its training via injecting poisoned samples. Early works showed that simple heuristics such as label flipping (Barreno et al., 2010; Paudice et al., 2019) can succeed in attack. However, these poisoning samples often look unnatural and are easy to detect (Papernot & McDaniel, 2018; Paudice et al., 2019). Consequently, clean-label attacks that only modify the poisoning samples’ features but not labels are preferred (Shafahi et al., 2018). Another drawback of heuristic attacks is the lack of performance guarantee as they do not directly solve the attack goals. In practice they may perform less effective. Bilevel optimization is widely used for data poisoning attacks (Bard & Falk, 1982; Biggio et al., 2012; Geiping et al., 2020; Jagielski et al., 2021; Koh et al., 2022). For convex victims such as logistic regression and support vector machine, the lower-level optimization is characterized by the KKT condition. This reduces the bilevel optimization to a constrained optimization that can be solved exactly (Mei & Zhu, 2015). For other victims, unfortunately, when their optimal solutions are NP-hard to identify, so are the bilevel optimization problems (Colson et al., 2007; Sinha et al., 2017) and inexact solutions are needed. When the second-order derivatives of the lower-level loss is cheap, using influence function to identify influential samples for the victim training and poisoning them can produce strong attacks (Koh & Liang, 2017). These attacks have been successfully applied to classical fair machine learning (Chang et al., 2020; Solans et al., 2021; Mehrabi et al., 2021), but the non-convexity of neural networks and expensive influence function computations make them unsuitable for poisoning FRL. Approximate solutions for attacking deep learning models have been proposed recently. For instance, inspired by model-agnostic meta-learning (MAML, Finn et al. (2017)), MetaPoison (Huang et al., 2020) back-propagated through a few unrolled gradient descent steps to capture dependency between the upper- and lower-level optimizations. GradMatch (Geiping et al., 2020) matched the gradient of upper- and lower-level losses and achieved state-of-the-art performance. However, it is unclear how to apply them to poison FRL. In this work, we propose the first work towards this goal and reduce it to an approximate optimization that can be handled. ### Table 1: Percentage of selected features and top robust features from Adult dataset, $P/N$ denotes the portion of poisoning samples, attacker is ENG-FLD. \| $P/N$ \| $\lambda_1$ ($\lambda_2 = 2\lambda_1$) \| Top Robust Features \| \|-------\|--------------------------------------\|---------------------\| \| \| 0.0025 0.005 0.01 \| \| \| ICARE-S \| 0.05 0.99 0.96 0.89 0.73 \| workclass, marital-status, occupation, native-country \| \| ICARE-US \| 0.05 0.95 0.89 0.84 0.79 \| native-country, occupation, workclass, education \| ### Conclusion and Future Works We develop the first data poisoning attack against FRL methods. Driven by MI-based fairness in FRL, we propose a new MI maximization attack goal and reveal its connection to existing fairness notion such as demographic parity. We derive an effective approximate solution to achieve this attack goal. Our attack outperforms baselines by a large margin and raises an alert of the vulnerability of existing FRL methods. We also theoretically analyze the difficulty of launching such an attack and establish an early success of principled defense. Motivated by promising results on tabular data, which is the primary focus of many FRL methods (Moyer et al., 2018; Zhao et al., 2019), we plan to extend our attack to fair machine learning on large-scale image and text datasets that also relies on deep neural networks, and delve into attacking these methods in the future. In addition, Jagielski et al. (2021) showed that an attack can be much more effective towards certain subpopulations and impossible to defend against, and we plan to explore this for further improvement of our attack. ACKNOWLEDGEMENT This work is supported in part by the US National Science Foundation under grant NSF IIS-2226108. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. REFERENCES Shotaro Akaho. A kernel method for canonical correlation analysis, 2007. David Barber and Felix Agakov. The im algorithm: a variational approach to information maximization. Advances in neural information processing systems, 16(320):201, 2003. Jonathan F Bard and James E Falk. An explicit solution to the multi-level programming problem. Computers & Operations Research, 9(1):77–100, 1982. Solon Barocas and Andrew D Selbst. Big data’s disparate impact. Calif. L. Rev., 104:671, 2016. Marco Barreno, Blaine Nelson, Anthony D Joseph, and J Doug Tygar. The security of machine learning. Machine Learning, 81:121–148, 2010. Amir Beck and Marc Teboulle. A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM journal on imaging sciences, 2(1):183–202, 2009. 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sW95puhphh	While the paper addresses the challenges of policy discoordination and privacy concerns, how does the proposed method handle issues related to credit assignment, especially when agents have conflicting goals or when their contributions to the global reward are imbalanced?	DECENTRALIZED MULTI-AGENT REINFORCEMENT LEARNING VIA ANTICIPATION SHARING Anonymous authors Paper under double-blind review ABSTRACT Centralized multi-agent reinforcement learning requires global policy access and coordination, often infeasible in decentralized applications. A challenge in decentralized MARL with individual rewards is misaligned local objectives without global coordination. Existing methods that share rewards, values or full policies have high overheads and coupled learning. We introduce a novel decentralized MARL method called Anticipation Sharing that induces coordination by sharing limited policy information. Agents update anticipations of peer action distributions, share these with neighbors, and identify deviations between individual and collective objectives. By exchanging anticipations, agents align behaviors without prohibitive overheads of full policy sharing. Our simulations demonstrate Anticipation Sharing enables decentralized cooperative learning using only local interactions. 1 INTRODUCTION Multi-agent reinforcement learning (MARL) enables collaborative decision-making in environments with distributed agents. It has diverse real-world applications including autonomous vehicles, robotics, and communications systems. Centralized MARL requires global information access and a central coordinator, often infeasible in decentralized settings. Without access to team rewards or objectives, decentralized agents face social dilemmas - prioritizing individual rewards can produce suboptimal collective outcomes. The Prisoner’s Dilemma exemplifies this tension. When agents act purely out of self-interest, they achieve lower returns compared to cooperating for the common good [Debreu, 1954]. Yet determining optimal collaborative strategies is challenging when only seeing a local viewpoint. Decentralized MARL tackles these cooperation challenges in distributed environments with individual rewards. By developing algorithms that align decentralized policies without global knowledge, agents can learn to optimize collective returns through only local interactions. This addresses real-world coordination problems where central controllers are infeasible. Several MARL methodologies have recently been proposed to enable decentralized learning, but they attribute a team reward to each agent, which is infeasible when an agent is privy only to its individual reward [Sun et al., 2022; Lauer & Riedmiller, 2000; Boutilier, 1996; Jiang & Lu, 2022]. To enhance cooperation among agents while keeping private individual rewards, several methods propose the exchange of information. For instance, some strategies involve sharing rewards to guide agents towards a collective optimum [Chu et al., 2020b; Yi et al., 2022; Chu et al., 2020a]. Others suggest sharing value function model parameters or values of the value functions through the aggregation from neighboring agents to achieve similar ends [Zhang et al., 2018a,b; 2020; Suttle et al., 2020; Du et al., 2022]. In these approaches, agents calculate a global value based on shared rewards or values, and subsequently, they adjust their policies to maximize this aggregated value. Some studies have explored consensus strategies focusing on policy rather than value [Zhang & Zavlanos, 2019; Stankovic et al., 2022a,b]. In real-world applications, the issue of privacy, particularly concerning rewards and values, becomes a significant hurdle. Agents often prioritize keeping this information confidential, posing a challenge to the practicality of methods that require such sharing. Additionally, sharing model parameters incurs substantial communication overhead and also privacy concerns, which can also result in the transfer of excessive and non-essential information, thereby slowing the learning process. In this paper, in response to the above challenges, we introduce a novel approach for decentralized cooperative policy learning when agents have individual rewards and no global perspectives. A key advantage of our method is achieving emergent collaboration without sharing sensitive information like actual rewards or model parameters between agents. The core concept we leverage is anticipation sharing (AS). Agents share anticipated action distributions, reflecting their preferences. The anticipations to other agents are solved by each agent to maximize its own return and then sent to corresponding agents for them to include as constraints when maximizing their returns. Such anticipations carry the information of individual returns implicitly. By exchanging these peer anticipations iteratively, agents can estimate their impacts on collective preferences while preserving individual privacy. We establish a theoretical lower bound that quantifies the discrepancy between an agent’s individual returns and the global collective returns. This enables formulating a surrogate objective for each agent aligned with the global goal while dependent only on local information. Our proposed decentralized MARL algorithm has agents optimize this surrogate through a dual-clipped policy update approach. It imposes constraints that penalize deviations between an agent’s policy and peer anticipated policies. This drives agents to converge not just to optimal local policies, but policies contributing to coordination. The iterative anticipation sharing process is central to enabling this decentralized collaborative learning. In essence, our method induces emergent cooperative behaviors without exposing sensitive individual rewards or models through a decentralized learning framework. Our empirical investigations reinforce the validity of the AS framework, demonstrating its competitive performance in specific tasks compared to traditional methods. This establishes AS not only as a theoretically sound but also practically effective avenue for harmonizing individual and collective objectives in decentralized cooperation. 2 RELATED WORK Centralised learning. Centralized learning in MARL typically involves a central unit that processes and coordinates actions across all agents. This approach, facilitating a comprehensive view of the environment, enables agents to optimize policies based on collective goals and shared information. Numerous contemporary MARL studies focus on optimizing multi-agent policies under the assumption of an evenly split shared team reward [Kuba et al., 2022; Su & Lu, 2022; Wu et al., 2021]. These studies often employ a blend of centralized learning and decentralized execution. For instance, some utilize centralized learning during policy development for optimal coordination, followed by decentralized execution allowing agents to act independently [Kuba et al., 2022; Wu et al., 2021]. Conversely, others adopt a decentralized learning approach while maintaining shared parameters across networks, a method that navigates between full centralization and independent agent operation [Sun et al., 2022]. In contrast to these methodologies, our research takes a distinct path by exploring decentralized MARL in environments where each agent operates based on individual rewards, without the reliance on a common team reward. This approach reflects a more realistic scenario in many real-world applications, where agents need to make autonomous decisions based on limited, individual information, and where centralized coordination is either impractical or undesirable due to privacy or scalability concerns. Value sharing. Value sharing methods use shared Q-values or state-values among agents to better align individual and collective goals. Many of these methods utilize consensus techniques to estimate the value of a joint policy and guide individual policy updates accordingly. For instance, a number of networked actor-critic algorithms exist based on value function consensus, wherein agents merge individual value functions towards a global consensus by sharing parameters [Zhang et al., 2018a,b, 2020; Suttle et al., 2020]. For communication efficiency, some algorithms reduce the parameters shared [Lin et al., 2019] while others emphasize sharing function values for global value estimation [Du et al., 2022]. However, these methods have an inherent limitation: agents modify policies individually, using fixed Q-values or state-values, making them less adaptive to immediate policy shifts from peers, which may introduce policy discouragement. In contrast, our approach enables more adaptive decentralized coordination by having agents directly share and respond to peer policy anticipations. Reward sharing. Reward sharing is about receiving feedback from a broader system-wise outcome perspective, ensuring that agents act in the group’s collective best interest. Some works have introduced a spatially discounted reward function (Chu et al., 2020b,a). In these approaches, each agent collaboratively shares rewards within its vicinity. Subsequently, an adjusted reward is derived by amalgamating the rewards of proximate agents, with distance-based discounted weights. Other methods advocate for the dynamic learning of weights integral to reward sharing, which concurrently evolve as agents refine their policies (Yi et al., 2022). In our research, we focus on scenarios where agents know only their individual rewards and are unaware of their peers’ rewards. This mirrors real-world situations where rewards are kept confidential or sharing rewards suffers challenges such as communication delays and errors. Consequently, traditional value or reward sharing methods fall short in these contexts. In contrast, our method induces coordination without requiring reward sharing. Policy sharing. Policy sharing strives to unify agents’ behaviors through an approximate joint policy. However, crafting a global policy for each agent based on its individual reward can lead to suboptimal outcomes. Consensus update methods offer a solution by merging individually learned policies towards an optimal policy. Several studies have employed such a strategy, focusing on a weighted sum of neighboring agents’ policy model parameters (Zhang & Zavlanos, 2019; Stankovic et al., 2022a,b). These methods are particularly useful when sharing individual rewards or value estimates is impractical. Yet, sharing policy model parameters risks added communication overheads and data privacy breaches. Whereas these methods share model parameters directly for policy consensus, we have agents share anticipations of policy outputs, avoiding parameter sharing. Social dilemmas. Social dilemmas highlight the tension between individual pursuits and collective outcomes. In these scenarios, agents aiming for personal gains can lead to compromised group results. For instance, one study has explored self-driven learners in sequential social dilemmas using independent deep Q-learning (Leibo et al., 2017). A prevalent research direction introduces intrinsic rewards to encourage collective-focused policies. For example, moral learners have been introduced with varying intrinsic rewards (Tennant et al., 2023) whilst other approaches have adopted an inequity aversion-based intrinsic reward (Hughes et al., 2018) or rewards accounting for social influences and predicting other agents’ actions (Jaques et al., 2019). Borrowing from economics, certain methods have integrated formal contracting to motivate global collaboration (Christoffersen et al., 2023). While these methods modify foundational rewards, we maintain the original objectives, emphasizing a collaborative, information-sharing strategy to nurture cooperative agents. Teammate modelling Teammate/opponent modeling in MARL often relies on agents having access to, or inferring, information about teammates’ goals, actions, or rewards. This information is then used to improve collective outcomes (Albrecht & Stone, 2018; He et al., 2016). Our approach differs from traditional team modeling. Rather than focusing on predicting teammates’ exact actions or strategies, our method involves each agent calculating and sharing anticipated action distributions that would benefit its own strategy. These anticipations are used by other agents (not the agent itself) to balance their own returns with the return of the agent sending the anticipation. This approach emphasizes anticipations that serve the agent’s own return optimization. Coordination occurs through strategic adaptation based on others’ anticipations that implicitly include information about their returns, rather than accurately modeling their behaviors. This key difference highlights our decentralized decision-making and coordination approach. It contrasts with conventional team modeling in MARL that focuses on modeling teammates’ behaviors directly. 3 BACKGROUND AND PROBLEM STATEMENT In this work, we approach the collaborative, decentralized multi-agent reinforcement learning problem with individual rewards using Networked Multi-agent Markov Decision Processes (Networked MMDPs). Specifically, we consider a Networked MMDP with $N$ agents, which can be represented as a tuple $\langle G, S, \{A^i\}_{i=1}^{N}, P, \{R^i\}_{i=1}^{N}, \gamma \rangle$, where $G = (V, E)$ denotes a communication graph, $S$ denotes a global state space, $A^i$ is the individual action space, $A = \Pi_{i=1}^{N} A^i$ is the joint action space, $P : S \times A \times S \rightarrow [0, 1]$ is the state transition function, $R^i : S \times A \rightarrow \mathbb{R}$ is the individual reward function, and $\gamma$ is a discount factor. Each agent $i$ selects action $a^i \in A^i$ based on its individual policy $\pi^i : S \times A^i \rightarrow [0, 1]$. The joint action of all agents is represented by $a \in A$, and the joint policy across these agents, conditioned on state $s \in S$, is denoted as $\pi(\cdot\|s) = \prod_{i=1}^{N} \pi^i(\cdot\|s)$. The primary objective in this setting is to maximize the cumulative discounted return for all agents, $$\eta(\pi) = \sum_{i=1}^{N} \mathbb{E}_{\tau \sim \pi} \left[ \sum_{t=0}^{\infty} \gamma^t r_t^i \right],$$ where the expectation, $\mathbb{E}_{\tau \sim \pi}[\cdot]$, is computed over trajectories with an initial state distribution $s_0 \sim d^\pi(s)$, action selection $a_t \sim \pi(\cdot\|s_t)$, and state transitions $s_{t+1} \sim P(\cdot\|s_t, a_t)$. The reward for an agent $i$ is $r_t^i = R^i(s, a)$. In our setup, agents must adjust their strategies in situations where rewards might conflict and without access to shared reward information. An individual advantage function is also introduced, $$A_i^\pi(s, a) = Q_i^\pi(s, a) - V_i^\pi(s)$$ which depends on the individual state-value and action-value functions, respectively $$V_i^\pi(s) = \mathbb{E}_{\tau \sim \pi} \left[ \sum_{t=0}^{\infty} \gamma^t r_t^i \| s_0 = s \right], \quad Q_i^\pi(s, a) = \mathbb{E}_{\tau \sim \pi} \left[ \sum_{t=0}^{\infty} \gamma^t r_t^i \| s_0 = s, a_0 = a \right].$$ 4 METHODOLOGY In decentralized settings with individual rewards, agents must balance personal objectives with collective goals, despite lacking global perspectives. Our approach, anticipation sharing (AS), facilitates this dual awareness without direct reward or objective sharing. Agents exchange anticipations about peer actions solved by maximizing their own return and take the anticipations from others into account when solving policies to maximize individual return, enabling each agent to infer collective objectives. This allows understanding broader impacts of actions through localized interactions. Unlike traditional methods that share explicit rewards or objectives, AS involves agents exchanging anticipations that implicitly include the information of others’ objectives. By observing how its actions align with aggregated anticipations, each agent can perceive the divergence between its individual interests and the inferred collective goals. This drives policy updates to reduce the identified discrepancy, bringing local and global objectives into closer alignment. Our constrained optimization approach leverages the identified divergences between individual and collective objectives to align decentralized policies. Agents iteratively share anticipated actions and adapt policies accounting for peer anticipations. This fosters continuous, adaptive refinement of strategies balancing both individual returns and collective dynamics inferred from shared anticipations. Our algorithm harnesses this divergence identification, ensuring decision-making integrates individual rewards and collective objectives surmised from interactions. 4.1 THEORETICAL DEVELOPMENTS We commence our technical developments by analyzing joint policy shifts in a centralized setting. This parallels foundational trust region policy optimization work [Schulman et al., 2015]. We prove the following bound on the expected return difference between new and old joint policies: Theorem 1 We establish a bound for the difference in expected returns between an old joint policy $\pi_{\text{old}}$ and a newer policy $\pi_{\text{new}}$: $$\eta(\pi_{\text{new}}) \geq \eta(\pi_{\text{old}}) + \zeta_{\pi_{\text{old}}}(\pi_{\text{new}}) - C \cdot D_{KL}^{\max}(\pi_{\text{old}} \|\| \pi_{\text{new}}),$$ where $$\zeta_{\pi_{\text{old}}}(\pi_{\text{new}}) = \mathbb{E}_{s \sim d^{\pi_{\text{old}}}(s), a \sim \pi_{\text{new}}(s)} \left[ \sum_i A_i^{\pi_{\text{old}}}(s, a) \right],$$ $$C = \frac{4 \max_{s, a} \|\sum_i A_i^{\pi_{\text{old}}}(s, a)\|}{(1 - \gamma)^2},$$ $$D_{KL}^{\max}(\pi_{\text{old}} \|\| \pi_{\text{new}}) = \max_s D_{KL}(\pi_{\text{old}}(\cdot\|s) \|\| \pi_{\text{new}}(\cdot\|s)).$$ The proof is given in Appendix A.1. The key insight is that the expected improvement in returns under the new policy depends on both the expected advantages it provides over the old policy, as well as the divergence between policy distributions. This quantifies the impact of joint policy changes on overall system performance given global knowledge, extending trust region concepts to multi-agent domains. However, this result relies on the strong assumption of centralized execution with full observability of joint policies. To address this limitation, we introduce the concept of an anticipated joint policy from each agent’s local perspective. As we will show, the anticipated joint policy is solved by optimizing individual objectives. Analyzing anticipated policies is crucial for assessing the discrepancy between individual objectives and the original collective one in decentralized learning. Definition 1 For each agent in a multi-agent system, we define the anticipated joint policy, denoted as $\tilde{\pi}^i$, formulated as $\tilde{\pi}^i(a\|s) = \prod_{j=1}^{N} \pi^{ij}(a^j\|s)$. Here, for each agent $i$, $\pi^{ij}$ represents the anticipation from agent $i$ to agent $j$’s policy when $j \neq i$. When $j = i$, we use $\pi^{ii} = \pi^i$ to indicate agent $i$’s own policy. To represent the collection of all such anticipated joint policies across agents, we use the notation $\tilde{\Pi} := (\tilde{\pi}^1, \ldots, \tilde{\pi}^i, \ldots, \tilde{\pi}^N)$. The anticipated joint policy represents an agent’s perspective of the collective strategy constructed from its own policy and anticipations to peers. We will present how to solve such anticipated joint policy in Section 4.2. Definition 2 The total expectation of individual advantages, considering the anticipated joint policies and a common state distribution, is defined as follows: $$\zeta_{\pi'}(\tilde{\Pi}) = \sum_i \mathbb{E}_{s \sim d^{\pi'}(s), a \sim \tilde{\pi}^i(a\|s)} \left[ A_{\pi'}(s, a) \right],$$ where $\zeta_{\pi'}(\tilde{\Pi})$ represents the sum of expected advantages for each agent $i$, calculated over their anticipated joint policy $\tilde{\pi}^i$ and a shared state distribution, $d^{\pi'}(s)$. The advantage $A_{\pi'}(s, a)$ for each agent is evaluated under a potential joint policy $\pi'$, which may differ from the actual joint policy $\pi$ in play. This definition captures the expected benefit each agent anticipates based on the anticipated joint actions, relative to the potential joint policy $\pi'$. This concept quantifies the expected cumulative advantage an agent could hypothetically gain by switching from some reference joint policy to the anticipated joint policies of all agents. It encapsulates the perceived benefit of the anticipated decentralized policies versus a centralized benchmark. Intuitively, if an agent’s anticipations are close to the actual policies of other agents, this expected advantage will closely match the actual gains. However, discrepancies in anticipations will lead to divergences, providing insights into the impacts of imperfect decentralized knowledge. Equipped with these notions of anticipated joint policies and total advantage expectations, we can analyze the discrepancy of the expectation of the total advantage caused by policy shift from the actual joint policy to the individually anticipated ones. Specifically, we prove the following bound relating this discrepancy: Theorem 2 The discrepancy between $\zeta_{\pi'}(\tilde{\Pi})$ and $\zeta_{\pi'}(\pi)$ is upper bounded as follows: $$\zeta_{\pi'}(\tilde{\Pi}) - \zeta_{\pi'}(\pi) \leq f_{\pi'} + \sum_i \frac{1}{2} \max_{s,a} \left\| A_{\pi'}(s, a) \right\| \cdot \sum_{s,a} \left( \tilde{\pi}^i(a\|s) - \pi(a\|s) \right)^2,$$ where $$f_{\pi'} = \sum_i \frac{1}{2} \max_{s,a} \left\| A_{\pi'}(s, a) \right\| \cdot \|A\| \cdot \\|d^{\pi'}\\|^2_2$$ The proof is given in Appendix A.2. This result quantifies the potential drawbacks of relying on imperfect knowledge in decentralized settings, where agents’ anticipations may diverge from actual peer policies. It motivates reducing the difference between anticipated and actual policies. Previous results bounded the deviation between total advantage expectations under the actual joint policy versus under anticipated joint policies. We now build on this to examine how relying too much on past policies can lead to misjudging the impact of new joint policy shifts over time. Specifically, we consider the relationship between $\zeta_{\pi_{\text{old}}}(\tilde{\Pi}_{\text{new}})$, the perceived benefit of the new anticipated joint policies $\tilde{\Pi}_{\text{new}}$, assessed from the perspective of the previous joint policy $\pi_{\text{old}}$, and $\eta(\pi_{\text{new}})$, which measures the performance of the new joint policy. The former represents a potentially myopic perspective informed heavily by the past policy and, as such, it may inaccurately judge the actual impact of switching to $\pi_{\text{new}}$ as quantified by $\eta(\pi_{\text{new}})$. The following result provides a lower bound of the expected return, $\eta(\pi_{\text{new}})$, of the newer joint policy. Theorem 3 The expected return of the newer joint policy is lower bounded as follows: $$ \eta(\pi_{\text{new}}) \geq \zeta_{\pi_{\text{old}}}(\tilde{\Pi}_{\text{new}}) + \eta(\pi_{\text{old}}) - C \cdot \sum_i D_{KL}^{\max}(\pi_{\text{old}}^{ii} \|\| \pi_{\text{new}}^{ii}) \\ - f_{\pi_{\text{old}}} - \sum_i \frac{1}{2} \max_{s,a} \|A_{\pi_{\text{old}}}^{ii}(s,a)\| \cdot \sum_{s,a} (\hat{\pi}_{\text{new}}^{ii}(a\|s) - \pi_{\text{new}}(a\|s))^2. $$ (9) The full proof is given in Appendix A.3. This theorem explains the nuanced dynamics of policy changes in decentralized multi-agent reinforcement learning, where agents learn separately. It sheds light on how uncoordinated local updates between individual agents affect the collective performance. At the same time, this result suggests a potential way to improve overall performance by leveraging the anticipated joint policies held by each agent. ### 4.2 A SURROGATE OPTIMIZATION OBJECTIVE Our preceding results established analytical foundations for assessing joint policy improvement in such settings. We now build upon these results to address the practical challenge of how agents can effectively optimize system-wide returns in a decentralized fashion. Directly maximizing the expected collective returns, $\eta(\pi)$ is intractable without a global view. However, Theorem 3 provides the insight that agents can optimize a more tractable localized surrogate objective, $\zeta_{\pi_{\text{old}}}(\tilde{\Pi})$. This simplifies the global objective into a decentralized form dependent only on an agent’s individual policy, denoted as $\pi^{ii}$, and its anticipations to others, $\pi^{ij}$, retaining the relevant complexities in a decentralized form. To this end, instead of using the original global objective, we leverage the lower bound given by Theorem 3 by maximizing the lower bound, the collective return can be maximized. Since the terms $\eta(\pi_{\text{old}})$ and $f_{\pi_{\text{old}}}$ featuring in Theorem 3 are not relevant to optimizing $\tilde{\Pi}$, they can be been omitted. Thus, we propose the following global constrained optimization problem as a surrogate objective of the original collective one: $$ \max_{\tilde{\Pi}} \zeta_{\pi_{\text{old}}}(\tilde{\Pi}) \\ \text{s.t. } \sum_i D_{KL}^{\max}(\pi_{\text{old}}^{ii} \|\| \pi^{ii}) \leq \delta, \sum_i \max_{s,a} \|A_{\pi_{\text{old}}}^{ii}(s,a)\| \cdot \sum_{s,a} (\hat{\pi}^{ii}(a\|s) - \pi(a\|s))^2 \leq \delta'. $$ (10) This global optimization objective captures the essence of coordinating joint policies to maximize localized advantages. However, it still assumes a centralized executor with full knowledge of $\tilde{\Pi}$. To make this feasible in decentralized MARL, we reformulate it from each agent’s limited perspective. Remarkably, we can distill the relevant components into a local objective and constraints for each individual agent, as follows: $$ \max_{\pi^{ii}} \mathbb{E}_{s \sim d_{\pi_{\text{old}}}^{ii}(s), a \sim \pi^{ii}(a\|s)} [A_{\pi_{\text{old}}}^{ii}(s,a)] \\ \text{s.t. } (a) \quad D_{KL}^{\max}(\pi_{\text{old}}^{ii} \|\| \pi^{ii}) \leq \delta_1, \quad (b) \quad \kappa_i \cdot \sum_{s,a_j} (\pi^{ij}(a_j\|s) - \pi^{jj}(a_j\|s))^2 \leq \delta_2, \forall j \neq i, \\ (c) \quad \kappa_i \cdot \sum_{s,a_i} (\pi^{ii}(a_i\|s) - \pi^{ji}(a_i\|s))^2 \leq \delta_2, \forall j \neq i, $$ (11) where $ \kappa_i = \max_{s,a} \|A_i^{\pi_{\text{old}}}(s, a)\| $. Note that the constraints in Eq. 11 depend on other agents’ policies $ \pi^{jj} $ as well as their anticipations of agent $ i $'s policy, $ \pi^{ji} $. To evaluate these terms, each agent $ j $ needs to share its action distribution $ \pi^{jj}(\cdot\|s) $ and the anticipated action distribution $ \pi^{ji}(\cdot\|s) $ to agent $ i $. This sharing allows each agent $ i $ to assess the constraint terms, which couple the individual advantage optimizations under local constraints. Such constraints reflect not only the differences between the true policy of others and the anticipations to them from an agent, but also the discrepancy between the agent’s own true policy and the anticipations from others. Distributing the optimization while exchanging critical policy information in this way balances autonomy for decentralized execution with maintaining global coordination between agents. This setup differs from teammate modeling where agent $ i $ tries to approximate peer policies $ \hat{\pi}^{ij} $ and use them when solving $ \pi^{ii} $, whereas Eq. 11 aims to optimize the anticipations $ \pi^{ij} $ together with $ \pi^{ii} $ and then $ \pi^{ij} $ is used by agent $ j $ to solve $ \pi^{jj} $. Therefore, the anticipations include the information about individual objectives implicitly. By exchanging the anticipations, individual agents can balance others’ objectives and thus the collective performance when optimizing its own objective. This setup also significantly differs from fully centralized learning where a coordinator has access to all policies. Here agents only share action distributions to evaluate coupling constraints, retaining decentralized computation. 4.3 A PRACTICAL ALGORITHM FOR LEARNING WITH AS We propose a structured approach to optimize the objective in Eq. 11. The derivation of the algorithm involves specific steps, each targeting different aspects of the optimization challenge. Note that in this practical algorithm, we present a general setup where the network topology of the system does not need to be fully-connected. Each agent only exchanges information with neighbours $ \{j\|j \in N_i\} $. This provides an approximation of the theoretical results. Step 1: Clipping Policy Ratio for KL Constraint. Addressing the KL divergence constraint (a) in Eq. 11 is crucial in ensuring our decentralized learning process remains effective. This constraint ensures that updates to an agent’s individual policy do not deviate excessively from its previous policy. To manage this, we incorporate a clipping mechanism, inspired by PPO-style clipping (Schulman et al., 2017), adapted for decentralized agents. We start by defining probability ratios for the individual policy and anticipated peer policies: \[ \xi_i = \frac{\pi^{ii}(a_i\|s; \theta^{ii})}{\pi_{\text{old}}^{ii}(a_i\|s; \theta_{\text{old}}^{ii})}, \quad \xi_N = \prod_{j \in N_i} \frac{\pi^{ij}(a_j\|s; \theta^{ij})}{\pi_{\text{old}}^{ij}(a_j\|s; \theta_{\text{old}}^{ij})}. \] These ratios measure the extent of change in an agent’s policy relative to its previous one and its anticipations to others. We then apply a clipping operation to $ \xi_i $, the individual policy ratio: \[ E_{s \sim d_{\pi_{\text{old}}}(s), a \sim \pi_{\text{old}}(a\|s)} \left[ \min \left( \xi_i \xi_N \hat{A}_i, \text{clip}(\xi_i, 1 - \epsilon, 1 + \epsilon) \xi_N \hat{A}_i \right) \right]. \] This method selectively restricts major changes to the individual policy $ \pi^{ii} $, while allowing more flexibility in updating anticipations of peer policies. It balances the adherence to the KL constraint with the flexibility needed for effective learning and adaptation in a decentralized environment. Step 2: Penalizing Anticipation Discrepancies. The objective of this step is to enforce constraints (b) and (c) in Eq. 11, which aim to penalize discrepancies between the anticipated and actual policies. Simply optimizing the advantage function may not sufficiently increase these discrepancies. Therefore, we introduce penalty terms that are activated when policy updates inadvertently increase these discrepancies. Specifically, we define states $ X^{ij} $ to identify when the policy update driven by the advantage exacerbates the discrepancies between the resulting anticipated policies and other agents’ current policies, and $ X^{ii} $ to identify the discrepancies between the resulting agent’s own policy and the ones anticipated from other agents. These are defined as \[ X^{ij} = \left\{ (s, a) \mid \frac{\pi^{ij}(a_j\|s; \theta^{ij})}{\pi^{jj}(a_j\|s)} \hat{A}_i > \hat{A}_i \right\}, \quad X^{ii} = \left\{ (s, a) \mid \frac{\pi^{ii}(a_i\|s; \theta^{ii})}{\pi^{ji}(a_i\|s)} \hat{A}_i > \hat{A}_i \right\}, \] where the pairs $(s, a)$ represent scenarios in which the gradient influenced by $\hat{A}_i$ increases the divergence between the two policies. The following indicator function captures this effect: \[ I_X(s, a) = \begin{cases} 1 & \text{if } (s, a) \in X, \\ 0 & \text{otherwise}. \end{cases} \] Step 3: Dual Clipped Objective. In the final step, we combine the clipped surrogate objective with coordination penalties to form our dual clipped objective: \[ \max_{\theta_i, \theta_j} \mathbb{E}_{s \sim d^{\text{old}}(s), a \sim \pi^{\text{old}}(a\|s)} [\min \left( \xi_i \xi_N, \hat{A}_i, \text{clip}(\xi_i, 1 - \epsilon, 1 + \epsilon) \xi_N, \hat{A}_i \right)] \\ - \kappa_i \cdot \sum_{j \in N_i} \rho_j I_{X^{ij}}(s, a) \\| \pi^{ij}(\cdot\|s; \theta^{ij}) - \pi^{ji}(\cdot\|s; \theta^{ji}) \\|_2^2 + \rho'_j I_{X^{ji}}(s, a) \\| \pi^{ji}(\cdot\|s; \theta^{ji}) - \pi^{ij}(\cdot\|s; \theta^{ij}) \\|_2^2. \] This step balances individual policy updates with the need for coordination among agents, thereby aligning individual objectives with collective goals. Implementation details. In our implementation, we use $\hat{\kappa}_i = \text{mean}_{s,a} \|\hat{A}_i\|$ to approximate $\kappa_i$ in order to mitigate the impact of value overestimation. Additionally, we adopt the same value for the coefficients $\rho_j$ and $\rho'_j$ across different $j$, and denote it as $\rho$. We also utilize the generalized advantage estimator (GAE) \cite{schulman2016apprenticeship} due to its well-known properties to obtain estimates \[ \hat{A}_i^t = \sum_{l=0}^{\infty} (\gamma \lambda)^l \delta_{V_i}^{t+l}, \quad \delta_{V_i}^{t+l} = r_i^{t+l} + \gamma V_i(s_{t+l+1}) - V_i(s_{t+l}), \] where $V_i$ is approximated by minimizing the following loss function, \[ L_{V_i} = \mathbb{E}[(V_i(s_t) - \sum_{l=0}^{\infty} \gamma^l r_i^{t+l})^2]. \] Algorithm 1 in Appendix F presents the detailed procedure used in our experimental section. Appendix E shows an illustration of our method. ## 5 EXPERIMENTS ### 5.1 TASKS AND BASELINES We evaluate the performance of our AS algorithm across a spectrum of tasks, spanning both discrete (Exchange and Cooperative Navigation) and continuous (Cooperative Predation) spaces and featuring diverse agent counts (from 3 to 20 agents). For a comprehensive assessment, we benchmark AS against three prominent baselines: Value Sharing (VS) \cite{du2022value}, Value Function Parameter Sharing (VPS) \cite{zhang2018value}, and Policy Model Sharing (PS) \cite{zhang2019policy}. A detailed description of the environments and baselines can be found in the Appendix. ### 5.2 RESULTS The training curves and final total returns of the different algorithms are shown in Figure 1. For the two discrete environments, "Exc." and "Navi.", there are 3 agents. The neighboring agents of each agent are enclosed within the dashed outline rectangles, as depicted in Figures 2(a) and (b) in Appendix B. In the continuous domain, we assess the algorithms using 6, 8, and 12 agents. Neighboring agents are defined as those within a normalized distance of 0.1. For each algorithm and task, we conduct 5 runs with different seeds. As seen in Figure 1, our AS algorithm performs the best consistently across all tasks, attaining policies that gain more total return than the baselines. This demonstrates the effectiveness and superiority of AS. It is important to note that the aim of our study is not to outperform the baseline algorithms but to provide a viable alternative in settings where agents cannot exchange values or rewards due to privacy constraints. For the baseline algorithms, VS and VPS exhibit unstable performance across tasks. This implies merely sharing values or value functions and achieving value consensus may be insufficient for cooperative policies. A hypothesis for the performance disparities is that despite approximating system-wide values, policy updates in these methods lack coordination, leading to inferior cooperation. Particularly in the Pred. task, VS and VPS exhibit better performance in some scenarios compared to Exc. and Navi. tasks. This difference can be explained by the nature of the tasks themselves. Exc. and Navi. demand a higher level of coordination, especially because agents are heterogeneous with unique individual objectives. Such environments intensify the need for precise and synchronized policy updates, making the coordination challenge more pronounced. In contrast, our method aims to address this discoordination by enabling more harmonized policy updates among agents, taking into consideration the anticipations of other agents’ policies, which leads to a more cohesive policy development process. PS also focuses on direct policy coordination rather than value consensus. However, results show PS has slow convergence on some tasks. Sharing policy parameters may entail redundant information unnecessary for effective coordination. In contrast, AS avoids sharing policy parameters, instead exchanging action distributions from policies. Furthermore, in AS each agent selectively shares anticipations only with corresponding agents, not indiscriminately with all neighbors. Our superior training efficiency and performance compared to PS showcases this benefit. As agent populations increase, PS convergence slows, while AS remains robust. We also conducted further studies regarding the scalability, impact of neighbourhood range, sensitivity to the penalty weight. Experimental results indicate AS’s robust performance with sparse network topology, different neighbour counts, and varying penalty weight. Details are given in Appendix. 6 CONCLUSIONS AND FUTURE WORK In this work, we tackled the challenge of decentralized multi-agent policy optimization under individual reward conditions, where individual interests can conflict with collective objectives. We introduced Anticipation Sharing (AS) as an alternative to traditional methods like intrinsic rewards, value sharing, and policy model sharing. AS enables agents to incorporate their individual interests into anticipations regarding the action distributions of other agents. In the process of exchanging their anticipations with each other, agents become aware of the collective interest implicitly, despite the fact that rewards, values, and policies are private to each agent. Theoretically, we established that the difference between agents’ actual action distributions and the anticipations from others bounds the difference between individual and collective objectives. We used this insight to create a novel individual objective that serves as a lower bound for the original collective objective, driving agents toward cooperative behaviors. Our decentralized MARL algorithm based on AS demonstrated the capability to produce pro-social agents in empirical experiments. In the future, several opportunities exist to enhance our understanding and application of the AS framework. We can refine individual objectives by investigating tighter bounds for measuring discrepancies between individual and collective interests, and delve deeper into alternative optimization strategies based on AS framework. Another prospective avenue involves exploring the integration of additional communication mechanisms into AS. It would be especially insightful to study these mechanisms within the context of dynamic topology structures that dictate cooperative information flows. Additionally, a thorough analysis of our algorithm’s convergence properties would be insightful. 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hp4yOjhwTs	The authors implement ALP-GMM by fixing the color. I don't think this is a good way to construct baseline. One can run ALP-GMM as a task sampler on a fixed set of tasks. ALP-GMM essentially assigns a probability for different tasks at the each round. If one task has the underlying U that is not aligned with the target task, ALP-GMM will adaptively reduce the weight for that task.	Causally Aligned Curriculum Learning Mingxuan Li and Junzhe Zhang and Elias Barcinboim Causal Artificial Intelligence Lab, Columbia University, USA {ml,junzhez,eb}@cs.columbia.edu Abstract A pervasive challenge in Reinforcement Learning (RL) is the “curse of dimensionality” which is the exponential growth in the state-action space when optimizing a high-dimensional target task. The framework of curriculum learning trains the agent in a curriculum composed of a sequence of related and more manageable source tasks. The expectation is that when some optimal decision rules are shared across source tasks and the target task, the agent could more quickly pick up the necessary skills to behave optimally in the environment, thus accelerating the learning process. However, this critical assumption of invariant optimal decision rules does not necessarily hold in many practical applications, specifically when the underlying environment contains unobserved confounders. This paper studies the problem of curriculum RL through causal lenses. We derive a sufficient graphical condition characterizing causally aligned source tasks, i.e., the invariance of optimal decision rules holds. We further develop an efficient algorithm to generate a causally aligned curriculum, provided with qualitative causal knowledge of the target task. Finally, we validate our proposed methodology through experiments in discrete and continuous confounded tasks with pixel observations. 1 Introduction As Roma was not built in one day, learning to achieve a complex task (e.g., cooking, driving) directly can be challenging. Instead, the human learning process is scaffolded with incremental difficulty to support acquiring progressively advanced knowledge and skills. The idea of training with increasingly complex tasks, known as curriculum learning, has been applied in reinforcement learning when Selfridge et al. (1985) used a carefully curated sequence of tasks to train agents to solve a modified Cart Pole system. In recent years, there has been a growing interest in automatically generating curricula tailored to the agent’s current capabilities, which opens up a new venue called “Automatic Curriculum Learning” (Portelas et al., 2020). An automatic curriculum generator requires two components: an encoded task space and a task characterization function (Narvekar et al., 2020; Wang et al., 2020). Task space encoding is often a bijective function that maps a task to a low dimensional vector (Parker-Holder et al., 2022; Klink et al., 2022; Florensa et al., 2018; Jiang et al., 2021; Portelas et al., 2019; Wang et al., 2019; 2020; Cho et al., 2023; Huang et al., 2022a). A proper task space encoding lays the foundation of a reasonable task characterization function measuring the fitness of tasks (Florensa et al., 2018; Dennis et al., 2020; Andreas et al., 2017; Sukhbaatar et al., 2018; Jiang et al., 2021). New training tasks, called source tasks, are generated by changing the target task’s state space or parameters of transition functions in the encoded task space. A system designer then determines in which order the agent should be trained in these source tasks, following the task characterization function. The set of generated source tasks and the training order defined upon this set defines a curriculum for the learning agent. Please see App. G for more related work. While impressive, most curriculum RL methods described so far rely on the assumption that generated source tasks are aligned with the target. Consequently, the agent could pick up some valuable skills by training in such source tasks, allowing it to behave optimally in certain situations in the target environment. However, this critical assumption does not necessarily hold in many real-world decision-making settings. For concreteness, consider a modified Sokoban game shown in Fig. 1 inspired by Schrader (2018) where an unobserved confounder $U_t$ randomly determines the box color $C_t$ (0 for yellow, 1 for blue) at every time step $t$. The agent receives a positive reward $Y_t$ only when it pushes the box to the goal state when the box color appears yellow ($U_t = 0$); otherwise, Figure 1: Examples of (a) full episode of a misaligned source task that intervenes in the box color, (b) full episode of an aligned source task that only changes the initial box location, and (c) an aligned curriculum where none of the source tasks intervenes in the box’s color. it gets penalized ($U_t = 1$). We apply several state-of-the-art curriculum generators that construct source tasks by fixing the box color to yellow or blue, including ALP-GMM (Portelas et al., 2019), PLR (Jiang et al., 2021), Goal-GAN (Florensa et al., 2018), and Currot (Klink et al., 2022). Fig. 1a shows an example of the generated source tasks. We evaluate agents’ performance trained by those generated curricula and compare it with the one directly trained in the target task. Surprisingly, simulation results shown in Fig. 2 reveal that agents trained by the curricula failed to learn to push the yellow box to the destination. This suggests source tasks generated by intervening in the box color are misaligned; that is, training in these source tasks harms the agents’ target task performance. Several observations follow from the Sokoban example. (1) A curriculum designer generates source tasks by modifying the data-generating mechanisms in the target tasks. (2) Such modifications could lead to a shift in system dynamics between the target task and source tasks. When this distribution shift is significant, training in source tasks may harm the agent’s learning. (3) The agent must avoid misaligned source tasks to achieve optimal learning performance. There exist methods attempting to address the challenges of misaligned source tasks leveraging a heuristic similarity measure between the target and source tasks (Svetlik et al., 2017; Silva & Costa, 2018). Yet, a systematic and theoretically justified approach for exploiting other types of knowledge, e.g., qualitative, about the target task is missing. This paper aims to address the challenges of misaligned source tasks in curriculum generation by exploring causal relationships among variables present in the underlying environment. To realize this, we formalize curriculum learning in the theoretical framework of structural causal models (SCMs) (Pearl, 2009). This formulation allows us to characterize misaligned source tasks by examining the structural invariance across the optimal policies obtained from the target and source tasks. More specifically, our contributions are summarized as follows. (1) We derive a sufficient graphical condition determining potentially misaligned source tasks. (2) We develop efficient algorithms for detecting misaligned source tasks and constructing source tasks that are guaranteed to align with the target task. (3) We introduce a novel augmentation procedure that enables state-of-the-art curriculum learning algorithms to generate aligned curricula to accelerate the agent’s learning. Finally, we validate the proposed framework through extensive experiments in various decision-making tasks. 1.1 Preliminaries This section introduces necessary notations and definitions that will be used throughout the discussion. We use capital letters ($X$) to denote a random variable, lowercase letters ($x$) to represent a specific value of the random variable, and $\Omega(\cdot)$ to denote the domain of a random variable. We use bold capital letters ($V$) to denote a set of random variables and use $\|V\|$ to denote its cardinality. The basic semantical framework of our analysis rests on structural causal models (SCMs) (Pearl, 2009; Bareinboim & Pearl, 2016). An SCM $ M $ is a tuple $(U, V, F, P)$, where $ U $ is a set of exogenous variables and $ V $ is a set of endogenous variables. $ F $ is a set of functions s.t. each $ f_V \in F $ decides values of an endogenous variable $ V \in V $ taking as argument a combination of other variables in the system. That is, $ V \leftarrow f_V(PA_V, U_V), PA_V \subseteq V, U_V \subseteq U $. Values of exogenous variables $ U $ are drawn from the exogenous distribution $ P(U) $. A policy $ \pi $ over a subset of variables $ X \subseteq V $ is a sequence of decision rules $\{\pi(X\|S_X)\}_{X \in X}$, where every $ \pi(X\|S_X) $ is a probability distribution mapping from domains of a set of covariates $ S_X \subseteq V $ to the domain of action $ X $. An intervention following a policy $ \pi $ over variables $ X $, denoted by $ \text{do}(\pi) $, is an operation which sets values of every $ X \in X $ to be decided by policy $ X \sim \pi(X\|S_X) $ (Correa & Bareinboim, 2020), replacing the functions $ f_X = \{f_X : \forall X \in X\} $ that would normally determine their values. For an SCM $ M $, let $ M_\pi $ be a submodel of $ M $ induced by intervention $ \text{do}(\pi) $. For a set $ Y \subseteq V $, the interventional distribution $ P(Y; \pi) $ is defined as the distribution over $ Y $ in the submodel $ M_\pi $, i.e., $ P_M(Y; \pi) \triangleq P_{M_\pi}(Y) $; restriction $ M $ is left implicit when it is obvious. Each SCM $ M $ is also associated with a causal diagram $ G $ (e.g., Fig. 3a), which is a directed acyclic graph (DAG) where nodes represent endogenous variables $ V $ and arrows represent the arguments $ PA_V, U_V $ of each structural function $ f_V \in F $. Exogenous variables $ U $ are often not explicitly shown by convention. However, a bi-directed arrow $ V_i \leftrightarrow V_j $ indicates the presence of an unobserved confounder (UC), $ U_{i,j} \in U $ affecting $ V_i, V_j $, simultaneously (Bareinboim et al., 2022). We will use standard graph-theoretic family abbreviations to represent graphical relationships, such as parents ($ pa $), children ($ ch $), descendants ($ de $), and ancestors ($ an $). For example, the set of parent nodes of $ X $ in $ G $ is denoted by $ pa(X)_G = \cup_{X \in X} pa(X)_G $. Capitalized versions $ Pa, Ch, De, An $ include the argument as well, e.g., $ Pa(X)_G = pa(X)_G \cup X $. A path from a node $ X $ to a node $ Y $ in $ G $ is a sequence of edges that does not include a particular node more than once. Two sets of nodes $ X, Y $ are said to be d-separated by a third set $ Z $ in a DAG $ G $, denoted by $(X \perp\!\!\!\perp Y\|Z)_G$, if every edge path from nodes in $ X $ to nodes in $ Y $ is “blocked” by nodes in $ Z $. The criterion of blockage follows Pearl (2009) Def. 1.2.3). For more details on SCMs, we refer readers to Pearl (2009); Bareinboim et al. (2022). For the relationship between (PO)MDPs and SCMs, please see App. H. 2 Challenges of Misaligned Source Tasks This section will formalize the concept of aligned source tasks and provide an efficient algorithmic procedure to find such tasks based on causal knowledge about the data-generating process. Formally, a planning/policy learning task (for short, a task) is a decision-making problem composed of an environment and an agent. We focus on the sequential setting where the agent determines values of a sequence of actions $ X = \{X_1, \ldots, X_H\} $ based on the input of observed states $ \{S_1, \ldots, S_H\} $. The mapping between states and actions defines the space of candidate policies, namely, Definition 1 (Policy Space). For an SCM $ M = \langle U, V, F, P \rangle $, a policy space $ \Pi $ is a set of policies $ \pi $ over actions $ X = \{X_1, \ldots, X_H\} $. Each policy $ \pi $ is a sequence of decision rules $ \{\pi_1(X_1\|S_1), \ldots, \pi_H(X_H\|S_H)\} $ where for every $ i = 1, \ldots, H $, (i) Action $ X_i $ is a non-descendent of future actions $ X_{i+1}, \ldots, X_H $, i.e., $ X_i \in V \setminus De(\bar{X}_{i+1:H}) $; (ii) States $ S_i $ are non-descendants of future actions $ X_i, \ldots, X_H $, i.e., $ S_i \subseteq V \setminus De(\bar{X}_{i:H}) $. Henceforth, we will consistently denote such a policy space by $ \Pi = \{\langle X_1, S_1 \rangle, \ldots, \langle X_H, S_H \rangle\} $. The agent interacts with the environment by performing intervention $ \text{do}(\pi), \forall \pi \in \Pi $ to optimize a reward function $ R(Y) $ taking a set of reward signals $ Y \subseteq V $ as input.$^1$ A policy space, a reward function, and an SCM environment formalize a target decision-making task. We will graphically describe a target task using an augmented causal diagram $ G $ constructed from the SCM $ M $; actions $ X $ are highlighted in blue; reward signals $ Y $ are highlighted in red; input states $ S_i $ for every action $ X_i \in X $ are shaded in light blue. For instance, Fig. 3a shows a causal diagram representing the decision-making task in the Sokoban game (Fig. 1). For every time step $ i = 1, \ldots, H $, $ L_i $ stands for the agent’s location, $ B_i $ for the box location, and $ C_i $ for the box color. --- $^1$For instance, a cumulative discounted reward is defined as $ R(Y) = \sum_{i=1}^{H} \gamma^{i-1} Y_i $ where $ Y_i \in V, i = 1, \ldots, H $, are endogenous variables, and $ \gamma \in (0, 1] $ is a discount factor. Definition 2 (Target Task). A target task is a tuple $ T = \langle M, \Pi, R \rangle $, where $ M = \langle U, V, F, P \rangle $ is an SCM, $ \Pi $ is a policy space over actions $ X \subseteq V $, and $ R $ is a reward function over signals $ Y \subseteq V $. The goal is to find an optimal policy $ \pi^* \in \Pi $ that maximizes the expected reward function $ \mathbb{E}[R(Y); \pi] $ evaluated in the underlying environment $ M $, i.e., \[ \pi^* = \arg\max_{\pi \in \Pi} \mathbb{E}_M[R(Y); \pi]. \] When the detailed parametrization of the SCM $ M $ is provided, the optimal policy $ \pi^* $ is obtainable by applying planning algorithms, e.g., dynamic programming (Bellman 1966) or influence diagrams (Koller & Milch 2003). However, when underlying system dynamics are complex or the state-action domains are high-dimensional, it might be challenging to solve an optimal policy even with state-of-the-art planning algorithms. We will then consider the curriculum learning approach (Selfridge et al. 1985), where the agent is not immediately trained in the target task but provided with a sequence of related yet simplified source tasks. Definition 3 (Source Task). For a target task $ T = \langle M, \Pi, R \rangle $, a source task $ T^{(j)} $ is a tuple $ \langle M^{(j)}, \Pi, R, \Delta^{(j)} \rangle $ where $ M^{(j)} $ is an SCM compatible with the same causal diagram as $ M $, i.e., $ G_M = G_{M^{(j)}} $; a set of variables $ \Delta^{(j)} \subseteq V $ is called edits where there might exist a discrepancy that $ f_V \neq f^{(j)}_V $ or $ P(U_V) \neq P^{(j)}(U_V) $ for every $ V \in \Delta^{(j)} $. In practice, source tasks are constructed from the target task by modifying parameters of the underlying structural functions $ F $ or exogenous distributions $ P(U) $. Consider again the Sokoban game described in Fig. 1. The system designer could generate a source task $ T^{(1)} $ by changing the agent and box’s initial location $ L_1, B_1 $. Fig. 1b shows a causal diagram $ G^{(1)} $ representing the source task $ T^{(1)} $; $ \tau^{(1)} $ is an edit indicator representing the domain discrepancies $ \Delta^{(1)} $ between the target $ T $ and source tasks $ T^{(1)} $. Here, arrows $ \tau^{(1)} \rightarrow L_1, \tau^{(1)} \rightarrow B_1 $ suggest that structural functions $ f_{L_1}, f_{B_1} $ or exogenous distributions $ P(U_{L_1}, U_{B_1}) $ have been changed in the source task $ T^{(1)} $ while other parts of the system remain the same as the target task $ T $. By simplifying the system dynamics, learning an optimal policy in the source task $ T^{(j)} $ could be easier than in the target task $ T $. The expectation here is that the optimal decision rules $ \pi^{(j)} $ over some actions $ X^{(j)} \subseteq X $ remain invariant across the source and target tasks. If so, we will call such source tasks as aligned. Training in an aligned source task thus guides the agent to move toward an optimal policy $ \pi^* $. For example, Fig. 1b shows an aligned source task for the Sokoban game where the agent and box’s locations are set close to the goal state. By training in the simplified task, the agent learns the optimal decision rule to push the yellow box to the goal state in this game. However, modifying the target task could lead to a misaligned source task whose system dynamics differ significantly from the target. Interestingly and more seriously, training in these source tasks may “harm” the agent’s performance, resulting in suboptimal decision rules, as illustrated next. We will consistently use the superscript $ (j) $ to indicate a diagram $ G^{(j)} \triangleq G_{M^{(j)}} $ associated with a source task $ T^{(j)} $. Similarly, we write $ P^{(j)}(Y; \pi) = P_{M^{(j)}}(Y; \pi) $ and $ \pi^{(j)} = \arg\max_{\pi \in \Pi} \mathbb{E}_{M^{(j)}}[R(Y); \pi] $. Example 1 (Misaligned Source Task). Consider the Sokoban game $ T = \langle M, \Pi, R \rangle $ described in Fig. 1. Fig. 3a shows its causal diagram $ G $. Specifically, the box color $ C_i $ (0 for yellow, 1 for blue) is determined by an unobserved confounder $ U_i \in \{0, 1\} $ randomly drawn from a distribution $ P(U_i = 1) = 3/4 $. Box location $ B_i $ and agent location $ L_i $ are determined following system dynamics in deterministic grid worlds (Chevalier-Boisvert et al., 2018). The reward signal $ Y_i $ is given by, \[ Y_i = \begin{cases} 10 & \text{if } B_i = \text{“next to goal”} \land X_i = \text{“push”} \land (U_i = 0) \\ -10 & \text{if } B_i = \text{“next to goal”} \land X_i = \text{“push”} \land (U_i = 1) \\ -0.1 & \text{otherwise} \end{cases}. \] If the agent pushes the box into the goal location (top right corner in Fig. 1), it receives a positive reward when the box appears yellow; it gets penalized when the box appears blue. Since $ C_i \leftarrow U_i $, evaluating the conditional reward $ E[Y_i \| b_i, c_i; do(x_i)] $ in the Sokoban environment $ M $ gives, \[ E[Y_i \| B_i = \text{“next to goal”}, C_i; do(X_i = \text{“push”})] = \begin{cases} 10 & \text{if } C_i = 0 \\ -10 & \text{if } C_i = 1 \end{cases}. \] Thus, the agent should aim to push yellow boxes to the goal location in the target. The curriculum designer now attempts to generate a source task $ T^{(2)} $ by fixing the box color to yellow, i.e., $ C_i \leftarrow 0 $. Fig. 3b shows the causal diagram $ G^{(2)} $ associated with the source environment $ M^{(2)} $ where edit indicators $ \tau^{(2)} $ denote the change in the structural function $ f_{C_i} $ determining the box color $ C_i $. Evaluating the conditional reward $ E[Y_i \| b_i; do(c_i, x_i)] $ in this manipulated environment $ M^{(2)} $ gives \[ E^{(2)}[Y_i \| B_i = \text{“next to goal”}; do(C_i = 0, X_i = \text{“push”})] = -5. \] Detailed computations are provided in App. B. Perhaps counter-intuitively, pushing the yellow box to the goal location in the source task $ T^{(2)} $ results in a negative expected reward. This is because box color $ C_i $ is only a proxy to the unobserved $ U_i $ that controls the reward. Fixing $ C_i $ won’t affect $ Y $ but only breaks this synergy, hiding the critical information of $ U_i $ from the agent. Consequently, when training in the source task $ T^{(2)} $, the agent will learn to never push the box even when it is next to the goal location, which is suboptimal in the target Sokoban game $ T $. 2.1 Causally Aligned Source Task Example 1 suggests that naively training in a misaligned source task may lead to suboptimal performance in the target task. The remainder of this section will introduce an efficient strategy to avoid misaligned source tasks, provided with the causal knowledge of the underlying data-generating mechanisms in the environment. For a target task $ T = \langle M, \Pi, R \rangle $, let $ G $ be the causal diagram associated with $ M $. Let $ G_\pi $ be an intervened diagram obtained from $ G $ by replacing incoming arrows if action $ X_i \in X $ with arrows from input states $ S_i $ for every action $ X_i \in X $. We first characterize a set of variables $ \Delta^{(j)} \subseteq V $ amenable to editing (for short, editable states) using independence relationships between edit indicators $ \tau^{(j)} $ and reward signals $ Y $. Formally, Definition 4 (Editable States). For a target task $ T = \langle M, \Pi, R \rangle $, let $ G $ be a causal diagram of $ M $ and $ X^{(j)} \subseteq X $ be a subset of actions. A set of variables $ \Delta^{(j)} \subseteq V \setminus X^{(j)} $ is editable w.r.t $ X^{(j)} $ if and only if $ \forall X_i \in X^{(j)} $, the following independence holds in the intervened diagram $ G_\pi $, \[ (\tau^{(j)} \perp \! \! \! \perp Y \cap De(X_i) \| X_i, S_i), \] where $ \tau^{(j)} $ is the set of added edit indicators pointing into nodes in $ \Delta^{(j)} $. For example, consider again the Sokoban game described in Example 1. The initial agent and box’s position $ \Delta^{(1)} = \{B_1, L_1\} $ is editable with regard to all actions $ X $ following Def. 4. Precisely, in the augmented diagram $ G^{(1)} $ of Fig. 3b, for every action $ X_i \in X $, the edit indicators $ \tau^{(1)} $ are d-separated the reward signals $ Y \cap De(X_i) = \{Y_i, \ldots, Y_H\} $ given input states $ \{L_i, B_i, C_i\} $. On the other hand, the set of box color variables $ \Delta^{(2)} = \{C_1, \ldots, C_H\} $ are not editable w.r.t. actions $ X $ since in the augmented diagram $ G^{(2)} $ of Fig. 3b, for every action $ X_i \in X $, there exists an active path between edit indicators $ \tau^{(2)} $ and reward signals $ \{Y_i, \ldots, Y_H\} $ given action $ X_i $ and input states $ \{L_i, B_i, C_i\} $, violating the criterion given by Def. 4. For a fixed policy $\pi \in \Pi$, for any subset $S \subseteq V$, we denote by $\Omega^{(j)}(S; \pi) = \{s \in \Omega(S) \| P_M(s; \pi) > 0\}$ the set of reachable values of $S$, which is the set of states that are possible to reach in a source task $T^{(j)}$ under intervention do$(\pi)$. The following proposition establishes that modifying functions and distributions over a set of editable states $\Delta^{(j)}$ leads to an aligned source task. Theorem 1 (Causally Aligned Source Task). For a target task $T = \langle M, \Pi, R \rangle$, let $T^{(j)} = \langle M^{(j)}, \Pi, R, \Delta^{(j)} \rangle$ be a source task of $T$ by modifying states $\Delta^{(j)} \subseteq V$. If $\Delta^{(j)}$ is editable w.r.t some actions $X^{(j)} \subseteq X$, then for every action $X_i \in X^{(j)}$, $$\pi^_i(X_i \| s_i) = \pi^{(j)}_i(X_i \| s_i), \quad \forall s_i \in \Omega^{(j)}(S_i; \pi^{(j)}) \cap \Omega(S_i; \pi^)$$ where $\pi^, \pi^{(j)} \in \Pi$ are optimal policies in the target $T$ and source $T^{(j)}$ tasks, respectively. Thm. 1 implies that whenever states $\Delta^{(j)}$ is editable w.r.t. some actions $X^{(j)}$, one could always construct an aligned source task $T^{(j)}$ such that the optimal decision rules $\pi^$ over $X^{(j)}$ is invariant across the target $T$ and source $T^{(j)}$ tasks. Consequently, one could transport these optimal decision rules trained in the source task $T^{(j)}$ without harming the agent’s performance in the target domain $T$. For example, in the Sokoban game of Example 1, since initial states $\Delta^{(1)} = \{B_1, L_1\}$ is editable w.r.t. actions $X$, moving the agent and box’s location leads to an aligned source task, which allows the agent to learn how to behave optimally when getting closer to the goal state. However, the performance guarantee in Thm. 1 does not necessarily hold when states $\Delta^{(j)}$ are not editable. For instance, recall that $\Delta^{(2)} = \{C_1, \ldots, C_H\}$ are not editable in the Sokoban game. Modifying the box’s color could lead to a misaligned source task $T^{(2)}$. An agent trained in this source task could pick up undesirable behaviors, as demonstrated in Example 1. Algo. 1 describes an algorithmic procedure, FindMaxEdit, to find a maximal editable set $\Delta^{(j)}$ in a causal diagram $G$ w.r.t. a set of actions $X^{(j)} \subseteq X$. A set of editable states $\Delta^{(j)}$ is maximal w.r.t. $X^{(j)}$ if there is no other editable states $\Delta^{(j)}_$ strictly containing $\Delta^{(j)}$. We always prefer a maximal editable set since it offers the maximum freedom to simplify the system dynamics in the target task. Particularly, FindMaxEdit iteratively adds endogenous variables $V \setminus (X \cup Y)$ to the editable states $\Delta^{(j)}$ and test the independence criterion in Def. 4. This procedure continues until it cannot add any more endogenous variables. Evidently, FindMaxEdit returns a maximal editable set $\Delta^{(j)}$ w.r.t. $X^{(j)}$. A natural question arising at this point is whether the ordering of endogenous variables $V$ changes the output. Fortunately, the next proposition shows that this is not the case. Theorem 2.* For a target task $T = \langle M, \Pi, R \rangle$, let $G_\pi$ be an intervened causal diagram of $M$ and let $X^{(j)} \subseteq X$ be a subset of actions. FindMaxEdit $(G_\pi, X^{(j)})$ returns a maximal editable set $\Delta^{(j)}$ w.r.t actions $X^{(j)}$; moreover, such a maximal set $\Delta^{(j)}$ is unique. Let $n$ and $m$ denote the number of nodes and edges in the intervened diagram $G_\pi$ and let $d$ be the number of actions $X$. Since testing d-separation has a time complexity of $O(n + m)$, FindMaxEdit has a time complexity of $O(d(n + m))$. We also provide other algorithmic procedures for directly deciding a set’s editability and constructing editable sets for a target task $T$ in App. C. ### 3 Curriculum Learning via Causal Lens Once a collection of source tasks is constructed, the system designer could organize them into an ordered list, called a curriculum, as defined next: --- 3 Causal aligned source tasks (Thm. 1) and editable states (Def. 4) are related to the concept of transportability in causal inference literature (Bareinboim & Pearl, 2016), which generalizes estimation of unknown causal effects from different domains. Here we study the generalizability of an optimal decision policy. Definition 5 (Curriculum). For a target task $ T = \langle M, \Pi, R \rangle $, a curriculum $ C $ for $ T $ is a sequence of source tasks $ \{T^{(j)}\}_{j=1}^N $, where $ T^{(j)} = \langle M^{(j)}, \Pi, R, \Delta^{(j)} \rangle $. For instance, Fig. 1c describes a curriculum in the Sokoban game where the agent and the box are placed increasingly further away from the goal location. Given a curriculum $ C $, a typical curriculum learning algorithm trains the agent sequentially in each source task, following the curriculum’s ordering. Algo. 2 shows the pseudo-code describing this training process. It first initializes an arbitrary baseline policy $ \pi^{(0)} $. For every source task $ T^{(j)} \in C $, the algorithm updates the current policy $ \pi^{(j-1)} $ such that the new policy $ \pi^{(j)} $ is optimal in the source task $ T^{(j)} $. This step could be performed using a standard gradient-based algorithm, e.g., the policy gradient (Sutton & Barto, 2018). The expectation is that, as the agent picks up more skills in the source tasks, it could consistently improve its performance in the target task or at least not regress. Definition 6 (Causally Aligned Curriculum). For a target task $ T = \langle M, \Pi, R \rangle $, let $ C = \{T^{(j)}\}_{j=1}^N $ be a curriculum for $ T $. Curriculum $ C $ is said to be causally aligned with $ T $ if for every $ j = 1, \ldots, N - 1 $, the set of invariant optimal decision rules across the source task and the target task expands, i.e., \[ \left( \pi^{(j)} \cap \pi^* \right) \subseteq \left( \pi^{(j+1)} \cap \pi^* \right), \] where $ \pi^* \in \Pi $ is an optimal policy in the target task $ T $. A naive approach to construct a causally aligned curriculum is to (1) construct a set of aligned source tasks by modifying editable states (Thm. 1), and (2) organize these tasks in an arbitrary ordering. However, the following example shows this is not a viable option. Example 2 (Overwriting in Curriculum Learning). Consider a two-stage target task where action $ X_1 $ takes input $ H $ and $ X_2 $ takes input $ Z $. The task SCM is, $ H = U_H, Z = \neg X_1 \oplus U_Z, Y_1 = 0.5 * (H \oplus X_1), Y_2 = \neg H \oplus X_2 \land Z $ where $ P(U_Z = 1) = 1/2, P(U_H = 1) = 1/10 $. Other than the reward $ Y_1, Y_2 $, all other variables are binary. The optimal policy for the target task is $ \pi^(X_1 = \neg H\|H) = 1, \pi^(X_2 = 0\|Z) = 1 $. We create two source tasks. For $ T^{(1)} $, let $ P(U_H = 1) = 9/10 $ while other parts stay the same as target task $ T $. For $ T^{(2)} $, let $ Z = \neg X_1 $ while other parts stay the same as target task $ T $. From the causal diagram, we see that $ \Delta^{(1)} = \{H\} $ is editable w.r.t $ X^{(1)} = \{X_1\} $ and $ \Delta^{(2)} = \{Z\} $ is editable w.r.t $ X^{(2)} = \{X_2\} $. Now if the agent is trained in a curriculum $ C = \{T^{(1)}, T^{(2)}\} $, its target task performance will deteriorate instead of improving. To witness, the optimal policy for $ X_2 $ in $ T^{(1)} $ is $ \pi^{(1)}(X_2 = 1\|Z) = 1 $ and the optimal policy for $ X_1 $ in $ T^{(2)} $ is $ \pi^{(2)}(X_1 = 0\|H) = 1 $. After training in $ T^{(1)} $, $ \pi^{(1)} $ has an expected target task reward of 0.55 since $ \pi^{(1)}(X_2 = 1\|Z) $ is not optimal in the target yet. So, the agent proceeds to train in $ T^{(2)} $. It will learn the optimal target policy for $ X_2, \pi^(X_2 = 0\|Z) = 1 $. But in the mean time, optimal policy of $ X_1 $ learned from $ T^{(1)}, \pi^(X_1 = \neg H\|H) = 1, $ is also overwritten by $ \pi^{(2)} $. The agent will only receive 0.5 in the target task, which is even worse than before training in $ T^{(2)} $. This suggests that curriculum $ C $ is not causally aligned. In the above example, the agent fails to learn an optimal policy due to “policy overwriting”. Fig. 4 provides a graphical representation of this phenomenon. Particularly, each source task $ T^{(1)}, T^{(2)} $ covers one of the optimal decision rules over action $ X_1, X_2 $, respectively. An agent trained in one of the source tasks, say $ T^{(1)} $, learns the optimal decision rule $ \pi^_1 $ for action $ X_1 $, but forgets the decision rule $ \pi^_2 $ for the other action $ X_2 $ learned previously in $ T^{(2)} $. The same overwriting also occurs when the agent moves from task $ T^{(2)} $ to $ T^{(1)} $. This means that regardless of how the system designer orders the curriculum, e.g., $ C = \{T^{(1)}, T^{(2)}, T^{(1)}, T^{(2)}, \ldots \} $, the agent will always forget useful skills it picked up from Algorithm 2: Curriculum Learning Input: A curriculum $ C $. Output: A policy $ \pi^{(N)} \in \Pi $. Initialize a baseline policy $ \pi^{(0)} $; for $ j = 1, \ldots, N $ do Update a policy $ \pi^{(j)} $ from $ \pi^{(j-1)} $ such that \[ \pi^{(j)} = \arg \max_{\pi \in \Pi} \mathbb{E}_{M^{(j)}} [R(Y); \pi] \] return $ \pi^{(N)} $; previous source tasks, thus making it unable to achieve satisfactory performance in the target task. This example implies that there are more conditions for a curriculum to be “causally aligned”. 3.1 Designing Causally Aligned Curriculum We will next introduce a novel algorithmic procedure to construct a causally aligned curriculum while avoiding the issue of overwriting. We will focus on a general class of soluble target tasks, which generalizes the perfect recall criterion (Koller & Friedman [2009] in the planning/decision-making literature (Lauritzen & Nilsson [2001]). Definition 7 (Soluble Target Task). A target task $ T = \langle M, \Pi, R \rangle $ is soluble if whenever $ j < i $, $ (Y \cap De(X_i)) \perp\!\!\!\perp \pi_j \| S_i, X_i) $ in $ G_\pi $, where $ \pi_j $ is a newly added regime node pointing to $ X_j $. In words, Def. 7 says that for a soluble target task $ T $, for every action $ X_i \in X $, the input states $ S_i $ summarizes all the states and actions’ history $ S_1, \ldots, S_{i-1}, X_1, \ldots, X_{i-1} $. If this is the case, an optimal policy $ \pi^* $ for task $ T $ is obtainable by solving a series of dynamic programs (Lauritzen & Nilsson [2001]; Koller & Milch [2003]). For instance, the Sokoban game $ T $ graphically described in Fig. 3a is soluble. For every time step $ i = 1, \ldots, H $, given input states $ S_i = \{ L_i, B_i, C_i \} $ and action $ X_i $, regime variables $ \pi_1, \ldots, \pi_{i-1} $ are d-separated from subsequent reward signals $ Y_i, \ldots, Y_H $. Theorem 3 (Causally Aligned Curriculum). For a soluble target task $ T = \langle M, \Pi, R \rangle $, a curriculum $ C = \{ T^{(j)} \}_{j=1}^N $ is causally aligned if the following conditions hold, (i) Every source task $ T^{(j)} \in C $ is causally aligned w.r.t. actions $ X^{(j)} $ (Def. 4); (ii) For every $ j = 1, \ldots, N - 1 $, actions $ X^{(j)} \subseteq X^{(j+1)} $. Consider again the Sokoban game described in Fig. 3a. Let $ C = \{ T^{(j)} \}_{j=1}^H $ be a curriculum such that for every source task $ T^{(j)} $ is obtained by editing the agent and box’s location $ \Delta^{(j)} = \{ L_i, B_i \} $ at time step $ i = H - j + 1 $. We now examine conditions in Thm. 3 and see if $ C $ is causally aligned. First, Condition (i) holds since every source task $ T^{(j)} $ is causally aligned w.r.t. actions $ X^{(j)} = \{ X_{H-j+1}, \ldots, X_H \} $ following discussion in the previous section. Also, Condition (ii) holds since for every $ j = 1, \ldots, H - 1 $, actions $ X^{(j)} \subseteq X^{(j+1)} $. This implies that one could construct a causally aligned curriculum in the Sokoban game by repeatedly editing the agent and box’ location following a reversed topological ordering; Fig. 1c describes such an example. The idea in Thm. 3 suggests a natural procedure for constructing a causally aligned curriculum, which is implemented in FindCausalCurriculum (Algo. 3). Particularly, it assumes access to a curriculum generator $ \text{GEN}(T, \Delta^{(j)}) $ which generates a source task $ T^{(j)} $ by editing a set of states $ \Delta^{(j)} \subseteq V $ in the target task $ T $. It follows a reverse topological ordering over actions $ X = \{ X_1, \ldots, X_H \} $. For every step $ j = H, \ldots, 1 $, the algorithm call the subroutine FindMaxEdit (Algo. 1) to find a set of editable states $ \Delta^{(j)} $ w.r.t. actions $ X^{(j)} = \{ X_j, \ldots, X_H \} $. It then calls the generator $ \text{GEN} $ to generate a source task $ T^{(j)} $ by editing states $ \Delta^{(j)} $. The conditions in Thm. 3 ensure that Algo. 3 returns a causally aligned curriculum. Corollary 1. For a soluble target task $ T = \langle M, \Pi, R \rangle $, let $ G_\pi $ be an intervened causal diagram of $ M $. FindCausalCurriculum $ (T, G_\pi) $ returns a causally aligned curriculum. A more detailed discussion on the additional conditions under which a combination of Algs. 2 and 3 is guaranteed to find an optimal target task policy is provided in App. D. 4 Experiments In this section, we build on Algo. 3 and different curriculum generators to evaluate causally aligned curricula for solving challenging tasks in which confounding bias is present and previous, non-causal generators cannot solve. In particular, we test four best-performing curriculum generators: --- Algorithm 3: FindCausalCurriculum Input: A target task $ T $, a causal diagram $ G_\pi $ Output: A causally aligned curriculum $ C $ Let $ C \leftarrow \emptyset $; for $ j = H, \ldots, 1 $ do Let $ X^{(j)} \leftarrow \{ X_j, \ldots, X_H \} $; Let $ \Delta^{(j)} \leftarrow \text{FindMaxEdit}(G_\pi, X^{(j)}) $; Let $ T^{(j)} \leftarrow \text{GEN}(T, \Delta^{(j)}) $; Let $ C = C \cup \{ T^{(j)} \} $; return $ C $; Figure 5: Target task performance of the agents at different training stages in Colored Sokoban (Row 1) and Button Maze (Row 2) using different curriculum generators (Columns). The horizontal green line shows the performance of the agent trained directly in the target. “original” refers to the unaugmented curriculum generator and “causal” refers to its causally augmented version. ALP-GMM (Portelas et al., 2019), PLR (Jiang et al., 2021), Goal-GAN (Florensa et al., 2018), and Currot (Klink et al., 2022) in two confounded environments with pixel observations: (a) Colored Sokoban, (b) Button Maze, (c) Continuous Button Maze (App. F). All experiments are conducted with five random seeds and reported in Interquartile Mean (IQM) normalized w.r.t the minimum and maximum rewards with 95% confidence intervals shown in shades. See App. F for more details. Colored Sokoban. Consider the same Sokoban game as shown in Example 1. The curriculum generators are allowed to vary the initial location of the agent, to vary the initial box location, and to intervene the box’s color. Without editing, the box color syncs with the true underlying rewards, i.e., pushing a yellow box always yields a positive reward. However, after intervening the box color, this sync is broken and the agent has no information on the right time to push the box. As shown in Fig. 5, agents trained by original curriculum generators failed to converge due to this. After causal augmentation, those misaligned source tasks with intervened box color are all eliminated from the search space during curriculum generation. The causal versions of those generators successfully train the agent to converge efficiently and surpass those trained directly in the target task. Button Maze. In this grid world environment (Chevalier-Boisvert et al., 2018), the agent must navigate to the goal location and step onto it at the right time. Specifically, after pushing the button, the goal region will turn green and yield a positive reward if the agent steps onto it. However, before pushing the button, there is only a 20% chance the agent gets a positive reward for reaching the goal, and the goal randomly blinks between red and green, independent of the underlying rewards. Curriculum generators can intervene the goal color and vary the agent’s initial location but intervening goal colors creates misaligned curricula (Thm. 3). As shown in Fig. 5, agents trained by vanilla curriculum generators failed to learn at all, while the agents trained by their causally-augmented versions all converged to the optimal, even surpassing the one trained directly in the target task. 5 CONCLUSION We develop a formal treatment for automatic curriculum design in confounded environments through causal lenses. We propose a sufficient graphical criterion that edits must conform with to generate causally aligned source tasks in which the agent is guaranteed to learn optimal decision rules for the target task. We also develop a practical implementation of our graphical criteria, i.e., FIND-MAXEDIT, that augments the existing curriculum generators into ones that generate aligned source tasks regardless of the existence of unobserved confounders. Finally, we analyze causally aligned curricula’ design principles with theoretical performance guarantees. The effectiveness of our approach is empirically verified in two high-dimensional pixel-based tasks. 6 ACKNOWLEDGEMENTS This research was supported in part by the NSF, ONR, AFOSR, DoE, Amazon, JP Morgan, and The Alfred P. Sloan Foundation. 7 REPRODUCIBILITY STATEMENT For all the theorems, corollaries, and algorithms, we provide proofs and correctness analysis in App. E. To implement the algorithms, we provide pseudo-code in the main text and App. C. We also provide experiment specifications, environment setup, and neural network hyperparameters in App. F. Colored Sokoban and Button Maze are implemented based on Sokoban (Schrader, 2018) and GridWorld (Chevalier-Boisvert et al., 2018), respectively. REFERENCES David Abel, Will Dabney, Anna Harutyunyan, Mark K Ho, Michael Littman, Doina Precup, and Satinder Singh. On the expressivity of markov reward. 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OkHHJcMroY	I am also a bit surprised by the fact that the mixing time of the original chain $(s_1,a_1,s_2,\ldots)$ does not pop up explicitly in the bounds. This would be a typical behavior for the optimization problems with dependent data. What is the explanation?	PILOT: AN $O(1/K)$-CONVERGENT APPROACH FOR POLICY EVALUATION WITH NONLINEAR FUNCTION APPROXIMATION Zhuqing Liu†, Xin Zhang‡, Jia Liu†, Zhengyuan Zhu‡, Songtao Lu∗ †Department of Electrical and Computer Engineering, The Ohio State University ‡Department of Statistics, Iowa State University ∗IBM Research, IBM Thomas J. Watson Research Center liu.9384@osu.edu, xinzhang@iastate.edu, liu@ece.osu.edu, zhuz@iastate.edu, songtao@ibm.com, ABSTRACT Learning an accurate value function for a given policy is a critical step in solving reinforcement learning (RL) problems. So far, however, the convergence speed and sample complexity performances of most existing policy evaluation algorithms remain unsatisfactory, particularly with non-linear function approximation. This challenge motivates us to develop a new path-integrated primal-dual stochastic gradient (PILOT) method, that is able to achieve a fast convergence speed for RL policy evaluation with nonlinear function approximation. To further alleviate the periodic full gradient evaluation requirement, we further propose an enhanced method with an adaptive-batch adjustment called PILOT+. The main advantages of our methods include: i) PILOT allows the use of constant step sizes and achieves the $O(1/K)$ convergence rate to first-order stationary points of non-convex policy evaluation problems; ii) PILOT is a generic single-timescale algorithm that is also applicable for solving a large class of non-convex strongly-concave minimax optimization problems; iii) By adaptively adjusting the batch size via historical stochastic gradient information, PILOT+ is more sample-efficient empirically without loss of theoretical convergence rate. Our extensive numerical experiments verify our theoretical findings and showcase the high efficiency of the proposed PILOT and PILOT+ algorithms compared with the state-of-the-art methods. 1 INTRODUCTION In recent years, reinforcement learning (RL) has achieved enormous successes in a large number of areas, including healthcare (Petersen et al., 2019), Raghu et al., 2017b), financial recommendation (Theocharous et al., 2015), ranking system (Wen et al., 2023), resources management (Mao et al., 2016) and robotics (Levine et al., 2016), Raghu et al., 2017a), to name just a few. In RL, an agent interacts with an environment and repeats the tasks of observing the current state, performing a policy-based action, receiving a reward, and transitioning to the next state. Upon collecting a trajectory of action-reward sample pairs, the agent updates its policy with the aim of maximizing its long-term accumulative reward. In this RL framework, a key step is the policy evaluation (PE) problem, which aims to learn the value function that estimates the expected long-term accumulative reward for a given policy. Value functions not only explicitly provide the agent’s accumulative rewards, but are also able to update the current policy so that the agent can visit valuable states more frequently (Lagoudakis & Parr, 2003). Regarding PE, two of the most important performance metrics are convergence rate and sample complexity. First, since PE is a subroutine of an overall RL task, developing fast-converging PE algorithms is of critical importance to the overall efficiency of RL. Second, due to the challenges in collecting a large number of training samples (trajectories of state-action pairs) for PEs in RL, reducing the number of samples (i.e., sample complexity) can significantly alleviate the burden of data collection for solving PE problems. These two important aspects motivate us to pursue a fast-converging PE algorithm with a low sample complexity in this work. Among various algorithms for PE, one of the simplest and most effective methods is the temporal difference (TD) learning approach \cite{Sutton1988}. In TD learning, instead of focusing on the predicted and actual outcomes, the key idea is to make the difference between temporally successive predictions small. Specifically, the TD learning approach learns the value function using the Bellman equation to bootstrap from the currently estimated value function. To date, there have been many algorithms proposed within the family of TD learning \cite{Dann2014}. However, most of these methods suffer from either unstable convergence performance, (e.g., TD($\lambda$) \cite{Sutton1988} for off-policy training) or high computational complexity (e.g., the least-squares temporal difference (LSTD) \cite{Boyan2002}) in training with massive features. One reason of the unstable convergence performance of these early attempts is that they do not leverage the gradient-oracle in PE. Thus, in recent years, gradient-based PE algorithms have attracted increasing attention. However, when working with nonlinear DNN models, the convergence performance of the conventional single-timescale TD algorithms may not be guaranteed \cite{Titsiklis1996}. To address this issue, some convergent two-timescale algorithms \cite{Maei2009, Chung2018} have been proposed at the expense of higher implementation complexity. Second, modern PE tasks could involve a large amount of state transition data. To perform PE, algorithms typically need to calculate full gradients that require all training data (e.g., gradient temporal difference (GTD) \cite{Sutton2008} and TD with gradient correction (TDC) \cite{Sutton2009}), which entails a high sample complexity. To the best of our knowledge, all existing works on PE either focus on linear approximation, such as GTD2 \cite{Sutton2009}, PDBG \cite{Du2017}, SVRG \cite{Du2017}, SAGA \cite{Du2017} or they exhibit slower theoretical convergence performance, as observed in STSG \cite{Qiu2020}, VR-STSG \cite{Qiu2020}, nPD-VR \cite{Wai2019} in the sense of achieving a convergence rate that is slower than $O(1/K)$, where $K$ is the number of iterations. Please see detailed discussions in Section 2. In light of the above limitations, in this paper, we ask the following critical question: Can we develop a fast-converging single-timescale algorithm for PE with nonlinear function approximation? In this paper, we give an affirmative answer to the above question. Specifically, we propose an efficient path-integrated primal-dual stochastic gradient algorithm (PILOT) to tackle the PE problem with nonlinear function approximation, which we recast as a minimax optimization problem. The proposed PILOT algorithm admits a simple and elegant single-timescale algorithmic structure. Besides, we further enhance PILOT by proposing PILOT$^+$, which uses adaptive batch sizes to avoid the periodic full gradient evaluation to further reduce sample complexity. The major contribution of this paper is that our proposed algorithms achieve the first $O(1/K)$ convergence rate ($K$ is the number of iterations) with constant step-sizes for PE with nonlinear function approximation, which is the best result in the literature so far. Our main results are highlighted below: - Utilizing a variance reduction technique, our PILOT algorithm facilitates the use of constant step-sizes while maintaining a low sample complexity. We demonstrate that, under reasonable mild assumptions and suitable parameter selections, PILOT attains an $O(1/K)$ convergence rate to a first-order stationary point for a class of nonconvex-strongly-concave (NCX-SCV) minimax problems encountered in RL. To establish this outcome, our convergence analysis employs new proof techniques and resolves an ambiguity present in the current state-of-the-art convergence analyses of Variance Reduction (VR)-based PE methods. - Our PILOT$^+$ algorithm leverages adaptive batch sizes, effectively integrating historical information throughout the optimization process without necessitating backtracking or condition verification. We demonstrate that PILOT$^+$ leads to a substantial reduction in both sample requirements and gradient computation loads. This reduction is made possible by our innovative adaptive batch size technique, which eliminates the need for full gradient evaluation. - Our comprehensive experimental results provide strong evidence of the superior performance of our algorithms compared to state-of-the-art gradient-based PE methods. Additionally, PILOT$^+$ exhibits the capability to further reduce the sample complexity of the PILOT algorithm. It is worth noting that while our primary focus is on PE, the design of our algorithms and the proof techniques developed also hold potential significance in the broader domain of minimax optimization, presenting independent theoretical interest. Table 1: PE algorithms comparison: $M$ is the size of the dataset and $K$ is the total iteration. \| Algorithm \| Function Approx. \| Problem \| Step-size \| Convergence Rate \| \|-----------\|------------------\|------------------\|-----------\|------------------\| \| GTD2 \| Linear \| - \| $\mathcal{O}(1)$ \| - \| \| PDBC \| Linear \| Convex-Concave \| $\mathcal{O}(1)$ \| $\mathcal{O}(1/K)$ \| \| SVRG \| Linear \| Convex-Concave \| $\mathcal{O}(1)$ \| $\mathcal{O}(1/K)$ \| \| SAGA \| Linear \| Convex-Concave \| $\mathcal{O}(1)$ \| $\mathcal{O}(1/K)$ \| \| TATD \| Linear \| Convex-Concave \| $\mathcal{O}(1)$ \| $\mathcal{O}(1/K)$ \| \| STSG \| Nonlinear \| Stochastic/ NCX-SCV \| $\mathcal{O}(1)$ \| $\mathcal{O}(1/K^{1/2})$ \| \| VR-STSG \| Nonlinear \| Stochastic/ NCX-SCV \| $\mathcal{O}(1)$ \| $\mathcal{O}(1/K^{2/3})$ \| \| nPD-VR \| Nonlinear \| Finite-Sum / NCX-SCV \| $\mathcal{O}(1/M)$ \| Slower than $\mathcal{O}(1/K)$ \| \| PILOT \| [Ours.] \| Nonlinear \| Finite-Sum / NCX-SCV \| $\mathcal{O}(1/K)$ \| \| PILOT+ \| [Ours.] \| Nonlinear \| Finite-Sum / NCX-SCV \| $\mathcal{O}(1/K)$ \| 1 The convergence rate of nPD-VR is ambiguous. See the detailed discussions in Sections 2 below and 4. 2 RELATED WORK 1) TD Learning with Function Approximation for PE: TD learning with function approximation plays a vital role in PE. The key idea of TD learning is to minimize the Bellman error for approximating the value function. However, most existing TD learning algorithms with theoretical guarantees focus on the linear approximation setting (e.g., (Sutton et al., 2008; Srikant & Ying, 2019; Xu et al., 2019; Touati et al., 2018; Patil et al., 2023; Li et al., 2021)). Existing works in (Doan et al., 2019; Liu et al., 2015; Macua et al., 2014; Zhang & Xiao, 2019; Patil et al., 2023; Li et al., 2021) provided a finite-time analysis for the distributed TD(0) and showed that the convergence rates of their algorithms are $\mathcal{O}(1/K)$. It was shown in (Du et al., 2017) that PE with linear function approximation by TD(0) can be formulated as a strongly convex-concave or convex-concave problem, and can be solved by a primal-dual method with a linear convergence rate. Unfortunately, the linearity assumption cannot be applied to a wide range of PE problems with nonlinear models. TD learning with nonlinear (smooth) function approximation is far more complex. The work in (Maei et al., 2009) was among the first to propose a general framework for minimizing the generalized mean-squared projected Bellman error (MSPBE) with smooth and nonlinear value functions. Despite their use of two-timescale step-sizes, it’s important to note that this approach yielded slow convergence performance. Other TD methods with nonlinear function approximations for PE include (Wang et al., 2017; 2016). Nonlinear TD learning was also investigated in (Qiu et al., 2020), which proposed two single-timescale first-order stochastic algorithms. However, the convergence rates of their STSG and VR-STSG methods are $\mathcal{O}(1/K^{1/4})$ and $\mathcal{O}(1/K^{1/3})$, while our PILOT algorithm achieves a much faster $\mathcal{O}(1/K)$ convergence rate, matching the standard one in the linear case. In PE with non-linear function approximation, the state-of-the-art and the most related work is (Wai et al., 2019), which showed that minimizing the generalized MSPBE problem is equivalent to solving a non-convex-strongly-concave (NCX-SCV) minimax optimization problem by applying the Fenchel’s duality. However, their best convergence results only hold for a small step-size that is $\mathcal{O}(1/M)$, where $M$ denotes the size of the dataset. This will be problematic for RL problems with a large state-action transition dataset. Furthermore, it is worth highlighting that their convergence rate bound takes the form of $\frac{F(K)}{K} + \text{Constant}$ (cf. Theorem 1, Eq. (26) in Wai et al., 2019). Here, the term $F(K)$ in the denominator inherently relies on the primal and dual values $\theta^{(K)}$ and $\omega^{(K)}$ at the $K$-th iteration. However, the bounding of $\omega^{(K)}$ in Wai et al., 2019 remains unclear, leading to ambiguity when attempting to guarantee an $\mathcal{O}(1/K)$ convergence rate. Therefore, whether an $\mathcal{O}(1/K)$ convergence rate is achievable in single-timescale PE with nonlinear function approximation and constant step-sizes remains an open question thus far. The key contribution and novelty in this paper is that we resolve the above open question by proposing two new algorithms, both achieving an $\mathcal{O}(1/K)$ convergence rate. To establish this result, we propose a new convergence metric (cf. Eq. (5) in Section 4), which necessitates new proof techniques and analysis. For easy comparisons, we summarize our algorithms in a comparison with the related existing works in Table 1. 2) Relations with NCX-SCV minimax Optimization: Although the focus of our paper is on PE, our algorithmic techniques are also related to the general area of NCX-SCV minimax optimization due to the primal-dual MSPBE formulation (cf. Eq. (1) in Section 3). Early attempts in (Nouiehed... et al., 2019; Lin et al., 2020b; Lu et al., 2019) developed gradient descent-ascent algorithms to solve the NCX-SCV minimax problems. However, these methods suffer from a high sample complexity and slow convergence rate. To overcome this limitation, two variance-reduction algorithms named SREDA (Luo et al., 2020) are proposed for solving NCX-SCV minimax problems, which share some similarities to our work. SREDA was later enhanced in Xu et al. (2020) to allow larger step-sizes. However, our algorithms differ from SREDA in the following key aspects: (i) Our algorithms are single-timescale algorithms (see Section 4 for the notions of single-timescale and two-timescale algorithms), which are much easier to implement. In comparison, SREDA is a two-timescale algorithm that also requires solving an inner concave maximization subproblem. To a certain extent, SREDA can be viewed as a triple-loop structure, and hence the implementation complexity of SREDA is higher than ours; (ii) In the initialization stage, SREDA uses a subroutine called PiSARAH to help the SREDA algorithm achieve the desired accuracy at the initialization step and can be seen as an additional step to solve an inner concave maximization subproblem. Thus, SREDA has a higher computation cost than our algorithm. (iii) The number of hyperparameters in SREDA is more than ours and it requires the knowledge of the condition number to set the algorithm’s parameters for good convergence performance. By contrast, our algorithms only require step-sizes $\alpha$ and $\beta$ to be sufficiently small (smaller than the upper bounds we provide in our theorems), which is easier to tune in practice. (iv) SREDA does not provide an explicit convergence rate in their paper (it is also unclear what their convergence rate is from their proof). Yet, we show that our PILOT algorithm has a lower sample complexity than that of SREDA. Another related work in NCX-SCV minimax optimization is Zhang et al. (2021), which provided sample complexity upper and lower bounds. However, there remains a gap between the sample complexity lower and upper bounds in Zhang et al. (2021). By contrast, the sample complexity of our PILOT algorithm is the first to match the lower bound $O(M + \sqrt{M} \epsilon^{-2})$ in Zhang et al. (2021). Furthermore, the algorithm in Zhang et al. (2021) contains an inner minimax subproblem (cf. Line 6 of Algorithm 1 in Zhang et al. (2021)). Solving such a subproblem in the inner loop incurs high computational costs. Due to this reason, the algorithm in Zhang et al. (2021) had to settle for an inexact solution, which hurts the convergence performance in practice. In contrast, our algorithm does not have such a limitation. 3 PRELIMINARIES AND PROBLEM STATEMENT We start by introducing the necessary background of RL, with a focus on the PE problem based on nonlinear function approximation. 1) Policy Evaluation with Nonlinear Function Approximation: RL problems are formulated using the Markov decision process (MDP) framework defined by a five-tuple $\{S, A, P, \gamma, R\}$, where $S$ denotes the state space and $A$ is the action space; $P : S \times A \rightarrow S$ represents state transition probabilities after taking actions; $R$ denotes the space of the received rewards, where each reward corresponds to the agent taking a specific action $a \in A$ from the set of possible actions when the system is in a particular state in the state space $s \in S$ (in this paper, we assume that the state and action spaces are finite but their dimensionality could be large); and $\gamma \in [0, 1)$ is a time-discount factor. For RL problems over an infinite discrete-time horizon $\{t \in \mathbb{N}\}$, the learning agent executes an action $a_t$ according to the state $s_t$ and some policy $\pi : S \times A \rightarrow [0, 1]$. The system then transitions into a new state $s_{t+1}$ in the next time slot, and the agent receives a reward $R(s_t, a_t)$ through its interaction with the environment. The trajectory generated by a policy $\pi$ is a sequence of state-action pairs denoted as $\{s_1, a_1, s_2, a_2, \ldots\}$. Specifically, for a policy $\pi$ (could be a randomized policy), the expected reward received by the agent at state $s$ in any given time slot can be computed as $R^\pi(s_t) = \mathbb{E}_{a \sim \pi(\cdot\|s)}[R^\pi(s_t, a)]$. The value function $V^\pi(s_0) = \mathbb{E}[\sum_{t=0}^{\infty} \gamma^t R(s_t) \| s_0, \pi]$ indicates the long-term discounted reward of policy $\pi$ over an infinite horizon with the initial state at $s_0 \in S$. Also, the Bellman equation implies that $V(\cdot)$ satisfies $V(s) = T^\pi V(s)$, where $T^\pi f(s) \triangleq \mathbb{E}[R^\pi(s) + \gamma f(s') \| a \sim \pi(\cdot\|s), s' \sim P(\cdot\|s, a)]$ denotes the Bellman operator. In RL, the agent’s goal is to determine an optimal policy $\pi^$ that maximizes the value function $V^\pi(s)$ from any initial state $s$. can be reformulated as an NCX-SCV minimax problem, which can be solved by our proposed VR-based single-timescale method efficiently. Here, the rate of $O(\epsilon^{-2})$ measured on the size of the primal objective function is equivalent to $O(1/K)$. However, the first obstacle in solving RL problems stems from evaluating $ V^\pi(\cdot) $ for a given $ \pi $ since $ P(\cdot\|s,a) $ is unknown. Moreover, it is often infeasible to store $ V^\pi(s) $ since the state space $ S $ could be extremely large. To address these challenges, one popular approach in RL is to approximate $ V^\pi(\cdot) $ using a family of parametric and smooth functions in the form of $ V^\pi(\cdot) \approx V_{\theta^\pi}(\cdot) $, where $ \theta^\pi \in \Theta \subseteq \mathbb{R}^d $ is a $ d $-dimensional parameter vector. Here, $ \Theta $ is a compact subspace. For notational simplicity, we will omit all superscripts “$ \pi $” whenever the policy $ \pi $ is clear from the context. In this paper, we focus on nonlinear function approximation, i.e., $ V_{\theta}(\cdot) : S \rightarrow \mathbb{R} $ is a nonlinear function with respect to (w.r.t.) $ \theta $. For example, $ V_{\theta}(\cdot) $ could be based on a $ \theta $-parameterized nonlinear DNN. We assume that the gradient and Hessian of $ V_{\theta}(\cdot) $ exist and are denoted as: $ g_\theta(s) := \nabla_\theta V_{\theta}(s) \in \mathbb{R}^d, H_\theta(s) := \nabla^2_\theta V_{\theta}(s) \in \mathbb{R}^{d \times d} $. Our goal is to find the optimal parameter $ \theta^ \in \mathbb{R}^d $ that minimizes the error between $ V_{\theta^}(\cdot) $ and $ V(\cdot) $. It has been shown in (Liu et al., 2018) that this problem can be formulated as minimizing the mean-squared projected Bellman error (MSPBE) of the value function (Liu et al., 2018, Proposition 1) as follows: \[ \text{MSPBE}(\theta) := \frac{1}{2} \\| \mathbb{E}_{s \sim D^\pi(\cdot)}[(T^\pi V_{\theta}(s) - V_{\theta}(s)) \nabla_\theta V_{\theta}(s)^\top] \\|_{D^{-1}}^2 \\ = \max_{\omega \in \mathbb{R}^d} \left( -\frac{1}{2} \mathbb{E}_{s \sim D^\pi(\cdot)}[(\omega^\top g_\theta(s))^2] + \langle \omega, \mathbb{E}_{s \sim D^\pi(\cdot)}[(T^\pi V_{\theta}(s) - V_{\theta}(s))g_\theta(s)] \rangle \right), \] where $ D^\pi(\cdot) $ is the stationary distribution under policy $ \pi $ and $ D \triangleq \mathbb{E}_{s \sim D^\pi}[g_\theta(s)g_\theta^\top(s)] \in \mathbb{R}^{d \times d} $ and $ \omega $ is referred to as the dual variable. 2) Primal-Dual Optimization for MSPBE: Minimizing $ \text{MSPBE}(\theta) $ in (1) is equivalent to solving a primal-dual minimax optimization problem: $ \min_{\theta \in \mathbb{R}^d} \max_{\omega \in \mathbb{R}^d} L(\theta, \omega) $, where $ L(\theta, \omega) \triangleq \langle \omega, \mathbb{E}_{s \sim D^\pi(\cdot)}[(T^\pi V_{\theta}(s) - V_{\theta}(s))g_\theta(s)^\top] \rangle - \frac{1}{2} \mathbb{E}_{s \sim D^\pi(\cdot)}[(\omega^\top g_\theta(s))^2] $. Since the distribution $ D^\pi(\cdot) $ is unknown and the expectation cannot be evaluated directly, most existing work in the literature (see, e.g., Liu et al., 2015; Wai et al., 2019; Du et al., 2017) considered the following empirical minimax problem by replacing the expectation in $ L(\theta, \omega) $ by a finite-sum sample average approximation based on an $ M $-step trajectory $ \{s_1, a_1, \ldots, s_M, a_M, s_{M+1}\} $ generated by some policy $ \pi $, i.e., \[ \min_{\theta \in \mathbb{R}^d} \max_{\omega \in \mathbb{R}^d} \frac{1}{M} \sum_{i=1}^{M} L_i(\theta, \omega) = \min_{\theta \in \mathbb{R}^d} \max_{\omega \in \mathbb{R}^d} L(\theta, \omega), \] where $ L_i(\theta, \omega) := \langle \omega, [R(s_i, a_i, s_{i+1}) + \gamma V_{\theta}(s_{i+1}) - V_{\theta}(s_i)] \times g_\theta(s_i) \rangle - \frac{1}{2} (\omega^\top g_\theta(s_i))^2 $. In Appendix B, we will also discuss the minimax problem with $ \theta \in \Theta, \omega \in \mathcal{W} $, where $ \Theta, \mathcal{W} $ are convex constrained sets. Solving Problem (2) for MSPBE constitutes the rest of this paper. Note that Problem (2) is non-convex in general (e.g., DNN-based nonlinear approximation). Let $ J(\theta) \triangleq \max_{\omega \in \mathbb{R}^d} L(\theta, \omega) $. Then, we can equivalently rewrite Problem (2) as follows: \[ \min_{\theta \in \mathbb{R}^d} \max_{\omega \in \mathbb{R}^d} L(\theta, \omega) = \min_{\theta \in \mathbb{R}^d} J(\theta). \] Note from (2) that $ L(\theta, \omega) $ is strongly concave w.r.t. $ \omega $, which guarantees the existence and uniqueness of the solution to the problem $ \max_{\omega \in \mathbb{R}^d} L(\theta, \omega), \forall \theta \in \mathbb{R}^d $. Then, given $ \theta \in \mathbb{R}^d $, we define the following notation: $ \omega^(\theta) := \arg\max_{\omega \in \mathbb{R}^d} L(\theta, \omega) $. Subsequently, $ J(\theta) $ can be further written as $ J(\theta) = L(\theta, \omega^(\theta)) $. We aim to minimize $ J(\theta) $ by finding the stationary point of $ L(\theta, \omega) $. For the sake of simplicity in notation, we use $ \omega^ $ to denote $ \omega^(\theta) $. Note that if $ D $ in Eq. (1) is positive definite, Problem (2) is strongly concave in $ \omega $, but non-convex in $ \theta $ in general due to the non-convexity of function $ V_{\theta} $. Thus, the Problem (2) is an NCX-SCV optimization problem. 3) Sample Complexity: In this paper, we adopt the following sample complexity metric to measure the data efficiency of an optimization algorithm, which is widely used in the literature (e.g., Luo et al., 2020; Zhang et al., 2021; Xu et al., 2020): Definition 1 (Sample Complexity).* The sample complexity is defined as the total number of required samplings from the dataset to evaluate incremental first-order oracle (IFO) until an algorithm converges, where one IFO call evaluates a pair of $ (L_i(\theta, \omega), \nabla L_i(\theta, \omega)), i \in [M] $. Although the finite-sum empirical loss is an approximation of the expected loss function for PE, as shown in Chen et al., 2021, under the conditions of bounded instantaneous loss and bounded derivatives (satisfied for most applications in practice), the approximation error of using empirical loss is small with high probability (cf. Chen et al., 2021, Lemma 2)). Thus, the empirical loss has been widely used as a proxy for the expected loss in the literature (Liu et al., 2015; Wai et al., 2019; Du et al., 2017). Algorithm 1 The path-integrated primal-dual stochastic gradient (PILOT). Input: An $M$-step trajectory of the state-action pairs $\{s_1, a_1, s_2, a_2, \ldots, s_M, a_M, s_{M+1}\}$ generated from a given policy; step sizes $\alpha, \beta \geq 0$; initialization points $\theta^0 \in \mathbb{R}^d$, $\omega^0 \in \mathbb{R}^d$. Output: $(\theta^{(\tilde{K})}, \omega^{(\tilde{K})})$, where $\tilde{K}$ is independently and uniformly picked from $\{1, \ldots, K\}$; 1: for $k = 0, 1, 2, \ldots, K - 1$ do 2: If mod($k, q) = 0$, compute full gradients $G_{\theta}^{(k)}, G_{\omega}^{(k)}$ as in Eq. (3). Otherwise, select $\|\mathcal{N}\|$ 3: samples independently and uniformly from $[M]$, and compute gradients as in Eq. (4). 4: Perform the primal-dual updates to obtain the next iterate $\theta^{(k+1)}, \omega^{(k+1)}$ as in Eq. (5). 5: end for 4 THE VARIANCE-REDUCED PRIMAL-DUAL METHOD (PILOT) In this section, we first present the variance-reduced primal-dual (PILOT) algorithm for PE, followed by the theoretical convergence results. Due to space limitation, we provide a proof sketch in the main text and relegate the detailed proofs to the supplementary material. 1) Algorithm Description: The full description of PILOT is illustrated in Algorithm 1. In PILOT, for every $q$ iterations, the algorithm calculates the full gradients as follows: $$G_{\theta}^{(k)} = \frac{1}{\|M\|} \sum_{i \in M} \nabla_{\theta} L_i(\theta^{(k)}, \omega^{(k)}), \quad G_{\omega}^{(k)} = \frac{1}{\|M\|} \sum_{i \in M} \nabla_{\omega} L_i(\theta^{(k)}, \omega^{(k)}), \text{if mod}(k, q) = 0.$$ (3) In all other iterations, PILOT selects a batch $\mathcal{N}$ and computes variance-reduced gradient estimators: $$G_{\theta}^{(k)} = \frac{1}{\|\mathcal{N}\|} \sum_{i \in \mathcal{N}} (\nabla_{\theta} L_i(\theta^{(k)}, \omega^{(k)}) - \nabla_{\theta} L_i(\theta^{(k-1)}, \omega^{(k-1)}) + G_{\theta}^{(k-1)}), \text{if mod}(k, q) \neq 0,$$ $$G_{\omega}^{(k)} = \frac{1}{\|\mathcal{N}\|} \sum_{i \in \mathcal{N}} (\nabla_{\omega} L_i(\theta^{(k)}, \omega^{(k)}) - \nabla_{\omega} L_i(\theta^{(k-1)}, \omega^{(k-1)}) + G_{\omega}^{(k-1)}), \text{if mod}(k, q) \neq 0.$$ (4a) The estimators in (4) are constructed iteratively based on the previous update information $\nabla_{\theta} L_i(\theta^{(k-1)}, \omega^{(k-1)})$ (resp. $\nabla_{\omega} L_i(\theta^{(k-1)}, \omega^{(k-1)})$ ) and $G_{\theta}^{(k-1)}$ (resp. $G_{\omega}^{(k-1)}$). PILOT updates the primal and dual variables as follows: $$\theta^{(k+1)} = \theta^{(k)} - \beta G_{\theta}^{(k)}, \quad \omega^{(k+1)} = \omega^{(k)} + \alpha G_{\omega}^{(k)},$$ (5) where parameters $\alpha$ and $\beta$ are the constant learning rates for the primal and dual updates. Remark 1. Single-Timescale Algorithm: Our PILOT algorithm is a single-timescale algorithm, which is much simpler to implement in practice compared to the two-timescale algorithms shown in Ma et al. (2009); Lin et al. (2020b). To see this, we first restate the notions of single- and two-timescale algorithms in the literature (see, e.g., Dalal et al. (2018)). Let $\alpha_t \geq 0$ and $\beta_t \geq 0$ represent the step-sizes at iteration $t$ for outer- and inner-variable updates, respectively. An algorithm is called a two-timescale algorithm if $\alpha_t / \beta_t \to 0$ or $\alpha_t / \beta_t \to +\infty$ as $t \to \infty$. On the other hand, an algorithm is called a single-timescale algorithm if $0 < C \leq \alpha_t / \beta_t \leq C'$ as $t \to \infty$, where $0 < C, C' < +\infty$ are two positive constants. In our proposed PILOT algorithm, since the step-sizes $\alpha_t$ and $\beta_t$ are constants, our PILOT algorithm is clearly a single-timescale algorithm. 2) Assumptions: Before showing the theoretical results, we first make the following assumptions: Assumption 1 ($\mu$-Strong Concavity). We assume that $L(\theta, \omega)$ is differentiable and $\mu$-strongly concave in $\omega$, where for any $\theta \in \mathbb{R}^d$, $L(\theta, \omega) \leq L(\theta, \omega') + \nabla_{\omega} L(\theta, \omega')^\top (\omega - \omega') - \frac{\mu}{2} \\|\omega - \omega'\\|^2$, $\forall \omega, \omega' \in \mathbb{R}^d$, $\mu > 0$. It can be shown that the condition in Assumption 1 is equivalent to: $\\|\nabla_{\omega} L(\theta, \omega) - \nabla_{\omega} L(\theta, \omega')\\| \geq \mu \\|\omega - \omega'\\|$, $\forall \omega, \omega' \in \mathbb{R}^d$ (see proofs in Zhou (2018) Lemmas 2 and 3)). Assumption 2 ($L_f$-Smoothness). We assume that for $i = 1, 2, \ldots, M$, both gradient $\nabla_{\theta} L_i(\theta, \omega)$ and $\nabla_{\omega} L_i(\theta, \omega)$ are $L_f$-smooth. That is, for all $\theta, \theta' \in \mathbb{R}^d$ and $\omega, \omega' \in \mathbb{R}^d$, there exists a constant $L_f > 0$ such that $\\|\nabla L_i(\theta, \omega) - \nabla L_i(\theta', \omega')\\| \leq L_f (\\|\theta - \theta'\\| + \\|\omega - \omega'\\|)$. Assumption 3 (Boundness from Below). There exists a finite lower bound $J^* = \inf_{\theta} J(\theta) > -\infty$. Assumption 4 (Bounded Variance). There exists a constant $\sigma > 0$ such that for all $\theta \in \mathbb{R}^d$, $\omega \in \mathbb{R}^d$, $\frac{1}{M} \sum_{i=1}^{M} \\|\nabla_{\theta} L_i(\theta, \omega) - \nabla_{\theta} L(\theta, \omega)\\|^2 \leq \sigma^2$ and $\frac{1}{M} \sum_{i=1}^{M} \\|\nabla_{\omega} L_i(\theta, \omega) - \nabla_{\omega} L(\theta, \omega)\\|^2 \leq \sigma^2$. We note that Assumption 1 is satisfied if the number of samples $ M $ is sufficiently large and the matrix $ D $ is positive definite. Assumption 3 is standard in the optimization literature. Assumption 4 is also commonly adopted for proving convergence results of SGD- and VR-based algorithms, or algorithms that draw a mini-batch of samples instead of all samples. Assumption 5 is guaranteed to hold under the compact set condition and common for stochastic approximation algorithms for minimax optimization (Qiu et al., 2020; Lin et al., 2020a). Assumptions 1–4 are also often used in temporal difference (TD) problems (see, e.g., Qiu et al., 2020; Wai et al., 2019). With these assumptions, we are now in a position to present our convergence performance results of PILOT. 3) Convergence Performance: In this paper, we propose a new metric for convergence analysis: \[ \mathcal{M}(\theta, \omega) := \\| \nabla J(\theta) \\|^2 + 2 \\| \omega - \omega^(\theta) \\|^2. \] The first term in (6) measures the convergence of the primal variable $ \theta $. As common in non-convex optimization analysis, $ \\| \nabla J(\theta) \\|^2 = 0 $ indicates that $ \theta $ is a first-order stationary point (FOSP) of Problem (2). The second term in (6) measures the convergence of $ \omega^{(k)} $ to the unique maximizer $ \omega^(\theta^{(k)}) $ for $ L(\theta, \cdot) $. Based on this new convergence metric, we can now introduce the notion of the approximate first-order stationary points. Definition 2. The point $ \{\theta, \omega\} $ is an $ \epsilon $-stationary point of function $ L(\theta, \omega) $ if $ \mathcal{M}(\theta, \omega) \leq \epsilon $. Several important remarks on the connections between our metric $ \mathcal{M}^{(k)} $ and the conventional convergence metrics in the literature are in order. A conventional convergence metric in the literature for NCX-SCV minimax optimization is $ \\| \nabla J(\theta^{(k)}) \\|^2 $ (Lin et al., 2020a; Luo et al., 2020; Zhang et al., 2021; Wu* et al., 2023; Huang et al., 2021; Wu et al., 2023), which is the first term of $ \mathcal{M}^{(k)} $. This is because $ \\| \nabla J(\theta) \\|^2 = 0 $ implies that $ \theta $ is a FOSP. Another conventional convergence metric in the literature of minimizing the empirical MSPBE problem is $ \\| \nabla_\theta L(\theta, \omega) \\|^2 + \\| \nabla_\omega L(\theta, \omega) \\|^2 $ (Tsitsiklis & Van Roy, 1996). Since the nonconvex-strong-concave minimax optimization problem is unconstrained in dual (i.e., $ \omega \in \mathbb{R}^d $), it follows from Lipschitz-smoothness in Assumption 2 and $ \\| \nabla_\omega L(\theta^{(k)}, \omega^(\theta^{(k)})) \\|^2 = 0 $ that $ \\| \omega^{(k)} - \omega^(\theta^{(k)}) \\|^2 \geq L_f^{-2} \\| \nabla_\omega L(\theta^{(k)}, \omega^{(k)}) \\|^2 $. Therefore, the second term in our $ \mathcal{M}^{(k)} $ (i.e., $ 2 \\| \omega^{(k)} - \omega^(\theta^{(k)}) \\|^2 $) is an upper bound of the second term in this conventional metric (i.e., $ \\| \nabla_\omega L(\theta, \omega) \\|^2 $). Thus, $ 2 \\| \omega^{(k)} - \omega^(\theta^{(k)}) \\|^2 $ is a stronger metric than $ \\| \nabla_\omega L(\theta, \omega) \\|^2 $ in the sense that an $ O(1/K) $ convergence rate under $ \mathcal{M}^{(k)} $ implies an $ O(1/K) $ convergence rate of the conventional metric, but the converse is not true. Moreover, the benefit of using $ 2 \\| \omega^{(k)} - \omega^(\theta^{(k)}) \\|^2 $ in our $ \mathcal{M}^{(k)} $ is that its special structure allows us to prove the $ O(1/K) $ convergence, while the second term in the conventional metric does not enjoy such a salient feature. With our proposed convergence metric in (6), we have the following convergence result: Theorem 1.* Under Assumptions 1–3, choose step-sizes: $ \alpha \leq \min\left\{ \frac{1}{4L_f}, \frac{2\mu}{34L_f^2 + 2\mu^2} \right\} $ and $ \beta \leq \min\left\{ \frac{1}{4L_f}, \frac{2(L_f + L_f^2/\mu)}{8\sqrt{7L_f^2}}, \frac{\mu^2}{8\sqrt{34L_f^2}} \right\} $. Let $ q = \|N\| = \lceil \sqrt{M} \rceil $, it holds that: \[ \frac{1}{K} \sum_{k=0}^{K-1} \mathbb{E}[\mathcal{M}^{(k)}] \leq \frac{1}{K \min\{1, L_f^2\}} \left[ \frac{16L_f^2}{\alpha \mu} C_2 + \frac{2}{\beta} C_1 \right] = O\left( \frac{1}{K} \right), \] where $ C_1 \triangleq \mathbb{E}[J(\theta^{(0)})] - \mathbb{E}[J(\theta^)], C_2 \triangleq (\mathbb{E}\\|\omega^(\theta^{(0)}) - \omega^{(0)}\\|^2) $ and $ \theta^* = \min_\theta J(\theta) $. Theorem 1 immediately implies the following result: Corollary 1. The overall sample complexity of PILOT is $ O(\sqrt{M} \kappa^3 \epsilon^{-1} + M) $, where $ \kappa = L_f/\mu $ denotes the condition number. Theorem 1 states that PILOT achieves an $ O(1/K) $ convergence rate to an $ \epsilon $-FOSP. The most challenging part in proving Theorem 1 stems from the fact that one needs to simultaneously evaluate the progresses of the gradient descent in the primal domain and the gradient ascent in the dual domain of the minimax problem. --- To see this, recall that $ D = \mathbb{E}_s [\nabla_\theta V_\theta(s) \nabla_\theta V_\theta(s)^\top] \in \mathbb{R}^{d \times d} $. Note that $ \mu = \lambda_{\min}(D) > 0 $ since $ D $ is positive definite and $ D $ tends to be full-rank as $ M $ increases. Thus, as soon as we find a $ \mu > 0 $ when $ M $ is sufficiently large, this $ \mu $ is independent of $ M $ as $ M $ continues to increase. Remark 2. It is worth noting that the nPD-VR method in (Wai et al., 2019) employs $ \\|\nabla_{\omega} L(\theta^{(k)}, \omega^{(k)})\\|^2 $ in their metric to evaluate convergence. However, this approach yields a term $ F(K) \triangleq \mathbb{E}[L(\theta^{(0)}, \omega^{(0)}) - L(\theta^{(K)}, \omega^{(K)})] $ in their convergence upper bound in the form of $ O(F(K)/K) $ (cf. Theorem 1, Eq. (26) in (Wai et al., 2019)). Since $ F(K) $ depends on $ K $, it is unclear whether the nPD-VR method in (Wai et al., 2019) can achieve an $ O(1/K) $ convergence rate or not. This ambiguous result motivates us to propose this new metric $ \mathfrak{M}(k) $ in Eq. (6) to evaluate the convergence of our PILOT algorithm. Consequently, we bound per-iteration change in $ J(\theta) $ instead of the function $ L(\theta^{(k)}, \omega^{(k)}) $. This helps us avoid the technical limitations of (Wai et al., 2019) and successfully establish the $ O(1/K) $ convergence rate. In addition to the $ O(1/K) $ convergence rate, our PILOT algorithm also enjoys the following salient features: a) Large and Constant Step-Sizes: It is worth noting that PILOT adopts a large $ O(1) $ (i.e., constant) step-size compared to the $ O(1/M) $ step-size of nPD-VR (Wai et al., 2019), where $ M $ represents the dataset size. This also induces a faster empirical convergence speed. Besides, PILOT’s estimator uses fresher information from the previous iteration (see Feature c) below), while VR-STSG (Qiu et al., 2020) and nPD-VR (Wai et al., 2019) only use the information from the beginning of $ q $-sized windows. Collectively, PILOT makes considerably larger progress than state-of-the-art algorithms (Qiu et al., 2020; Wai et al., 2019). b) A Recursive Path-Following VR Approach for minimax Problems: In the literature, most existing single-timescale methods adopt vanilla stochastic gradients as gradient estimators $ G_{\theta,t} $ and $ G_{\omega,t} $, which suffer from slow convergence rates. To the best of our knowledge, the only VR-based single-timescale method is (Qiu et al., 2020). However, (Qiu et al., 2020) is based on the SVRG-type VR technique, which achieves a slower $ O(1/K^{3/2}) $ convergence rate. In comparison, our work is based on an advanced recursive path-following VR-based update, which enables the use of constant step-sizes to achieve the first $ O(1/K) $ convergence rate in the literature. 5 THE ADAPTIVE-BATCH PILOT METHOD (PILOT$^+$) Note that PILOT still requires full gradients every $ q $ iterations. This motivates us to propose an adaptive-batch method called PILOT$^+$ to further lower the sample complexity. 1) Algorithm Description: The full description of PILOT$^+$ is illustrated in Algorithm 2. In PILOT$^+$, our key idea is to use the gradients calculated in the previous loop to adjust the batch size $ N_s $ of the next loop. Specifically, PILOT$^+$ chooses $ N_s $ at the $ k $-th iteration as: \[ \|N_s\| = \min\{c_\gamma \sigma^2 (\gamma^{(k)})^{-1}, c_\epsilon \sigma^2 \epsilon^{-1}, M\}, \] where $ c_\gamma, c_\epsilon > c $ for certain constant $ c $, $ M $ denotes the size of the dataset, and $ \gamma^{(k+1)} = \sum_{i=(n_k-1)q}^{n_k} \\|G_{\theta,i}\\|^2/q $ is the stochastic gradients calculated in the previous iterations. In PILOT$^+$, for every $ q $ iterations, we select $ N_s $ samples independently and uniformly from $[M]$ and compute gradient estimators as follows: \[ G_{\theta}^{(k)} = \frac{1}{\|N_s\|} \sum_{i \in N_s} \nabla_{\theta} L_i(\theta^{(k)}, \omega^{(k)}), \quad G_{\omega}^{(k)} = \frac{1}{\|N_s\|} \sum_{i \in N_s} \nabla_{\omega} L_i(\theta^{(k)}, \omega^{(k)}). \] For other iterations, PILOT$^+$ is exactly the same as PILOT. Next, we will theoretically show that such an adaptive batch-size scheme still retains the same convergence rate, while achieving an improved sample complexity. 2) Convergence Performance: For PILOT$^+$, we have the following theoretical convergence result: Theorem 2. Under Assumptions 1–4, choose step-sizes: $ \alpha \leq \min\{\frac{1}{4L_f}, \frac{2\mu}{34L_f^2 + 2\mu^2}\} $ and $ \beta \leq \min\{\frac{1}{4L_f}, \frac{1}{2(L_f + L_f^2/\mu)}, \frac{\mu}{8\sqrt{7}L_f^2}, \frac{\mu^2}{8\sqrt{34}L_f^2}\} $. Let $ \|N_s\| = \min\{c_\gamma \sigma^2 (\gamma^{(k)})^{-1}, c_\epsilon \sigma^2 \epsilon^{-1}, M\} $, $ q = \lceil \sqrt{M} \rceil $, $ \|\mathcal{N}\| = \lceil \sqrt{M} \rceil $ and $ c_\gamma \geq (288L_f^2/\mu^2 + 8) $ in PILOT$^+$, where $ c_\gamma \geq c $ for some constant $ c > 4K + \frac{68K}{\beta \mu^2} $. With constants $ C_1 \triangleq \mathbb{E}[J(\theta^{(0)})] - \mathbb{E}[J(\theta^)] $ and $ C_2 \triangleq (\mathbb{E}[\\|\omega^(\theta^{(0)}) - \omega^{(0)}\\|^2]) $, it holds that: \[ \frac{1}{K} \sum_{k=0}^{K-1} \mathbb{E}[\mathfrak{M}(k)] \leq \frac{1}{K \min\{1, L_f^2\}} \left[ K \cdot \frac{\epsilon}{2} + \frac{16L_f^2}{\alpha \mu} C_2 + \frac{2}{\beta} C_1 \right] = O\left(\frac{1}{K}\right) + \frac{\epsilon}{2}. \] Theorem 2 immediately implies the following result: Algorithm 2 Adaptive-batch PILOT method (PILOT$^+$). Input: A trajectory of the state-action pairs $\{s_1, a_1, s_2, a_2, \ldots, s_M, a_M, s_{M+1}\}$ generated from a given policy; step sizes $\alpha, \beta \geq 0$; initialization points $\theta^0 \in \Theta$, $\omega^0 \in \mathbb{R}^d$. Output: $(\theta^{(\tilde{K})}, \omega^{(\tilde{K})})$, where $\tilde{K}$ is independently and uniformly picked from $\{1, \ldots, K\}$; 1: for $k = 0, 1, 2, \ldots, K - 1$ do 2: \hspace{1em} If mod$(k, q) = 0$, select $\|N_k\|$ indices independently and uniformly from $[M]$ as in Eq. (7) and calculate stochastic gradients as in Eq. (8); 3: \hspace{1em} If mod$(k, q) \neq 0$, select $\|N\|$ samples independently and uniformly from $[M]$; Compute gradients as in Eq. (4); 4: \hspace{1em} Perform the primal-dual updates as in Eq. (5). 5: end for Corollary 2. The overall sample complexity of PILOT$^+$ is $O(\sqrt{MK^3/\epsilon} + M)$, where $\kappa = L_f/\mu$ denotes the condition number. 6 EXPERIMENTAL RESULTS In this section, we conduct our numerical experiments to verify our theoretical results. We compare our work with the basic stochastic gradient (SG) method [Lin et al., 2020b] and three state-of-the-art algorithms for PE: nPD-VR [Wai et al., 2019], STSG [Qiu et al., 2020] and VR-STSG [Qiu et al., 2020]. Due to space limitation, we provide our detailed experiment settings in the Appendix. ![Figure 1: MountainCar-v0 environment.](image1) ![Figure 2: Cartpole-v0 environment.](image2) ![Figure 3: MountainCar-v0 environment.](image3) ![Figure 4: Cartpole-v0 environment.](image4) Numerical Results: First, we compare the loss value and gradient norm performance based on MountainCar-v0 and Cartpole-v0 with nPD-VR, SG, STSG, and VR-STSG in Figs. 1 and 2. We initialize all algorithms at the same point, which is generated randomly from the normal distribution. We can see that VR-STSG and nPD-VR slowly converge after 40 epochs, while STSG and SG fail to converge. PILOT converges faster than all the other algorithms with the same step-size values. As for Cartpole-v0, we clearly see a trend of approaching zero-loss with PILOT. These results are consistent with our theoretical result that one can use a relatively large step-size with PILOT, which leads to a faster convergence performance. Also, we compare the sample complexity of PILOT and PILOT$^+$ in MountainCar-v0 and Cartpole-v0, and the results are shown in Figs. 3 and 4, respectively. We can see that PILOT$^+$ converges to the same level with much fewer samples than PILOT does. 7 CONCLUSION In this paper, we proposed two algorithms called PILOT and PILOT$^+$ for PE with nonlinear approximation and performed the theoretical analysis of their convergence and sample complexity. The PILOT algorithm is based on a single-timescale framework by utilizing VR techniques. The PILOT algorithm allows the use of constant step-sizes and achieves an $O(1/K)$ convergence rate. The PILOT$^+$ algorithm improves the sample complexity of PILOT by further applying an adaptive batch size based on historical stochastic gradient information. Our experimental results also confirmed our theoretical findings in convergence and sample complexity. ACKNOWLEDGMENTS AND DISCLOSURE OF FUNDING This work has been supported in part by NSF grants CAREER CNS-2110259 and CNS-2112471. REFERENCES Justin A Boyan. Technical update: Least-squares temporal difference learning. Machine Learning, 49:233–246, 2002. Greg Brockman, Vicki Cheung, Ludwig Pettersson, Jonas Schneider, John Schulman, Jie Tang, and Wojciech Zaremba. Openai gym. arXiv preprint arXiv:1606.01540, 2016. Tianyi Chen, Kaiqing Zhang, Georgios B Giannakis, and Tamer Basar. 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R1crLHQ4kf	Moreover, can the adversarial optimization problem be formulated to reduce divergence from the benign data distribution, while still fooling the ASR system? What are the challenges in constructing such	LEVERAGING CHARACTERISTICS OF THE OUTPUT DISTRIBUTION FOR IDENTIFYING ADVERSARIAL AUDIO EXAMPLES Anonymous authors Paper under double-blind review ABSTRACT Adversarial attacks can mislead automatic speech recognition (ASR) systems into producing an arbitrary desired output. This is easily achieved by adding imperceptible noise to the audio signal, thus posing a clear security threat. To prevent such attacks, we propose a simple but efficient adversarial example detection strategy applicable to any ASR system that predicts a probability distribution over output tokens in each time step. We measure a set of characteristics of this distribution: the median, maximum, and minimum over the output probabilities, the entropy of the distribution, as well as the Kullback-Leibler and the Jensen-Shannon divergence with respect to the distributions of the subsequent time step. Then, by leveraging the characteristics observed for both benign and adversarial data, we apply binary classifiers, including simple threshold-based classification, ensembles of these simple classifiers, and neural networks. In an extensive analysis of different state-of-the-art ASR systems and language data sets, we demonstrate the supreme performance of this approach, receiving a mean area under the receiving operator characteristic (AUROC) for distinguishing adversarial examples against clean and noisy data higher than 99% and 98%, respectively. To assess the robustness of our method, we propose adaptive attacks that are constructed with an awareness of the defense mechanism in place. This results in a decrease in the AUROC, but at the same time, the adversarial clips become noisier, which makes them easier to detect through filtering and creates another avenue for preserving the system’s robustness. 1 INTRODUCTION Voice recognition technologies are widely used in the devices that we interact with daily—in smartphones or virtual assistants—and are also being adapted for more safety-critical tasks like self-driving cars (Wu et al., 2022) and healthcare applications. Safeguarding these systems from malicious attacks thus plays a more and more critical role, e.g., manipulated erroneous transcriptions can potentially lead to breaches in customer security. Another example involves the targeting of commercial speech recognition devices like Google Assistant, Google Home, Microsoft Cortana, and Amazon Echo using over-the-air attacks. They use substitute models to mimic the unknown target model, aiming to make the system recognize their desired inputs (Chen et al., 2020). By modifying an audio signal for the Kaldi ASR system, for example, the system could output a false transcription containing the command to purchase a product (Schönherr et al., 2019). State-of-the-art ASR systems are based on deep learning (Kahn et al., 2020; Chung et al., 2021). Unfortunately, deep neural networks (NN) are highly vulnerable to adversarial attacks, since the inherent properties of the model make it easy to generate an input that is necessarily mislabeled, simply by incorporating a low-level additive perturbation (Szegedy et al., 2014; Goodfellow et al., 2015; Ilyas et al., 2019; Du et al., 2020). A well-established method to generate adversarial examples (AE), which is also applicable to ASR systems, is the Carlini & Wagner (C&W) attack (Carlini & Wagner, 2018). It aims to minimize a perturbation $\delta$ that—when added to a benign audio signal $x$—induces the system to recognize a phrase chosen by the attacker. The psychoacoustic attack (Schönherr et al., 2019; Qin et al., 2019) specifically developed for ASR systems goes one step further than the C&W attack. By considering principles of acoustic perception, it creates an inconspicuous disturbance $\delta$ utilizing time-frequency masking, i.e., it shapes the perturbations to fall below the estimated time-frequency masking threshold of human listeners, rendering $\delta$ hardly perceptible, and sometimes even inaudible to humans. Motivated by the security gap of ASR in the presence of adversarial attacks, in this work, we introduce a novel detection technique to distinguish benign from adversarial data by analyzing the distribution of tokens generated by an ASR system at each output step. Our method relies on the observed statistical characteristics of attacked samples and trains Gaussian classifiers (GC), ensemble models, and NN using these as features. To assess the generality of our findings, we evaluate our method’s performance across diverse state-of-the-art ASR models and datasets that cover a range of languages. Empirical results confirm that the proposed detection technique effectively differentiates between benign and targeted adversarial data, achieving an AUROC exceeding 99% in all tested end-to-end (E2E) models. To assess the effectiveness of our defense in more challenging scenarios, we test our classifiers when faced with noisy audio data, untargeted attacks, and create adaptive adversarial samples, assuming the attacker to have complete knowledge about the defense mechanism. While the classifiers demonstrate robustness w.r.t. noise, they are vulnerable to adaptive attacks. However, as the resulting adversarial audio files are more distorted, they are easier to spot for human ears and identifiable using filtering techniques. We demonstrate that our approach surpasses the leading temporal dependency technique and the noise flooding method by achieving an improvement in all test data. Moreover, our method is suitable for use with any ASR system that forecasts a probability distribution over output tokens at each time step, and it eliminates the need for supplementary data preprocessing, adversarial training augmentation, or model fine-tuning. 2 RELATED WORK When it comes to mitigating the impact of adversarial attacks, there are two main research directions. On the one hand, there is a strand of research dedicated to enhancing the robustness of models. On the other hand, there is a separate research direction that focuses on designing detection mechanisms to recognize the presence of adversarial attacks. Concerning the robustness of models, there are diverse strategies, one of which involves modifying the input data within the ASR system. This concept has been adapted from the visual to the auditory domain. Examples of input data modifications include quantization, temporal smoothing, down-sampling, low-pass filtering, slow feature analysis, and auto-encoder reformation (Meng & Chen [2017], Guo et al. [2018], Pizarro et al. [2021]). However, these techniques become less effective once integrated into the attacker’s deep learning framework (Yang et al. [2019]). Another strategy to mitigate adversarial attacks is to accept their existence and force them to be perceivable by humans (Eisenhofer et al. [2021]), with the drawback that the AEs can continue misleading the system. Adversarial training (Madry et al. [2018]), in contrast, involves employing AEs during training to enhance the NN’s resiliency against adversarial attacks. Due to the impracticality of covering all potential attack classes through training, adversarial training has major limitations when applied to large and complex data sets, such as those commonly used in speech research (Zhang et al. [2019]). Additionally, this approach demands high computational costs and can result in reducing the accuracy on benign data. A recent method borrowed from the field of image recognition is adversarial purification, where generative models are employed to cleanse the input data prior to inference. However, only a few studies have investigated this strategy within the realm of audio. Presently, its ASR applications are confined to smaller vocabularies, and it necessitates substantial computational resources, while also resulting in decreased accuracy when applied to benign data (Wu et al., 2023). In the context of improving the discriminative power against adversarial attacks, Rajaratnam & Kalita (2018) introduced a noise flooding (NF) method that quantifies the random noise needed to change the model’s prediction, with smaller levels observed for AEs. However, NF was only tested against a specific untargeted attack on a 10-word speech classification system. A prominent non-differentiable approach uses the inherent temporal dependency (TD) in raw audio signals (Yang et al., 2019). This strategy requires a minimal length of the audio stream for optimal performance. Unfortunately, Zhang et al. (2020) successfully evaded the detection mechanism of TD by preserving the necessary temporal correlations, leading to the generation of robust AEs once again. Däubener et al. (2020) proposed AEs detection for hybrid ASR systems based on uncertainty measures. They applied their method to a limited vocabulary tailored for digit recognition. Two of these uncertainty metrics—the mean Kullback-Leibler divergence (KLD) and mean entropy—are also among those characteristics of the output distribution that we investigate, next to many others, for constructing defenses against AEs in this paper. It’s worth noting that Meyer et al. (2016) also utilized the averaged KLD between the output distributions of consecutive time-steps (which they referred to as mean temporal distance), but to assess the reliability of an ASR output over time. 3 BACKGROUND Adversarial attacks In order to keep things convenient, we assume that the label transcript $ y $ and the input audio signal $ x $ are related by $ y = f(x) $, where $ f(\cdot) $ refers to the ASR system’s function, which maps an audio input to the sequence of words it most likely contains. To create a targeted AE, we need to find a small perturbation $ \delta $ of the input that causes the ASR system to predict the desired transcript $ \hat{y} $ given $ x + \delta $, i.e., $ f(x + \delta) = \hat{y} \neq y = f(x) $. This perturbation $ \delta $ is usually constructed by gradient descent-based minimization of the following function \[ l(x, \delta, \hat{y}) = l_t(f(x + \delta), \hat{y}) + c \cdot l_a(x, \delta), \] (1) which includes two loss functions: (1) a task-specific loss, $ l_t(\cdot) $, to find a distortion that induces the model to output the desired transcription target $ \hat{y} $, and (2) an acoustic loss, $ l_a(\cdot) $, that is used to make the noise $ \delta $ smaller in energy and/or imperceptible to human listeners. In the initial steps of the iterative optimization procedure, the weighting parameter $ c $ is usually set to small values to first find a viable AE. Later, $ c $ is often increased, in order to minimize the distortion, to render it as inconspicuous as possible. The most common targeted attacks for audio are the C&W Attack and Qin’s Imperceptible Attack, two well-established optimization-based adversarial algorithms. These techniques have proven successful in targeted attacks and offer a publicly available PyTorch implementation. In the C&W attack (Carlini & Wagner, 2018), $ l_t $ is the negative log-likelihood of the target phrase and $ l_a = \|\delta\|^2_2 $. Moreover, $ \|\delta\| $ is constrained to be smaller than a predefined value $ \epsilon $, which is decreased step-wise in an iterative process. The Imperceptible Attack (Qin et al., 2019) is divided into two stages. The first stage of the attack follows the approach outlined by C&W. The second stage of the algorithm aims to decrease the perceptibility of the noise by using frequency masking, following psychoacoustic principles. Moreover, several untargeted attacks have been proposed. These include the projected gradient descent (PGD) (Madry et al., 2018), a well-known optimization-constrained method, as well as two model-independent attacks—the Kenansville attack (Abdullah et al., 2020, 2021) utilizing signal processing methods, and the genetic attack (Alzantot et al., 2018), a gradient-free optimization algorithm. End-to-end ASR systems An E2E ASR system (Prabhavalkar et al., 2023) can be described as a unified ASR model that directly transcribes a speech waveform into text, as opposed to orchestrating a pipeline of separate ASR components. Here, the system directly converts a sequence of acoustic input features into a sequence of tokens (e.g., phonemes, characters, or words). Ideally, E2E ASR models are fully differentiable and thus can be trained end-to-end by maximizing the conditional log-likelihood with respect to the desired output. Various E2E ASR models follow an encoder-only or an encoder-decoder architecture and typically are built using recurrent neural network (RNN) or transformer layers. Special care is taken of the unknown temporal alignments between the input waveform and output text, where the alignment can be modeled explicitly (e.g., CTC (Graves et al., 2006), RNN-T (Graves, 2012)), or implicitly using attention (Watanabe et al., 2017). Furthermore, language models can be integrated in order to improve prediction accuracy by considering the most probable sequences (Toshniwal et al., 2018). 4 OUTPUT DISTRIBUTION-BASED DEFENSE APPROACH We propose to leverage the probability distribution over the tokens from the output vocabulary in order to identify adversarial attacks. A schematic of our approach is displayed in Fig. 1. An audio clip—either benign or malicious—is fed to the ASR system. The system then generates probability distributions over the output tokens in each time step. The third step is to compute pertinent characteristics of these output distributions, as detailed below. Then, we use a function (i.e., the mean, median, maximum, or minimum) to aggregate the values of the characteristics to a single score per utterance. Lastly, we employ a binary classifier for differentiating adversarial instances from test data samples. Characteristics of the output distribution For each time step $t$, the ASR system produces a probability distribution $p^{(t)}$ over the tokens $i \in V$ of the output vocabulary $V$. For an output utterance of length $T$ we compute the following quantities of this distribution for every $t \in \{1, \ldots, T\}$: - the median of $p^{(t)}(i), i = 1, 2, \ldots, \|V\|$, the minimum $\min_{i \in \{1,\ldots,\|V\|\}} p^{(t)}(i)$, the maximum $\max_{i \in \{1,\ldots,\|V\|\}} p^{(t)}(i)$, - the Shannon entropy $H(p^{(t)}) = -\sum_{i=1}^{\|V\|} p^{(t)}(i) \cdot \log p^{(t)}(i)$, - the Kullback–Leibler divergence (KLD) between the output distributions in two successive time steps $D_{KL}(p^{(t)} \|\| p^{(t+1)}) = \sum_{i=1}^{\|V\|} p^{(t)}(i) \cdot \log \frac{p^{(t)}(i)}{p^{(t+1)}(i)}$, and the Jensen-Shannon divergence (JSD) between the output distributions in two successive time steps, which is obtained as a symmetrized alternative of the Kullback-Leibler divergence $$D_{JSD}(p^{(t)}, p^{(t+1)}) = \frac{1}{2} D_{KL}(p^{(t)} \|\| M) + \frac{1}{2} D_{KL}(p^{(t+1)} \|\| M), \text{ where } M = \frac{1}{2}(p^{(t)} + p^{(t+1)}).$$ We aggregated the step-wise median, minimum, maximum, and entropy over all steps $t = 1, \ldots, T$ of the output sequence into a single score by taking the mean, median, minimum, or maximum w.r.t. the respective values for different time steps $t$. Binary classifier The extracted characteristics of the output distribution can then be used as features for a binary classifier. An option to obtain simple classifiers is to fit a Gaussian distribution to each score computed for the utterances from a held-out set of benign data. If the probability of a new audio sample is below a chosen threshold, this example is classified as adversarial. For illustration, Fig. 2 displays histograms of the mean entropy values for the LSTM-LAS-CTC model’s predictive distribution over benign and adversarial data using LibriSpeech. A more sophisticated approach is to employ ensemble models (EM), in which multiple Gaussian distributions, fitted to a single score each, produce a unified decision by a majority vote. Another option is to construct an NN that takes all the characteristics described above as input. Adaptive attack An adversary with complete knowledge of the defense strategy can implement so-called adaptive attacks. In order to show the advantage of our proposed defense, we analyze the options for adaptive AEs. For this, we construct a new loss $l_k$ by adding a penalty $l_s$ to the loss function in equation (1) weighted with some factor $\alpha$: $$l_k(x, \delta, \hat{y}) = (1 - \alpha) \cdot l(x, \delta, \hat{y}) + \alpha \cdot l_s(x).$$ When attacking a Gaussian classifier that is based on characteristic $c$, $l_s$ corresponds to the $L_1$ norm of the difference between the mean $\overline{s}^c$ of the Gaussian fitted to the respective scores of benign data (resulting from aggregating $c$ over each utterance) and the score of $x$. When attacking an EM, $l_s$ is set to $$l_s(x) = \sum_{i=1}^{T} \|\overline{s}^{c_i} - s^{c_i}(x)\|,$$ where $c_1 \ldots c_T$ corresponds to the characteristics used by the Gaussian classifiers of the ensemble is composed of. In the case of NNs, $l_s(x)$ is simply the $L_1$ norm, quantifying the difference between the NN’s predicted outcome (a probability value) and one (indicating the highest probability for the benign category). 5 EXPERIMENTS 5.1 EXPERIMENTAL SETTINGS Datasets We use the LibriSpeech dataset (Panayotov et al., 2015) comprises approximately 1,000 hours of English speech, sampled at a rate of 16KHz, extracted from audiobooks. We further use Aishell (Bu et al., 2017), an open-source speech corpus for Mandarin Chinese. Since Chinese is a tonal language, the speech of this corpus exhibits significant and meaningful variations in pitch. Additionally, we consider the Common Voice (CV) corpus (Ardila et al., 2020), one of the largest multilingual open-source audio collections available in the public domain. Created through crowdsourcing, CV includes additional complexities within the recordings, such as background noise and reverberation. ASR systems We analyzed fully integrated Pytorch-based deep learning end-to-end speech engines. In order to assess the versatility of our method, which relies on identifying specific characteristics in the system response to attacked samples, we trained various ASR models on different datasets and languages such as English, German, Italian, and Mandarin. These models generate diverse output formats, depending on their tokenizer selection, which can encode either characters or subwords. Specifically, the models we use produce output structures with neuron counts of 32, 500, 1,000, 5,000, or 21,128. We investigate three different models. The first employs a wav2vec2 encoder (Baevski et al., 2020) and a CTC decoder. The second integrates an encoder, a decoder, and an attention mechanism between them, as initially proposed with the Listen, Attend, and Spell (LAS) system (Chan et al., 2016), employing a CRDNN encoder and a LSTMs decoding (Chorowski et al., 2015). The third model implements a transformer architecture relying on attention mechanisms for both encoding and decoding (Vaswani et al., 2017; Wolf et al., 2020). The models are shortly referred to as wav2vec, LSTM, and Trf, respectively, in our tables. To improve generalization, we applied standard data augmentation techniques provided in SpeechBrain: corruption with random samples from a noise collection, removing portions of the audio, dropping frequency bands, and resampling the audio signal at a slightly different rate. Adversarial attacks To generate the AEs, we utilized a repository that contains a PyTorch implementation of all considered attacks (Olivier & Raj, 2022). We randomly selected 200 samples from the test set, with 100 of them designated for testing purposes. For targeted attacks, each of these samples was assigned a new adversarial target transcript sourced from the same dataset. Our selection process adhered to four guiding principles: (1) the audio file’s original transcription cannot be used as the new target description, (2) there should be an equal number of tokens in both the original and target transcriptions, (3) each audio file should receive a unique target transcription, and (4) audio clips must be no longer than five seconds. We reduced the audio clip Table 1: Comparison of the performance of ASR systems on benign and noisy data, in terms of word and sentence error rate on 100 utterances. LM denotes the language model. \| Model \| Language \| LM \| Benign data \| Noisy data \| \|-----------\|----------------\|----\|-------------\|------------\| \| \| \| \| WER SER \| SER SNRSeg \| SNR \| \| LSTM \| Italian (It) \| x \| 15.65% 52% \| 31.74% 72% \| -3.65 6.52 \| \| LSTM \| English (En) \| x \| 5.37% 31% \| 8.46% 45% \| 2.75 6.67 \| \| LSTM \| English (En-LM)\| ✓ \| 4.22% 24% \| 5.90% 27% \| 2.75 6.67 \| \| wav2vec \| Mandarin (Ma) \| ✓ \| 4.37% 28% \| 8.49% 43% \| 5.25 4.50 \| \| wav2vec \| German (Ge) \| x \| 8.65% 33% \| 16.08% 51% \| -2.66 7.85 \| \| Trf \| Mandarin (Ma) \| x \| 4.79% 29% \| 7.40% 40% \| 5.25 4.50 \| \| Trf \| English (En) \| ✓ \| 3.10% 20% \| 11.87% 44% \| 2.75 6.67 \| Table 2: Quality of 100 generated C&W, Psychoacoustic, and adaptive attacks, measured by the average performance of the ASR systems across all models w.r.t the target utterances as well as the SNRs. Adaptive attack customized to target a GC optimized for the most effective characteristic. \| Model \| WER SER \| C&W attack SNRSeg \| SNR \| WER SER \| Psychoacoustic attack SNRSeg \| SNR \| WER SER \| Adaptive attack SNRSeg \| SNR \| \|-----------\|---------\|-------------------\|-----\|---------\|-------------------------------\|-----\|---------\|------------------------\|-----\| \| LSTM (It) \| 0.84% 3.00% \| 17.79 \| 44.51 \| 0.84% 3.00% \| 18.17 \| 38.52 \| 0.84% 3.00% \| -1.47 \| 18.36 \| \| LSTM (En) \| 1.09% 2.00% \| 14.91 \| 33.29 \| 1.09% 2.00% \| 15.14 \| 31.92 \| 0.30% 1.00% \| 0.23 \| 14.01 \| \| LSTM (En-LM)\| 1.19% 2.00% \| 17.50 \| 36.46 \| 1.19% 2.00% \| 17.82 \| 33.93 \| 0.40% 1.00% \| 3.18 \| 16.82 \| \| wav2vec (Ma)\| 0.08% 1.00% \| 22.22 \| 31.35 \| 0.08% 1.00% \| 22.73 \| 30.66 \| 0.08% 1.00% \| -4.30 \| 4.09 \| \| wav2vec (Ge)\| 0.00% 0.00% \| 20.58 \| 50.86 \| 0.00% 0.00% \| 21.08 \| 41.46 \| 0.00% 0.00% \| -12.96 \| 10.88 \| \| Trf (Ma) \| 0.00% 0.00% \| 31.93 \| 49.35 \| 0.00% 0.00% \| 29.47 \| 32.69 \| 0.00% 0.00% \| -1.09 \| 8.01 \| \| Trf (En) \| 0.00% 0.00% \| 27.85 \| 53.54 \| 0.00% 0.00% \| 28.70 \| 37.68 \| 0.00% 0.00% \| -0.19 \| 14.69 \| length to save time and resources, as generating AEs for longer clips can take up to an hour, depending on the computer and model complexity (Carlini & Wagner [2018]). A 5-sec length was a favorable trade-off between time/resources, and the number of AEs created per model. A selection of benign, adversarial, and noisy data employed in our experiments are available online at https://confunknowm.github.io/characteristics_demo_AES/. We initialize the adaptive attack with inputs that are already misleading the system. Then, to generate adaptive AEs, we follow the approach of minimizing the loss function described in equation 2 and execute 1,000 additional iterations on 100 randomly chosen AEs. We evaluate the adaptive attacks by keeping the $\alpha$ value constant at a value of 0.3, while the $\delta$ factor, which is gradually reduced in an iterative manner to reduce noise, remains unchanged during the initial 500 iterations. This approach noticeably diminishes the discriminative capability of our defense across all models. However, this reduction in discriminative power comes at the expense of generating noisy data that, as evidenced by our experimental results in Subsection 5.3, can be easily detected through filtering. Further experiments were carried out using different configurations; these changes resulted in data with lower noise levels but also led to weaker attacks. Detailed outcomes of these experiments are available in the App. A.1 Adversarial example detectors We construct three kinds of binary classifiers: Based on the 24 single scores, we obtain 24 simple Gaussian classifiers (GC) per model. To construct an ensemble model, we implement a majority voting technique, utilizing a total of $T \in \{3, 5, 7, 9\}$ GCs. The choice of which GCs to incorporate is determined by evaluating the performance of each characteristic across all models and ranking them in descending order based on the results of the validation set. The outcome of the ranking can be found in App. A.2. The neural network architecture consists of three fully connected layers, each with 72 hidden nodes, followed by an output layer. We employ a sigmoid activation function to generate a probability output in the range of 0 to 1 that can be converted to class values. The network is trained using ADAM optimization (Kingma & Ba [2015]) with a learning rate of 0.0001 for 250 epochs. Running the assessment with our detectors took approx. an extra 18.74 ms per sample, utilizing an NVIDIA A40 with a memory capacity of 48 GB, see App. A.3 for more details. 5.2 Quality of ASR systems and adversarial attacks To assess the quality of the trained models as well as the performance of the AEs, we measured the word error rate (WER), the character error rate (CER), the sentence error rate (SER), the Signal- Table 3: Quality of 100 generated PGD, Genetic, and Kenansville attacks, measured by the average performance of the ASR systems across all models w.r.t the true labels as well as the SNRs. \| Model \| WER \| PGD attack SER \| SNR \| Genetic attack SER \| SNR \| Kenansville attack SER \| SNR \| \|------------\|-----\|----------------\|-----\|-------------------\|-----\|------------------------\|-----\| \| LSTM (lt) \| 121%\| 100% \| 7.39\| 25.76 \| 41.6%\| 83.0% \| 3.04\| \| \| \| \| \| \| \| \| \| \| LSTM (En) \| 95% \| 100% \| 15.13\| 25.59 \| 24.5%\| 85.0% \| 6.49\| \| \| \| \| \| \| \| \| \| \| LSTM (En-LM)\| 100%\| 100% \| 15.19\| 26.21 \| 23.3%\| 83.0% \| 6.63\| \| \| \| \| \| \| \| \| \| \| wav2vec (Ma)\| 90% \| 100% \| 20.09\| 23.68 \| 36.2%\| 94.0% \| 6.24\| \| \| \| \| \| \| \| \| \| \| wav2vec (Ge)\| 102%\| 100% \| 6.88 \| 26.79 \| 30.7%\| 78.0% \| 1.72\| \| \| \| \| \| \| \| \| \| \| Trf (Ma) \| 126%\| 100% \| 19.49\| 26.41 \| 44.1%\| 96.0% \| 4.36\| \| \| \| \| \| \| \| \| \| \| Trf (En) \| 102%\| 100% \| 14.88\| 26.58 \| 17.8%\| 77.0% \| 8.79\| Signal-to-Noise Ratio (SNR), and the Segmental Signal-to-Noise Ratio (SNR\textsubscript{seg}). The latter measures the adversarial noise energy in Decibels and considers the entire audio signal. Thus, a higher SNR\textsubscript{seg} indicates less additional noise. Specific information about each of these formulas is available in the App.\[A.4\] Quality of ASR systems Tab.\[16\] in the App. reports the results achieved with different SpeechBrain recipes on all datasets. The performance is consistent with those documented by Ravanelli et al.$2021$, where you can also find detailed hyperparameter information for all these models. To determine the classifier’s effectiveness in a situation that better mimics reality, 100 benign audio clips are contaminated with background noise. This involves introducing random samples from a noise dataset into the speech signal. The noise instances are randomly sampled from the Freesound section of the MUSAN corpus $Snyder et al., 2015$ $Ko et al., 2017$, which includes room impulse responses, as well as 929 background noise recordings. We utilize SpeechBrain’s environmental corruption function to add noise to the input signal. Tab.\[1\] presents the performance of the ASR systems for noisy data, utilizing a total of 100 utterances. The impact on system performance is evident, resulting in a significant rise in WER due to the low SNR ratio. Quality of adversarial attacks To estimate the effectiveness of the targeted adversarial attacks we measured the error w.r.t. the target utterances, reported in Tab.\[2\]. We achieved nearly 100% success in generating targeted adversarial data for all attack types across all models. The model with the lowest average SNR distortion registered at 31.35 dB, while the highest—i.e., the least distorted—was 53.54 dB. In a related study by Carlini & Wagner$2018$, they reported a mean distortion of 31 dB. In contrast, for untargeted attacks, we measured the error relative to the true label, the higher the WER the stronger the attack. We consider it a genuine threat as one where the attack produces a WER of at least 50%, surpassing the impact influence of background noise. Diverse settings were explored in our experiments, and these are detailed in the App.\[A.6\]. Both PGD and Kenansville regulate the distortion of the attack using an SNR factor to limit the perturbations, but in different ways, with PGD achieving optimal results at a factor of 25, while Kenansville performed best at a factor of 10. In the case of a genetic attack, we found a minimal effect on the WER, failing to reach 50% across all models, these results are presented in Tab.\[3\]. In general, our findings are in line with the results discussed by Olivier & Raj$2022$. When generating AEs with the proposed adaptive adversarial attack, we also managed to achieve an almost 100% success rate, see Tab.\[2\]. However, the AEs turned out to be much noisier, as displayed by a maximum average SNR value of 18.36 dB when comparing all models. This makes the perturbations more easily perceptible to humans. 5.3 Performance of Adversarial Example Detectors Detecting C&W and Psychoacoustic attacks To distinguish benign audio clips from malicious inputs, we calculate the characteristic scores and use them to train binary classifiers as described in Sec.\[4\]. The detection performance of our classifiers w.r.t C&W and Psychoacoustic attacks are quite similar. Therefore, we present the C&W results in Tab.\[4\] and include the Psychoacoustic results in the App.\[A.7\]. We contrast our binary classifiers with NF and TD. For the GC, we report for each model the performance for the characteristic best-performing on the validation set (detailed results for all other characteristics can be found in the App.\[A.8\]). Our findings show that the proposed binary classifiers consistently outperform NF and TD across all models when distinguishing between benign and adversarial data, achieving an impressive discrimination accuracy of over 99% in every case, regardless of the deep learning architecture used by the ASR system, the data it was trained on, Table 4: Comparing classifiers on clean and noisy data, evaluating AUROC for all models using 100 samples from the clean test set and 100 C&W AEs. () denotes best-performing score-characteristic. \| Model \| Score-Characteristic() \| Noisy vs. C&W adversarial data \| Benign vs. C&W adversarial data \| \|-------------\|--------------------------\|-------------------------------\|--------------------------------\| \| \| \| NF \| TD \| GC \| NN \| NF \| TD \| GC \| NN \| \| LSTM (lt) \| Mean-Median \| 0.9218 \| 0.8377 \| 0.9686 \| 0.9557 \| 0.8762 \| 0.8923 \| 0.9980 \| 0.9962 \| \| LSTM (En) \| Mean-Median \| 0.9289 \| 0.9695 \| 0.9996 \| 0.9868 \| 0.9697 \| 0.9993 \| 0.9992 \| 0.9992 \| \| LSTM (En-LM)\| Max-Max \| 0.9680 \| 0.9622 \| 0.9835 \| 0.9875 \| 0.9345 \| 0.9293 \| 0.9828 \| 0.9903 \| \| wav2vec (Ma)\| Mean-Hotspot \| 0.9408 \| 0.9294 \| 0.9816 \| 0.9830 \| 0.9383 \| 0.9319 \| 0.9941 \| 0.9910 \| \| wav2vec (Ge)\| Max-Min \| 0.9372 \| 0.9557 \| 0.9652 \| 0.9992 \| 0.8725 \| 0.9836 \| 0.9941 \| 0.9910 \| \| Trf (Ma) \| Median-Max \| 0.9572 \| 0.9790 \| 0.9864 \| 0.9803 \| 0.9243 \| 0.9828 \| 0.9978 \| 0.9969 \| \| Trf (En) \| Max-Median \| 0.9702 \| 0.9448 \| 0.9287 \| 0.9844 \| 0.8998 \| 0.9828 \| 1.0000 \| 1.0000 \| Average AUROC across all models: 0.9462 / 0.9386 / 0.9695 / 0.9821 / 0.8990 / 0.9620 / 0.9952 / 0.9953 Table 5: Classification accuracies for classifiers, based on a threshold for a maximum 1% FPR (if possible) and a minimum 50% TPR, using 100 benign data and 100 C&W AEs. \| Model \| TD \| GC \| EM=3 \| EM=5 \| EM=7 \| EM=9 \| NN \| \|-------------\|-------------\|-------------\|--------------\|--------------\|--------------\|--------------\|-------------\| \| LSTM (lt) \| 72.50% / 0.05 \| 98.00% / 0.01 \| 94.00% / 0.01 \| 92.50% / 0.01 \| 90.50% / 0.01 \| 91.50% / 0.01 \| 95.00% / 0.01 \| \| LSTM (En) \| 85.00% / 0.02 \| 98.50% / 0.01 \| 99.50% / 0.00 \| 99.50% / 0.00 \| 99.50% / 0.00 \| 99.50% / 0.00 \| 98.50% / 0.00 \| \| LSTM (En-LM)\| 74.00% / 0.02 \| 90.50% / 0.01 \| 91.00% / 0.01 \| 92.00% / 0.01 \| 97.50% / 0.01 \| 97.50% / 0.01 \| 97.50% / 0.01 \| \| wav2vec (Ma)\| 97.00% / 0.01 \| 98.00% / 0.01 \| 98.50% / 0.01 \| 96.00% / 0.01 \| 91.50% / 0.01 \| 90.50% / 0.01 \| 98.50% / 0.01 \| \| wav2vec (Ge)\| 94.00% / 0.03 \| 98.00% / 0.01 \| 97.00% / 0.01 \| 98.00% / 0.00 \| 96.50% / 0.00 \| 98.00% / 0.00 \| 97.00% / 0.01 \| \| Trf (Ma) \| 96.50% / 0.01 \| 98.00% / 0.01 \| 96.00% / 0.01 \| 96.50% / 0.00 \| 96.00% / 0.00 \| 96.00% / 0.00 \| 95.50% / 0.01 \| \| Trf (En) \| 93.00% / 0.01 \| 99.50% / 0.01 \| 100.0% / 0.00 \| 100.0% / 0.00 \| 100.0% / 0.00 \| 100.0% / 0.00 \| 100.0% / 0.00 \| Avg. accuracy / FPR: 87.43% / 0.02 / 97.14% / 0.01 / 96.43% / 0.01 / 96.43% / 0.00 / 95.86% / 0.00 / 95.86% / 0.00 / 97.07% / 0.01 and whether it employs a language model during decoding or not. In a more challenging context, where distinguishing between noisy and adversarial data, our proposed defense still surpasses NF and TD for all models except for one. We observe that among all classifiers, the NN stands out as the most robust when comparing noisy and benign data scenarios, showing only a minimal decrease of 1.42% in the average AUROC across all models. It’s worth noting that the performance of TD on noisy data hasn’t been analyzed before, and former investigations were limited to the English language (Yang et al., 2019). Similarly, NF was solely tested against the untargeted genetic attack in a 10-word classification system. Some characteristics perform consistently well, independently of the adversarial data, and only benign data is needed for choosing the threshold. This is displayed by the results for GCs based on the mean-median for both targeted attacks in the first two columns of Tab. 6. Moreover, even the neural network solely trained on C&W attacks performs equally well against Psychoacoustic AEs. These results indicate a good transferability to other kinds of targeted attacks. To evaluate the goodness-of-fit performance of our classifiers, we adopted a conservative threshold selection criterion: the highest false positive rate (FPR) below 1% (if available) while maintaining a minimum true positive rate (TPR) of 50%. This evaluation considers EMs with different total voting values $T \in \{3, 5, 7, 9\}$. Hence, our classifiers consistently achieve a high average accuracy exceeding 95%, surpassing the performance of TD, as indicated in Tab. 5. We suggest opting for an EM approach, which tends to minimize variance, or an NN that apart from minimizing variance has the potential for enhanced generalization with further refinements. Additional goodness-of-fit measurements across all models are available in the App. A.9. Detecting untargeted attacks To assess the transferability of our detectors to untargeted attacks, we investigated the defense performance of GCs based on the mean-median characteristic and NNs trained on C&W AEs when exposed to PGD, Genetic, or Kenansville attacks. Results are reported in Tab. 6. While the detection performance decreases in comparison to targeted attacks, our methods are still way more efficient than TD, with AUROCs even exceeding 90% for the Kenansville attack. In general, the Genetic attack proves challenging to detect, which may be attributed to its limited impact on the WER (compare Tab. 3). Advantageously, limited research addresses untargeted attacks in large-vocabulary ASR systems; in general, they are less threatening and all instances we investigated are characterized by noise, making them easily noticeable by human hearing. Table 6: AUROC assessment to detect AEs using GCs and NNs across various attacks. \| Model \| TD \| C&W GC \| NN \| TD \| Psychoacoustic GC \| NN \| TD \| PGD GC \| NN \| TD \| Genetic GC \| NN \| TD \| Kenansville GC \| NN \| \|----------------\|----\|--------\|----\|----\|-------------------\|----\|----\|--------\|----\|----\|------------\|----\|----\|---------------\|----\| \| LSTM (En) \| 0.89\| 1.00 \| 1.00\| 0.89\| 1.00 \| 1.00\| 0.71\| 0.94 \| 0.94\| 0.54\| 0.58 \| 0.68\| 0.68\| 0.82 \| 0.92\| \| LSTM (En-LM) \| 0.93\| 0.95 \| 0.99\| 0.94\| 0.96 \| 0.99\| 0.83\| 1.00 \| 0.78\| 0.55\| 0.64 \| 0.69\| 0.76\| 0.89 \| 0.89\| \| wav2vec (Ma) \| 0.99\| 0.99 \| 0.99\| 0.99\| 0.99 \| 0.99\| 0.77\| 0.84 \| 0.84\| 0.65\| 0.67 \| 0.73\| 0.89\| 0.97 \| 0.97\| \| wav2vec (Ge) \| 0.98\| 1.00 \| 0.99\| 0.98\| 1.00 \| 0.99\| 0.82\| 0.78 \| 0.40\| 0.55\| 0.65 \| 0.58\| 0.77\| 0.93 \| 0.85\| \| Trf (Ma) \| 0.98\| 0.99 \| 1.00\| 0.99\| 0.99 \| 0.99\| 0.86\| 0.76 \| 0.91\| 0.59\| 0.65 \| 0.84\| 0.90\| 1.00 \| 1.00\| \| Trf (En) \| 0.98\| 1.00 \| 1.00\| 0.99\| 1.00 \| 1.00\| 0.75\| 0.75 \| 0.59\| 0.53\| 0.46 \| 0.64\| 0.73\| 0.89 \| 0.94\| \| Avg. \| 0.96\| 0.99 \| 1.00\| 0.97\| 0.99 \| 0.99\| 0.78\| 0.86 \| 0.78\| 0.56\| 0.59 \| 0.69\| 0.79\| 0.90 \| 0.92\| Table 7: Evaluating filtering to preserve system robustness in accuracy with 100 clean test set samples and 100 adaptive C&W AEs, using a threshold aiming for a maximum 1% FPR when feasible. \| Model \| Adaptive AE attack performance pre-filtering \| Filtering AEs aiming a GC \| Filtering AEs aiming an EM=9 \| Filtering AEs aiming a NN \| \|----------------\|---------------------------------------------\|---------------------------\|----------------------------\|--------------------------\| \| \| GC \| EM=9 \| NN \| LPF: WER / CER \| SG: WER / CER \| LPF: WER / CER \| SG: WER / CER \| LPF: WER / CER \| SG: WER / CER \| \| LSTM (En) \| 33.50\| 50.50\| 53.50\| 76.00 / 78.50\| 82.00 / 94.00\| 74.00 / 79.00\| 74.50 / 88.00\| 79.50 / 86.00\| 88.50 / 97.50\| \| LSTM (En-LM) \| 29.50\| 50.50\| 67.50\| 80.50 / 85.50\| 98.00 / 100.00\| 86.50 / 96.00\| 86.50 / 96.00\| 99.50 / 100.00\| \| wav2vec (Ma) \| 42.50\| 50.50\| 67.50\| 80.50 / 85.50\| 99.00 / 100.00\| 76.50 / 80.50\| 94.50 / 95.00\| 81.00 / 85.50\| 99.50 / 99.50\| \| wav2vec (Ge) \| 37.50\| 58.00\| 49.50\| 99.50 / 99.50\| 96.00 / 96.00\| 100.0 / 100.0\| 96.00 / 96.00\| 100.0 / 100.0\| 99.50 / 99.50\| \| Trf (Ma) \| 25.50\| 71.00\| 60.50\| 96.00 / 98.00\| 97.50 / 96.00\| 96.50 / 98.50\| 98.00 / 98.00\| 97.50 / 98.00\| 98.00 / 97.50\| \| Trf (En) \| 28.50\| 40.00\| 26.00\| 74.00 / 74.00\| 79.50 / 79.50\| 71.50 / 71.50\| 74.50 / 74.50\| 82.00 / 82.00\| 89.50 / 89.50\| \| Avg. accuracy \| 31.71\| 48.50\| 50.86\| 83.86 / 85.64\| 91.86 / 94.07\| 80.86 / 82.43\| 88.21 / 91.64\| 89.00 / 91.14\| 96.29 / 97.64\| Detecting adaptive adversarial attacks The accuracy of our classifiers experiences a substantial decline across all models due to adaptive attacks when evaluated with a threshold aiming for a maximum FPR of 1% (where feasible). That means, that the defense provided gets ineffective if its usage is known to the attacker. However, one can leverage the fact, that the adaptive attack results in much noisier examples. To do so, we compare the predicted transcription of an input signal with the transcription of its filtered version using metrics like WER and CER. We employed two filtering methods: a low-pass filter (LPF) with a 7 kHz cutoff frequency, eliminating high-frequency components (Monson et al., 2014) and a PyTorch-based Spectral Gating (SG) (Sainburg, 2019; Sainburg et al., 2020), an audio-denosing algorithm that calculates noise thresholds for each frequency band and generates masks to suppress noise below these thresholds. We then tried to distinguish attacks from benign data based on the resulting WER and CER values. When contrasting the accuracy results in Tab. 5 for AEs that have not been tailored to the classifier type with those in Tab. 7 for adaptive AEs, SG proves highly effective in distinguishing between adversarial and benign data across most models. This is especially evident with the NN classifier, which consistently matches or even surpasses the accuracy achieved by the non-tailored AEs, leading to an average accuracy boost from 97.07% to 97.64%, a gain of 0.57%. 6 DISCUSSION & CONCLUSION We have demonstrated that characteristics of the distribution over the output tokens can serve as features of binary classifiers, turning them into an effective tool for identifying targeted adversarial attacks against ASR systems. As an example of such characteristics, the mean (w.r.t. the distributions from different time steps) of the median of the probabilities holds the greatest discriminative power across different models. Even on challenging data, these characteristics allow us to distinguish adversarial examples from benign data with high reliability. Our empirical findings strongly support employing a combination of these characteristics either in an ensemble of simple Gaussian classifiers or as input to a neural network to yield the best performance. This approach showcases exceptional discriminative power across a variety of modern ASR systems trained on different language corpora. It will be interesting to evaluate if the use of these characteristics of output distributions can also serve as indicators of other pertinent aspects, such as speech quality and intelligibility, which is a target for future work. REFERENCES Hadi Abdullah, Kevin Warren, Vincent Bindschaedler, Nicolas Papemot, and Patrick Traynor. Sok: The faults in our ASRs: An overview of attacks against automatic speech recognition and speaker identification systems. 2021 IEEE Symposium on Security and Privacy (SP), pp. 730–747, 2020. 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SQFDJLyJNB	Given that a Gaussian Mixture Model (GMM) is employed, does the assumption of similarity between learning stages in conventional continual learning scenarios, which may vary greatly, potentially limit the model's capabilities?	PROMPTCCD: LEARNING GAUSSIAN MIXTURE PROMPT POOL FOR CONTINUAL CATEGORY DISCOVERY Anonymous authors Paper under double-blind review ABSTRACT In this paper, we address the challenging open-world learning problem of continual category discovery (CCD). Initially, a labelled dataset consisting of known categories is provided to the model. Subsequently, unlabelled data arrives continuously at different time steps, which may contain objects from known or novel categories. The primary objective of CCD is to automatically assign labels to unlabelled objects, regardless of whether they belong to seen or unseen categories. However, the crucial challenge in continual category discovery is to automatically discover new categories in the unlabelled stream without experiencing catastrophic forgetting, which remains an open problem even in conventional, fully supervised continual learning. To address this challenge, we propose PromptCCD, a simple yet effective approach that utilizes Gaussian mixture model as a prompting method for CCD. At the core of PromptCCD is our proposed Gaussian Mixture Prompt Module (GMP), which acts as a dynamic pool updating over time to provide guidance for embedding data representation and avoid forgetting during continual category discovery. Additionally, our GMP provides the unique advantage of enabling on-the-fly estimation of category numbers, which enables it to discover categories in the unlabelled stream without prior knowledge of category numbers. Finally, we extend the standard evaluation metric for generalized category discovery to CCD and benchmark state-of-the-art methods using different datasets. Our PromptCCD significantly outperforms other methods, demonstrating the effectiveness of our approach. 1 INTRODUCTION The human visual system has the remarkable ability to learn and reason about novel concepts over time. For instance, humans can learn about newly discovered animals and extinct ones in different timelines with ease. This ability also extends to other concept axes such as arts, products, and more. Hence, the challenge of discovering novel visual concepts within unlabelled images over a period while retaining previously seen visual concepts becomes a critical aspect in the design of artificial visual systems. Continual category discovery (CCD) (Zhang et al., 2022) aims to empower artificial visual systems with this ability by extending the challenging open-world learning problems of novel category discovery (NCD) (Han et al., 2019) and generalized category discovery (GCD) (Vaze et al., 2022) to a continual learning scenario (see Fig. 1). By enabling artificial visual systems to learn and reason about novel concepts over time, CCD represents an essential step towards developing more intelligent and adaptive visual systems that can operate effectively in dynamic environments. Advancements in vision foundation models have shown promise in various computer vision tasks, from image classification and object detection to more complex tasks like scene understanding (Caron et al., 2021; Oquab et al., 2023). State-of-the-art models like transformers have demonstrated their strong performance in static environments where they are trained on a fixed set of categories. Given the progress and capabilities of these foundation models, we are interested in investigating how these models can be repurposed to continually adapt to dynamic environments where they must discover and learn from new visual data categories over time. There are two major challenges in CCD. The first challenge is catastrophic forgetting, a well-known issue in continual learning settings (De Lange et al., 2021). Traditional techniques for mitigating forgetting, such as rehearsal-based (Rebuffi et al., 2017), distillation-based (Li & Hoiem, 2017), architecture-based (Li et al., 2019), and prompting-based methods (Wang et al., 2022b,a), assume fully labelled data at each stage, which is incompatible with the CCD framework where the goal is to work with unlabelled data streams. The second challenge is the discovery of novel visual concepts. While Generalized Category Discovery (GCD) is a related task, most existing methods focus mainly on static unlabelled data, making them unsuitable for the continually evolving nature of CCD. To tackle these challenges in adapting foundational vision models for CCD, we introduce a Gaussian Mixture Prompt learning framework. This framework employs a Gaussian Mixture Model (GMM) to model the data distribution at each learning stage dynamically. By enriching the visual feature representation with adaptive queried Gaussian Mixture Prompts (GMP), our method excels at identifying new visual concepts across successive learning stages. Concurrently, these prompts facilitate the model’s seamless adaptation to emerging data while preserving its performance on previously acquired categories, thus preventing catastrophic forgetting. In addition to outperforming existing CCD solutions, our framework provides the unique advantage of enabling on-the-fly estimation of category numbers — often assumed to be predetermined in prior works (Zhang et al., 2022). We summarize our main contributions as follows: (1) We introduce Gaussian Mixture Prompt Module (GMP), a new prompt learning technique that leverages Gaussian mixture component(s) to generate better representation and mitigate the catastrophic forgetting problem on previously learned data. (2) We propose the first prompt learning framework tailored for CCD, PromptCCD, which can be coupled with our proposed GMP and existing prompt learning techniques for effective continual category discovery. (3) We extensively experiment with benchmarking datasets and compared our method with baseline methods under both known and unknown category number scenarios, significantly outperforming the state-of-the-art. 2 METHOD In GCD, given a labelled and unlabelled set of images, the task is to recognize and discover all known and novel classes in the unlabelled set. The CCD task extends this task into the continual setup where the unlabelled data stream keeps coming in different time steps. Thus, the main objective of CCD task is to discover novel classes in a dynamic setting without forgetting learned knowledge from the previous streamed data, i.e. a decrease in the model’s performance on known categories. In this section, we briefly describe how CCD task is formulated. Given dataset $D = D^l \cup D^u$ consisting of labelled and unlabelled data respectively. $D^l = \{(x_i, y_i)\}_{i=1}^{N}$ contains tuples of the input $x_i \in X$ and its corresponding labels $y_i \in Y$. The labelled dataset of known categories will be used for the model to learn in the initial stage. In the subsequent (discovery) stages, assuming the total number of stages is $T$, the unlabelled data stream $D^u$ is divided into $T$ subsets such that $D^u = \{D^u_t\}_{t=1}^{T}$ where each unlabelled set at stage $t$, $D^u_t = \{D^u_{t,o}, D^u_{t,n}\}$, consists of unlabelled instances from known and novel categories, respectively. Our goal in CCD is to train a model $H_\theta : X \rightarrow Z$ parameterized by $\theta$ that first, learns from labelled $D^l$ and in the discovery stages, learns from unlabelled data $D^u_t$ for time steps $T$ such that $H_\theta$ can be used to discover novel classes and assign class labels to all unlabelled instances utilizing representative feature $z_i \in Z$ without forgetting previously learned knowledge from old streamed data. In the following, we first elaborate on the design of our baseline and proposed methods, followed by an explanation of how our model learns during the initial and continual discovery stages. 2.1 PROMPT POOL LEARNING FOR CONTINUOUS CATEGORY DISCOVERY Vision foundation models are pretrained representations trained on a large-scale dataset and are task-agnostic. These models can achieve remarkable performance across certain downstream tasks even with minimal fine-tuning. Prompt tuning (Wang et al., 2022a; b) has emerged as a powerful method for adapting these foundation models to supervised continual learning settings. However, directly utilizing these prompt learning techniques is unsuitable for CCD task, as all these works assume the incoming data stream has label information. We start our exploration by constructing a prompt learning baseline for CCD. Inspired by Wang et al. (2022b; a), we design a baseline model for CCD that leverages a shared memory pool of prompts. The model extracts a feature from a query example using a frozen pretrained model, and the feature will be used to retrieve the top-k most relevant prompts from the fixed-size $M$ prompts in the shared pool. These prompts are then used to guide the model’s representation learning by prepending them with the input’s embeddings, optimised with contrastive learning at each learning stage. The formulation of the method is presented below (as depicted in Fig. 2). Given a model $H_\theta : \{\phi, f_\theta\}$, $\phi$ is an MLP projection head, and $f_\theta = \{f_e, f_b\}$ is the transformer-based feature backbone which consists of input embedding layer $f_e$ and self-attention blocks $f_b$. An input image $x \in \mathbb{R}^{H \times W \times 3}$ where $H, W$ represent the height and width of the image, is first split into $L$ tokens (patches) such that $x_q \in \mathbb{R}^{L \times (h \times w \times 3)}$ where $h, w$ represent the height and width of the image patches. These patches are then projected to the input embedding layer such that $x_e = f_e(x_q) \in \mathbb{R}^{L \times z}$. To construct the prompt learning technique, a learnable prompt pool is initialized as $V = \{V_m\}_{m=1}^M$ where $V_m \in \mathbb{R}^{L \times z}$ and $M$ is the total number of prompts (which is fixed, across stages). Additionally, a query function $f_{\theta^} : \mathbb{R}^{H \times W \times 3} \rightarrow \mathbb{R}^{z[\text{CLS}]}$ is initialized to map $x \rightarrow$ classification token. To form the key-value memory query function, these prompts $V_i$ are then paired with learnable key $k_i$ so that, given query $f_{\theta^}(x)$ and $\{k_m\}_{m=1}^M$ set of keys, we calculate their similarity using cosine distance $\gamma$ and take the top-k keys. With these selected keys, we can return the set of associated prompts called $V_{\text{top-k}}$. Then, a set of embeddings $x_{\text{total}} = [V_{\text{top-k}}; x_e]$ is formed by prepending the selected prompts with the patch embeddings. Finally, we feed the embeddings to the self-attention blocks $f_b(x_{\text{total}})$. As our baseline adopts the contrastive learning strategy, let $\{x_i, x'_i\}$ be the two views of randomly augmented image $x_i$. These two pairs are then fed to $H_\theta$ such that $z_i, z'_i = \phi(f_\theta(x_i, x'_i))$. We optimize our baseline by combining contrastive learning losses, Sec. 2.3 and the surrogate loss, Eq. (1) to pull selected keys closer to corresponding query features. Finally, when stage $t$ training is finished, we transfer current prompt pool $V$ to the next stage. $$L_{\text{surrogate}}(x_i, k_m) = \gamma(f_{\theta^}(x_i), k_m).$$ Although this prompt learning technique by design works in our baseline, several limitations arise when applied to CCD task. First, the representation learning process only considers the unlabelled data in the current time step, which can cause representation bias towards current data and disrupt the representation learned for the previous data. Additionally, the category discovery process is separate from the representation learning process, which means that there is no proper mechanism for transferring knowledge from old classes to new classes. This transfer of knowledge is essential for the category discovery task. Second, the fixed-size prompt module can lead to parameter inefficiency and restrict the model’s ability to discover new categories and avoid forgetting. Figure 3: The design paradigm of our PromptCCD framework. PromptCCD continually discovers new categories while retaining previously discovered ones by learning a dynamic Gaussian mixture prompt (GMP) pool to guide the self-supervised vision foundation model for CCD. To prevent catastrophic forgetting, we generate replay samples from the previously fitted GMM at time step $t-1$ and fit them into the current GMM at time step $t$. 2.2 Gaussian Mixture Prompt Pool Learning for Continual Category Discovery By referring to the aforementioned limitations, there is a need for a prompting technique that requires minimal to almost zero supervision, and design-wise, its parameter has to be dynamic and flexible. With that goal in mind, here we propose the Gaussian Mixture Prompt Module (GMP), a novel prompt learning technique that uses the Gaussian mixture model (GMM) as a prompt pool. Here, we listed several key advantages that our prompt module offers; First, GMP’s prompt serves a dual role, namely (1) as a task prompt to instruct the model (like in Wang et al. (2022a;b)) and (2) as class prototypes (see Appendix I for details) to act as parametric replay sample distribution for discovered classes. The second role, which is unique and important for CCD/GCD, not only allows the model to draw unlimited replay samples to facilitate the representation tuning and class discovery in the next time step but also allows the model to transfer knowledge of previously discovered and novel categories and incorporate this information when making the decision to discover a novel category. Second, our GMP module enables easy adjustment of parameters and dynamic expansion across stages. This flexibility is particularly valuable in CCD tasks where the number of classes can change over time. Finally, GMP can be seamlessly combined with a category number estimator to tackle the open-world nature of CCD, where the number of categories within the unlabelled data stream is unknown. Next, we will show how we formulate our framework, Fig. 3 utilizing GMP, Fig. 4. Figure 4: Our proposed Gaussian mixture prompt module (GMP) estimates the probability of the input query $z\|\text{CLS}$ by calculating the log-likelihood of Eq. (2). We then use the top-k mean components as prompts to guide our model for CCD. Gaussian Mixtures Prompt Module (GMP). We build a probabilistic multivariate GMM representing the presence of a sub-population within an overall population. As the Gaussian Mixture distribution is a linear superposition of Gaussian’s, thus we can formulate GMM as: $$p(z) = \sum_{c=1}^{C} \pi_c N(z\|\mu_c, \Sigma_c) \quad \text{s.t.} \quad \sum_{c=1}^{C} \pi_c = 1.$$ Eq. (2) represents the Gaussian probability density function in GMM, consisting of $C$ Gaussian components, a set of learnable mixture weights components $\{\pi_1, \pi_2, ..., \pi_C\}$, component’s mean $\{\mu_1, \mu_2, ..., \mu_C\}$, and component’s covariance $\{\Sigma_1, \Sigma_2, ..., \Sigma_C\}$. When we initialize our GMM, we assume that $ C $ is known (see Sec. 2.2 for the case when $ C $ is unknown). For every learning stage, we do not directly use GMM for prompting as it needs to be fit using EM algorithm by strong features, i.e., $ Z[\text{CLS}] = \sum_{i=1}^{\|X\|} f_\theta(x_i) $, where $ X $ is the set of images in $ D_t $. Given feature $ z[\text{CLS}] = f_\theta(x) $ i.e., the classification token queried by our model, to find the component $ \pi_i $, which is associated with $ z[\text{CLS}] $, the model calculates the log probability density value, Eq. (2) and returns a set $ W $ of log-likelihood probability value for different $ \pi_i $. Then, we pick the top-k component’s mean indexes such that $ \text{top-k} = \arg\max_{w' \in W, \|W'\| = k} \sum_{w \in W'} w $. With these selected top-k indexes, we can return the set of associated prompts as $ \mu_{\text{top-k}} $ mean components. Similar to our baseline, a set of embeddings $ x_{\text{total}} = [\mu_{\text{top-k}}; x_e] $ is formed by prepending the selected prompts with the patch embeddings. As our method adopts the contrastive learning strategy, let $ \{x_i, x'_i\} $ be the two views of randomly augmented image $ x_i $. These two pairs are then fed to $ H_\theta $ such that $ z_i, z'_i = \phi(f_\theta(x_i, x'_i)) $. We optimize our model with contrastive learning losses, Sec. 2.3. Since we aim to use GMM dynamically across stages, once the training process is complete, we further make use of the learned GMM to draw a set of random samples $ Z^_t $ by Eq. (2) where the set has $ S $ samples for each component $ c $. This is done to handle forgetting previously learned knowledge as the generated GMM samples $ Z^_t $ from the current stage will be used to fit the next GMM. By combining samples from the previous stage and the current features i.e., $ Z[\text{CLS}]_t = \sum_{i=1}^{\|X\|} f_\theta(x) + Z^_{t-1} $, GMM learns rich features, which leads to better prompt embeddings. The pseudo-code of the overall method is provided in Appendix A. Unknown number of classes in unlabelled data. In the real open-world scenario, the number of categories $ C $ is often unknown. Estimating the number of categories from unlabelled data is an important question. Existing work i.e., GPC (Zhao et al., 2023) has addressed this problem using the Semi-Supervised Gaussian Mixture Model (SS-GMM). As our prompt module is also based on GMM, we can seamlessly combine our prompt learning method with the GPC category estimator. In general, the GMM is first fit with $ Z[\text{CLS}] $ with an initial value of $ C $, then an automatic splitting-and-merging strategy based on the Metropolis-Hastings ratio framework is used to measure the compactness and separability of clusters formed by the model. Clusters are split into two if they are separable, and two clusters are merged into one if they are cluttered. This process will continue until the optimization is finished. See Appendix D for details and experiment results. 2.3 Optimization objectives for different learning stages Both supervised, Eq. (3) and unsupervised, Eq. (4) contrastive losses are formulated as follows: \[ L^s_i = -\frac{1}{\|\mathbb{N}(i)\|} \sum_{q \in \mathbb{N}(i)} \log \frac{\exp(z_i \cdot z_q/\tau)}{\sum_{j=1}^{n} \mathbb{I}_{[n \neq i]} \exp(z_i \cdot z_j/\tau)}, \] \[ L^u_i = -\log \frac{\exp(z_i \cdot z'_i/\tau)}{\sum_{j=1}^{n} \mathbb{I}_{[n \neq i]} \exp(z_i \cdot z'_j/\tau)}, \] where $ \mathbb{I}_{[n \neq i]} $ is an indicator so that the same image index will not be considered a negative pair, $ \tau $ is the temperature value, and $ \mathbb{N}(i) $ is the set of images with the same label $ y $ in a mini-batch $ B $. Optimization during initial learning from labelled data. Given labelled data stream $ D^l $ in the initial stage, the model optimizes both supervised, Eq. (3) and unsupervised, Eq. (4) contrastive losses. The total loss over the batch is formalized as Eq. (5), where $ B^L $ denotes the labelled images in $ B $ and $ \lambda $ is the weighting coefficient. Optimization during class discovery from unlabelled data. After learning feature representation in the initial stage, the model proceeds to the discovery stage, where the incoming data stream is unlabelled $ D^u_t = \{D^u_{tn}, D^u_{tn}\} $. Similar to the initial stage, the model adopts the self-supervised learning strategy, and we use unsupervised contrastive learning with a loss formulated in Eq. (4). Thus, the loss over the batch is formalized as Eq. (5) without the supervised contrastive loss, Eq. (3). \[ L_{\text{total}} = (1 - \lambda) \sum_{i \in B^L} L^u_i + \lambda \sum_{i \in B^L} L^s_i. \] 3 Experiments To assess the performance of our proposed framework compared with other models, we evaluate and compare PromptCCD framework with the state-of-the-art continual category discovery, generalized category discovery and continual learning models on generic image datasets and the more challenging fine-grained image dataset. Thus, in this section, we describe our experimental setups in Sec. 3.1; then, we present our main experimental results in Sec. 3.2. Finally, ablation studies in Sec. 3.3 are conducted to verify our model’s effectiveness. ### 3.1 Experimental Setups Datasets. We conduct our experiments on various benchmark datasets, namely CIFAR-100 (C 100) (Krizhevsky & Hinton, 2009), ImageNet-100 (IN 100) (Russakovsky et al., 2015), TinyImageNet (Tiny 200) (Le & Yang, 2015), and the Caltech-UCSD Birds-200-2011 (CUB-200) (Wah et al., 2011). (1) CIFAR-100 contains 100 classes with 600 images per class. It is divided into 500 training and 100 testing images per class. (2) ImageNet100, contains 100 classes with 1350 images per class. It is divided into 1300 training and 50 testing images per class. (3) In TinyImageNet, there are 100,000 images divided into 200 classes. Each class has 500 training images, 50 validation images, and 50 test images. We use its training and test images in our experiments. (4) Lastly, CUB-200 is a fine-grained visual categorization dataset with 11,788 images of 200 bird species. Algorithm 1 CCD evaluation metric \| Input: \| $ f(.) $ models for each stage in $\{1,\ldots,T\}$ and datasets $\{D^L_i, D^U_i\}$ \| \| Output: \| The ACC outputs for every stage. \| 1. Initialize set $ A^L = \{D^L_1\} $ 2. for $ t \in \{1,\ldots,T\} $ do 3. $ AC_{t-1} = SS\text{-Kmeans(Model: } f_t(.), \text{ Labelled set: } A^L, \text{ Unlabelled set: } \{D^U_t\}) $ 4. Use labels assigned by SS-Kmeans such that $ D^L_t \leftarrow D^U_t $ 5. $ A^L \leftarrow A^L \cup D^L_t $ 5. return $\{ACC_t \| t=1,\ldots,T\}$ Implementation details. We use ViT-B/16 backbone (Dosovitskiy et al., 2021) initialized with DINO self-supervised vision foundation features (Caron et al., 2021) for all experiments. Please note that Wang et al. (2022b,a) utilized a well-pretrained model with supervision, which is suitable for the standard supervised continual learning task. However, it is not allowed to use such models for CCD task due to label information leakage. During training, only the final block of the vision transformer is finetuned 200 epochs with a batch size of 128, using an SGD optimizer and a cosine decay learning rate scheduler with an initial learning rate of 0.1 and minimum learning rate of 0.0001, and weight decay of 0.00005. For the mixture prompt module, we optimize the GMM every 30 epochs and start the prompt learning when the epoch is greater than 30. We set top-$k$ to be 5, and the number of GMM samples to 100. We pick the final model by selecting the best performing model on "old acc" using the validation set (evaluated every 10 epochs). All input images are resized to $224 \times 224$ and normalized to match the DINO pretrained model settings. For our proposed method, we follow the standard practice of self-supervised learning training procedure by training a base encoder/backbone $f_b$ and a projection head $\phi$ to maximize the agreement using a contrastive loss with $\lambda = 0.35$. For other compared methods, we chose the right hyper-parameters following their original papers. Finally, for the class number estimation, we follow the procedures proposed by GCD (Vaze et al., 2022) i.e., by utilizing GCD’s class number estimation method on DINO features with a binary search algorithm within the range of $[\|\mathcal{Y}_L\|, 1000]$ across all datasets and GPC (Zhao et al., 2023) dynamic class number estimation; We build our proposed framework with PyTorch library, trained in a single NVIDIA RTX 3090 GPU. Experiment settings and evaluation metrics. CCD task consists of several stages. We set the number of stages to 4 with a specific CCD’s split ratio presented in Table 1 following the setup of Zhang et al. (2022). The model is fine-tuned in each stage. During test time, the output classification token $[\text{CLS}]$ features are used for clustering. For the clustering algorithm and label assignment, we use semi-supervised k-means (Vaze et al., 2022) on the training sets at stage $t$ and measure the clustering quality given the ground truth labels $y_i$ and the model’s clustering prediction $\hat{y}_i$ such that: \[ ACC = \max_{g \in G(\mathcal{Y}_U)} \frac{1}{\|D^U_t\|} \sum_{i=1}^{\|D^U_t\|} I\{y_i = g(\hat{y}_i)\}, \] where $G(\mathcal{Y}_U)$ represents set of all permutations of class labels in the unlabelled set $D^U_t$. For the evaluation metrics across stages, we use the clustering accuracy ACC consisting of ‘All’, ‘Old’ and ‘New’ sets of metrics. ‘All’ indicates the overall accuracy on the entire set $D^U_t$, ‘Old’ and ‘New’ indicate the accuracy from instances of unlabelled data from $D^L_t$ and $D^U_t$ respectively. To properly measure the performance on the CCD task, we extend the commonly used ACC for static data into the CCD setting, as shown in Algorithm 1. Here, we use labelled data from $\{D^L_i, \sum_{i=1}^{t-1} D^U_i\}$ to help guide SS-Kmeans clustering algorithm. We set $u^* \leftarrow u$ to indicate that we use predicted labels on previously unlabelled data $D^U_t$ from the previous stage. Baselines. We compare our method with the other representative CCD, GCD and continual learning models approaches, including 1) Grow and Merge (GM) (Zhang et al., 2022); 2) ORCA (Cao --- Table 1: Data distribution in CCD task. \| Class splits \| Stage 0 \| Stage 1 \| Stage 2 \| Stage 3 \| \|--------------\|---------\|---------\|---------\|---------\| \| $[\mathcal{Y}_L] \leq \|\mathcal{Y}_L\| < 0.7 \times \|\mathcal{Y}_L\|$ \| 87% \| 7% \| 3% \| 3% \| \| $[\mathcal{Y}_L] \geq 0.7 \times \|\mathcal{Y}_L\| < 0.8 \times \|\mathcal{Y}_L\|$ \| 0% \| 70% \| 20% \| 10% \| \| $[\mathcal{Y}_L] \geq 0.8 \times \|\mathcal{Y}_L\| < 0.9 \times \|\mathcal{Y}_L\|$ \| 0% \| 0% \| 90% \| 10% \| \| $[\mathcal{Y}_L] \geq 0.9 \times \|\mathcal{Y}_L\|$ \| 0% \| 0% \| 0% \| 100% \| et al., 2022); 3) GCD (Vaze et al., 2022); 4) SimGCD (Wen et al., 2023); 5) L2P (Wang et al., 2022b); and 6) DualPrompt (DP) (Wang et al., 2022a). As GM’s encoder is based on ResNet-18 network (He et al., 2016), we re-implement their dynamic branch mechanism with the vision transformer backbone network and observe improved performance for their method compared to their original results; see Appendix E for details. We also re-implement GCD and SimGCD to suit the continual learning settings further. We integrate a replay-based method into these model where, for each stage, the model saves several samples for each discovered class and mix these samples with the next incoming streamed images. Lastly, we adopt L2P’s and Dual Prompt’s prompt pool module and their corresponding surrogate loss, for our baseline model and integrate it with our framework. ### 3.2 Main Results Table 2: Results on various coarse and fine-grained datasets where C is known in each unlabelled set. \| Model \| Stage 1 ACC (%) \| Stage 2 ACC (%) \| Stage 3 ACC (%) \| Average ACC (%) \| \|------------------------\|-----------------\|-----------------\|-----------------\|-----------------\| \| \| All Old New \| All Old New \| All Old New \| All Old New \| \| ORCA (Cao et al., 2022) \| 62.05 71.55 55.40 \| 63.21 67.14 62.45 \| 55.79 65.05 54.17 \| 60.35 67.91 57.34 \| \| GCD (Vaze et al., 2022) \| 85.11 88.61 82.66 \| 72.18 69.33 72.73 \| 63.59 63.14 63.67 \| 73.62 73.69 73.02 \| \| SimGCD (Wen et al., 2023) \| 65.33 89.68 48.29 \| 54.89 67.36 52.51 \| 32.21 52.77 28.61 \| 50.81 69.94 43.14 \| \| GCD w/replay \| 71.28 82.00 63.77 \| 66.52 72.48 65.38 \| 57.45 69.52 55.33 \| 65.08 74.67 61.49 \| \| SimGCD w/replay \| 50.97 80.23 62.19 \| 40.42 62.19 40.42 \| 37.49 52.99 37.49 \| 49.09 69.04 36.52 \| \| Grow and Merge (Zhang et al., 2022) \| 84.77 70.49 60.77 \| 68.31 62.95 57.42 \| 48.82 56.00 48.75 \| 77.30 63.84 55.25 \| \| PromptCCD w/L2P \| 86.77 79.76 91.69 \| 85.05 64.10 89.05 \| 73.45 56.95 76.33 \| 81.75 66.94 85.69 \| \| PromptCCD w/DP \| 76.55 82.98 72.06 \| 65.05 75.33 63.09 \| 61.08 73.53 58.90 \| 67.56 77.26 64.68 \| \| PromptCCD w/GMP (Ours) \| 90.20 90.73 92.51 \| 85.83 75.62 87.78 \| 76.64 67.14 78.30 \| 84.22 77.83 86.20 \| \| ORCA (Cao et al., 2022) \| 79.03 78.29 79.54 \| 71.53 77.05 70.47 \| 68.77 77.33 67.27 \| 73.11 77.56 72.43 \| \| GCD (Vaze et al., 2022) \| 82.45 83.51 81.71 \| 82.27 78.57 82.98 \| 81.39 79.11 81.78 \| 82.03 80.40 82.15 \| \| SimGCD (Wen et al., 2023) \| 85.70 84.94 83.70 \| 76.77 78.29 76.77 \| 70.37 75.47 69.03 \| 79.01 79.01 79.01 \| \| GCD w/replay \| 79.75 80.82 79.00 \| 71.07 68.38 69.57 \| 64.40 78.29 61.97 \| 71.74 79.16 70.18 \| \| SimGCD w/replay \| 59.78 80.00 45.63 \| 49.36 64.10 46.55 \| 41.35 58.48 38.35 \| 50.16 67.53 43.51 \| \| Grow and Merge (Zhang et al., 2022) \| 75.45 76.86 74.46 \| 72.52 75.24 72.00 \| 68.23 74.38 67.15 \| 72.07 75.49 71.20 \| \| PromptCCD w/L2P \| 81.93 80.69 82.84 \| 63.77 73.81 64.24 \| 66.32 73.06 63.38 \| 71.41 79.35 70.82 \| \| PromptCCD w/DP \| 77.87 78.57 77.17 \| 63.44 68.80 63.44 \| 58.10 68.20 58.10 \| 68.03 76.45 68.03 \| \| PromptCCD w/GMP (Ours) \| 84.62 84.29 84.86 \| 80.06 79.62 80.15 \| 82.75 77.62 83.65 \| 82.47 80.51 82.88 \| \| ORCA (Cao et al., 2022) \| 59.98 66.90 55.14 \| 53.69 60.52 52.39 \| 55.51 55.95 55.43 \| 56.39 61.12 54.32 \| \| GCD (Vaze et al., 2022) \| 65.81 70.73 62.36 \| 59.34 58.00 59.59 \| 51.01 54.92 50.39 \| 58.72 61.08 57.44 \| \| SimGCD (Wen et al., 2023) \| 49.41 68.92 35.76 \| 37.00 57.76 33.75 \| 32.75 52.76 29.25 \| 39.92 59.81 39.29 \| \| GCD w/replay \| 63.83 65.98 62.33 \| 58.03 58.81 57.88 \| 55.16 58.48 54.58 \| 59.01 61.09 58.26 \| \| SimGCD w/replay \| 41.82 64.45 37.37 \| 31.82 32.32 34.50 \| 31.82 32.32 34.50 \| 33.57 35.73 33.87 \| \| Grow and Merge (Zhang et al., 2022) \| 69.02 64.14 73.96 \| 68.09 59.76 70.40 \| 56.96 52.81 57.68 \| 65.19 58.90 67.34 \| \| PromptCCD w/L2P \| 69.36 69.31 69.40 \| 67.57 60.48 69.36 \| 56.08 57.71 55.79 \| 64.33 62.50 64.71 \| \| PromptCCD w/DP \| 72.75 72.65 72.81 \| 62.01 59.71 62.45 \| 65.16 56.76 67.19 \| 66.64 63.04 67.48 \| \| ORCA (Cao et al., 2022) \| 49.79 66.43 38.66 \| 31.50 65.71 24.24 \| 43.71 70.00 38.58 \| 41.67 67.38 38.83 \| \| GCD (Vaze et al., 2022) \| 59.66 78.21 47.36 \| 49.38 72.14 44.55 \| 57.34 72.14 54.46 \| 55.46 74.16 48.79 \| \| SimGCD (Wen et al., 2023) \| 41.08 65.00 30.07 \| 39.07 63.57 26.34 \| 32.67 62.71 27.58 \| 36.19 64.76 27.85 \| \| GCD w/replay \| 56.71 71.14 48.38 \| 38.03 57.14 34.58 \| 35.81 67.50 37.47 \| 47.05 69.42 46.42 \| \| SimGCD w/replay \| 38.82 62.86 30.79 \| 34.88 52.14 31.21 \| 38.08 46.79 34.68 \| 37.26 53.94 32.23 \| \| Grow and Merge (Zhang et al., 2022) \| 38.64 70.71 27.92 \| 29.25 65.71 21.52 \| 44.29 56.07 39.69 \| 37.53 64.16 29.71 \| \| PromptCCD w/L2P \| 50.63 73.57 42.96 \| 52.38 72.14 48.18 \| 60.12 69.29 56.55 \| 54.38 71.67 49.23 \| \| PromptCCD w/DP \| 50.41 73.64 46.35 \| 49.63 73.09 46.13 \| 61.12 73.09 57.94 \| 57.79 77.79 60.65 \| \| PromptCCD w/GMP (Ours) \| 50.39 82.86 51.55 \| 56.25 70.29 51.36 \| 65.43 73.21 62.40 \| 60.36 78.45 55.10 \| Quantitative analysis. We evaluate our method in two scenarios: when the class number $ C $ is known (Table 2) in each unlabelled set at different stages, and when the class number is unknown (Table 3). (1) Table 2 shows the CCD evaluation results on generic and fine-grained datasets where each unlabelled set’s class number, $ C $, is known at different stages. Overall, PromptCCD w/GMP outperforms the other methods in all datasets across all instances (‘All’, ‘Old’, ‘New’) accuracy. As our base model is based on GCD (Vaze et al., 2022), we show that simply integrating our Gaussian mixture prompt module can sufficiently improve a static GCD model and can adapt in the CCD setting. We argue that not all prompting techniques effectively solve CCD task. By comparing our model with PromptCCD w/\{L2P, DP\} (our baselines), we observe that our model can handle class scaling better, as shown in Table 2, where our model performs better in both ‘Old’ and ‘New’ accuracy while the baselines suffer from performance loss at the later stage. We hypothesize that this performance drop is because their prompt pool parameters are not scalable, which limits the model’s prompt technique to "instruct" the model when the number of parameters needed to learn or preserve is growing over time. Unlike our baseline, our prompting technique is scalable as we build our pool of prompts based on the Gaussian mixture models. To prevent forgetting, we can preserve previous knowledge by sampling each learned mixture component and using these samples to fit the next GMM. (2) To show the performance comparison for each model in a more realistic setting where $ C $ is unknown, we also report the benchmark results in Table 3, where we only show three representative models i.e., GCD, Grow and Merge, and our model. Our method consistently outperforms all other methods by a large margin across the board, demonstrating the superior performance of our approach in the more realistic case when the class number is unknown. Table 3: Results on various coarse and fine-grained datasets where C is unknown in each unlabelled set. Here, we estimate C for all methods using (Vaze et al., 2022) C-est algorithm on DINO features. \| Model \| Stage 1 ACC (%) \| Stage 2 ACC (%) \| Stage 3 ACC (%) \| Average ACC (%) \| \|-------\|-----------------\|-----------------\|-----------------\|----------------\| \| \| All \| Old \| New \| All \| Old \| New \| All \| Old \| New \| All \| Old \| New \| \| C100 \| \| \| \| \| \| \| \| \| \| \| \| \| \| Estimated C \| C^EST, 84, C^GT, 80 \| C^EST, 84, C^GT, 90 \| C^EST, 84, C^GT, 100 \| \| GCD (Vaze et al., 2022) \| 85.26 \| 84.12 \| 82.66 \| 71.92 \| 71.11 \| 72.09 \| 63.39 \| 59.32 \| 63.05 \| 72.80 \| 71.63 \| 72.93 \| \| Grow and Merge (Zhang et al., 2022) \| 63.43 \| 72.29 \| 57.23 \| 57.56 \| 57.52 \| 57.56 \| 54.51 \| 51.05 \| 55.12 \| 58.50 \| 60.29 \| 56.64 \| \| PromptCCD w/GMP (Ours) \| 90.13 \| 90.45 \| 91.60 \| 78.32 \| 78.81 \| 79.18 \| 75.89 \| 64.76 \| 77.83 \| 81.44 \| 76.34 \| 82.87 \| \| IN100 \| \| \| \| \| \| \| \| \| \| \| \| \| \| Estimated C \| C^EST, 90, C^GT, 80 \| C^EST, 90, C^GT, 90 \| C^EST, 91, C^GT, 100 \| \| GCD (Vaze et al., 2022) \| 76.88 \| 76.80 \| 76.80 \| 73.33 \| 73.33 \| 73.33 \| 67.83 \| 67.83 \| 67.83 \| 71.10 \| 70.26 \| 68.17 \| \| Grow and Merge (Zhang et al., 2022) \| 64.61 \| 64.73 \| 56.11 \| 47.18 \| 42.10 \| 47.18 \| 57.13 \| 57.19 \| 45.50 \| 57.64 \| 73.67 \| 52.59 \| \| PromptCCD w/GMP (Ours) \| 78.21 \| 77.62 \| 78.57 \| 76.40 \| 72.29 \| 78.00 \| 69.83 \| 76.67 \| 68.63 \| 74.81 \| 75.53 \| 75.07 \| \| Tiny200 \| \| \| \| \| \| \| \| \| \| \| \| \| \| Estimated C \| C^EST, 169, C^GT, 160 \| C^EST, 169, C^GT, 180 \| C^EST, 172, C^GT, 200 \| \| GCD (Vaze et al., 2022) \| 65.09 \| 73.10 \| 60.14 \| 57.18 \| 58.71 \| 56.39 \| 48.82 \| 53.07 \| 47.97 \| 57.00 \| 60.83 \| 55.00 \| \| Grow and Merge (Zhang et al., 2022) \| 57.77 \| 63.00 \| 54.11 \| 41.16 \| 53.57 \| 38.00 \| 51.00 \| 50.43 \| 51.10 \| 49.98 \| 53.43 \| 47.74 \| \| PromptCCD w/GMP (Ours) \| 66.96 \| 72.86 \| 63.43 \| 61.96 \| 59.14 \| 62.50 \| 58.94 \| 58.14 \| 59.08 \| 62.62 \| 63.38 \| 61.67 \| \| CUB200 \| \| \| \| \| \| \| \| \| \| \| \| \| \| Estimated C \| C^EST, 166, C^GT, 160 \| C^EST, 192, C^GT, 180 \| C^EST, 220, C^GT, 200 \| \| GCD (Vaze et al., 2022) \| 52.51 \| 68.28 \| 42.16 \| 45.36 \| 70.07 \| 40.15 \| 54.20 \| 70.71 \| 50.97 \| 50.69 \| 69.69 \| 44.42 \| \| Grow and Merge (Zhang et al., 2022) \| 43.20 \| 62.50 \| 30.31 \| 31.02 \| 67.14 \| 24.09 \| 33.49 \| 50.83 \| 28.14 \| 36.10 \| 60.16 \| 27.51 \| \| PromptCCD w/GMP (Ours) \| 57.94 \| 77.50 \| 44.87 \| 58.00 \| 76.45 \| 48.03 \| 63.99 \| 77.14 \| 61.42 \| 58.31 \| 77.02 \| 51.44 \| Figure 5: TSNE visualization of CIFAR100 with features from our model PromptCCD w/GMP and Grow and Merge with DINO encoder in each stage following Table 1 distribution. Qualitative analysis. Lastly, to visualize the feature representation generated by our method, we use t-SNE algorithm (Van der Maaten & Hinton, 2008) to project the high-dimensional features of $ \{D^l_1, D^u_1\} $ in each stage into low-dimensional space. For the sake of comparison, we also provide the visualization for the feature representation generated by Grow and Merge (Zhang et al., 2022). The qualitative visualization can be seen in Fig. 5; nodes of the same colour indicate that the instances belong to the same category. Moreover, for stage $ t > 0 $, we only highlight the feature’s node belonging to unknown categories. It is observed that across stages, our cluster features are more discriminative. 3.3 Ablation studies Table 4: Ablation study on different components of our approach \| Covariance Type \| No. Prompt \| No. GMM Sampling \| Sup.Contrastive \| C100 Avg ACC (%) \| CUB200 Avg ACC (%) \| \|-----------------\|------------\|------------------\|-----------------\|------------------\|-------------------\| \| \| \| \| \| All \| Old \| New \| All \| Old \| New \| \| N/A \| 0 prompt \| 0 sample \| ✓ \| 73.62 \| 73.69 \| 73.02 \| 55.46 \| 74.16 \| 48.79 \| \| Diagonal \| 5 prompts \| 100 samples \| ✓ \| 57.86 \| 65.18 \| 54.59 \| 33.29 \| 53.87 \| 26.69 \| \| Diagonal \| 2 prompts \| 100 samples \| ✓ \| 79.02 \| 75.21 \| 80.00 \| 57.24 \| 77.26 \| 51.50 \| \| Diagonal \| 5 prompts \| 100 samples \| ✓ \| 80.69 \| 76.26 \| 81.48 \| 59.16 \| 78.21 \| 53.65 \| \| Diagonal \| 10 prompts \| 100 samples \| ✓ \| 80.54 \| 73.92 \| 83.56 \| 60.28 \| 77.73 \| 54.06 \| \| Diagonal \| 5 prompts \| 0 samples \| ✓ \| 80.33 \| 72.23 \| 83.17 \| 57.84 \| 75.05 \| 51.91 \| \| Diagonal \| 5 prompts \| 20 samples \| ✓ \| 80.18 \| 73.89 \| 80.60 \| 58.87 \| 76.67 \| 52.44 \| \| Full \| 5 prompts \| 100 samples \| ✓ \| 78.59 \| 76.81 \| 78.56 \| 60.36 \| 78.45 \| 55.10 \| \| Spherical \| 5 prompts \| 100 samples \| ✓ \| 84.22 \| 77.83 \| 86.20 \| 60.06 \| 75.71 \| 54.01 \| To investigate the effectiveness of our Gaussian mixture-based prompt, we analyzed each component in our prompt module and present the results in Table 4. The results show a clear advantage of adopting the Gaussian mixture prompt into our model. The number of prompts, type of covariance, and number of GMM sampling are identified as important factors. For the CIFAR-100 and CUB-200 datasets, the optimal number of prompts is five, and the number of GMM samples is 100. Regarding GMM's covariance type, "Spherical" is found to be better for CIFAR-100, while "FULL" covariance type is better for CUB-200. The default configurations are "Diagonal" for covariance type, top 5 for prompt selection, and 100 samples for GMM sampling, which appears to be a good trade-off. 4 RELATED WORK Novel/Generalized category discovery is proposed to address the setting where there could be novel categories in the unlabelled dataset, and the goal is to automatically cluster those novel categories together (Han et al., 2019; 2021). Novel category discovery (NCD) assumes no overlap between the unlabelled and labelled data (Han et al., 2019; Zhao & Han, 2021; Fini et al., 2021), while generalized category discovery (GCD) (Vaze et al., 2022) is proposed to consider the setting where the categories in the unlabelled set can come from both the known and novel categories. It has been shown that self-supervised pretrained representations (Caron et al., 2021) greatly aid category discovery (Vaze et al., 2022). Vaze et al. (2022) further finetunes the model pretrained using one SSL contrastive loss (Chen et al., 2020) and one supervised contrastive loss (Khosla et al., 2020). Label assignment is done using a semi-supervised $k$-means algorithm. SimGCD (Wen et al., 2023) investigated the performance of parametric classifiers of different design choices, providing a strong baseline for GCD. Other works have proposed to focus on fine-grained categories (Fei et al., 2022), automatic category estimation (Hao et al., 2023; Zhao et al., 2023), and prompt learning (Zhang et al., 2023). Continual learning aims to train models that can learn to perform on a sequence of tasks, with the restriction of the model can only see the data for the current task it is trained on (De Lange et al., 2021). Catastrophic forgetting (McCloskey & Cohen, 1989) is a phenomenon that when the model is trained on a new task, it will quickly forget the knowledge on the task it has been trained on before, resulting in a catastrophic reduction of performance on the old tasks. There exists a rich literature on designing methods that enable the model to both learn to do the new task and maintain the knowledge of old tasks (Rebuffi et al., 2017; Li & Hoiem, 2017; Li et al., 2019; Wang et al., 2022b; Graves et al., 2016; Boschini et al., 2022; Buzzeo et al., 2020). However, these works all assume that the incoming tasks have all the labels for the data. In contrast, in our considered setting, we assume that the new data is fully unlabelled and can have category overlap with previous tasks. Continual category discovery (CCD) is a newly proposed setting with limited explorations (Zhang et al., 2022; Joseph et al., 2022; Liu et al., 2023; Roy et al., 2022). A setting termed class-iNCD is proposed by Roy et al. (2022), which is a two-stage setting where the model is first trained on a set of labelled data and then a set of only unlabelled data is provided where there is no class overlap between the two sets. Roy et al. (2022) proposed FRoST that performs replay using the feature prototypes learned on the labelled data during the discovery phase to prevent forgetting. Feature distillation and mutual information-based regularizers have also been shown to be effective for this task in NCDwF (Joseph et al., 2022). MSc-iNCD (Liu et al., 2023) extends this setting to multiple stages, and it is shown that a large pretrained model could greatly improve the performance of the discovery performance of novel categories in each of the multiple stages. Grow and Merge (Zhang et al., 2022) also tackles a similar multi-stage discovery setting, and a method containing a growing phase and a merge phase is proposed; the growing phase will use novelty detection to detect the novel categories and train the model to perform NCD; the merge phase combines the learned knowledge of the novel categories with the previous categories into a single model. A recently proposed setting termed IGCD (Zhao & Mac Aodha, 2023) considers a similar setting with MSc-iNCD, and a dataset based on the iNaturalist website is created. The most related work to ours is Zhang et al. (2022). In our paper, we adopt the data splits from Zhang et al. 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wM01y5BPM9	Beyond knowing where the object is (for which one could also use object detection in general, for this simplified case with fixed background the computer vision literature is also providing various method for background subtraction), in which way are these representations interpretable? It seems to me that these representation are	IDENTIFIABLE REPRESENTATION LEARNING VIA ARCHITECTURE EQUIVARIANCES Anonymous authors Paper under double-blind review ABSTRACT Despite their immense success and usefulness, current deep learning systems are still lacking in interpretability, robustness, and out of distribution generalisation. In this work we propose a method that helps address some of these issues in image and video data, by exploiting equivariances naturally present in the data. It enables learning latent representations that are identifiable and interpretable, and that can be intervened on to visualise counterfactual scenarios. The latent representations naturally correspond to positions of objects subject to image transformations, and so our method trains object detectors completely unsupervised, without object annotations. We prove that the learned latent variables are identifiable up to permutations and small shifts up to the size of model’s receptive fields, and perform experiments demonstrating this in practice. We apply it to real world videos of balls moving in mini pool (translational equivariance), cars driving around a roundabout (rotational equivariance) and objects approaching the camera on a conveyor belt (scale equivariance). In all cases, transformation-equivariant representations are learned unsupervised. We show that intervening on the learned latent space results in successful generalisation out of the training distribution, and visualise realistic counterfactual videos never observed at training time. The method has natural applications in industry, such as inspection and surveillance, with static cameras. 1 INTRODUCTION Some challenges facing current deep learning systems are interpretability, robustness, and out of distribution generalisation (Gilpin et al., 2018; Schölkopf et al., 2021). This is especially important in high-stakes domains such as healthcare and law, where for ML systems to be adopted, they need to be explainable, interpretable, and have guarantees about how they operate (Davenport & Kalakota, 2019; Bibal et al., 2021). One way to address this problem is to exploit the knowledge of equivariances naturally present in the data. For example, in an object detection task, one might want to detect an object regardless of its position and so one might use a neural network architecture that is equivariant to translation. In another example, in a system for monitoring traffic at a roundabout, one might exploit the circular structure of the system and design an architecture that is equivariant to rotation. Or, if for example, one deals with egocentric footage of highway traffic where vehicles become smaller as they drive further, one might want to make use of equivariance to scale for recognising the vehicles. In all of these cases, making a network equivariant to the right transformations has multiple benefits, including making the latent space more interpretable, obtaining extra guarantees about the structure of the latent space, and better generalisation to unseen data that obey the same set of equivariances. Additionally, an equivariant network requires a smaller number of training samples as well as a smaller memory footprint due to weight sharing, thus reducing the time for data collection and training the network. In this paper we propose a method to achieve this using an autoencoder-based architecture, where the encoder and decoder consist of blocks that make the latent representation equivariant to a specified transformation. This transformation is defined via a warping grid that can encode equivariances (e.g. to translations, rotations or scaling). The grid only needs to be specified once for each video scene, thus making it useful for inspection or surveillance applications, where cameras are typically static. Specifically, the encoder consists of a warping function followed by a standard CNN and a soft argmax function, and these operations are approximately inverted by the decoder. We prove that this configuration produces equivariant representations and also prove that the latent representation recovers the true variables (in this case, the objects’ positions) up to small shifts. After training we \| Work \| Generative \| Any equivariance \| Multiple objects \| Identifiable \| \|-----------------------\|------------\|------------------\|------------------\|--------------\| \| Ours \| ✓ \| ✓ \| ✓ \| ✓ \| \| Henriques & Vedaldi (2017) \| X \| ✓ \| X \| X \| \| Jakab et al. (2018) \| ✓ \| X \| ✓ \| X \| Table 1: Comparison between the characteristics of our work and of two relevant related works. can intervene on the latent variables and decode them into realistic counterfactual images and videos to visualise hypothetical scenarios never observed at training time. Making networks equivariant to different transformations has been studied before (e.g., Cohen & Welling (2014); Sosnovik et al. (2020); Han et al. (2021), and others), however many works achieve this by focusing on the properties of kernels and on discrete transformations, while we focus on equivariance to continuous transformations via input image warps. Equivariant and invariant networks were studied in different areas (Dieleman et al. (2015); Han et al. (2021); Lee et al. (2022); Pielawski et al. (2020); Musallam et al. (2022); Gupta et al. (2020)), however most works focus on discriminative problems (classification or regression), while our focus is to generate counterfactual images and videos never seen at training time. Further, differently from previous works studying identifiability of neural networks (Hyvarinen & Morioka (2016, 2017); Klindt et al. (2021); Khemakhem et al. (2020a,b); Zimmermann et al. (2021); Gresele et al. (2020)), we obtain guarantees for the identifiability of the learned latent representation by imposing equivariances on the model architecture. Concretely, our contributions in this paper are: 1. A novel generative, multi-object, equivariance-based method for learning latent representations of videos that are identifiable, interpretable, generalise out of the training distribution, and can be intervened on to generate counterfactual videos. 2. A proof of identifiability of the learned latent representation, showing that the latent variables are identifiable up to translations on the order of the model’s receptive fields. 3. Various experiments demonstrating the method on real world videos, including balls moving in mini pool (translational equivariance), cars driving around a roundabout (rotational equivariance), and objects on a conveyor belt under perspective (scale equivariance). The experiments demonstrate identifiability in practice, as well as the ability to generate realistic counterfactual videos never seen at training time, by intervening on the learned latent space. A direct comparison between our work and selected related works is shown in Table 1. 2 RELATED WORK Equivariances to different transformations in deep learning have been studied before. Cohen & Welling (2016) generalise CNNs to group equivariant CNNs (G-CNNs), however for many transformations this may require storing many filters. Gens & Domingos (2014) aims to achieve the same goal using Symmetry Networks. Cohen & Welling (2017) generalise G-CNNs to steerable CNNs which removes the memory scaling issue and allows working with infinite element groups. Cohen et al. (2019) propose gauge equivariant CNNs where the equivariance is to local gauge transformations on the surface of a sphere. Weiler & Cesa (2019) use E(2)-equivariant convolutions with steerable CNNs. Henriques & Vedaldi (2017) propose warped convolutions which achieve equivariance by warping the input image before passing it through a CNN. Focusing on specific transformations, Marcos et al. (2016, 2017); Li et al. (2018); Dieleman et al. (2015, 2016); Han et al. (2021); Pielawski et al. (2020); Gupta et al. (2020); Worrall et al. (2017) deal with equivariance and invariance to rotations and Kanazawa et al. (2014); Sosnovik et al. (2020) deal with equivariance and invariance to scale. In our work we deal with equivariances to continuous transformations (i.e. equivariance to a group with infinite number of elements), but we achieve this by warping the images, unlike for example steerable CNNs (Cohen & Welling (2017)) which achieves this using kernel properties. The closest work to ours is probably Henriques & Vedaldi (2017), however our method is generative while theirs is discriminative, and it has no guarantees of identifiability. Equivariant networks have been applied to different areas. For example, Dieleman et al. (2015) use rotational invariance for galaxy classification; Han et al. (2021) use rotational equivariance for aerial object detection; Lee et al. (2022) use equivariance for keypoint detection in images; Pielawski et al. (2020) use rotational equivariance for image registration; Musallam et al. (2022) use equivariant features for pose regression, and Gupta et al. (2020) use rotation equivariance for tracking. While most of the applications of equivariances have been discriminative (i.e. classification, regression), in this work we focus on generative modeling where we use equivariances to generate realistic data never observed at training time (counterfactuals). Identifiability of learned representations has been studied in the field of causal representation learning (Scholkopf et al., 2021). Locatello et al. (2019) have shown that learning identifiable latent variables is not possible in general without making assumptions about the model and the data. Thus, different works have made different assumptions about the distribution of the latent variables and about the mechanisms relating them (Hyvarinen & Morioka, 2016; 2017; Kindt et al., 2021; Khemakhem et al., 2020a,b; Zimmermann et al., 2021; Gresele et al., 2020); for an overview of identifiability assumptions in different works see Ahuja et al. (2022). Unlike previous works, we achieve identifiability by imposing grid-based spatial equivariances on the encoder and decoder architectures. 3 METHOD In this section we present our method, which is based on an autoencoder architecture whose latent representation is equivariant to different transformations of the input images (fig. 1). We start with a brief discussion of translational equivariance in CNNs (sec. 3.1), followed by the description of the warping process we use to obtain different types of equivariances (sec. 3.2) and finally describing the representational bottleneck (sec. 3.3). 3.1 CNNs AND TRANSLATIONAL EQUIVARINCE Depending on the data, one might want to choose different parametrisations for the encoder and the decoder of an autoencoder. For example, without any prior knowledge one might parametrise $\psi$ and $\phi$ by MLPs, as they have been shown to be universal function approximators (Hornik et al., 1989). However, if one knows that e.g. translating an input image $x_t$ should result in a proportional shift in the latent variables $z_t$, one might choose to parametrise $\psi$ and $\phi$ by CNNs. This is referred to as translational equivariance and it can be generalised to a broader class of transformations such as rotations or scaling. In general, a network $\psi$ is equivariant to transformation $T$ if applying the transformation $T$ to the data before passing it through the network is equivalent to passing the data through the network and applying a transformation $T'$ afterwards, i.e. $$\psi(T \circ x) = T' \circ \psi(x)$$ where $T$ and $T'$ may or may not be the same. CNNs consist of layers computing the convolution between a feature map $x$ and a filter $F$, defined in one dimension as $$(x \star F)[i] = \sum_j x[j]F[j - i]$$ Intuitively, this corresponds to sliding the filter $F$ across the feature map $x$ and at each position of the filter $i$ computing the dot product between the feature map $x$ and the filter $F$. Convolutional layers... \| Experiment \| Inverse Warp \| Forward Warp \| \|------------\|--------------\|--------------\| \| Translation \| $ x = u_1, \ y = u_2 $ \| $ u_1 = x, \ u_2 = y $ \| \| Rotation \| $ x = a_1 + b_1 \cdot c^{u_1} \cos(u_2) $ $ y = a_2 + b_2 \cdot c^{u_1} \sin(u_2) $ \| $ u_1 = \frac{1}{2} \log(c)^{-1} \log\left(\frac{x-a_1}{b_1}\right)^2 + \frac{y-a_2}{b_2}^2 $ $ u_2 = \arctan_2\left(\frac{y-a_2}{b_2}, \frac{x-a_1}{b_1}\right) $ \| \| Scale \| $ x = a_1 + b_1 \cdot c^{u_1} $ $ y = a_2 + b_2 \cdot c^{u_2} $ \| $ u_1 = \log(c_1)^{-1} \log\left(\frac{x-a_1}{b_1}\right) $ $ u_2 = \log(c_2)^{-1} \log\left(\frac{y-a_2}{b_2}\right) $ \| Table 2: Summary of expressions used to perform forward and inverse warp for different experiments, expressed in terms of the original image coordinates $ x, y $ and warped image coordinates $ u_1, u_2 $. are equivariant to translations, i.e. \[ ((\tau \circ x) \star F)[i] = \sum_j x[j-t]F[j-i] = \sum_j x[j]F[j-(i-t)] = \tau \circ (x \star F)[i] \] (3) where $ \tau $ is the translation operator that translates a feature map by $ t $ pixels, and we have used the substitution $ j \rightarrow j + t $ at the second equality. However, CNNs are not equivariant to other types of transformations such as rotations or scaling. We will now discuss one solution, using warping. ### 3.2 Generalised Equivariances via Warping In order to achieve equivariance to a broader class of transformations, we can change the variables of the data from cartesian coordinates to a new set of coordinates that achieves the desired equivariance when shifted (similar to Henriques & Vedaldi (2017)). Formally, we define the forward warp $ f_w $ as the invertible transformation that is applied to an image to change its coordinates to a new set of coordinates $(u_1, u_2)$ in which translation $ \tau $ corresponds to the desired transformation $ T $ in the original space (table 2, third column), and we define the inverse warp $ f_w^{-1} $ as the inverse of this transformation (table 2, second column), i.e. \[ [f_w^{-1} \circ \tau \circ f_w](x) = T(x) \] (4) For example, to obtain translational equivariance, $ T = \tau $, one can set $ f_w = I $ which means that the warped coordinates are identical to the original ones (table 2, first row; fig. 2, left column). To achieve equivariance to rotation transformations $ T $, one can change the variables to polar coordinates using a polar warp $ f_w $, where shifts along the angular dimension correspond to rotations in the original space (table 2, rows 2-3; fig. 2, middle column). Similarly, to achieve equivariance to scaling transformations $ T $, one can use a logarithmic warping map $ f_w $ to change the variables to log coordinates where the shifts correspond to scaling in the original space (table 2, rows 4-5; fig. 2, right column). Using this definition, we can prove that the warp $ f_w $ post-composed with the encoder CNN $ \psi $ is equivariant to the desired transformation $ T $ on the input and to the translation $ \tau $ on the output as \[ \psi \circ f_w(T \circ x) = \psi \circ f_w \circ (f_w^{-1} \circ \tau \circ f_w) \circ x = \psi \circ \tau \circ f_w \circ x = \tau \circ (\psi \circ f_w \circ x) \] (5) where at the first equality we have used the definition of $ T $ (eq. 4), at the second equality we have used the fact that $ f_w \circ f_w^{-1} = I $ as $ f_w $ is invertible, and at the third equality we have used the fact that the CNN $ \psi $ is equivariant to translations $ \tau $ (eq. 3). Note that Henriques & Vedaldi (2017) prove this equivariance only for exponential maps $ f_w $, while our assumption is weaker, namely that $ f_w $ has to be an invertible function that obeys $ f_w^{-1} \circ \tau \circ f_w = T $ (eq. 4), or equivalently, $ \tau \circ f_w = f_w \circ T $, thus generalising their proof.$^1$ We can prove a similar equivariance result for the decoder, namely that the decoder CNN $ \phi $ post-composed with the inverse warp $ f_w^{-1} $ is equivariant to the translation $ \tau $ on the input and to the desired transformation $ T $ on the output, i.e. \[ f_w^{-1} \circ \phi \circ (\tau \circ x) = f_w^{-1} \circ \tau \circ \phi \circ x = f_w^{-1} \circ \tau \circ (f_w \circ f_w^{-1}) \circ \phi \circ x = T \circ (f_w^{-1} \circ \phi \circ x) \] (6) where at the first equality we have used the fact that the CNN $ \phi $ is equivariant to translations $ \tau $ (eq. 3), at the second equality we have inserted the identity $ f_w \circ f_w^{-1} = I $, and at the third equality we --- $^1$For example, we can let $ f_w $ be both a polar coordinate warp $(x = u_1 \cos u_2, y = u_1 \sin u_2)$ and a log-polar coordinate warp $(x = e^{u_1} \cos u_2, y = e^{u_1} \sin u_2)$, while the results of Henriques & Vedaldi (2017) only apply to the log-polar warp because it is an exponential map, and not to the standard polar warp. have used the definition of $T$ (eq. [4]). In practice, we implement the forward and inverse warps $f_w$ and $f_w^{-1}$ by computing the forward and inverse warping grids $G_w$ and $G_w^{-1}$ offline by $$G_w = \{ f_w^{-1}(u_1, u_2) : (u_1, u_2) \in \{0, 1, ..., U_1\} \times \{0, 1, ..., U_2\} \}$$ $$G_w^{-1} = \{ f_w(x, y) : (x, y) \in \{0, 1, ..., X\} \times \{0, 1, ..., Y\} \}$$ where $f_w$ and $f_w^{-1}$ are obtained from table 2 (columns 2-3), $X, Y$ are the image dimensions, and $U_1, U_2$ are the dimensions of the warped space (Henriques & Vedaldi, 2017). Note the correspondence of inverses between $G_w$ and $f_w^{-1}$, and between $G_w^{-1}$ and $f_w$. These grids are then used online to warp the images as $f_w(x) = x[G_w]$ and $f_w^{-1}(x) = x[G_w^{-1}]$ where $x[G]$ denotes sampling an image $x$ at the points defined by the grid $G$ using bilinear interpolation, which is a fast operation. Note that the warping grids only need to be defined once for every video scene, making it practical for applications where the camera is static. In the next section we discuss how these equivariances of feature maps relate to the learned latent representation. ### 3.3 From Feature Maps to Variables So far we have only worked with images and feature maps, but ultimately we would like to obtain scalar latent variables that are equivariant to transformations applied to the images. This is because dealing with scalars is more natural and interpretable than dealing with feature maps – for example, it is natural to think about an object’s position in terms of its coordinates instead of a feature map. To do this, we first define a translation $\tau$ of a (1D) feature map $x$ and a translation $\tau'$ of a scalar $z$ as $$\tau(x)[i] = x[i - t], \quad \tau'(z) = z + t$$ where $i$ is the position in the feature map $x$, $\tau$ shifts an image by $t$ pixels, and $\tau'$ shifts a scalar by $t$ units. To relate translations in feature maps to translations in latent variables, we can use a function that computes a scalar property of a feature map $x$, such as argmax, defined as $\text{argmax}(x) = \{i : x[j] \leq x[i] \forall j\}$. Using these definitions we can now prove the equivariance of argmax, i.e. that shifting the feature map $x$ by $\tau$ corresponds to shifting the latent variable $\text{argmax}(x)$ by $\tau'$: $$\text{argmax}(\tau \circ x) = \{i : \tau \circ x[j] \leq \tau \circ x[i] \forall j\} = \{i : x[j - t] \leq x[i - t] \forall j\}$$ $$= \{i + t : x[j] \leq x[i] \forall j\} = \text{argmax}(x) + t = \tau' \circ \text{argmax}(x)$$ where at the first equality we have used the definition of argmax, at the second equality we have used the definition of $\tau$ (eq. 9 left), at the third equality we have used the substitution $i \rightarrow i + t$, at the fourth equality we have used the definition of argmax, and at the last equality we have used the definition of $\tau'$ (eq. 9 right). Similarly, to now relate shifts in latent variables $z$ to shifts of feature maps $x$, we can invert the action of the argmax operation. Because argmax is a many-to-one function, finding an exact inverse is not possible, but we can obtain a pseudo-inverse using the delta function defined as $\delta(z)[i] = \delta(i - z)$ where $\delta$ is the Dirac delta function. We can show that delta is a pseudo-inverse of argmax because $\text{argmax} \circ \delta \circ z = \{i : \delta(x - z)[j] \leq \delta(x - z)[i] \forall j\} = z$. Now, similar to the argmax function, we can prove that the delta function is equivariant to the latent variable shift $\tau'$ on the input and the feature map shift $\tau$ on the output, i.e. $$\text{delta}(\tau' \circ z)[i] = \delta(i - \tau' \circ z) = \delta(i - z - t) = \text{delta}(z)[i - t] = \tau \circ \text{delta}(z)[i]$$ (11) where at the first equality we have used the definition of delta, at the second equality we have used the definition of $\tau'$ (eq. 9 right), at the third equality we have used the definition of delta, and at the last equality we have used the definition of $\tau$ (eq. 9 left). Now we have the tools to convert between equivariances in feature maps and latent variables via the functions argmax and delta. However, because these operations are not differentiable, for neural network training we approximate argmax via a differentiable function softargmax, defined in two dimensions as $$\text{softargmax}(x) = \left( \frac{1}{I} \sum_{i=0}^{I-1} \sum_{j=0}^{J-1} \sigma_1 \left( \frac{x}{\Theta} \right)[i,j], \frac{1}{J} \sum_{i=0}^{I-1} \sum_{j=0}^{J-1} \sigma_2 \left( \frac{x}{\Theta} \right)[i,j] \right)$$ (12) where $\sigma$ is the softmax function defined in one dimension as $\sigma(x)[i] = \exp(x[i]) / \sum_j \exp(x[j])$, $\sigma_1(x)$ and $\sigma_2(x)$ is the softmax function evaluated along the first and second dimensions of $x$, $\Theta$ is a temperature hyperparameter, $[i,j]$ is the image index, $I$ is the image width, and $J$ is the image height. As the temperature $\Theta$ in (12) approaches zero, softargmax reduces to the classical argmax function. Similarly, we can approximate the hard delta function using a differentiable render function as $$\text{render}(z)[i] = N(i - z, \sigma^2)$$ (13) where $N(i - z, \sigma^2)$ is a normal distribution evaluated at $i - z$ with variance given by the hyperparameter $\sigma^2$. As the variance $\sigma^2$ in eq. (13) approaches zero, the render function reduces to the hard delta function. Therefore, now we have all the elements we need to create an equivariant architecture where the encoder and decoder are defined, respectively, by $$z_t = \text{softargmax} \circ \psi \circ f_w \circ x_t, \quad \hat{x}_t = f_w^{-1} \circ \phi \circ \text{render} \circ z_t.$$ (14) This is illustrated in fig. 1. In the next section we prove identifiability of the learned latent variables. ### 4 THEORETICAL RESULTS In this section we show that the learned latent variables are identifiable with respect to the ground truth physical variables, up to permutations and small shifts. Theorem 1 (Identifiability of latent representation). Consider an image $x_t$ with objects of size $s_O$, warping map $f_w$, CNN encoder $\psi$ with receptive field size $s_\psi$, CNN decoder $\phi$ with receptive field size $s_\phi$, soft argmax function softargmax, Gaussian rendering function render, and latent variables $z_t$, composed as $z_t = \text{softargmax} \circ \psi \circ f_w \circ x_t$ and $\hat{x}_t = f_w^{-1} \circ \phi \circ \text{render} \circ z_t$ (fig. 1). Assuming (A1) The reconstruction loss is minimised, $\min_{\psi, \phi} L(\hat{x}, x)$. (A2) Each object has at least two distinct positions in the training set. (A3) The warping map $f_w$ is a diffeomorphism. (A4) There are no two identical objects in any image $x_t$. (A5) Each image $x_t$ has the same background. (A6) The Gaussian rendered by the render function is a delta function. Then the latent variables $z_t$ are identified up to permutations and maximum shifts of $\min(s_\psi + f_w(s_O), s_\phi)/2$. For the special case that $s_\psi = s_\phi = s_{RF}$, the shifts reduce to $s_{RF}/2$. Here we present a proof sketch; for a full proof see Appendix A. First, minimising the reconstruction loss (A1) means that the objects in the predicted image have to be reconstructed at the same positions as in the original image. Then, the dataset having each object present at a minimum of 2 different positions (A2) ensures that the latent variables used by the decoder must contain information about each object, and thus the encoder must learn to match all objects. Next, the warp being a diffeomorphism (A3) the encoder being equivariant to the transformation that generated the data (eq. 5), and each image containing distinct objects (A4) on a static background (A5) ensure that each different object is mapped to a unique latent variable. This variable is correct up to a small shift, because any part of the receptive field of the encoder can match any part of the (warped) object, $(s_\psi + f_w(s_O))/2$, not just the center. Similarly, when decoding there is possibly another small shift because any part of the decoder filter may be convolved with the rendered delta function (A6) i.e. $s_\phi/2$. Because the \| \| MLP \| CNN \| Keypoint CNN \| Proposed Method \| \|----------------\|--------------\|--------------\|--------------\|-----------------\| \| \| MSE Acc. \| MSE Acc. \| MSE Acc. \| MSE Acc. \| \| Translation \| 2.05 98.3% \| 1.6 · 10^5 99.0% \| – – \| 8.2 · 10^{-3} 99.6% \| \| Rotation \| 1.92 97.3% \| 9.6 · 10^3 95.5% \| 0.197 96.9% \| 1.7 · 10^{-2} 97.3% \| \| Scale \| 7.25 96.9% \| 4.3 · 10^4 92.2% \| 0.192 97.1% \| 1.9 · 10^{-2} 97.5% \| Table 3: Results showing the mean squared error of the predicted latent variables w.r.t. estimated ground truth physical variables (MSE, lower is better) and the image reconstruction accuracy of the decoded video frames w.r.t. input video frames (Acc., higher is better). Results are reported for the proposed method and for MLP, CNN, and keypoint CNN baselines for each experiment. predicted and original objects must have the same position $A_1$, the shifts from the encoder and the decoder have to cancel each other, and thus the latent variables are shifted by a maximum amount of $\min((s_\psi + f_w(s_O))/2, s_\phi/2)$. Additionally, because the objects can be mapped to the variables in an arbitrary order, there is additional non-identifiability due to object permutations. ## 5 EXPERIMENTS In this section we present 3 experiments validating our method from sec. 3, one using translational equivariance (sec. 5.1), one using rotational equivariance (sec. 5.2), and one using scale equivariance (sec. 5.3). In each experiment we demonstrate that making the network architecture equivariant to a transformation naturally present in each dataset allows one to identifiably learn latent variables corresponding to the ground truth physical variables (table 3, MSE), and to intervene on the learned latent variables (fig. 3) to generate realistic counterfactual videos never seen at training time (fig. 4). We implement the method described in sec. 3 using the architecture in fig. 1 with the warps summarised in table 2. For comparison, in each experiment we also train 3 analogous baseline models: MLP, CNN and keypoint CNN (Jakab et al., 2018). For implementation details see appendix B. ### 5.1 TRANSLATION Setup. The training and test sets for this experiment consist of 15 and 11 frames respectively from a video of two balls moving on a mini pool table, visualised in fig. 2 upper left plot. Because the table naturally extends horizontally and vertically, we seek to employ an autoencoder architecture that is equivariant to horizontal and vertical translations. Because a standard CNN is already translationally equivariant, we use a standard CNN encoder and decoder with an identity warp (table 2, first row) visualised in fig. 2 (first column). Identifiability results. The latent variables corresponding to the training data are visualised in fig. 3 (left plot) in blue and purple for the first and second balls respectively, resulting in straight lines for the moving balls as expected. When compared to the estimated ground truth variables describing the balls’ position, the latent variables mean squared error on the test set is very small, orders of magnitude smaller than the baselines (table 3, MSE, top row). This is to be expected as an MLP architecture does not exhibit any equivariances and so performs poorly on the test set, where the balls are now at positions never encountered at training time. Similarly, the CNN baseline achieves a comparably poor performance, because while the network contains convolutional layers, the translational equivariance is broken by the linear layer mapping features to latent variables. In this case, the keypoint CNN baseline is equivalent to our method due to the warp being an identity, and so is not included in table 3. Our method also achieves the best test set reconstruction accuracy (table 3, Acc., top row) as the translational equivariance allows it to successfully generalise to the test set. Counterfactual results. Once the mapping between the images and the latent space has been learned, we can use the translational equivariance property of the network to intervene on the latent variables and generate videos of counterfactual scenarios that were never observed at training time. For example, one can visualise the balls moving in opposite directions at a constant speed (fig. 3 left plot, red and orange; fig. 4 upper middle plot) and the white ball bouncing off one of the table edges while slowing down (fig. 3 left plot, green; fig. 4 upper right plot). Note that none of these scenarios were observed at training time, demonstrating that the model successfully generalises out of the training distribution. It also allows controlled generation with interpretable latent variables. Figure 3: Latent space showing the training data and two different counterfactuals for each experiment that are out of the training distribution. $z_x$ and $z_y$ denotes the horizontal and vertical position, $z_\theta$ and $z_{\log r}$ are the angular and (log) radial position, and $z_{10x}$ and $z_{10y}$ are the horizontal and vertical position on a log scale. The colour intensity denotes the arrow of time (light to dark). 5.2 Rotation Setup. The training and test sets consist of 35 and 15 frames each from a video of two cars driving around a quarter of a roundabout, visualised in fig. 4, middle left plot. Because the cars at the roundabout can move in an angular (forward or backward) or radial (change lanes) direction, we would like to employ an autoencoder architecture which is equivariant to rotation and radial shifts around the center of the roundabout. We achieve this by using a log-polar warp (table 2, middle row), visualised in fig. 2 (second column), together with a standard CNN autoencoder. Identifiability results. The latent variables corresponding to the training data are visualised in fig. 3, middle plot, in blue and purple for the two cars respectively. The data forms two approximately straight lines with a steadily increasing angular position (and a slightly increasing radial position), as expected. When compared to the estimated ground truth variables describing the car’s position, the latent variables’ mean squared error on the test set is an order of magnitude smaller than the best baseline (table 3, MSE, middle row), which reflects the fact that none of the baselines exhibit equivariance to rotation and radial position. Consequently, our method also achieves the best test set reconstruction accuracy (table 3, Acc., middle row) as the rotational equivariance allows it to generalise successfully to the test set. Although the MLP baseline achieves a comparable reconstruction accuracy to our method, this is misleading because the MLP renders the objects at incorrect positions and the high accuracy arises due to better reconstruction of the background, whereas our method reconstructs the cars at the correct positions albeit with slightly more noise. Counterfactual results. Once the encoder and decoder have been learned, we can use the rotational and radial equivariance property of the network to intervene on the latent variables and generate videos of counterfactual scenarios that were never observed at training time. For example, one can make the first latent variable have a constant radial distance and an increasing angular position (fig. 3, middle plot, red) to visualise the white car continuing to drive around the whole roundabout (fig. 4, center plot), or have the second variable increase its angular position and decrease its radial position (fig. 3, middle plot, yellow) to visualise the blue car moving forward while changing lanes at the same time (fig. 4, top right plot). Because none of these scenarios were observed at training time, this demonstrates that the model successfully generalises out of the training distribution. 5.3 Scale Setup. The training and test sets consist of 79 and 40 frames respectively from a video of two sushi bowls moving closer to the camera on a conveyor belt (visualised in fig. 4, bottom left plot). Because the bowls have a different scale depending on their position, we would like to employ an autoencoder architecture that is equivariant to scale. We achieve this by using a scale warp (table 2, bottom row) visualised in fig. 2 (right column), together with a standard CNN autoencoder. Identifiability results. The latent variables corresponding to the training data are visualised in fig. 3 (right plot, blue and purple for the two bowls respectively). The data forms a diagonal line in the latent space in logarithmic coordinates, showing an approximately exponential relationship between Figure 4: Training data (left column) and two different counterfactuals (middle and right columns) for each experiment (rows). Our method learns to detect the objects unsupervised, with no object annotations. Additionally, it can generate images and extrapolate them to new situations. The counterfactuals are obtained by intervening on and decoding the latent variables to obtain out-of-distribution data never seen during training. the bowls’ position and scale. When compared to the estimated ground truth variables describing the bowls’ position, the latent variables’ mean squared error on the test set is an order of magnitude smaller than the best baseline (table 3, MSE, bottom row), which is to be expected as none of the baselines exhibit scale equivariance. Our method also achieves the best reconstruction accuracy (table 3, Acc., bottom row) as the scale equivariance allows it to successfully generalise to the test set. Counterfactual results. Once the encoder and decoder have been learned, we can use the scale equivariance property of the network to intervene on the latent variables and generate videos of counterfactual scenarios that were never observed at training time. For example, one can extrapolate the latent variables for the first object (fig. 3 right plot, orange) to visualise where the orange bowl has been in the past (fig. 4 bottom middle plot), or extrapolate the variables for the second object in both directions (fig. 4 right plot, green) to visualise where the blue bowl was in the past and where it will be in the future, assuming constant speed (fig. 4 bottom right plot). Because none of these scenarios were observed at training time this demonstrates the model successfully generalising out of the training distribution. We note that it is naturally easier to extrapolate from larger to smaller scales (orange bowl, fig. 4 bottom middle plot) than it is to extrapolate in the opposite direction (blue bowl, fig. 4 bottom right plot) since more details are required to extrapolate to larger than smaller scales. 6 CONCLUSION In this work we presented a method for learning an identifiable and interpretable latent representation of images and videos by exploiting equivariances naturally present in the data. We achieved this using an autoencoder architecture where the image is warped by a map corresponding to a specified equivariance before being passed through a CNN and a softargmax operation, and it is reconstructed by inverting this process. We proved that the learned latent representation is identifiable with respect to the ground truth variables and demonstrated this experimentally. We then applied the method to real world videos with multiple objects and different naturally present equivariances, and showed that by intervening on the latent representation we can generate realistic counterfactual videos that were never observed at training time. It also works as an unsupervised object detector, trained using raw video footage. In future work we would like to expand the current class of equivariance transformations and consider dealing with non-static backgrounds. Reproducibility Statement: we will make all source code available upon publication. REFERENCES Kartik Ahuja, Jason Hartford, and Yoshua Bengio. Properties from mechanisms: an equivariance perspective on identifiable representation learning. In International Conference on Learning Representations, 2022. URL https://openreview.net/forum?id=g5ynW-jMq4M. Adrien Bibal, Michael Lognoul, Alexandre Streel, and Benoît Frénay. Legal requirements on explainability in machine learning. 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z7K2faBrDG	Following the above comment, the proposed method would be unable to give a measure to predict the Mean Opinion Score (MOS) on distortion in databases such as TID [Ponomarenko et al. 13] or KADID [Lin et al. 19]. If this is not the case, the authors should mention how to infer this MOS.	Perceptual Scales Predicted by Fisher Information Metrics Jonathan Vacher∗ MAP5, Université Paris Cité, CNRS, F-75006, Paris, France jonathan.vacher@u-paris.fr Pascal Mamassian LSP, Département d’études cognitives, École normale supérieure, PSL University, CNRS, 75005 Paris, France pascal.mamassian@ens.fr Abstract Perception is often viewed as a process that transforms physical variables, external to an observer, into internal psychological variables. Such a process can be modeled by a function coined perceptual scale. The perceptual scale can be deduced from psychophysical measurements that consist in comparing the relative differences between stimuli (i.e. difference scaling experiments). However, this approach is often overlooked by the modeling and experimentation communities. Here, we demonstrate the value of measuring the perceptual scale of classical (spatial frequency, orientation) and less classical physical variables (interpolation between textures) by embedding it in recent probabilistic modeling of perception. First, we show that the assumption that an observer has an internal representation of univariate parameters such as spatial frequency or orientation while stimuli are high-dimensional does not lead to contradictory predictions when following the theoretical framework. Second, we show that the measured perceptual scale corresponds to the transduction function hypothesized in this framework. In particular, we demonstrate that it is related to the Fisher information of the generative model that underlies perception and we test the predictions given by the generative model of different stimuli in a set of difference scaling experiments. Our main conclusion is that the perceptual scale is mostly driven by the stimulus power spectrum. Finally, we propose that this measure of perceptual scale is a way to push further the notion of perceptual distances by estimating the perceptual geometry of images i.e. the path between images instead of simply the distance between those. 1 Introduction Difference Scaling Difference scaling methods allow us to measure the relative perceptual differences of multiple stimuli in human observers. Such methods have been used as early as in the 1960s to measure the relative differences of perceived color, contrast or loudness (see Maloney & Yang (2003) and references therein). This is only at the beginning of our century that a fitting method, called Maximum Likelihood Difference Scaling (MLDS), was developed (Maloney & Yang 2003; Knoblauch & Maloney 2008) to infer the function that maps the physical to the perceptual space. This function is called perceptual scale. The critical assumption behind the fitting methods dates back to Thurstone’s law of comparative judgment (see case V Thurstone (1927)): the difference between two values along a psychological dimension is corrupted by noise that has a constant variance. The perceptual scale informs us about how a stimulus is perceived when modified along a continuous physical scale (e.g. color, contrast, …). When the slope of the perceptual scale is steep, perception changes rapidly with small physical changes i.e. the observer is highly sensitive to physical variations. When the slope is shallow, perception is stable even for large physical variations i.e. the observer is weakly sensitive to physical variations. Recently, the MLDS method has been used to measure the perceptual scales of surface texture (Emrith et al., 2010), watercolor effect (Devinc & Knoblauch, 2012), slant-from-texture (Aguilar et al., 2017), lightness (Aguilar & Maertens, 2020) or probabilities (Zhang et al., 2020). However, perceptual scales of more fundamental physical variables such as orientation and spatial frequency have not been measured nor related to existing probabilistic ∗https://jonathanvacher.github.io/ theory of perception. Additionally, while relations between standard Two-alternative Forced Choice (2AFC) measurements and perceptual scales have been studied (Aguilar et al., 2017), it was a theory of perception that was previously described (Wei & Stocker, 2017) that has been used to derive predictions. Probabilistic modeling of perception Thurstone’s law of comparative judgment is a first brick in the history of probabilistic modeling of perception (Thurstone, 1927). Indeed, it introduced the incipient concept of random variable (Chebyshev, 1867; Kolmogoroff, 1933) in psychophysics. Then, the development of computer science and information theory had major impact in perception studies, bringing concepts such as redundancy reduction and information maximization (Atneave, 1954; Barlow et al., 1961). More specifically, when applied to texture perception, these concepts led to Julesz’ hypothesis that perception of textures is statistical (Victor et al., 2017) i.e. textures with similar statistics are indistinguishable. Later on, together with advances in image processing, Julesz’ hypothesis led to modern texture synthesis algorithms (Portilla & Simoncelli, 2000; Gatys et al., 2015). In parallel, a core theorem of probabilities, namely Bayes rule, was found to efficiently predicts human perceptual behaviors (Knill & Richards, 1996). Further works have been dedicated to solve the inverse problem of identifying observers’ prior that best explain their perception (Stocker & Simoncelli, 2006; Girshick et al., 2011; Vacher et al., 2018; Manning et al., 2023). Inspired by neural population coding models, an optimal observer theory is now described in details by Wei & Stocker (2017). The main consequence of this theory is the existence of a simple relation between perceptual bias and sensitivity. Yet, the theory is limited to a hypothesized scalar perceptual variable while it is established that only part of the neurons of the primary visual cortex is tuned to scalar variables such as spatial and temporal frequencies or orientation (Olshausen & Field, 2005). In higher visual areas, it is more and more difficult to identify scalar variables that uniquely drive single neurons as they respond to increasingly complex patterns (Bashivan et al., 2019). In some previous work, Wainwright (Wainwright, 1999) have used higher dimensional natural image statistics (auto-correlation and power spectrum) to explain various psychophysical observations. Though, it is unclear how this approach relates to the univariate Bayesian framework. Fisher information in neural populations The theory behind the work of Wei & Stocker (2017) is largely inspired by previous work on neural population coding (Brunel & Nadal, 1998). In this and subsequent works (Yarrow et al., 2012; Kanitscheider et al., 2015; Bethge et al., 2002; Wei & Stocker, 2016), it is often ultimately assumed that neurons are Poisson firing neurons parameterized by the tuning curve of a scalar variable. Variants of this optimal coding framework have been explored to explain the response of single neurons (Laughlin, 1983; von der Twer & MacLeod, 2001). As stated previously, it is a quite restrictive framework as all neurons are not tuned to a scalar variable (Olshausen & Field, 2005). By applying this framework to perception, Wei and Stocker remove these unnecessary assumptions. They derive the relation between perceptual bias and sensitivity which at its core comes from the relation between Fisher information and prior under the optimal coding assumption (Brunel & Nadal, 1998) \[ P_S(s) \propto \sqrt{I(s)} \] where $ S $ is a stimulus variable. Fisher information is used to quantify the variance of a stimulus estimator from a neural population encoding (Cramer-Rao lower bound). In contrast, priors are introduced in observer models to explain perceptual biases. Therefore, Equation (1) links neural population models and perceptual models. However, less attention is dedicated to the underlying encoding model, where a stimulus variable $ S $ is non-linearly related to an internal measurement $ M $ through a function $ \psi $ plus an additive Gaussian noise $ N $ with constant variance, \[ M = \psi(S) + N. \] Interestingly, such an encoding model is very similar to the assumptions behind the observer model underlying the MLDS method. However, the precise nature of these internal measurements has so far remained abstract. Perceptual distance A perceptual distance is a score of image quality used to quantify and to compare the performances of image restoration or generation methods. Perceptual distances have been introduced to overcome the limitation of the Signal-to-Noise Ratio (also known as SNR). Indeed, images with similar SNR could vary subjectively in quality when presented to human observers (Wang et al., 2004). The Structural SIMilarity index (SSIM) is a score that is popular to provide a better account of perceptual similarity as compared to SNR. Since then, variations of SSIM have been proposed to more specific purposes such as estimating photo retouching (Kee & Farid, 2011). However, these scores require to compare the image to be rated to a reference image. In recent years, the success of deep generative modeling have led to the emergence of new scores such as the Inception score (Salimans et al., 2016) or the Fréchet Inception Distance (FID) (Heusel et al., 2017). These scores have in common that they compare the generated image distribution to the true empirical image distribution instead of a generated image to a reference image. In addition, they are based on Deep Neural Network (DNN) features. Overall, it has been shown that such DNN features-based scores are better aligned with human perception than SSIM or SNR for example (Zhang et al., 2018). One possible explanation is that DNN are able to capture high-order image statistics and that as hypothesized in vision, our perception is deeply related to image statistics (see Hepburn et al., (2022) and references in previous paragraphs). Yet, these coined perceptual distances are not exempt of limitations as they could be subject to bias when classes specific features are present or not (Kykkänniemi et al., 2023). Overcoming these biases will likely require to move away from training by measuring higher-order statistics on the image directly without relying on some learned or random filters (Amir & Weiss, 2021). One other limitation of perceptual distances is that they do not provide any information about how well a model has captured the perceptual geometry of images. Providing a full account of perceptual geometry is more demanding, it requires to compare the path when moving from one image to another and not only their distance along this path. Contributions Our work brings several contributions to overcome the limitations introduced above. First, we explain that a convergence theorem of discrete spot noises is a way to resolve the tension between univariate Bayesian theories of perception (Wei & Stocker, 2017) and the high dimensionality of images (Wainwright, 1999). More precisely, the hypothesis that an observer has a univariate representation of the distribution of the parameter of interest (e.g. spatial frequency), is compatible with the assumption that an observer is measuring the spectral energy distribution of the image in that both assumptions leads to similar predictions. Second, we show that the function $\psi$, introduced in Equation 2, can be interpreted as the perceptual scale as measured by a difference scaling experiment. Then, we demonstrate again that this function $\psi$ can be predicted from the Fisher information of the stimulus when using the true distribution of the noisy internal stimulus representation knowing the presented stimulus i.e. the distribution of the measurements $M$ that we give explicitly. Therefore, we provide a clear link between theory and experiment. Third, we propose to go further in exploring perceptual distances by estimating how well the geometry of natural images captured by models matches the perceptual geometry. For this purpose, we empirically test the prediction given by the Fisher information of Gaussian vectors and processes in a series of experiments (code and data[^1], texture interpolation code[^2]) involving stochastic stimuli characterized by their power spectrum or their higher-order statistics captured by VGG-19 (Gatys et al., 2015). In particular, we collapse the high dimensionality of these statistics by interpolating between single textures (Vacher et al., 2020) and we measure the corresponding perceptual scale when going from one texture to another. Finally, we propose the Area Matching Score (AMS) score to quantify the mismatch between the predicted and the measured perceptual scales providing a clear method to evaluate the perceptual alignment between generative image models and human vision. Notations Unless stated differently, upper case letters (e.g. $X$) are random variables and lower case letters (e.g. $x$) are samples or realizations of those random variables. The probability density at $X = x$ is denoted $P_X(x)$. Similarly, the conditional probability density at $X = x$ knowing $Y = y$ is denoted $P_{X\|Y}(x, y)$. The set $S = [s_{\text{init}}, s_{\text{final}}]$ is the stimulus segment. 2 METHODS 2.1 STOCHASTIC VISUAL STIMULATION We recall some theoretical results about the artificial textures we use in the current work. Firstly, we use textures that are stationary Gaussian Random Fields (GRFs) fully characterized by their scalar mean and their auto-correlation function (or equivalently their power spectrum i.e. the auto-correlation Fourier transform). Interestingly, such GRFs can be seen as the limit of high intensity discrete spot noises. This result allows one to relate the densities of local image features such as orientation and scales to the power spectrum of the image (seen as a GRF). In summary, it provides a link between scalar densities and the high-dimensional Gaussian distribution of the image. We will [^1]: https://github.com/JonathanVacher/perceptual_metric [^2]: https://github.com/JonathanVacher/texture-interpolation see in later sections that this result leads both approach to similar predictions for perceptual scales. Secondly, we use naturalistic textures that are obtained by imposing high-order and high-dimensional statistics obtained using VGG-19. However, the result mentioned above and detailed below does not hold for naturalistic textures. It is unknown at this stage if similar results could be obtained with some feature under some (non-linear) transformation. Asymptotic Discrete Spot Noise Let $ \xi_0 = (1, 0) $. Let $ g_\sigma $ be a Gabor function defined for all $ \sigma > 0 $ and for all $ x \in \mathbb{R}^2 $ by $ g_\sigma(x) = \frac{1}{2\pi} \cos(x \cdot \xi_0) e^{-\frac{\sigma^2}{2} \\|x\\|^2} $. In addition, let $ \varphi_{z,\theta} $ be a scaled rotation defined for all $ (z,\theta) \in \mathbb{R}_+ \times [0, \pi] $ by $ \varphi_{z,\theta}(x) = z R_{-\theta}(x) $ where $ R_\theta $ is the rotation of angle $ \theta $. Now, let $ F_{\lambda,\sigma} $ be a discrete spot noise of intensity $ \lambda > 0 $ defined as the following random field for all $ x \in \mathbb{R}^2 $, $ F_{\lambda,\sigma}(x) = \frac{1}{\sqrt{\lambda}} \sum_{k \in \mathbb{N}} g_\sigma(\varphi_{Z_k,\Theta_k}(x - X_k)) $ where $ (X_k, Z_k, \Theta_k)_{k \in \mathbb{N}} $ are iid random variables. Specifically $ (X_k)_{k \in \mathbb{N}} $ is a 2-D Poisson process of intensity $ \lambda > 0 $ and $ (Z_k, \Theta_k)_{k \in \mathbb{N}} $ have densities $ (\mathbb{P}_Z, \mathbb{P}_\Theta) $. Proposition 1 (Convergence and Power Spectrum). In the limit of infinite intensity $ (\lambda \to +\infty) $ and pure wave $ (\sigma \to 0) $, $ F_{\lambda,\sigma} $ converges towards a Gaussian field $ F $ with the following power spectrum for all $ \xi \in \mathbb{R}^2 $, \[ \hat{\gamma}(\xi) = \frac{1}{\\|\xi\\|^2} \mathbb{P}_Z(\\|\xi\\|) \mathbb{P}_\Theta(\angle \xi) \] where $ \xi = (\\|\xi\\| \cos(\angle \xi), \\|\xi\\| \sin(\angle \xi)) $. Proof. This is a special case of Proposition 2 in Vacher et al. (2018). The general result is Theorem 3.1 in Galerne (2010). In practice, the distribution $ \mathbb{P}_Z $ and $ \mathbb{P}_\Theta $ are parametrized by $ (Z_0, \Sigma_Z) $ and $ (\Theta_0, \Sigma_\Theta) $ respectively. By providing a relation between local feature statistics (orientation and scale) and the image power spectrum, Proposition 1 will allow us to justify the common assumption made when modeling psychophysical data that is, the feature of interest is directly used by the observer instead of the image (Knill & Richards, 1996; Stocker & Simoncelli, 2006; Girshick et al., 2011). See Section 2.3. Interpolation of Naturalistic Textures Even though for experimental purposes the GRFs described above can be parameterized by just a few scalar variables (Vacher et al., 2018), naturalistic textures depend on the statistics of high-dimensional features extracted at different layers of VGG-19 (Gatys et al., 2015; Vacher et al., 2020). Previous algorithms widely used in vision studies (Portilla & Simoncelli, 2000; Vacher & Briand, 2021) were using fewer parameters, but the number was still too large to derive clear and interpretable results (Okazawa et al., 2015). One way to efficiently collapse the dimension parameterizing those textures is to use interpolation (Vacher et al., 2020). As a consequence the texture features of an interpolation of textures extracted at layers $ k $ are interpreted as realizations of a random variable $ A_k(s) $ with mean $ \mu_W(s) $ and covariance $ \Sigma_W(s) $ (see Appendix C). Assuming Gaussianity, it will become possible to derive predictions for the perceptual scale measured along the interpolation path. ### 2.2 Thurstone Scale, Fisher Information and MLDs First, we define more precisely the encoding model given in Equation 2 as follows \[ M = R + N \quad \text{where} \quad R = \psi(S) \quad \text{with} \quad \psi : S \to \mathbb{S}. \] We use the above description to highlight the fact that $ M, R $ and $ N $ are random variables that are internal to the observer while $ S $ is external to them, it is an environment variable, an external stimulus. In practice, the noise $ N $ is often assumed to be Gaussian with variance $ \sigma^2 $. It corrupts the internal representation of the stimulus $ R $ to give what we call the internal measurement $ M $. Then, we define Fisher information for two abstract unidimensional random variables. Definition 1 (One-dimensional Fisher information). Let $ X $ and $ Y $ be two random variables defined respectively on two abstract spaces $ \mathbb{X} $ and $ \mathbb{Y} $ and let $ \mathbb{P}_{X\|Y} $ be the conditional density of $ X $ knowing $ Y $. The Fisher information carried by $ X $ about $ Y $ is a function $ I : \mathbb{Y} \to \mathbb{R} $ defined for all $ y \in \mathbb{Y} $ by \[ I_Y(y) = \mathbb{E}_{X\|Y} \left( \left( \frac{\partial \log(\mathbb{P}_{X\|Y})}{\partial y}(X,y) \right)^2 \right). \] Figure 1: Texture samples and predicted perceptual scales for the spatial frequency mode ($z_0$), the spatial frequency bandwidth ($b_z$) and the orientation bandwidth ($\sigma_\theta$). Bottom-right: prediction obtained by combining Appendix A and Equation (5). In statistics, Fisher information is used as an upper bound of the precision of an estimation (see Cramér-Rao bound). This is also how we interpret it for an observer, namely the maximal precision of their estimation of a stimulus $S$. The precise definition given above is helpful to realize that the Fisher information carried by $M$ about $S$ ($I_S$) is different from the one carried by $M$ about $R$ ($I_R$). We can go one step further though, and establish the following relation between those two $$I_S(s) = \psi'(s)^2 I_R(\psi(s)). \quad (5)$$ A reformulation of Thurstone law of comparative judgment (Thurstone, 1927) is to assume that the Fisher information of an observer’s internal representation $I_R$ is constant. It is not so obvious to understand why this assumption is relevant. The idea is that an observer has only access to her internal states, she never observes any realization of an external stimulus $S$. Every external variable is transformed to an internal one through the psychological function $\psi$. Therefore, without any knowledge about the external world, a fair assumption is to allocate equal resources to every possible internal state in order to be equally precise in our estimates of different states (without knowing what they correspond to in the external world). This assumption is also equivalent to assuming that internal observer’s noise (a common notion used in psychophysics) is constant. If the internal Fisher information is constant we can now express the psychological function simply in terms of external Fisher information. This is summarized in the following proposition. Proposition 2. Assume Equation (5), the internal Fisher information $I_R$ is constant if and only if for all $s \in S$ the psychological function $\psi$ verifies $$\psi(s) \propto \int_{s_{init}}^s \sqrt{I_S(t)} dt. \quad (6)$$ Proof. See Appendix D. --- Relation to the MLDS observer model. In the MLDS framework, an observer has to judge which pair of stimuli is more similar to another. Assuming three stimuli $(s_i, s_j, s_k)$, those are transformed through the psychological scale $\psi$ and the observer responds by comparing the difference of differences between the pairs $d_{i,j,k} = \|\psi(s_i) - \psi(s_j)\| - \|\psi(s_j) - \psi(s_k)\|$. This difference is assumed to be corrupted by an internal noise $N_{mlds}$ of constant variance $\sigma^2_{mlds}$, i.e., $\Delta_{i,j,k} = d_{i,j,k} + N_{mlds}$. Those assumptions are sufficient to recover an estimate of $\psi$ (Knoblauch & Maloney, 2008). In addition, it is often assumed that there is no specific internal ordering of the variables so that the difference $d_{i,j,k}$ can be written without absolute value. In that case, assuming the encoding model (5) is enough to recover the MLDS observer model, we have the following relation between the noise variances $\sigma^2_{mlds} = 4\sigma^2$. Figure 2: Texture samples and predicted perceptual scales for various interpolation between arbitrary textures. Red corresponds to the early sensitivity group (i.e. shallow-to-steep slope). Blue corresponds to the late sensitivity group (i.e. steep-to-shallow slope). Yellow corresponds to conflicting prediction across VGG-19 layers. Bottom-right: prediction obtained by combining Proposition 4 and Equation (5). For pixels, images and wavelet, we also assume Gaussianity as in Equation (8) (pixel and wavelet) and as in Equation (7) (images). See details in Appendix F. 2.3 Fisher Information Carried by the Image vs by the Local Features In the previous section, we have introduced an observer model based on an abstract random internal measurement $M$. It is often unclear what those measurements are. Ideally inspired by neurophysiology, the measurements are responses of neurons to the image often modeled by Linear/Non-linear operations, even though those modeling stages are often dropped in perceptual studies. In the case of GRFs parameterized by spatial frequency (or scale) and orientation distributions, it is often accepted to assume that the measurements are samples of an appropriate distribution e.g. a Log-Normal distribution for the spatial frequency or a Von-Mises distribution for the orientation. Using the notation of the previous section, these cases correspond to measurement $M = Z$ with stimulus $S = Z_0$ and measurement $M = \Theta$ with stimuli $S = \Theta_0$. We will see that in both cases this is equivalent to consider that measurements are the image itself $M = F$ with $S = Z_0$ or $S = \Theta_0$. This is because Fisher information is given in closed-form and that perceptual scale can be predicted using Proposition 2. Similar results hold for $S = B_Z$ and $S = \Sigma_\Theta$ (note that $(Z_0, B_Z)$ and $(\Theta_0, \Sigma_\Theta)$ are parameters of $\mathbb{P}_Z$ and $\mathbb{P}_\Theta$ introduced in Section 2.1). The predictions are given in Figure 1. Fisher Information of Log-Normal and Von-Mises Distributions We give the precise parametrization and the corresponding Fisher information of the Log-Normal and Von-Mises distributions in Appendix A. Fisher Information of Parametric GRFs Now, we consider a GRF texture $F$ with mean $\mu \in \mathbb{R}$ and autocorrelation function $\gamma$ (or equivalently power spectrum $\hat{\gamma}$) parameterized by $s \in S$. Mathematically, the texture can be expressed for all $x \in \mathbb{R}^2$ and $s \in S$, as $$F(x, s) = \mu + \int_{\mathbb{R}^2} k(x - y, s) dW(y)$$ (7) where $k(\cdot, s) = \mathcal{F}^{-1}(\sqrt{\hat{\gamma}(\cdot, s)})$ and $W$ is a classical Wiener process. Proposition 3. The Fisher Information carried by $F$ about $S$ is $$I(s) = \frac{1}{2} \int_{\mathbb{R}^2} \frac{1}{\|\hat{\gamma}(\xi, s)\|^2} \left\| \frac{\partial \hat{\gamma}(\xi, s)}{\partial s} \right\|^2 d\xi = \frac{1}{2} \int_{\mathbb{R}^2} \left\| \frac{\partial \log(\hat{\gamma}(\xi, s))}{\partial s} \right\|^2 d\xi.$$ Proof. This is a specific case of Whittle formula (Whittle, 1953, Theorem 9). □ We combine Proposition [3] with Proposition [1] using the Log-normal and the Von-Mises distributions to express the power spectrum $\hat{\gamma}$ parameterized by $S = Z_0$, $S = BZ$, $S = \Theta_0$ or $S = \Sigma_\theta$. Therefore, the Fisher information carried by measurements $M = F$ comes down to the Fisher information carried by measurements $M = Z$ (spatial frequency) or $M = \Theta$ (orientation) as described above up to a multiplicative constant of $1/2$. As a consequence both approaches lead to similar predictions about the perceptual scales measured for these parameters. Fisher Information of Parametric Gaussian Vectors In the case of interpolation between naturalistic textures, we do not have a direct generative model of the texture conditionally on the interpolation parameter $s$. Instead, the texture is generated using a gradient descent to impose the statistics of VGG-19 features at multiples layers for which we have a generative model. Therefore, at layer $k$ and for $s \in S$ the activation $A_k(s)$ of texture $F_k(s)$ is $$A_k(s) = \mu_k(s) + \Sigma_k(s)N \quad \text{with} \quad \mu_k \in C^1(S, \mathbb{R}^{q_k}) \quad \text{and} \quad \Sigma_k \in C^1(S, \mathbb{R}^{d_k \times d_k})$$ (8) where $N \sim \mathcal{N}(0, I_{d_k})$ is a standard normal random vector and $d_k$ is the feature dimension of layer $k$. Proposition 4. The Fisher Information carried by $A_k$ about $S$ is $$I(s) = \mu'_k(s)\Sigma_k(s)^{-1}\mu_k(s) + \frac{1}{2} \text{Tr}(\Sigma_k(s)^{-1}\Sigma'_k(s)\Sigma_k(s)^{-1}\Sigma'_k(s)) .$$ Proof. See Appendix [B]. As stated earlier in the manuscript, no link can be made with interpretable feature distributions as it is the case for GRFs. Though, precise expressions for $\mu_k$ and $\Sigma_k$ are available in close forms in the Gaussian case as assumed here (see Appendix [C]). In practice, the feature activations are not Gaussian (Vacher et al., 2020). ### 2.4 Predictions and Experimental Methods The calculated Fisher informations together with Proposition [2] allows us to predict the perceptual scales corresponding to the parameters described in the previous section. These predictions hold under the assumed generative models for measurements. Predictions In the case of GRFs textures, we recall that assuming that spatial frequency or orientation are directly measured or that the image as a whole is measured makes no differences in the prediction (see bottom-right of Figure [1]). In the case of naturalistic textures, the measurements are assumed to be the feature activations of the texture in VGG-19 at layer 2 to 5 (see bottom-right of Figure [2]). For the naturalistic, we alternatively propose that measurements are the single pixel gray levels, the image itself (i.e. the power spectrum as for GRFs) or the wavelet activations. Experimental Methods The experiment consists of trials where participants have to make a similarity judgment. Participants are presented with 3 stimuli with parameters $s_1 < s_2 < s_3$ and are required to choose which of the two pairs with parameters $(s_1, s_2)$ and $(s_2, s_3)$ is the most similar. We used four sets of textures (see Figure [1] and [2]): (i) the first set consists of parameterized artificial textures where we measured the perceptual scales of spatial frequency, spatial frequency bandwidth and orientation bandwidth (see Appendix [E] for details); (ii) the other three sets consist of interpolations between arbitrary textures. A set of textures for which the perceptual scale corresponds to an early sensitivity (i.e. steep-to-shallow slope, see top-left of Figure [2]). Another one for which the perceptual scale corresponds to late sensitivity (i.e. shallow-to-steep slope, see top-right of Figure [2]). And a last set for which the predictions are inconsistent from one layer of VGG-19 to another (bottom-left of Figure [2]). All stimuli had an average luminance of 128 (range $[0, 255]$) and an RMS contrast of 39.7. For each texture pair, we use 13 equally spaced ($\delta_s = 0.083$) interpolation weights. To ensure that stimulus comparisons are around the discrimination threshold we only use triplets such that $\|s_{1,3} - s_2\| \leq 3\delta_s$. For each texture pair, a group of 5 naive participants performed the experiment. Participants were recruited through the platform prolific (https://www.prolific.com), performed the Figure 4: Measured and predicted (power spectrum) perceptual scales for the early (top row) and late (bottom row) sensitivity pairs. Error bars represent 99.5% bootstrapped confidence intervals. experiments online, and were paid 9£/hr. Monitor gamma was measured using a psychometric estimation and corrected to 1. The MLDS model is described at the end of Section 2.2. The protocol was approved by the CER U–Paris (IRB 00012020–54). 3 RESULTS 3.1 ORIENTATION AND SPATIAL FREQUENCY The perception of spatial frequencies is well-known in vision studies, its perceptual scale is expected to be logarithmic. Such a scale is also predicted by Fisher information as integrating the square-root (Proposition 2) of a squared inverse (Proposition 5) leads to a logarithm. The measured perceptual scale of the spatial frequency mode matches correctly this prediction (left of Figure 3). The spatial frequency and orientation bandwidths are less studied, the predictions are qualitatively the same as for the spatial frequency mode (Proposition 2 and Appendix Proposition 6). The Fisher information of the orientation bandwidth is more complex but leads to a similar curve. The measured perceptual scale is more variable for the orientation bandwidth (larger error bars) but is still in line with the prediction (predicted offset from linear behavior is exaggerated, see right of Figure 3). In contrast, the measured perceptual scales of spatial frequency bandwidth is approximately linear for low values while its gets supra-linear at intermediate values and even saturate for the highest values (center of Figure 3). 3.2 INTERPOLATION BETWEEN NATURALISTIC TEXTURES We present the perceptual scales measured for the different groups of natural textures in Figure 4 and 5. On these figures, the prediction given by the auto-correlation (i.e. when considering the textures as GRFs) is shown. Predictions assuming alternative measurements (pixel, wavelet, and VGG-19) are quantitatively compared in Figure 5 using the following Area Matching Score: \[ \text{AMS} = \int_0^1 \frac{\text{sign}(f_m(x) - x)(f_h(x) - x)}{\|f_m(x) - x\|} \, dx \] where $ f_m $ and $ f_h $ are respectively the measured and predicted scales. Intuition about score values is given in Figure 6. Early and Late Sensitivity For the set of early sensitivity texture pairs, measured perceptual scales are inline with the predictions (Figure 4), in the sense that a linear (pair01, pair04–05) or a supra-linear (pair02–03) perceptual scale is measured in all texture pairs. The same hold for the set of late sensitivity texture pairs (Figure 4), but this time in the sense that a linear (pair06 and pair08) or a sub-linear (pair07 and pair09–10) perceptual scale is measured in all texture... pairs. Such a result is also valid for the predictions based on alternative measurement assumptions as shown in Figure 6 by the fact that all scores are positive for pair01-10. Conflicting predictions For the set of textures with conflicting predictions (Figure 5), for both texture pairs, we observe that the GRF measurement assumption predicts a late sensitivity while the measured perceptual scale corresponds to an early sensitivity with a late saturation. Other measurement assumptions are not providing better prediction as their score is either close to 0 or negative. However, there is one exception for pair11 under the wavelet measurement assumption which has an ideal score, close to 1 (up to score limitation, see Section 4). Measurement Assumption Scores As previously stated, all assumptions predict correctly whether the scale corresponds to late or early sensitivity (positive scores) except for the conflicting prediction texture pairs. Note that pair05 has also a score close to 0 under all assumptions (though this might be due to score limitation, see Section 4). On average, the GRF assumption is the best with an average score ($\pm99.5\%$ CI) close to 1 ($0.92 \pm 0.69$). The single pixel distribution assumption only predicts a linear behavior and has therefore a score close to 0. In contrast, the wavelet and VGG-19 assumptions often overestimate the early or late sensitivity (average scores above 1). We conducted additional experiments in which we fixed the power spectrum of all textures along a path between a pair to be the average of the pair’s. In this case the power spectrum cannot explain the measured perceptual scale the operation is indeed deteriorating discriminability (see Appendix H). ### 4 DISCUSSION AND CONCLUSION In the case of GRFs, we have shown that the univariate assumption behind the Bayesian theories of perception and the absence of this assumption (i.e. the observer is using all the information in the image) lead to the same prediction for the perceptual scales of spatial frequencies, orientations and their bandwidths. Such a result is due to the fact that these local feature distributions directly appear in the power spectrum (the Fourier transform of the auto-correlation) of GRFs (Proposition 1) and to our second result that is the perceptual scale is related to the Fisher information of the feature distribution (Proposition 2). In the case of naturalistic textures, it is unknown if such a result relating a (non-linear) transform and some feature distribution holds. Therefore, it is necessary to make new hypotheses about the measurements in order to predict the perceptual scale of an observer. We tested this issue in a series of difference scaling experiments involving GRF and naturalistic textures. Our main result is that the perceptual scale is mainly driven by the auto-correlation (or the power spectrum). However, it does not perfectly explain the measured perceptual scales, and in particular, the perceptual scale of pair11 appears to be driven by the wavelet representation. A highly interesting future directions is to compare the perceptual scale to a neurometric scale, an equivalent scale but deduced from neurophysiological recordings as the equivalent exists for the psychometric function (Newsome et al., 1989; Berens et al., 2011). Other limitations lie in the MLDS method. Usually, running a difference scaling experiment requires to know ahead of time an approximation of the observer’s sensitivity to the parameter that one would like to test. Here, we have not estimated the sensitivity of each participant and, therefore, have not adapted the stimuli accordingly. Yet, it seems that we were near the participants sensitivity (see Appendix C). In addition, the MLDS method is limited to the study of a single stimulus dimension while extensions can still be developed (Knoblauch et al., 2012) and compared to higher-dimensional theories (Malo & Gutiérrez, 2006; Laparra & Malo, 2015). All these questions demonstrate the ambition of our approach and the work that remains to be done to understand, beyond perceptual distance, perceptual metrics. REPRODUCIBILITY STATEMENT The reproducibility of our work will be ensured by the links provided to the data and code. Theoretical results are supported by proofs or references to proof. 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MhzKwuvpm6	`This reward structure allows us to utilize any single-agent reinforcement learning algorithm, instead of using supervised learning to optimize over loss functions defined in Equations 11 and 12.` What is the difference to GAIL? GAIL also allows any single-agent reinforcement learning algorithm. And why should we conduct `supervised learning over loss functions defined in Equations 11 and 12`? What are you specifying?	RILE: Reinforced Imitation Learning Anonymous authors Paper under double-blind review Abstract Learning to imitate behaviors from a limited set of expert trajectories is a promising way to acquire a policy. In imitation learning (IL), an expert policy is trained directly from data in a computationally efficient way, but requires vast amounts of data. On the other hand, inverse reinforcement learning (IRL) deduces a reward function from expert data and then learns a policy with reinforcement learning via this reward function. Although this mitigates the data inefficacy problem of imitation learning, IRL approaches suffer from efficiency issues because of sequential learning of the reward function and the policy. In this paper, we combine the strengths of imitation learning and inverse reinforcement learning and introduce RILE: Reinforced Imitation Learning. Our novel dual-agent framework enables joint training of a teacher agent and a student agent. The teacher agent learns the reward function from expert data. It observes the student agent’s behavior and provides it with a reward signal. At the same time the student agent learns a policy by using reward signals given by the teacher. Training the student and the teacher jointly in a single learning process offers scalability and efficiency while learning the reward function helps to alleviate data-sensitivity. Experimental comparisons in reinforcement learning benchmarks against imitation learning baselines highlight the superior performance offered by RILE particularly when the number of expert trajectories is limited. 1 Introduction Learning to achieve human-level performance in complex tasks with artificial agents is a long-pursued goal in machine learning research. Reinforcement learning (RL) offers a solution to this problem by maximizing a utility/reward function through diverse interactions with the environment. However, this reward function must be meticulously tailored to the task to ensure that its maximization leads to optimal actions [Sutton & Barto, 2018]. This becomes infeasible for complex tasks, where an agent needs to sequentially perform multiple subtasks. To bypass the need for reward engineering, one can learn task execution from expert demonstrations. Research proposed two primary approaches for this purpose: Imitation learning and Inverse Reinforcement Learning. Imitation learning (IL) aims to learn a mapping from the current observation of the environment to action from given expert demonstrations. Imitation learning algorithms are developed to recover expert-like policies for tasks with large state-action spaces [Hussein et al., 2017]. While imitation learning approaches tackle the reward engineering limitation of RL, they struggle in generalizing beyond the provided expert trajectories. Thus, an extensive collection of high-quality expert demonstrations is essential for achieving good performance [Zheng et al., 2022]. On the other hand, Inverse Reinforcement Learning (IRL) approaches aim to learn the intrinsic reward function of the expert. This reward function is used to guide an RL agent, enabling it to reproduce expert-like behavior. IRL generally suffers from scalability and inefficiency issues [Zheng et al., 2022], since it relies on sequentially learning the reward function and the policy. These problems get exaggerated when the task gets complex, due to larger observation and action spaces. In this work, we aim to bridge the gap between imitation learning and inverse reinforcement learning. We propose RILE, a novel approach for learning a reward function and a policy simultaneously. Our framework comprises two interacting agents: a student agent and a teacher agent. The teacher agent observes the student and provides a reward, which the student aims to maximize simultaneously. In return, the teacher is rewarded based on the similarity between the behavior of the student agent. and the behavior observed in a limited set of samples of expert trajectories. This setting enables the student to replicate expert behavior without being trained on expert trajectories nor directly observing the similarity of its policy’s behavior to that of an expert. This architecture allows our framework to leverage the strengths of both IL and IRL, resulting in a hybrid approach that effectively compensates their respective limitations. Specifically, introducing the teacher agent as an intermediary between the policy learning and reward acquisition stages enables training the student agent in a standard RL setting while ensuring that its policies mimic expert behavior. This breaks the data-policy connection common to existing IL solutions [Ho & Ermon (2016)] and facilitates a less data-sensitive learning process that retains the generalization capabilities of standard RL, since RL agent does not try to overfit to data directly or via some similarity metric. Consequently, it can generalize over the specific state-action pairs of expert trajectories. In addition, the dual-agent setting enables simultaneous learning of the intrinsic reward function of the expert and the policy that replicates their behavior. This surpasses the limitation of iterative sequential learning of reward and policy common to IRL approaches. Our framework is capable of acquiring the reward function and policy in a single learning process. To demonstrate efficacy, we compare our method to state-of-the-art of imitation learning and inverse reinforcement learning on two different benchmarks: Atari games [Bellemare et al. (2013)] and MuJoCo control tasks [Todorov et al. (2012)]. Experimental results reveal that our approach outperforms baselines especially when the available expert data is limited. This indicates a better data-efficacy of our method compared to baselines. 2 RELATED WORK We review the literature on learning expert behavior from demonstrations. Commonly, expert demonstrations are sourced either through direct queries to the expert in any observable state or by collecting sample trajectories demonstrated by the expert. We present related work that aligns with the most prevalent approaches of the latter setting, namely Imitation Learning and Inverse Reinforcement Learning. Both, IL and IRL, form the conceptual foundation of RILe. Offline reinforcement learning also learns policies from data, which may include expert demonstrations. In contrast to our setting, its main goal is to learn a policy without any online interactions with the environment. We refer the reader to [Levine et al. (2020)] for an overview of offline RL. Furthermore, hierarchical reinforcement learning (HRL) splits tasks into subtasks at different levels of temporal and functional abstraction [Sutton et al. (1999), Dayan & Hinton (1992)]. While HRL has been combined with imitation learning [Le et al. (2018)], its goal is different to our setting as it abstracts long-horizon tasks to render them learnable instead of replicating expert behavior. Imitation Learning The earliest work on imitation learning introduced Behavioral cloning (BC) [Bain & Sammut (1995)], which aims to learn a policy congruent with expert demonstrations through supervised learning. SEARN introduces a classifier to BC that facilities exploring the observation space in continued training after cloning the expert policy [Daumé et al. (2009)]. DAgger proposes the aggregation of expert demonstrations with policy experiences during the training of the policy for improving generalization over expert demonstrations [Ross et al. (2011)]. [Ho & Ermon (2016)] introduced Generative Adversarial Imitation Learning (GAIL) where a discriminator aims to understand whether queried behavior stems from a policy or from expert demonstrations, while a generator tries to fool the discriminator by learning a policy that exhibits expert-like behavior. InfoGAIL extends upon GAIL by extracting latent factors from expert behavior and employing them during imitation learning [Li et al. (2017)]. [Hester et al. (2018)] proposed Deep Q-learning from Demonstrations (DQfD) where the learning agent is first pre-trained using expert demonstrations, followed by a subsequent policy optimization through interactions with the environment. Similarly, expert data is leveraged in [Le et al. (2018); Kostrikov et al. (2019)] to reduce number of environment interactions and increase learning efficacy. Zero-Shot Visual Imitation first learns a policy without considering expert demonstrations, and then uses expert data in a goal-conditioned setting to fine-tune the policy [Pathak et al. (2018)]. ValueDice proposes an off-policy imitation learning method using a distribution-matching objective between policy and expert behavior [Kostrikov et al. (2020)]. Although the field of imitation learning has seen innovative advancements, the requirement for high-quality expert data and the need for data efficacy remain open challenges [Zheng et al. (2022)]. Moreover, the limited generalization capability of IL approaches persists (Toyer et al., 2020). We address these limitations related to IL’s data sensitivity by introducing an intermediary teacher agent, thereby breaking the direct connection between the policy and the expert demonstrations. Inverse Reinforcement Learning In inverse reinforcement learning, Ng & Russell (2000) introduced three algorithms to learn the intrinsic reward function of an expert and acquire the expert policy from it. Apprenticeship learning builds on IRL and proposed to represent the reward function as a linear combination of features (Abbeel & Ng, 2004). Maximum Entropy Inverse Reinforcement Learning is proposed to deal with the noise in expert demonstrations and recover the expert reward function better (Ziebart et al., 2008). Several works extended IRL to include negative examples into the learning process (In Lee et al., 2016; Shiarlis et al., 2016; Bogert et al., 2016). Guided Cost Learning approximates the reward function with a neural network and makes maximum entropy methods applicable to continuous state-action spaces (Finn et al., 2016). An adversarial reward learning framework is proposed by Fu et al. (2018) to address the scalability issues of classical approaches. Chen et al. (2021) introduces a pipeline that makes IRL work with unstructured, real-world data. Cross-embodiment scenarios are considered in XIRL, opening up a new direction in IRL (Zakka et al., 2022). Despite the advancements in IRL, the efficacy of the learning process and scalability to complex problems remain open challenges (Arora & Doshi, 2021). The main reason for these limitations is the iterative sequential learning framework employed in IRL. We solve this efficacy problem by learning the policy and reward function, via training a student agent and a teacher agent, in a single joint learning process. ### 3 BACKGROUND #### 3.1 PRELIMINARIES Our work considers an imitation learning problem from expert trajectories. Each trajectory comprises states $ s \in S $ and actions $ a \in A $, where $ S $ and $ A $ are state and action spaces respectively. The set of expert trajectories is defined as $ \tau_E = \{[(s_0, a_0), (s_1, a_1), \ldots], [(s_0, a_0), (s_1, a_1), \ldots]\} $, which are sampled from an expert policy $ \pi_E \in \Pi $, where $ \Pi $ is the set of all possible policies. $ P(s'\|s, a) $ is an unknown state transition probability function. The reward function $ R(s, a) $ generates a reward given a state-action pair $(s, a)$. In this work, we consider $ \gamma $-discounted infinite horizon settings. Following Ho & Ermon (2016), expectation with respect to the policy $ \pi \in \Pi $ refers to the expectation when actions are sampled from $ \pi(s) $: \[ E_\pi[R(s, a)] = E_\pi[\sum_{t=0}^{\infty} \gamma^t R(s_t, a_t)], \] where $ s_0 $ is sampled from an initial state distribution $ \rho(s) $, $ a_t $ is given by $ \pi(\cdot\|s_t) $ and $ s_{t+1} $ is determined by the transition model as $ P(\cdot\|s_t, a_t) $. #### 3.2 REINFORCEMENT LEARNING (RL) Reinforcement learning seeks to find an optimal policy that maximizes the discounted cumulative reward. The reinforcement learning problem is defined as \[ RL(R_\theta) = \pi^* = \arg\max_\pi E_\pi[R_\theta(s, a)] = E_\pi[\sum_{t=0}^{\infty} \gamma^t R_\theta(s_t, a_t)], \] (1) where $ R_\theta \in \mathbb{R} $ is a reward function parameterized by $ \theta $ and the optimal policy is indicated by $ \pi^* $. Regularization can be introduced with the entropy function $ H(\pi) $. In this work $ \gamma $-discounted casual entropy function is considered, which defined as $ H(\pi) = E_\pi[-\log \pi(a\|s)] $ (Ho & Ermon, 2016; Bloem & Bambozzi, 2014). Incorporating entropy regularization into the problem transforms it into \[ RL(R_\theta) = \pi^* = \arg\max_\pi H(\pi) + E_\pi[R_\theta(s, a)]. \] (2) 3.3 Inverse Reinforcement Learning (IRL) Given sample trajectories $\tau_E$ of an expert policy $\pi_E$, inverse reinforcement learning, $IRL(\tau_E)$, tries to recover the reward function, $R^$ that would result in expert behavior when optimized in a reinforcement learning training, $RL(R^)$. In other words, the goal can be defined as $$RL(R^) = \pi^ = \arg\min_\pi E_{\tau_E}[L(\pi, \pi_E)]$$ (3) where $L(\pi, \pi_E)$ is a loss function that measures difference between given policies. Inverse reinforcement learning seeks to find the reward function in which the expert policy performs better than any other policy. $$IRL(\tau_E) = \arg\max_{R \in \mathbb{R}} \left(E_{\pi_E}[R(s,a)] - \max_\pi E_\pi[R(s,a)]\right)$$ (4) With entropy regularization $H(\pi)$, maximum casual entropy inverse reinforcement learning [Ziebart et al., 2008] can be defined as $$IRL(\tau_E) = \arg\max_{R \in \mathbb{R}} \left(E_{\pi_E}[R(s,a)] - \max_\pi \left(E_\pi[R(s,a)] + H(\pi)\right)\right)$$ (5) 3.4 Adversarial Imitation Learning (AIL) In contrast to inverse reinforcement learning, imitation learning aims to directly acquire the expert policy from given expert trajectory samples. It can be formulated as $$IL(\tau_E) = \arg\min_\pi E_{\tau_E}[L(\pi(s), \pi_E(s))].$$ (6) GAIL [Ho & Ermon, 2016] extends imitation learning to an adversarial setting by quantifying the similarity between policies of the agent and the expert with a discriminator $D_\phi(s,a)$, parameterized by $\phi$. Its goal is to find the optimal policy that minimizes this difference metric while maximizing an entropy constraint by training the discriminator and the policy at the same time. The optimization problem can be formulated as a zero-sum game between the discriminator $D_\phi(s,a)$ and the policy $\pi$, represented by $$\min_\pi \max_\phi E_\pi[\log D_\phi(s,a)] + E_{\tau_E}[\log(1 - D_\phi(s,a))] - \lambda H(\pi).$$ (7) In other words, the reward function that is maximized by the policy is defined as a similarity function, expressed as $R(s,a) = -\log(D_\phi(s,a))$. 3.5 Problem Formulation An Standard MDP is defined as $MDP_S : (S,A,R,T,K,\gamma)$ where $S$ is state-space, consist of all possible environment states, and $A$ is action space consists of all possible environment actions. $R = R(s,a) : SxA \rightarrow \mathbb{R}$ is the reward function. $T = \{P_{sa}\}$ is transition dynamics where $P_{sa}$ is defined as the state distribution upon taking action $a$ in state $s$. $K$ is initial state distribution, i.e. $s_0 \sim K$ and $\gamma$ is the discount factor. Another MDP is also defined, which can be stated as $MDP_T : (S_T,A_T,R_T,T_T,K_T,\gamma)$, where $S_T$ is state space defined as $SxA$, so consisting all possible state action pairs from $MDP_S$. $A_T$ is action space, a mapping from $S_T = (SxA) \rightarrow \mathbb{R}$, so the action is a scalar value. $R_T : S_T \rightarrow \mathbb{R}$ is only state-based reward. $T_T = \{P_{s_t,a_t}\}$ is transition dynamics where $P_{s_t,a_t}$ is defined as the state distribution upon taking action $a_t$ in state $s_t$. $K_T$ is initial state distribution, i.e. $s_{t,0} \sim K$. We assume that we have an access to $m$ expert trajectories, all of which have $n$ time-steps, $\zeta = \{s^{E,i}_0, s^{E,i}_1, \ldots, s^{E,i}_n\}_{i=1}^m$. 4 RILe: Reinforced Imitation Learning We propose Reinforced Imitation Learning (RILe) to combine the strengths of adversarial imitation learning and inverse reinforcement learning. The goal of the hierarchical framework is to learn the reward function of an expert and recover a policy that emulates expert-like behavior simultaneously in one learning process, without directly assessing the similarity between the behavior of the trained agent and the expert. Our framework consists of three key components: a discriminator, a student agent, and a teacher agent (Figure 1). Discriminator The discriminator aims to understand whether a given state-action pair comes from an expert trajectory or not. It is defined as a feed-forward deep neural network, parameterized by $\phi$. Given expert state-action pairs $(s, a) \sim \zeta$ and other state-action pairs whose source is different than the expert data, $(s, a) \not\in \zeta$, the discriminator aims to discriminate expert pairs from others. Thus, the optimization problem is defined as $$\max_{\phi} E_{(s,a)\sim\zeta}[\log(D_\phi(s,a))] + E_{(s,a)\not\sim\zeta}[\log(1 - D_\phi(s,a))].$$ \hspace{1cm} (8) Student Agent The student agent aims to learn a policy $\pi_S$ by interacting with an environment in a standard RL setting within $MDP_S$, where for each of its actions $a^S$ the environment returns a new state $s^E$. However, rather than from a hand-crafted reward function, the student agent receives its reward from the policy of the teacher agent $\pi_T$. Therefore, in $MDP_S$, the reward function is represented by the teacher policy $R = \pi_T$. The student agent is guided by the actions of the teacher agent, i.e., the action of the teacher is the reward of the student: $r^S = \pi_T((s^E, a^S))$. The optimization problem of the student agent is defined as $$\min_{\pi_S} -E_{(s^E,a^S)\sim\pi_S}[\pi_T((s^E,a^S))].$$ \hspace{1cm} (9) The student agent aims to recover the optimal policy $\pi^_S$ defined as $$\pi^_S = \arg\max_{\pi_S} E_{(s^E,a^S)\sim\pi_S}\left[\sum_{t=0}^{\infty} \gamma^t \pi_T((s^E_t, a^S_t))\right].$$ \hspace{1cm} (10) Teacher Agent The teacher agent aims to guide the student to mimic expert behavior by operating as its reward mechanism. Therefore, the teacher agent learns a policy $\pi_T$ that produces adequate reward signals to guide the student agent, by interacting with an environment in a standard RL setting within $MDP_T$. Since the state space of $MDP_T$ is defined over state-action pairs of $MDP_S$, the state of the teacher comprises the state-action pair of the student $s^T = (s^E, a^S)$. It generates a scalar action $a^T$ which is given to the student agent as reward $r^S$. The teacher agent’s reward function, which depends only on its state, is defined as $R^T = Y$, where $Y$ is a reward approximating network. Therefore, the optimization problem of the teacher can be defined as $$\min_{\pi_T} E_{s^T\sim\pi_S}[Y].$$ \hspace{1cm} (11) The teacher agent aims to recover the optimal policy $\pi^_T$ by maximizing the cumulative reward yielded through function $Y$: $$\pi^_T = \arg\max_{\pi_T} E_{(s^T)\sim\pi_S}\left[\sum_{t=0}^{\infty} \gamma^t[Y((s^T_t, ))]\right] = E_{(s^E,a^S)\sim\pi_S}\left[\sum_{t=0}^{\infty} \gamma^t[Y((s^E_t, a^S_t))]\right].$$ \hspace{1cm} (12) RILe RILe combines the three key components defined previously in order to converge to a student policy, which mimics expert behaviors presented in $\zeta$. To achieve this goal, the discriminator optimization problem is tweaked as \[ \max_{\phi} E_{(s,a) \sim \zeta} [\log(D_\phi(s,a))] + E_{(s,a) \sim \pi_S} [\log(1 - D_\phi(s,a))]. \tag{13} \] In other words, the discriminator aims to discriminate state-action pairs from expert and student agent. This reformulated discriminator is employed as the reward function of the teacher $ Y = \log(D_\phi) $, which translates the teacher’s optimization problem to \[ \min_{\pi_T} E_{(s,a) \sim \pi_S} [\log(D_\phi(s,a))]. \tag{14} \] In RILe, the student policy $ \pi_S $ is trained with soft actor-critic (SAC) [Haarnoja et al., 2018], with the aim of maximizing the cumulative rewards obtained from the teacher agent. Concurrently, the teacher agent $ \pi_T $ is trained with proximal policy optimization (PPO) [Schulman et al., 2017] to maximize the cumulative reward derived from the discriminator. Consequently, to increase its rewards, the teacher agent must encourage the student to generate state-action pairs that deceive the discriminator into perceiving them as originating from an expert. SAC is chosen to train the student policy to leverage past experiences and guidance from the teacher. PPO is utilized to train the teacher to enable fast adaptations to the changing feedback of the learning discriminator. The training algorithm is given in Appendix B. To prove that the student agent can learn expert-like behavior, we need to show that the teacher agent learns to give higher rewards to student experiences that match with the expert state-action pair distribution, as this would enable a student policy to eventually mimic expert behavior. Lemma 1: Given the discriminator $ D_\phi $, the teacher agent optimizes its policy $ \pi^{\theta_T} $ via policy gradients to provide rewards that guide the student agent to match expert’s state-action distributions. However, since the teacher is guided by a discriminator, we also need to show that the discriminator successfully learns to discriminate expert state-action pairs, i.e., understand whether the given state-action pair is generated by the expert or not. Lemma 2: The discriminator $ D_\phi $, parameterized by $ \phi $ will converge to a function that estimates the probability of a state-action pair being generated by the expert policy, when trained on samples generated by both a student policy $ \pi^{\theta_S} $ and an expert policy $ \pi_E $. The proofs of these lemmas are presented in Appendix C. ### 4.1 Intuition Behind RILe In AIL, the learning agent, is guided by a discriminator that follows the definition presented in Eq. (13). However, in AIL, the student tries to satisfy the discriminator directly. Since the discriminator just aims to minimize a step-based cross entropy loss, it cannot consider the long-term effects of generated rewards. This myopic discriminator consequently leads to an agent that can mimic expert state-action pairs but cannot consider if its choices are optimal for long-horizon tasks. Moreover, such myopic strategies may also lead to failure in understanding connections between different possible states. In contrast, IRL incrementally updates the reward function and, at each iteration, re-trains a policy from scratch. Through this approach, IRL can learn the effect of reward signals on the behavior of a policy. However, iterative reward and policy training are inefficient, rendering IRL computationally infeasible for most real-world problems. In RILe, we synergize advantages of IRL and AIL. Specifically, similar to IRL, we are learning the reward function via the teacher agent, and train a policy via the student agent that reflects updates in the reward function. However, we guide the teacher learning via an adversarial discriminator, inspired from AIL, and learn the reward function via RL to consider long-horizon effects of produced rewards. By introducing the adversarial discriminator, we can simultaneously learn a reward function and a policy simultaneously, rendering RILe computationally feasible, in contrast to IRL. Furthermore, the teacher agent learns to act based on long term effects of the produced reward signals and relations between different states by minimizing long-horizon costs within the standard RL setting. Figure 1: Framework overview. The framework consists of three key components: a student agent, a teacher agent, and a discriminator. The student agent learns a policy $\pi_S$ by interacting with an environment where for each of its actions $a^S$, the environment returns a new state $s^E$. It receives its reward from the teacher’s policy $\pi_T$, which evaluates the state action pair of the student agent $s^T = (s^E, a^S)$ and chooses an action $a^T$ that then becomes the reward of the student agent $r^S = r^T$. The teacher agent is rewarded $r^T$ by a discriminator $D$ that tries to distinguish if a state stems from an agent ($s^T$) or from expert demonstrations ($s^D$). In a single learning process, our framework can learn policies that exhibit expert behavior without having direct access to expert demonstrations. 5 EXPERIMENTAL EVALUATION 5.1 EXPERIMENTAL SETUP We evaluate RILe against baselines on different tasks from two different reinforcement learning benchmarks: (1) Atari games (Bellemare et al., 2013) and (2) MuJoCo control tasks (Todorov et al., 2012). For the MuJoCo benchmark, we purposely evaluated on control tasks of varying complexity, encompassing both low- and high-dimensional state spaces. For all experiments, OpenAI Gym is used as the simulation framework (Brockman et al., 2016). All the tasks are described in detail in supp. material. To obtain expert trajectories, we utilize the experts from RL-Zoo3 (Raffin, 2020). Their policies were trained on the true cost function of each task, which are defined by Brockman et al. (2016). Different numbers of trajectories are sampled (e.g., 1 or 100 for Atari) from these trained experts to assess performance across a range of available expert demonstrations. To ensure the visitation of states that are not present in expert demonstrations, all experiments are initialized randomly. This is further reinforced by the stochastic nature of the actions taken by the learning agents. RILe is tested against an imitation learning-, an adversarial imitation learning- and an inverse reinforcement learning baseline, which are: - Behavioral cloning (BC): Employed as the supervised imitation learning baseline. - Generative Adversarial Imitation Learning (GAIL): GAIL is utilized as the adversarial imitation learning baseline. - Adversarial Inverse Reinforcement Learning (AIRL): Utilized as the inverse reinforcement learning baseline. For all baselines, we use their respective implementation of stable-baselines3 (Gleave et al., 2022). Networks are randomly initialized at the start. In all tasks of both benchmarks, the policy of BC is trained for a 1000 epochs via supervised learning. All baselines are trained for 2 million time-steps, and RILe is trained for 1 million time-steps. Figure 2: Mean ± std. error of the reward achieved on evaluation by RILe and baselines in MuJoCo control tasks. 5.2 ATARI For the Atari benchmark, all methods are evaluated using two sets of expert demonstrations, comprising one expert trajectory and 100 expert trajectories, respectively. Instead of using an image-based observation state, all approaches use a vector representation of the RAM of the Atari emulator as state space. Moreover, frame-stacking is avoided and single frame observations are used, which changed the hardness of games. In the case of discrete Atari tasks, we employ PPO (Schulman et al., 2017) as the learning agent for baselines and as the teacher and student agents in RILe. Hyperparameter sweeps and selected hyperparameters can be found in Appendix D. The hyperparameters for PPO agents follow the default settings of stable-baselines3. Table 1: Mean ± std. err. of the attained reward on test trajectories of RILe and baselines in Atari environments. Traj. stands for the number of available expert trajectories during training. \| \| Asteroids \| BeamRider \| Qbert \| SpaceInv. \| \|-------\|-----------\|-----------\|---------\|-----------\| \| RILe \| 1960±33.4 \| 458±18.7 \| 270±34.8 \| 279.8±6.5 \| \| GAIL \| 1402.8±19.8 \| 330±34 \| 125±3.4 \| 222.1±10.9 \| \| AIRL \| 140±2.5 \| 0±0 \| 0±0 \| 270±13.5 \| \| BC \| 1550±20.3 \| 264±6.8 \| 150±5.9 \| 180±1.2 \| \| RILe \| 1904±45.9 \| 498.4±15.8 \| 315±17.3 \| 295.59±12.7 \| \| GAIL \| 1729±38.5 \| 409.2±22.3 \| 125±2.6 \| 235±3.1 \| \| AIRL \| 140±0.5 \| 0±0 \| 0±0 \| 270±2.9 \| \| BC \| 1440±56.3 \| 616±45.3 \| 820.75±11.8 \| 180±4.4 \| Table 1 presents the performance of RILe alongside the baselines. RILe performs better than baselines in all of the tasks as presented. An exception is Qbert, where behavioral cloning outperforms all other approaches when trained on 100 expert trajectories. 5.3 MuJoCo In the case of the MuJoCo benchmark, methods are evaluated with five different sets of expert demonstrations: 1, 5, 10, 15, 20 expert trajectories. SAC (Haarnoja et al., 2018) is used as learning agent for baselines and for the student agent in RILe. RILe’s teacher agent employs a PPO policy (Schulman et al., 2017). Hyperparameter sweeps and selected hyperparameters can be found in Appendix D. The performance of RILe and the baselines in MuJoCo-based control tasks is presented in Figure 2. RILe outperforms baselines in all three tasks. This holds true even in the case of the Humanoid task, which involves a larger state-action space and greater complexity. The consistently superior performance of RILe across all three sets of expert demonstrations demonstrates that our method performs effectively even with a limited amount of data. In order to compare the sample efficacy between methods, at the end of each 10000th time-step during training, methods are evaluated on their own environment, and results are presented in Figure 3 for Humanoid-v3 environment with different sizes of expert data. RILe is significantly more sample efficient when compared to AIRL and GAIL. Behavioral cloning also demonstrated superior sample efficacy, however, as presented in Figure 2, this doesn’t translate into a better performance in tests, because of over-fitting. 6 DISCUSSION We have demonstrated in the experiments that our method beats the baselines in different settings with different data availability and can perform well even with just one expert demonstration. This shows the data-efficacy of our method when compared to imitation learning and inverse reinforcement learning baselines. The experiments are conducted in ten different tasks, and all experiments are initialized randomly. RILe generalizes better than the baselines in states which are not included in the expert demonstrations. Since the policies of all approaches, including the student agent and the teacher agent are stochastic, the training eventually covers states which are not included in expert demonstrations, especially when the number of trajectories are small. Hence, the reported results indicate how robust policies are towards deviations from the expert demonstrations. Although combining imitation learning and inverse reinforcement learning in RILE offers advantages, it also suffers from limitations. The main challenge is learning the reward function along with a policy, which means training the policy with a changing reward function. This inherently unstable setting can make the student agent get stuck in local minima, which results in sub-optimal behavior. To overcome this, we update the teacher agent less frequently by using a higher batch size compared to the student agent. Moreover, balancing the learning rates of the discriminator and the policies is difficult. For example, we have observed that for some training runs on the more challenging tasks of the MuJoCo benchmark, the teacher agent fails to satisfy the discriminator, since the latter converges exceptionally fast. This in turn makes it difficult for the teacher agent to find a reward for the student agent that tricks the discriminator. In such cases, the problem can be tackled by adjusting the learning rate of the discriminator or updating the discriminator less frequently. However, a more fundamental solution is required to optimally balance the different components of the architecture. Future work should focus on improving the stability and unbalanced learning issues in RILe. One promising approach could consider using a learning curriculum or to learn an adaptable update frequency or learning rate for the discriminator. REPRODUCIBILITY STATEMENT For the reproducibility of the results presented in this paper, all trained models along with scripts to generate results are provided as supplementary material. For the details of the experiments and used expert policies, refer to Appendix C. Detailed experimental results are presented in Appendix D. REFERENCES Pieter Abbeel and Andrew Y Ng. Apprenticeship learning via inverse reinforcement learning. In Proceedings of the twenty-first international conference on Machine learning, pp. 1, 2004. Saurabh Arora and Prashant Doshi. A survey of inverse reinforcement learning: Challenges, methods and progress. Artificial Intelligence, 297:103500, 2021. Michael Bain and Claude Sammut. A framework for behavioural cloning. In Machine Intelligence 15, pp. 103–129, 1995. Marc G Bellemare, Yavar Naddaf, Joel Veness, and Michael Bowling. The arcade learning environment: An evaluation platform for general agents. 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Peter Dayan and Geoffrey E. Hinton. Feudal reinforcement learning. In Advances in Neural Information Processing Systems 5, [NIPS Conference], pp. 271–278, San Francisco, CA, USA, 1992. Morgan Kaufmann Publishers Inc. ISBN 1558602747. Chelsea Finn, Sergey Levine, and Pieter Abbeel. Guided cost learning: Deep inverse optimal control via policy optimization. In International conference on machine learning, pp. 49–58. PMLR, 2016. Justin Fu, Katie Luo, and Sergey Levine. Learning robust rewards with adversarial inverse reinforcement learning. In International Conference on Learning Representations, 2018. Adam Gleave, Mohammad Taufeeque, Juan Rocamonde, Erik Jenner, Steven H. Wang, Sam Toyer, Maximilian Ernestus, Nora Belrose, Scott Emmons, and Stuart Russell. imitation: Clean imitation learning implementations. arXiv:2211.11972v1 [cs.LG], 2022. URL https://arxiv.org/abs/2211.11972 Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial nets. Advances in neural information processing systems, 27, 2014. Tuomas Haarnoja, Aurick Zhou, Pieter Abbeel, and Sergey Levine. Soft actor-critic: Off-policy maximum entropy deep reinforcement learning with a stochastic actor. In International conference on machine learning, pp. 1861–1870. PMLR, 2018.
V8aD5pUcVX	The main concern is that the pretraining image-text pairs contain a significant amount of object-level annotations. This leads an unfair comparison when the downstream tasks are most object-centric questions.	What Makes for Good Visual Tokenizer Supervision for Large Language Models? Anonymous authors Paper under double-blind review Abstract We empirically investigate proper pre-training supervision to build good visual tokenizers, making Large Language Models (LLMs) powerful Multimodal Large Language Models (MLLMs). In our benchmark, which is curated to evaluate MLLM’s visual semantic understanding and fine-grained perception capabilities, we discussed different visual tokenizers pre-trained with dominant methods (i.e., DeiT, CLIP, MAE, DINO and DINOv2), and observed that: i) Fully/weakly supervised models capture more semantics than self-supervised models, but the gap is narrowed by scaling up the pre-training dataset. ii) Self-supervised models are better at fine-grained perception, where patch-level supervision is particularly effective. iii) Tuning the visual tokenizer leads to the loss of semantics obtained from large-scale pretraining, which is unfavorable with the relatively small-scale instruction-tuning dataset. Given the findings, we reviewed methods that attempted to unify semantics and fine-grained visual understanding, e.g., patch-level feature distillation with semantically-rich targets. We obtain an intriguing insight: without further modification, mask-based strategies that were once all the rage may not be good visual tokenizer supervision. Based on this critical observation, we obtain a new MLLM equipped with a tailored Good Visual Tokenizer – GVT, which exhibits strong visual comprehension capability at multiple scales. In particular, without introducing extra parameters and task-specific fine-tuning, GVT achieves superior performance on visual question answering, image captioning, and other fine-grained visual understanding tasks such as object counting and multi-class identification. 1 Introduction Large Language Models (LLMs) (Brown et al., 2020; Touvron et al., 2023; Radford et al., Ouyang et al., 2022) have demonstrated remarkable performance for various downstream tasks without task-specific fine-tuning. Recently, based on the powerful LLMs, there has been a surge of research (Li et al., 2023b; Alayrac et al., 2022; Zhu et al., 2023; Liu et al., 2023; Ye et al., 2023; Huang et al., 2023; Yang et al., 2023b; Driess et al., 2023) that successfully adapt LLMs to vision-language tasks, resulting in powerful Multimodal LLMs (MLLMs), e.g., BLIP-2 (Li et al., 2023b). When properly fed with visual data, they are shown to be capable of understanding the visual world and responding to instructions accordingly. Such vision-language understanding capability makes LLM a universal interface for multimodal tasks, contributing towards a tentative yet promising direction towards Artificial General Intelligence (AGI) (Bubeck et al., 2023; OpenAI, 2023). Within this framework, images are projected to the linguistic space for the LLMs to understand, where the common practice employs an image-text pre-trained visual tokenizer with contrastive supervisor$^1$, i.e., CLIP. However, even though CLIP has shown strong capacity for image representations, to the best of our knowledge, it is yet to be explored whether CLIP is the optimal visual tokenizer for MLLMs. The absence of such investigation calls for a comprehensive comparison of existing visual tokenizers under the MLLMs’ framework. However, recent MLLMs have mostly investigated their performance in terms of generation quality (Zhu et al., 2023; Liu et al., 2023) or on a small set of questions (Ye et al., 2023), leaving an in-depth quantitative evaluation untouched. $^1$In this work, we study visual tokenizers that map images into a continuous latent space. Figure 1: Different tasks require a visual understanding of different perspectives. Mainstream vision-language tasks (VQA and image captioning) mainly focus on the general and overall semantics of the image. In this work, to investigate the fine-grained visual understanding of a model, we also study two tasks: (c) Object Counting (OC) and (d) Multi-Class Identification (MCI), focusing on region and instance level understandings, respectively. To this end, we curated a new benchmark to study what pretraining supervision makes for a Good Visual Tokenizer (GVTBench). It is specially designed to evaluate an MLLM’s visual understanding capability from two important perspectives: semantic understanding and fine-grained visual perception capabilities. As shown in Figure 1, the former is evaluated on Visual Question Answering (VQA) and image captioning. While the latter is tested on two new tasks: Object Counting (OC) and Multi-Class Identification (MCI), which requires an in-depth understanding of fine-grained visual information. Based on this benchmark, we comprehensively evaluated existing visual tokenizers with identical architecture but different pretraining supervisions, including fully supervised DeiT (Touvron et al., 2021), text-guided weakly supervised CLIP (Radford et al., 2021) and self-supervised MAE (He et al., 2022), DINO (Caron et al., 2021), DINOv2 (Oquab et al., 2023) models (Section 2). Our main observations are i) fully supervised and text-guided weakly supervised visual tokenizers demonstrate better semantic representation capacity than their self-supervised counterparts, but the gap is narrowed by scaling up the pretraining dataset (i.e., CLIP vs. DINOv2). ii) Self-supervised visual tokenizers show better fine-grained visual perception capacity, where patch-level supervision leads to superior region-level understanding. iii) On instruction tuning datasets which are often smaller than visual tokenizer pretraining dataset (Liu et al., 2023; Zhu et al., 2023), jointly tuning the visual tokenizer leads to noticeable semantic loss (i.e., frozen CLIP performs much better than tunable CLIP on semantic understanding tasks). Given the fact that none of the previous visual tokenizers exhibit both good semantic and fine-grained visual perceptual capabilities, we reviewed existing methods that integrate semantic and regional supervision and questioned whether they bring the best of the two worlds into a single visual tokenizer. Existing methods can be mainly divided into two categories. Methods in the first group (Zhong et al., 2022; Ye et al., 2023) enhance a pretrained CLIP with region-level supervision, which comes from a pretrained Region Proposal Network (RPN) or bounding box annotations. However, we found that this leads to the loss of original semantics, which cannot be justified by the limited improvements in fine-grained visual perception capabilities. The other group of methods (Fang et al., 2023; Wei et al., 2022b) utilizes patch features from a pretrained CLIP as region supervision to train a new model, intending to enhance its fine-grained visual perceptual capability while maintaining the rich semantics. Specifically, Fang et al. (2023) and Wei et al. (2022a) use CLIP features to supervise the training of Masked-Image-Modeling (MIM), while Feature Distillation (Wei et al., 2022b) directly distills the CLIP feature into a new model without patch masking. Nonetheless, the introduction of \texttt{[MASK]} token in MIM leads to train-test mismatch, requiring the visual tokenizer to be jointly optimized in the instruction-tuning process, which again leads to semantic loss with the small-scale instruction tuning dataset. As such, we argue that, without architectural modification, the mask-based strategies that were once all the rage may not be good visual tokenizer supervision under MLLM’s framework. Based on these insights, we seek a new visual tokenizer with both strong semantic understanding and fine-grained visual perception capabilities via Feature Distillation (Wei et al., 2022b). Specifically, given a pretrained CLIP with rich semantics, we distill it into a new model by using the patch features as supervision, without patch masking. In this way, the rich semantics from large-scale image-text contrastive pretraining is preserved, and the fine-grained visual perceptual capability is greatly enhanced with patch supervision. With our new visual tokenizer and the language model Vicuna (FastChat, 2023), we obtain a new MLLM with Good Visual Tokenizer (GVT). Benefiting from the versatile visual tokenizer, GVT is able to perform well vision language tasks that require visual understanding at multiple levels. Without introducing extra parameters, we achieve superior Table 1: Detailed Statistics of GVT-Bench. \| Task \| Dataset \| Evaluation Dimension \| #Questions \| Question Type \| Answer Type \| \|---------------\|---------------\|--------------------------\|------------\|--------------------------------\|------------------\| \| VQA \| VQAv2 \| General semantics \| 440k \| Multiple \| Free-form Text \| \| Image Captioning \| MS-COCO \| Overall semantics \| 25k \| What does the image describe? \| Free-form Text \| \| OC \| MS-COCO&VCR \| Region understanding \| 20k \| How many obj are there in the image? \| Number \| \| MCI \| MS-COCO&VCR \| Instance understanding \| 20k \| Does obj exist in the image? \| Yes/No \| performance on semantic understanding tasks, i.e., VQA and image captioning, as well as fine-grained visual understanding tasks: instance counting and multi-class identification. To summarize, our contributions are as follows: - To effectively evaluate MLLM’s visual understanding capacity at different levels, we curate a new benchmark (GVTBench) which includes both semantic understanding tasks (VQA and image captioning) as well as fine-grained visual understanding tasks (Object Counting and Multi-Class Identification). Based on GVTBench, we perform extensive experiments to study what makes for good visual tokenizer supervision for MLLMs and make three main observations. - We reviewed methods that combine CLIP with fine-grained supervision to see if they can achieve the best of both worlds in terms of visual semantics and fine-grained understanding. We found that the SOTA pre-trained models (i.e., EVA) are inapplicable due to the train-test mismatch caused by MIM. Such mask-based visual tokenizers rely on further tuning with instructions, which leads to the loss of pre-trained rich semantics. - Based on the insights, we tailor a new visual tokenizer by distilling the patch-level semantics of a pre-trained CLIP without masking. With our visual tokenizer and Vicuna, we arrive at a superior MLLM (GVT) with strong visual understanding capability, achieving state-of-the-art performance on our curated benchmark. 2 GVTBench for Empirical Study To comprehensively study what makes for good visual tokenizer supervision for MLLMs, we conduct a series of experiments to study the properties of various visual tokenizers with the same architecture but different pretraining methods. In this work, we mainly investigate MLLMs’ visual understanding capability from two important perspectives: semantic understanding and fine-grained visual perception. 2.1 Experimental Setup GVTBench. A comprehensive evaluation requires a benchmark that suitably quantifies MLLM’s visual understanding capability. Nonetheless, existing vision-language tasks mainly focus on general and overall semantics [Farhadi et al., 2010; Goyal et al., 2017], leaving a special focus on fine-grained visual perception untouched. To this end, we curated a new benchmark – GVTBench. It evaluates the semantic understanding capability of an MLLM on VQA [Goyal et al., 2017] and Image Captioning (IC) [Lin et al., 2014]. We report accuracy for the former and CIDEr [Vedantam et al., 2015] for the latter. For fine-grained visual perception capability evaluation, we specially designed two new tasks for MLLMs: - Object Counting (OC). We ask the model to count the number of certain objects appearing in the image with the prompt “Question: How many {obj} are there in the image? Answer:”. We regard it as a classification task and report a model’s prediction accuracy. - Multi-Class Identification (MCI). We ask the model if a certain object exists in the image with the prompt “Question: Does {obj} exist in the image? Answer:”. The model is expected to answer “Yes/No”, resulting in a binary classification problem. We report accuracy for this task. Notably, in the VQAv2 [Goyal et al., 2017] benchmark, there are also questions related to numbers and small-scale objects. Nevertheless, these questions are of high diversity and are often coupled with other semantic relations, making it unsuitable to strictly evaluate fine-grained visual understanding capabilities. For example, to answer a typical question “How many people are sitting on the bench?” in VQAv2, the model should first understand the relation (sit_on), which is thus not suitable for evaluating fine-grained visual understanding solely. In contrast, our OC and MCI tasks evaluate MLLM’s understanding of individual objects, which is decoupled from semantic relations and thus a more appropriate test bed for fine-grained visual understanding evaluation. To summarize, there are a total of 4 tasks in our GVTBench. (1) VQAv2 covers questions of various types. We thus take this benchmark to evaluate the general semantic understanding capability of a model. This task requires the model to have a good understanding of various high-level semantics in the image, including relatively abstract concepts such as actions and relations. (2) We use image captioning to quantify the capability of overall semantic understanding, which requires the model to understand the global information of the image. It requires the model to have a proper comprehension of the image and grasp the overall information such as the main activity and theme. Furthermore, we curated (3) OC and (4) MCI to evaluate MLLM’s region-level and instance-level understanding capability, respectively. Compared to the former two tasks, the latter two tasks are totally decoupled from other semantics such as actions and relations, resulting in a better focus on fine-grained visual understanding. The details of GVTBench are shown in Table 1. Experimental Setting. We use visual tokenizers with different supervision to encode an image into a set of visual tokens. Then, we follow Flamingo [ml-foundations, 2023] to use the Perceiver Resampler [Jaegle et al., 2021] to reduce the number of visual tokens to a fixed length, which are fed into LLM (i.e., Vicuna). The models are trained on an instruction-tuning dataset which contains about 5M image-text pairs. In the training process, the language model is always frozen, while the visual tokenizer can be frozen or jointly optimized. More details are deferred to the appendix. 2.2 Comparing Visual Tokenizers On GVTBench, we evaluate visual tokenizers with the same architecture ViT-B [Dosovitskiy et al.] but different pretraining strategies, including fully supervised DeiT [Touvron et al., 2021], self-supervised DINO [Caron et al., 2021], DINOv2 [Oquab et al., 2023], MAE [He et al., 2022] and text-guided weakly supervised CLIP [Radford et al., 2021]. To further investigate the effect of pretraining dataset size, we also compared a CLIP pretrained with 20M image-text pairs, using the checkpoint provided by [Yang et al., 2023a]. Based on the results in Table 2, we arrive at the following observations: Fully/weakly supervised models capture more semantics than self-supervised ones, but the gap is narrowed or even mitigated by scaling up the pre-training dataset. With tokenizers pretrained on relative small-scale dataset (i.e., ImageNet-1k [Russakovsky et al., 2015]) with 1.28M images), DeiT demonstrates better image captioning performance (65.8 CIDEr) than self-supervised models DINO (45.0) and MAE (37.3), without jointly tuning the visual tokenizer. However, with 142M images for pretraining, the self-supervised model – DINOv2 outperforms the supervised DeiT on image captioning (67.9) and VQA (51.3), and is only inferior to CLIP which is pretrained with weak supervision from a large-scale dataset with 400M image-text pairs. Self-supervised models are better at fine-grained perception, where patch-level supervision is particularly effective. On fine-grained visual understanding tasks, i.e., OC and MCI, self-supervised models demonstrate consistently better performance than those with supervision. When they are jointly tuned on the instruction dataset, their OC and MCI performance are mostly boosted, indicating their fine-grained visual perception capability gets improved. Among all the self-supervised models, MAE achieves the best performance, indicating the patch-based supervision is particularly effective for improving fine-grained visual understanding. Tuning semantic-rich visual tokenizer leads to semantic loss on small-scale instruction tuning dataset. When the tokenizer is jointly optimized on the instruction tuning dataset, the rich semantics obtained from large-scale pretraining in CLIP and DINOv2 have noticeably dropped (e.g., CLIP VQA 52.2 → 47.7 and DINOv2 captioning 67.9 → 49.6). We conjecture this is due to the relatively small scale of our instruction dataset (~5M ≪ 142M). As such, for modern MLLMs that are often --- Note these strategies adopt diverse protocols for pretraining, due to their inherent disparities. We thus adopt the off-the-shelf checkpoints for a fair comparison. tuned on small-scale and high-quality instruction datasets (Zhu et al., 2023; Liu et al., 2023), jointly tuning the visual tokenizer may not be a good option. Table 2: Comparison of visual tokenizers with different pretraining strategies. The best result is bold while the second best is underlined. \| Tuning \| Supervision \| Visual Tokenizer \| # Images \| VQA \| COCO-Caption \| COCO-OC \| COCO-MCI \| VCR-OC \| VCR-MCI \| Avg \| \|--------\|-------------\|------------------\|----------\|-----\|--------------\|---------\|----------\|--------\|---------\|-----\| \| \| Fully \| DeiT \| 1.28M \| 48.3\| 65.8 \| 37.5 \| 83.6 \| 29.7 \| 62.5 \| 54.6\| \| × \| Self \| DINO \| 1.28M \| 50.1\| 45.0 \| 46.5 \| 80.8 \| 33.1 \| 56.3 \| 52.0\| \| \| \| MAE \| 1.28M \| 48.4\| 37.3 \| 47.5 \| 82.7 \| 24.2 \| 60.3 \| 50.1\| \| \| \| DINO2 \| 142 M \| 51.3\| 67.9 \| 47.0 \| 86.0 \| 33.3 \| 61.5 \| 57.8\| \| \| Weakly \| CLIP-20M \| 20 M \| 48.2\| 60.9 \| 42.5 \| 79.1 \| 26.5 \| 58.3 \| 52.6\| \| \| \| CLIP \| 400 M \| 52.2\| 69.3 \| 42.5 \| 86.0 \| 33.4 \| 71.2 \| 59.1\| \| \| Fully \| DeiT \| 1.28M \| 50.7\| 38.4 \| 41.0 \| 86.9 \| 31.2 \| 63.6 \| 52.0\| \| ✓ \| Self \| DINO \| 1.28M \| 47.3\| 54.1 \| 44.5 \| 86.6 \| 30.2 \| 57.3 \| 53.3\| \| \| \| MAE \| 1.28M \| 48.9\| 48.0 \| 47.5 \| 88.7 \| 34.8 \| 71.4 \| 56.7\| \| \| \| DINO2 \| 142 M \| 50.5\| 49.6 \| 43.5 \| 84.1 \| 33.2 \| 68.9 \| 55.0\| \| \| Weakly \| CLIP-20M \| 20 M \| 49.6\| 61.2 \| 37.0 \| 84.5 \| 30.0 \| 62.2 \| 54.7\| \| \| \| CLIP \| 400 M \| 47.7\| 64.2 \| 45.5 \| 88.0 \| 34.5 \| 68.8 \| 58.1\| 3 UNIFYING SEMANTIC AND FINE-GRAINED VISUAL UNDERSTANDING 3.1 CLIP WITH REGION-BASED TRAINING The generalist MLLMs call for a versatile visual tokenizer that could properly represent an image’s content at multiple levels. However, based on the results in Table 2, none of existing pretraining methods leads to a good visual tokenizer that excels at both semantic and fine-grained visual perception capabilities. This motivates us to explore whether the best of the two worlds can be achieved by any other method. Fine-tuning CLIP with region supervision. One stream of work (Zhong et al., 2022; Minderer et al., 2022) attempted to improve region representation capability of a pretrained CLIP by fine-tuning it with region supervision, which has demonstrated improved performance for open-vocabulary object detection. This motivates us to study if this also enhances CLIP as a visual tokenizer. We mainly investigated RegionCLIP (Zhong et al., 2022) and Owl-ViT (Minderer et al., 2022). The former finetune a CLIP with region-level supervision from bounding boxes generated by a pretrained RPN, while the latter utilizes the region annotation from an object detection dataset. We compared these methods with CLIP, and show the results in Table 3. It can be observed that without joint tuning the visual tokenizer, both RegionCLIP and Owl-ViT show severe performance drop on image captioning and VQA, indicating the rich semantics in the original CLIP is lost during their region fine-tuning process. On the other hand, when the visual tokenizers are jointly tuned on the instruction-tuning dataset, their fine-grained representation capability improves by a margin (on OC and MCI performance), but this cannot justify the loss of semantic representation capability, resulting in inferior overall performance compared to the original CLIP. Table 3: Comparing CLIP with its region-tuned counterparts. \| Tuning \| Visual Tokenizers \| VQAv2 \| COCO-Caption \| COCO-OC \| COCO-MCI \| VCR-OC \| VCR-MCI \| Avg \| \|--------\|-------------------\|-------\|--------------\|---------\|----------\|--------\|---------\|-----\| \| × \| CLIP Radford et al., 2021 \| 52.2 \| 69.3 \| 42.5 \| 86.0 \| 33.4 \| 71.2 \| 59.1 \| \| × \| RegionCLIP Zhong et al., 2022 \| 48.7 \| 28.5 \| 41.0 \| 86.0 \| 34.1 \| 70.9 \| 51.5 \| \| × \| Owl-ViT Minderer et al., 2022 \| 44.0 \| 32.5 \| 43.0 \| 80.8 \| 33.5 \| 68.3 \| 50.4 \| \| ✓ \| CLIP Radford et al., 2021 \| 47.7 \| 64.2 \| 45.5 \| 88.0 \| 34.5 \| 68.8 \| 58.1 \| \| ✓ \| RegionCLIP Zhong et al., 2022 \| 49.7 \| 65.5 \| 47.5 \| 86.4 \| 34.1 \| 69.1 \| 58.7 \| \| ✓ \| Owl-ViT Minderer et al., 2022 \| 50.8 \| 61.2 \| 38.5 \| 87.1 \| 34.2 \| 71.3 \| 57.2 \| Semantic Feature as Region Supervision. Another stream of work utilized CLIP’s patch feature as region-level supervision for pretraining, aiming to obtain a model with both strong semantics and better region representations. Specifically, EVA (Fang et al., 2023) and MVP (Fang et al., 2023) use CLIP’s patch feature as regression target for Masked Image Modeling (MIM) pretraining, while FD (Wei et al., 2022b) does not employ the masking strategy and directly distills CLIP’s patch feature into a new model. We compared these methods in Table 4. Without jointly tuning the visual tokenizer, FD results in performance improvement on both semantic and fine-grained visual understanding upon CLIP. However, when a patch masking strategy is adopted, the performance of EVA significantly drops. This can be attributed to the introduction of the [MASK] token for MIM, which is only used for pretraining the visual tokenizer but discarded afterward. In this way, the train-test mismatch arises without tuning the visual tokenizer, leading to unsatisfactory performance for downstream tasks. On the other hand, when the visual tokenizer is jointly optimized with the instruction data, they are inferior to the original CLIP on VQA and image captioning, indicating semantic loss occurs. Given the fact that modern MLLMs are often trained on high-quality and small-scale instruction datasets (Zhu et al., 2023; Liu et al., 2023), our observation suggests that visual tokenizer should be frozen to maintain the powerful semantic representation capability from large-scale pretraining. Nonetheless, for visual tokenizers pretrained with MIM, the introduction of the $[\text{MASK}]$ token inevitably leads to train-test mismatch, necessitating it to be jointly tuned on the instruction data. This contradiction indicates that mask-based pretraining may not lead to a good visual tokenizer under MLLM’s framework. As such, even though the results in Table 2 suggest that region-level supervision is effective for fine-grained visual understanding, it should be carefully utilized under the MLLMs framework. To properly utilize it to improve CLIP’s fine-grained visual perceptual capabilities, the results in Table 3 demonstrate that, with its current architecture, the mask-based strategies that were once all the rage may not lead to good visual tokenizer supervision. ### Table 4: Comparison of different strategies that utilize CLIP features as region supervision. \| Method \| Tuning \| Mask \| VQAv2 \| COCO-Caption \| COCO-OC \| COCO-MCI \| VCR-OC \| VCR-MCI \| Avg \| \|-----------------\|--------\|------\|-------\|--------------\|---------\|----------\|--------\|---------\|-----\| \| CLIP (Radford et al., 2021) \| × \| - \| 52.2 \| 69.3 \| 42.5 \| 86.0 \| 33.4 \| 71.2 \| 59.1\| \| FD (Wei et al., 2022b) \| × \| × \| 49.4 \| 72.1 \| 46.5 \| 86.7 \| 34.2 \| 72.3 \| 60.2\| \| EVA (Fang et al., 2023) \| × \| ✓ \| 42.9 \| 27.0 \| 46.9 \| 70.5 \| 21.6 \| 59.9 \| 44.8\| \| CLIP (Radford et al., 2021) \| ✓ \| - \| 47.7 \| 64.2 \| 45.5 \| 88.0 \| 34.5 \| 68.8 \| 58.1\| \| FD (Wei et al., 2022b) \| ✓ \| × \| 49.3 \| 53.3 \| 40.5 \| 85.8 \| 32.1 \| 70.2 \| 55.2\| \| EVA (Fang et al., 2023) \| ✓ \| ✓ \| 51.4 \| 61.6 \| 45.9 \| 87.1 \| 31.4 \| 69.8 \| 57.9\| ### 3.2 MLLM with Good Visual Tokenizer Based on the insights above, we found the patch supervision introduced by feature distillation is helpful in maintaining the semantic representation capability of CLIP while improving its fine-grained perceptual capabilities. As such, we tune a new visual tokenizer that unifies the advantages of semantic representation and fine-grained visual perception capabilities. In particular, we achieve this objective by utilizing a visual tokenizer pretrained on large-scale datasets and properly integrating it with patch-level supervision. Motivated by the findings in Table 4, we do not use any mask-based strategy, so the rich semantics could be preserved by freezing it in the instruction tuning process. To achieve stronger performance, we take the powerful EVA-CLIP (Sun et al., 2023) based on ViT-L as the teacher model and randomly initialize another model with identical architecture as the student. During training, each image is fed into the teacher and student model, obtaining the representation $t$ and $s \in \mathbb{R}^D$ for each image patch, respectively. Then, we perform feature distillation with the following objective: \[ L_{\text{distill}}(s, t) = \begin{cases} \frac{1}{2} (g(s) - \text{whiten}(t))^2 / \beta, & \text{if } \|g(s) - \text{whiten}(t)\| \leq \beta \\ \|g(s) - \text{whiten}(t)\| - \frac{1}{2} \beta, & \text{otherwise} \end{cases} \] The patch features from the student model are first passed through a learnable function $g(\cdot)$, which is a $1 \times 1$ convolution layer. The whitening operation is utilized to stabilize the training process, which is implemented as a non-parametric layer normalization without scaling and bias (Wei et al., 2022b). In the FD process, only the student model and the projector $g(\cdot)$ are used for training, while the teacher model is frozen. Based on the tuned visual tokenizer, we construct a new MLLM with Good Visual Tokenizer (GVT). The framework of GVT is shown in Figure 2. Following ml-foundations (2023), we also random initialize a Receiver Resampler (Jaegle et al., 2021) with 32 learnable queries to attend to the features from the visual tokenizer. Then, the features from the Perceiver Resampler are fed into the LLM (Vicuna-7B, FastChat, 2023) together with the language prompts. The whole model is trained by the language modeling loss, and only the Perceiver Resampler is optimized in this process. Figure 2: The framework of our GVT. We first distill the features of a pretrained CLIP via smoothed $L_1$ loss. Then, we use it to encode images into a set of tokens, which are fed into the Perceiver Resampler (Jaegle et al., 2021) as soft prompts. Together with language instructions, these prompts are fed into LLM to generate responses. Only the Perceiver Resampler is optimized in this process. 4 EXPERIMENTS 4.1 EXPERIMENTAL SETUP We train our model on a joint dataset of image-text pairs, including CC3M (Sharma et al., 2018), SBU (Vicente et al., 2016), Visual Genome (Krishna et al., 2017) and MS-COCO (Lin et al., 2014). We formulate these datasets as image captioning task, and use "what does the image describe?" as prompt during training. Besides, we also use two object detection datasets – Object365 (Shao et al., 2019) and OpenImagesV6 (Kuznetsova et al., 2020) to design a set of object-centric tasks following (Piergiovanni et al., 2022). The LLaVA-150k (Liu et al., 2023) dataset is also utilized for joint training. This results in a total of 15M image-text pairs. The images are resized to $224 \times 224$, and we adopt random resized crop and horizontal flipping for data augmentation during training. The model is trained for 50k steps with 2k steps for linear warmup. We use AdamW (Loshchilov & Hutter, 2017) optimizer with a learning rate of $1e^{-4}$ and batch size 1024. The training process takes about 2 days on 32 Tesla V100 GPUs. For feature distillation, we followed the training protocol in Wei et al. (2022b) except that we trained the model for a total of 50 epochs on the ImageNet-1k (Russakovsky et al., 2015) dataset due to its high quality. The $\beta$ is set to 2.0 through the process. For more implementation details, please refer to our appendix. 4.2 COMPARISON WITH OTHER MLLMs We evaluate GVT on our GVTBench, and compare it with recent MLLMs, including Flamingo (ml_foundations, 2023), KosMos-1 (Huang et al., 2023), BLIP-2 (Li et al., 2023b), LLaVa (Liu et al., 2023), MiniGPT4 (Zhu et al., 2023). The results are shown in Table 5. Our GVT achieves the best overall performance across competitors. Specifically, on tasks requiring fine-grained visual perception, i.e., OC and MCI on both COCO and VCR datasets, GVT surpasses models with larger visual tokenizer and more curated data. This indicates our visual tokenizer can better capture the fine-grained visual information, providing representations with better details. For semantic understanding tasks including VQA and image captioning, GVT achieves the second-best result. It is only inferior to BLIP-2, which utilized a much larger instruction dataset with high-quality image captions filtered by (Li et al., 2022). \| Model \| #Vis. Tok. Size \| VQAv2 \| COCO-Caption \| COCO-OC \| COCO-MCI \| VCR-OC \| VCR-MCI \| Avg \| \|----------------\|----------------\|-------\|--------------\|---------\|----------\|--------\|---------\|-----\| \| Flamingo-9B \| 438 M \| 51.8 \| 79.4 \| - \| - \| - \| - \| - \| \| Kosmos-1 \| 307 M \| 51.0 \| 84.7 \| - \| - \| - \| - \| - \| \| LLaVa \| 307 M \| 39.0 \| 48.3 \| 22.2 \| 52.0 \| 24.6 \| 66.9 \| 44.7\| \| MiniGPT4+ \| 1.0 B \| 58.2 \| 80.6 \| 21.5 \| 76.8 \| 25.1 \| 70.1 \| 55.4\| \| BLIP-2 \| 1.0 B \| 62.4 \| 93.3 \| 48.0 \| 81.9 \| 20.2 \| 68.9 \| 62.5\| \| GVT (Ours) \| 307 M \| 60.4 \| 89.9 \| 56.2 \| 89.3 \| 40.3 \| 78.9 \| 69.2\| Table 5: Comparison with MLLMs. The best results are bold and the second best is underlined. 4.3 Ablation Study Effect of Feature Distillation. To further validate the effectiveness of Feature Distillation, we compared the visual tokenizer before and after in Table 6. It can be observed that the distilled visual tokenizer achieves comparable performance on semantic understanding tasks (VQA and Image Captioning), while greatly improving fine-grained visual perception tasks (OC and MCI), resulting in improved overall performance. This observation is aligned with our findings in Section 3, where feature distillation consistently improves model performance across different architectures. We also provide an evaluation on SEED-Bench [Li et al., 2023a], which is a recently released MLLM benchmark focusing on visual understanding. In Table 7, FD improves performance more on fine-grained understanding tasks such as instance identity, location, and counting. Table 6: Comparison between visual tokenizer with and without FD. \| Visual \| VQAv2 \| COCO-Caption \| COCO-OC \| COCO-MCI \| VCR-OC \| VCR-MCI \| Avg \| \|--------\|-------\|--------------\|---------\|----------\|--------\|---------\|-----\| \| EVA-CLIP \| 60.5 \| 90.8 \| 43.5 \| 85.6 \| 37.6 \| 71.1 \| 64.9 \| \| EVA-CLIP-FD \| 60.4 \| 89.9 \| 56.2 \| 89.3 \| 40.3 \| 78.9 \| 69.2 \| Table 7: Comparison between with and without FD on SEED-Bench [Li et al., 2023a]. \| Visual \| Scene \| Inst.Id \| Inst.Loc \| Inst.Attr \| Inst.Count \| Spatial \| Interaction \| Reason \| Avg \| \|--------\|-------\|---------\|----------\|-----------\|------------\|---------\|-------------\|--------\|-----\| \| EVA-CLIP \| 41.26 \| 34.30 \| 31.40 \| 29.84 \| 34.81 \| 32.98 \| 31.96 \| 50.75 \| 35.93 \| \| EVA-CLIP-FD \| 41.74 \| 35.50 \| 31.79 \| 29.45 \| 36.17 \| 31.96 \| 31.96 \| 51.06 \| 36.20 \| Table 8: Comparison of visual tokenizer with different LLMs. \| LLM \| Visual Tokenizer \| VQAv2 \| COCO-Caption \| COCO-OC \| COCO-MCI \| VCR-OC \| VCR-MCI \| Avg \| \|-----\|------------------\|-------\|--------------\|---------\|----------\|--------\|---------\|-----\| \| Flant5-xxl \| EVA-CLIP \| 55.8 \| 68.1 \| 42.5 \| 70.6 \| 19.9 \| 66.6 \| 53.9 \| \| Flant5-xxl \| EVA-CLIP-FD \| 55.4 \| 67.2 \| 43.6 \| 71.4 \| 20.3 \| 66.8 \| 54.1 \| \| LLaMa-7B \| EVA-CLIP \| 54.2 \| 66.3 \| 42.9 \| 68.3 \| 17.3 \| 54.4 \| 50.6 \| \| LLaMa-7B \| EVA-CLIP-FD \| 53.9 \| 67.5 \| 43.2 \| 70.3 \| 18.9 \| 56.2 \| 51.7 \| Effectiveness with Different LLMs. Our GVT is trained with our distilled visual tokenizer and Vicuna-7B as LLM. In fact, our distilled visual tokenizer is also effective with different LLMs. In Table 8, our distilled visual tokenizer can generally improve the overall performance when using Flant5-xxl and LLaMa-7B as LLM, with the performance on OC and MCI particularly improved. 4.4 Visualizations Attention Maps. To further understand how FD improves fine-grained understanding, we selected one query in the perceiver resampler, and visualized the attentions in two heads in Figure 3. It can be observed that, without FD, the attention mostly focuses on the salient areas of the image, and the attention maps in two different heads are generally similar. In contrast, with FD, the attention maps exhibit higher diversity, which is aligned with [Wei et al., 2022b]. Also, the attention may focus more on informative but non-salient regions (e.g., broccoli and bike in the last column). Qualitative Results. We show some qualitative comparison of OC and MCI between our GVT and BLIP-2 in Figure 4. It can be observed that our method demonstrates better fine-grained visual understanding capabilities than the baseline method. Take the first example in OC as an example, our method not only recognizes the 3 people in the foreground but also takes the fourth person who is far away from the camera into consideration. Besides, GVT also successfully recognizes non-salient or small-sized objects in the image, such as the bicycle and broccoli in MCI. 5 Related Work Multimodal Large Language Models. Recently, with the open source of Large Language Models [Touvron et al., 2023; FastChat, 2023; Radford et al., 2022; Chung et al., 2022], a lot of large multimodal models are constructed based on them. Mini-GPT4 [Zhu et al., 2023] is built on the instruction-tuned Vicuna [FastChat, 2023] and the visual encoder from BLIP-2 [Li et al., 2023b], with only a linear layer trained to bridge the two modules. This simple design results in a powerful multi-modal chatbot, with noticeable vision-language understanding capability. LLaVa [Liu et al., 2023] adopts CLIP as visual tokenizer, and trains the projector with a curated dataset with balanced concepts. The model then can be finetuned for downstream tasks, e.g., ScienceQA [Lu et al., 2022]. Apart from using frozen visual tokenizer, mPLUG-OWL [Ye et al., 2023] tunes the Perceiver Resampler with large-scale image-text data in the first stage, followed by the finetuning of the language model with LoRA [Hu et al., 2021] in the second stage. Although these generalist models have demonstrated impressive capability on multimodal tasks, we find that they mostly focus on the general or overall semantic understanding of the image, ignoring more fine-grained visual perception. Visual Tokenizer Pretraining. Visual encoders have been shown to benefit from large-scale pretraining for downstream tasks. The most common approach first pretrains the model on a large dataset with annotations, e.g., ImageNet (Russakovsky et al., 2015), and finetunes it for downstream tasks such as semantic segmentation (Zhou et al., 2019) and object detection (Lin et al., 2014). Recently, self-supervised pre-training have also shown to improve model’s representation capability. The typical contrastive-based methods (Caron et al., 2021; Chen et al., 2020; Chen & He, 2021) trains the model by aligning views from the same image. Inspired by the idea of mask-language-modeling for pretraining language models (Kenton & Toutanova, 2019), masked-image-modeling has also evolved for visual encoder pretraining. These methods mask a proportion of image patches before feeding them into the model, and ask the model to recover the masked patches. Some methods (Bao et al.) discretize the masked patches via a pretrained tokenizer (Ramesh et al., 2021). Recently, auto-encoder based (He et al., 2022) methods ask the model to directly generate the masked patch in the continuous space. Another stream of visual encoders is pretrained on massive image-text pairs via contrastive learning (Radford et al., 2021), achieving strong zero-shot understanding. 6 CONCLUSION AND FUTURE WORK We comprehensively studied various visual tokenizer supervisions through the lens of MLLM. Our investigation reveals that i) fully/weakly supervised models perform generally better than self-supervised ones on semantic representation, ii) Self-supervised models are better at fine-grained visual perception, where patch-level supervision is particularly effective. iii) jointly tuning the visual tokenizer on the small-scale instruction dataset leads to the loss of rich semantics from large-scale pretraining. Then, we seek a visual tokenizer supervision that excels at both semantic understanding and fine-grained visual perception. We reviewed existing methods and found that directly fine-tuning CLIP with region supervision does not lead to a versatile visual tokenizer. Besides, the masking strategy for pretraining is not suitable due to the train-test mismatch. Based on the insights above, we tune a new visual tokenizer, which distills the CLIP patch feature into a new model without masking. With our visual tokenizer, Vicuna can better understand images at multiple levels, resulting in superior performance on various vision-language tasks. For future work, we would like to explore a more versatile visual tokenizer that can handle more challenging visual understandings. REFERENCES Jean-Baptiste Alayrac, Jeff Donahue, Pauline Luc, Antoine Miech, Iain Barr, Yana Hasson, Karel Lenc, Arthur Mensch, Katherine Millican, Malcolm Reynolds, et al. Flamingo: a visual language model for few-shot learning. NeurIPS, 2022. Hangbo Bao, Li Dong, Songhao Piao, and Furu Wei. Beit: Bert pre-training of image transformers. In ICLR. 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ekz1hN5QNh	In the re-scaling procedure, you assume that the variance direction is along the geodesic intersecting the origin, however, this may not be the case, therefore is not an accurate formulation. Can you elaborate if I have mis-understood.	Fully Hyperbolic Convolutional Neural Networks for Computer Vision Ahmad Bdeir\textsuperscript{1,}, Kristian Schwethelm\textsuperscript{2,,\dagger} & Niels Landwehr\textsuperscript{1} \textsuperscript{1} Data Science Department, University of Hildesheim \textsuperscript{2} Chair for Artificial Intelligence in Medicine, Technical University of Munich \{bdeira, schwethelm, landwehr\}@uni-hildesheim.de Abstract Real-world visual data exhibit intrinsic hierarchical structures that can be represented effectively in hyperbolic spaces. Hyperbolic neural networks (HNNs) are a promising approach for learning feature representations in such spaces. However, current HNNs in computer vision rely on Euclidean backbones and only project features to the hyperbolic space in the task heads, limiting their ability to fully leverage the benefits of hyperbolic geometry. To address this, we present HCNN, a fully hyperbolic convolutional neural network (CNN) designed for computer vision tasks. Based on the Lorentz model, we generalize fundamental components of CNNs and propose novel formulations of the convolutional layer, batch normalization, and multinomial logistic regression. Experiments on standard vision tasks demonstrate the promising performance of our HCNN framework in both hybrid and fully hyperbolic settings. Overall, we believe our contributions provide a foundation for developing more powerful HNNs that can better represent complex structures found in image data. Our code is publicly available at https://github.com/kschwethelm/HyperbolicCV. 1 Introduction Representation learning is a fundamental aspect of deep neural networks, as obtaining an optimal representation of the input data is crucial. While Euclidean geometry has been the traditional choice for representing data due to its intuitive properties, recent research has highlighted the advantages of using hyperbolic geometry as a geometric prior for the feature space of neural networks. Given the exponentially increasing distance to the origin, hyperbolic spaces can be thought of as continuous versions of trees that naturally model tree-like structures, like hierarchies or taxonomies, without spatial distortion and information loss (Nickel \& Kiela, 2018; Sarkar, 2012). This is compelling since hierarchies are ubiquitous in knowledge representation (Noy \& Hafner, 1997), and even the natural spatial representations in the human brain exhibit a hyperbolic geometry (Zhang et al., 2023). Leveraging this better representative capacity, hyperbolic neural networks (HNNs) have demonstrated increased performance over Euclidean models in many natural language processing (NLP) and graph embedding tasks (Peng et al., 2022). However, hierarchical structures have also been shown to exist in images. Mathematically, Khrulkov et al. (2020) have found high $\delta$-hyperbolicity in the final embeddings of image datasets, where the hyperbolicity quantifies the degree of inherent tree-structure. Extending their measurement to the whole model reveals high hyperbolicity in intermediate embeddings as well (see Appendix D.1). Intuitively, hierarchies that emerge within and across images can be demonstrated on the level of object localization and object class relationships. A straightforward example of the latter is animal classification hierarchy, where species is the lowest tier, preceded by genus, family, order, etc. Similarly, on a localization level, humans are one example: the nose, eyes, and mouth are positioned on the face, which is a part of the head, and, ultimately, a part of the body. This tree-like localization forms the basis of part-whole relationships and is strongly believed to be how we parse visual scenes (Biederman, 1987; Hinton, 1979; Kahneman et al., 1992). In light of these findings, recent works have begun integrating hyperbolic geometry into vision architectures (Mettes et al., 2023; Fang et al., 2023). Specifically, they rely on the Poincaré ball Equal contribution. †Work done while at University of Hildesheim. and the Lorentz model as descriptors of hyperbolic space and formalize hyperbolic translations of neural network layers. This is challenging due to ill-defined hyperbolic analogs of, e.g., addition, multiplication, and statistical measures. Currently, most HNN components are only available in the Poincaré ball as it supports the gyrovector space with basic vector operations. However, due to its hard numerical constraint, the Poincaré ball is more susceptible to numerical instability than the Lorentz model (Mishne et al., 2022), which motivates introducing the missing layers for the Lorentz model. Moreover, HNNs in computer vision have been limited to hybrid architectures that might not fully leverage the advantages of hyperbolic geometry as they rely on Euclidean encoders to learn hyperbolic representations. Until now, hyperbolic encoder architectures are missing in computer vision, although prevalent in NLP and graph applications (Peng et al., 2022). In this work, we present HCNN, a fully hyperbolic framework for vision tasks that can be used to design hyperbolic encoder models. We generalize the ubiquitous convolutional neural network (CNN) architecture to the Lorentz model, extend hyperbolic convolutional layers to 2D, and present novel hyperbolic formulations of batch normalization and multinomial logistic regression. Our methodology is general, and we show that our components can be easily integrated into existing architectures. Our contributions then become three-fold: 1. We propose hybrid (HECNN) and fully hyperbolic (HCNN) convolutional neural network encoders for image data, introducing the fully hyperbolic setting in computer vision. 2. We provide missing Lorentzian formulations of the 2D convolutional layer, batch normalization, and multinomial logistic regression. 3. We empirically demonstrate the performance potential of deeper hyperbolic integrations in experiments on standard vision tasks, including image classification and generation. 2 RELATED WORK Hyperbolic image embeddings Previous research on HNNs in computer vision has mainly focused on combining Euclidean encoders and hyperbolic embeddings. This approach involves projecting Euclidean embeddings onto the hyperbolic space in the task heads and designing task-related objective functions based on hyperbolic geometry. Such simple hybrid architectures have been proven effective in various vision tasks like recognition (Yu et al., 2022; Khrulkov et al., 2020; Liu et al., 2020; Guo et al., 2022), segmentation (Hsu et al., 2020; Atigh et al., 2022), reconstruction/generation (Mathieu et al., 2019; Nagano et al., 2019; Ovinnikov, 2019; Qu & Zou, 2022), and metric learning (Ermolov et al., 2022; Yan et al., 2021; Yue et al., 2023). However, there remains the discussion of whether the single application of hyperbolic geometry in the decoder can fully leverage the present hierarchical information. In contrast, HE/HCNN also learns latent hyperbolic feature representations in the encoder, potentially magnifying these benefits. We also forgo the typically used Poincaré ball in favor of the Lorentz model, as it offers better stability and optimization (Mishne et al., 2022). For a complete overview of vision HNNs and motivations, refer to (Mettes et al., 2023; Fang et al., 2023). Fully hyperbolic neural networks Designing fully hyperbolic neural networks requires generalizing Euclidean network components to hyperbolic geometry. Notably, Ganea et al. (2018) and Shimizu et al. (2020) utilized the Poincaré ball and the gyrovector space to generalize various layers, including fully-connected, convolutional, and attention layers, as well as operations like split, concatenation, and multinomial logistic regression (MLR). Researchers have also designed components in the Lorentz model (Nickel & Kiela, 2018; Fan et al., 2022; Chen et al., 2021; Qu & Zou, 2022), but crucial components for vision, like the standard convolutional layer and the MLR classifier, are still missing. Among the hyperbolic layer definitions, fully hyperbolic neural networks have been built for various tasks in NLP and graph applications (Peng et al., 2022). However, no hyperbolic encoder architecture has yet been utilized in computer vision. Our work provides formulations for missing components in the Lorentz model, allowing for hyperbolic CNN vision encoders. Concurrently, van Spengler et al. (2023) proposed a fully hyperbolic Poincaré CNN. Normalization in HNNs There are few attempts at translating standard normalization layers to the hyperbolic setting. To the best of our knowledge, there is only a single viable normalization layer for HNNs, i.e., the general Riemannian batch normalization (Lou et al., 2020). However, this method is not ideal due to the slow iterative computation of the Fréchet mean and the arbitrary re-scaling operation that is not based on hyperbolic geometry. The concurrent work on Poincaré CNN (van Spengler et al., 2023) only solved the first issue by using the Poincaré midpoint. In contrast, we propose an efficient batch normalization algorithm founded in the Lorentz model, which utilizes the Lorentzian centroid (Law et al., 2019) and a mathematically motivated re-scaling operation. Numerical stability of HNNs The exponential growth of the Lorentz model’s volume with respect to the radius can introduce numerical instability and rounding errors in floating-point arithmetic. This requires many works to rely on 64-bit precision at the cost of higher memory and runtime requirements. To mitigate this, researchers have introduced feature clipping and Euclidean reparameterizations (Mishne et al., 2022; Guo et al., 2022; Mathieu et al., 2019). We adopt these approaches to run under 32-bit floating point arithmetic and reduce computational cost. 3 BACKGROUND This section summarizes the mathematical background of hyperbolic geometry (Cannon et al., 2006; Ratcliffe, 2006). The $n$-dimensional hyperbolic space $\mathbb{H}^n_K$ is a Riemannian manifold $(\mathcal{M}^n, g^K)$ with constant negative curvature $K < 0$, where $\mathcal{M}^n$ and $g^K$ represent the manifold and the Riemannian metric, respectively. There are isometrically equivalent models of hyperbolic geometry. We employ the Lorentz model because of its numerical stability and its simple exponential/logarithmic maps and distance functions. Additionally, we use the Poincaré ball for baseline implementations. Both hyperbolic models provide closed-form formulae for manifold operations, including distance measures, exponential/logarithmic maps, and parallel transportation. They are detailed in Appendix A. Lorentz model The $n$-dimensional Lorentz model $\mathbb{L}^n_K = (\mathcal{L}^n, g^K)$ models hyperbolic geometry on the upper sheet of a two-sheeted hyperboloid $\mathcal{L}^n$, with origin $\mathbf{0} = [\sqrt{-1/K}, 0, \cdots, 0]^T$ and embedded in $(n + 1)$-dimensional Minkowski space (see Figure 2). Based on the Riemannian metric $g^K = \text{diag}(-1, 1, \ldots, 1)$, the manifold is defined as $$\mathcal{L}^n := \{x \in \mathbb{R}^{n+1} \| \langle x, x \rangle_{\mathcal{L}} = \frac{1}{K}, x_t > 0\},$$ with the Lorentzian inner product $$\langle x, y \rangle_{\mathcal{L}} := -x_t y_t + x_s^T y_s = x^T \text{diag}(-1, 1, \ldots, 1)y.$$ When describing points in the Lorentz model, we inherit the terminology of special relativity and call the first dimension the time component $x_t$ and the remaining dimensions the space component $x_s$, such that $x \in \mathbb{L}^n_K = [x_t, x_s]^T$ and $x_t = \sqrt{\\|x_s\\|^2 - 1/K}$. 4 FULLY HYPERBOLIC CNN (HCNN) We aim to give way to building vision models that can fully leverage the advantages of hyperbolic geometry by learning features in hyperbolic spaces. For this, we generalize Euclidean CNN components to the Lorentz model, yielding one-to-one replacements that can be integrated into existing architectures. In the following, we first define the cornerstone of HCNNs, i.e., the Lorentz convolutional layer, including its transposed variant. Then, we introduce the Lorentz batch normalization algorithm and the MLR classifier. Finally, we generalize the residual connection and non-linear activation. 4.1 LORENTZ CONVOLUTIONAL LAYER Hyperbolic feature maps The convolutional layer applies vector operations to an input feature map containing the activations of the previous layer. In Euclidean space, arbitrary numerical values can be combined to form a vector. However, in the Lorentz model, not all possible value combinations represent a point that can be processed with hyperbolic operations ($\mathbb{L}_K^n \subset \mathbb{R}^{n+1}$). We propose using channel-last feature map representations throughout HCNNs and adding the Lorentz model’s time component as an additional channel dimension. This defines a hyperbolic feature map as an ordered set of $n$-dimensional hyperbolic vectors, where every spatial position contains a vector that can be combined with its neighbors. Additionally, it offers a nice interpretation where an image is an ordered set of color vectors, each describing a pixel. Formalization of the convolutional layer We define the convolutional layer as a matrix multiplication between a linearized kernel and a concatenation of the values in its receptive field, following Shimizu et al. (2020). Then, we generalize this definition by replacing the Euclidean operators with their hyperbolic counterparts in the Lorentz model. Given a hyperbolic input feature map $x = \{x_{h,w} \in \mathbb{L}_K^n\}_{h,w=1}^{H,W}$ as an ordered set of $n$-dimensional hyperbolic feature vectors, each describing image pixels, the features within the receptive field of the kernel $K \in \mathbb{R}^{m \times n \times H \times W}$ are $\{x_{h'+\delta h,w'+\delta w} \in \mathbb{L}_K^n\}_{h',w'=1}^{H',W'}$, where $(h', w')$ denotes the starting position and $\delta$ is the stride parameter. Now, we define the Lorentz convolutional layer as $$y_{h,w} = \text{LFC}(\text{HCat}(\{x_{h'+\delta h,w'+\delta w} \in \mathbb{L}_K^n\}_{h',w'=1}^{H',W'})),$$ where $\text{HCat}$ denotes the concatenation of hyperbolic vectors, and $\text{LFC}$ denotes a Lorentz fully-connected layer performing the affine transformation and parameterizing the kernel and bias, respectively (see Appendix A). Additionally, we implement padding using origin vectors, the analog of zero vectors in hyperbolic space. The LFC layer is similar to Chen et al. (2021) but does not use normalization as it is done through the hyperbolic batch normalization formulated below. Extension to the transposed setting The transposed convolutional layer is usually used in encoder-decoder architectures for up-sampling. A convolutional layer carries out a transposed convolution when the correct local connectivity is established by inserting zeros at certain positions. Specifically, when stride $s > 1$, then $s - 1$ zero vectors are inserted between the features. We refer to Dumoulin & Visin (2016) for illustrations. Under this relationship, the Lorentz transposed convolutional layer is a Lorentz convolutional layer with changed connectivity through origin padding. 4.2 LORENTZ BATCH NORMALIZATION Given a batch $B$ of $m$ features $x_i$, the traditional batch normalization algorithm (Ioffe & Szegedy, 2015) calculates the mean $\mu_B = \frac{1}{m} \sum_{i=1}^{m} x_i$ and variance $\sigma_B^2 = \frac{1}{m} \sum_{i=1}^{m} (x_i - \mu_B)^2$ across the batch dimension. Then, the features are re-scaled and re-centered using a parameterized variance $\gamma$ and mean $\beta$ as follows $$\text{BN}(x_i) = \gamma \odot \frac{x_i - \mu_B}{\sqrt{\sigma_B^2 + \epsilon}} + \beta.$$ At test time, running estimates approximate the batch statistics. They are calculated iteratively during training: $\mu_t = (1 - \eta)\mu_{t-1} + \eta \mu_B$ and $\sigma_t^2 = (1 - \eta)\sigma_{t-1}^2 + \eta \sigma_B^2$, with $\eta$ and $t$ denoting momentum and the current iteration, respectively. We generalize batch normalization to the Lorentz model using the Lorentzian centroid and the parallel transport operation for re-centering, and the Fréchet variance and straight geodesics at the origin’s tangent space for re-scaling. Re-centering To re-center hyperbolic features, it is necessary to compute a notion of mean. Usually, the Fréchet mean is used (Lou et al., 2020), which minimizes the expected squared distance between a set of points in a metric space (Pennec, 2006). Generally, the Fréchet mean must be solved iteratively, massively slowing down training. To this end, we propose to use the centroid with respect to the squared Lorentzian distance, which can be calculated efficiently in closed form (Law et al., 2019). The weighted Lorentzian centroid, which solves $ \min_{\mu \in L^n_K} \sum_{i=1}^{m} \nu_i d^2_L(x_i, \mu) $, with $ x_i \in L^n_K $ and $ \nu_i \geq 0, \sum_{i=1}^{m} \nu_i > 0 $, is given by \[ \mu = \frac{\sum_{i=1}^{m} \nu_i x_i}{\sqrt{-K \|\| \sum_{i=1}^{m} \nu_i x_i \|\|_L}}. \] (5) In batch normalization, the mean is not weighted, which gives $ \nu_i = \frac{1}{m} $. Now, we shift the features from the batch’s mean $ \mu_B $ to the parameterized mean $ \beta $ using the parallel transport operation $ PT^K_{\mu_B \rightarrow \beta}(x) $. Parallel transport does not change the variance, as it is defined to preserve the distance between all points. Finally, the running estimate is updated iteratively using the weighted centroid with $ \nu_1 = (1 - \eta) $ and $ \nu_2 = \eta $. Re-scaling For re-scaling, we rely on the Fréchet variance $ \sigma^2 \in \mathbb{R}^+ $, defined as the expected squared Lorentzian distance between a point $ x_i $ and the mean $ \mu $, and given by $ \sigma^2 = \frac{1}{m} \sum_{i=1}^{m} d^2_L(x_i, \mu) $ (Köbler et al., 2022). In order to re-scale the batch, features must be moved along the geodesics connecting them to their centroid, which is generally infeasible to compute. However, geodesics intersecting the origin are very simple, as they can be represented by straight lines in tangent space $ T_0 L^n_K $. This is reflected by the equality between the distance of a point to the origin and the length of its corresponding tangent vector $ (d_L(x, 0) = \|\| \log^K_0(x) \|\|) $. Using this property, we propose to re-scale features by first parallel transporting them towards the origin $ PT^K_{\mu_B \rightarrow 0}(\log^K_{\mu_B}(x)) $, making the origin the new centroid and straightening the relevant geodesics. Then, a simple multiplication re-scales the features in tangent space. Finally, parallel transporting to $ \beta \in L^n_K $ completes the algorithm and yields the normalized features. The final algorithm is formalized as \[ LBN(x) = \exp^K_{\beta} \left( PT^K_{0 \rightarrow \beta} \left( \gamma \cdot \frac{PT^K_{\mu_B \rightarrow 0}(\log^K_{\mu_B}(x))}{\sqrt{\sigma^2_B + \epsilon}} \right) \right). \] (6) 4.3 Lorentz MLR Classifier In this section, we consider the problem of classifying instances that are represented in the Lorentz model. A standard method for multi-class classification is multinomial logistic regression (MLR). Inspired by the generalization of MLR to the Poincaré ball (Ganea et al., 2018; Shimizu et al., 2020) based on the distance to margin hyperplanes, we derive a formulation in the Lorentz model. Hyperplane in the Lorentz model Analogous to Euclidean space, hyperbolic hyperplanes split the manifold into two half-spaces, which can then be used to separate instances into classes. The hyperplane in the Lorentz model is defined by a geodesic that results from the intersection of an $ n $-dimensional hyperplane with the hyperboloid in the ambient space $ \mathbb{R}^{n+1} $ (Cho et al., 2019). Specifically, for $ p \in L^n_K $ and $ w \in T_p L^n_K $, the hyperplane passing through $ p $ and perpendicular to $ w $ is given by \[ H_{w,p} = \{ x \in L^n_K \| \langle w, x \rangle_L = 0 \}. \] (7) This formulation comes with the non-convex optimization condition $ \langle w, w \rangle_L > 0 $, which is undesirable in machine learning. To eliminate this condition, we use the Euclidean reparameterization of Mishne et al. (2022), which we extend to include the curvature parameter $ K $ in Appendix B.1. In short, $ w $ is parameterized by a vector $ z \in T_0 L^n_K = [0, az/\|\|z\|\|] $, where $ a \in \mathbb{R} $ and $ z \in \mathbb{R}^n $. As $ w \in T_p L^n_K $, $ z $ is parallel transported to $ p $, which gives \[ w := PT^K_{0 \to p} (\bar{z}) = [\sinh(\sqrt{-K}a)\|\|z\|\|, \cosh(\sqrt{-K}a)z]. \] (8) Inserting Eq. 8 into Eq. 7, the formula of the Lorentz hyperplane becomes \[ \tilde{H}_{z,a} = \{ x \in \mathbb{L}_K^n \| \cosh(\sqrt{-K}a)\langle z, x_s \rangle - \sinh(\sqrt{-K}a)\|\|z\|\| x_t = 0 \}, \] (9) where $ a $ and $ z $ represent the distance and orientation to the origin, respectively. Finally, we need the distance to the hyperplane to quantify the model’s confidence. It is formulated by the following theorem, proven in Appendix B.2. Theorem 1* Given $ a \in \mathbb{R} $ and $ z \in \mathbb{R}^n $, the minimum hyperbolic distance from a point $ x \in \mathbb{L}_K^n $ to the hyperplane $ \tilde{H}_{z,a} $ defined in Eq. 9 is given by \[ d_L(x, \tilde{H}_{z,a}) = \frac{1}{\sqrt{-K}} \sinh^{-1} \left( \frac{\cosh(\sqrt{-K}a)\langle z, x_s \rangle - \sinh(\sqrt{-K}a)\|\|z\|\| x_t}{\sqrt{\|\|\cosh(\sqrt{-K}a)z\|\|^2 - (\sinh(\sqrt{-K}a)\|\|z\|\|)^2}} \right). \] (10) MLR in the Lorentz model Lebanon & Lafferty (2004) formulated the logits of the Euclidean MLR classifier using the distance from instances to hyperplanes describing the class regions. Specifically, given input $ x \in \mathbb{R}^n $ and $ C $ classes, the output probability of class $ c \in \{1, ..., C\} $ can be expressed as \[ p(y = c \| x) \propto \exp(v_{w_c}(x)), \quad v_{w_c}(x) = \text{sign}(\langle w_c, x \rangle)\|\|w_c\|\|d(x, H_{w_c}), \quad w_c \in \mathbb{R}^n, \] (11) where $ H_{w_c} $ is the decision hyperplane of class $ c $. We define the Lorentz MLR without loss of generality by inserting the Lorentzian counterparts into Eq. 11. This yields logits given by the following theorem, proven in Appendix B.3. Theorem 2 Given parameters $ a_c \in \mathbb{R} $ and $ z_c \in \mathbb{R}^n $, the Lorentz MLR’s output logit corresponding to class $ c $ and input $ x \in \mathbb{L}_K^n $ is given by \[ v_{z_c,a_c}(x) = \frac{1}{\sqrt{-K}} \text{sign}(\alpha)\beta \sinh^{-1} \left( \frac{\alpha}{\beta} \right), \] (12) \[ \alpha = \cosh(\sqrt{-K}a)\langle z, x_s \rangle - \sinh(\sqrt{-K}a), \] \[ \beta = \sqrt{\|\|\cosh(\sqrt{-K}a)z\|\|^2 - (\sinh(\sqrt{-K}a)\|\|z\|\|)^2}. \] ### 4.4 LORENTZ RESIDUAL CONNECTION AND ACTIVATION Residual connection The residual connection is a crucial component when designing deep CNNs. As vector addition is ill-defined in the Lorentz model, we add the vector’s space components and concatenate a corresponding time component. This is possible as a point $ x \in \mathbb{L}_K^n $ can be defined by an arbitrary space component $ x_s \in \mathbb{R}^n $ and a time component $ x_t = \sqrt{\|\|x_s\|\|^2 - 1/K} $. Our method is straightforward and provides the best empirical performance compared to other viable methods for addition we implemented, i.e., tangent space addition (Nickel & Kiela, 2018), parallel transport addition (Chami et al., 2019), Möbius addition (after projecting to the Poincaré ball) (Ganea et al., 2018), and fully-connected layer addition (Chen et al., 2021). Non-linear activation Prior works use non-linear activation in tangent space (Fan et al., 2022), which weakens the model’s stability due to frequent logarithmic and exponential maps. We propose a simpler operation for the Lorentz model by applying the activation function to the space component and concatenating a time component. For example, the Lorentz ReLU activation is given by \[ y = \begin{bmatrix} \sqrt{\|\|\text{ReLU}(x_s)\|\|^2 - 1/K} \\ \text{ReLU}(x_s) \end{bmatrix}. \] (13) 5 EXPERIMENTS We evaluate hyperbolic models on image classification and generation tasks and compare them against Euclidean and hybrid HNN counterparts from the literature. To ensure a fair comparison, in every task, we directly translate a Euclidean baseline to the hyperbolic setting by using hyperbolic modules as one-to-one replacements. All experiments are implemented in PyTorch (Paszke et al., 2019), and we optimize hyperbolic models using adaptive Riemannian optimizers (Bécigneul & Ganea, 2018) provided by Geoopt (Kochurov et al., 2020), with floating-point precision set to 32 bits. We provide detailed experimental configurations in Appendix C and ablation experiments in Appendix D. 5.1 IMAGE CLASSIFICATION Experimental setup We evaluate image classification performance using ResNet-18 (He et al., 2015b) and three datasets: CIFAR-10 (Krizhevsky, 2009), CIFAR-100 (Krizhevsky, 2009), and Tiny-ImageNet (Le & Yang, 2015). All these datasets exhibit hierarchical class relations and high hyperbolicity (low $\delta_{rel}$), making the use of hyperbolic models well-motivated. For the HCNN, we replace all components in the ResNet architecture with our proposed Lorentz modules. Additionally, we experiment with a novel hybrid approach (HECNN), where we employ our Lorentz decoder and replace only the ResNet encoder blocks with the highest hyperbolicity ($\delta_{rel} < 0.2$), i.e., blocks 1 and 3 (see Appendix D.1). To establish hyperbolic baselines we follow the literature (Atigh et al., 2022; Guo et al., 2022) and implement hybrid HNNs with a Euclidean encoder and a hyperbolic output layer (using both the Poincaré MLR (Shimizu et al., 2020) and our novel Lorentz MLR). Additionally, we report classification results for the concurrently developed fully hyperbolic Poincaré ResNet (van Spengler et al., 2023). For all models, we adopt the training procedure and hyperparameters of DeVries & Taylor (2017), which have been optimized for Euclidean CNNs and yield a strong Euclidean ResNet baseline. Main results Table 1 shows that hyperbolic models using the Lorentz model achieve the highest accuracy across all datasets, outperforming both the Euclidean and Poincaré baselines. In contrast, the Poincaré HNNs are consistently worse than the Euclidean baseline, aligning with the results of Guo et al. (2022). Notably, only in the case of CIFAR-10, all models exhibit equal performance, which is expected due to the dataset’s simplicity. We also notice that the hybrid encoder model outperforms the fully hyperbolic model, indicating that not all parts of the model benefit from hyperbolic geometry. Overall, our findings suggest that the Lorentz model is better suited for HNNs than the Poincaré ball. This may be attributed to the better numerical stability causing fewer inaccuracies (Mishne et al., 2022). Furthermore, we achieve a notable improvement (of up to 1.5%) in the accuracy of current HNNs. This shows the potential of using our HCNN components in advancing HNNs. Table 1: Classification accuracy (%) of ResNet-18 models. We estimate the mean and standard deviation from five runs. The best performance is highlighted in bold (higher is better). \| Model \| CIFAR-10 ($\delta_{rel} = 0.26$) \| CIFAR-100 ($\delta_{rel} = 0.23$) \| Tiny-ImageNet ($\delta_{rel} = 0.20$) \| \|------------------------------\|----------------------------------\|----------------------------------\|-------------------------------------\| \| Euclidean (He et al., 2015b) \| 95.14±0.12 \| 77.72±0.15 \| 65.19±0.12 \| \| Hybrid Poincaré (Guo et al., 2022) \| 95.04±0.13 \| 77.19±0.50 \| 64.93±0.38 \| \| Hybrid Lorentz (Ours) \| 94.98±0.12 \| 78.03±0.21 \| 65.63±0.10 \| \| Poincaré ResNet (van Spengler et al., 2023) \| 94.51±0.15 \| 76.60±0.32 \| 62.01±0.56 \| \| HECNN Lorentz (Ours) \| 95.16±0.11 \| 78.76±0.24 \| 65.96±0.18 \| \| HCNN Lorentz (Ours) \| 95.14±0.08 \| 78.07±0.17 \| 65.71±0.13 \| Adversarial robustness Prior works have demonstrated the robustness of hyperbolic models against adversarial attacks (Yue et al., 2023; Guo et al., 2022). We expect better performance for HCNNs/HECNNs due to the bigger effect fully hyperbolic models have on the embedding space as can be seen in Figure 3. We believe the benefit could come from the increased inter-class separation afforded by the distance metric which allows for greater slack in the object classification. To study this, we employ the trained models and attack them using FGSM (Goodfellow et al., 2015) and PGD (Madry et al., 2019) with different perturbations. The results in Table 2 show that our HCNN is more Table 2: Classification accuracy (%) after performing FGSM and PGD attacks on CIFAR-100. We estimate the mean and standard deviation from attacking five trained models (higher is better). \| Max. perturbation $\epsilon$ \| FGSM \| PGD \| \|------------------------------\|---------------\|--------------\| \| \| 0.8/255 \| 1.6/255 \| 3.2/255 \| 0.8/255 \| 1.6/255 \| 3.2/255 \| \| Euclidean (He et al., 2015b) \| 65.70±0.28 \| 54.98±0.39 \| 39.97±0.43 \| 64.43±0.29 \| 49.76±0.42 \| 26.30±0.40 \| \| Hybrid Poincaré (Guo et al., 2022) \| 64.68±0.40 \| 53.32±0.60 \| 37.52±0.50 \| 63.43±0.44 \| 48.41±0.60 \| 23.78±0.75 \| \| Hybrid Lorentz (Ours) \| 65.27±0.52 \| 53.82±0.49 \| 40.53±0.31 \| 64.15±0.53 \| 49.05±0.68 \| 27.17±0.40 \| \| HECNN Lorentz (Ours) \| 66.13±0.41 \| 55.71±0.43 \| 42.76±0.37 \| 65.01±0.49 \| 50.82±0.37 \| 30.34±0.22 \| \| HCNN Lorentz (Ours) \| 66.47±0.27 \| 57.14±0.30 \| 43.51±0.35 \| 65.04±0.28 \| 52.25±0.34 \| 31.77±0.55 \| robust, achieving up to 5% higher accuracy. In addition, and contrary to Guo et al. (2022), we observe that hybrid decoder HNNs can be more susceptible to adversarial attacks than Euclidean models. Low embedding dimensionality HNNs have shown to be most effective for low-dimensional embeddings (Peng et al., 2022). To this end, we reduce the dimensionality of the final ResNet block and the embeddings and evaluate classification accuracy on CIFAR-100. The results in Figure 3 verify the effectiveness of hyperbolic spaces with low dimensions, where all HNNs outperform the Euclidean models. However, our HCNN and HECNN can leverage this advantage best, suggesting that hyperbolic encoders offer great opportunities for dimensionality reduction and designing smaller models with fewer parameters. The high performance of HECNN is unexpected as we hypothesized the fully hyperbolic model to perform best. This implies that hybrid encoder HNNs might make better use of the combined characteristics of both Euclidean and hyperbolic spaces. 5.2 IMAGE GENERATION Experimental setup Variational autoencoders (VAEs) (Kingma & Welling, 2013; Rezende et al., 2014) have been widely adopted in HNN research to model latent embeddings in hyperbolic spaces (Nagano et al., 2019; Mathieu et al., 2019; Ovinnikov, 2019; Hsu et al., 2020). HNNs have shown to generate more expressive embeddings under lower dimensionalities which would make them a good fit for VAEs. In this experiment, we extend the hyperbolic VAE to the fully hyperbolic setting using our proposed HCNN framework and, for the first time, evaluate its performance on image generation using the standard Fréchet Inception Distance (FID) metric (Heusel et al., 2017). Building on the experimental setting of Ghosh et al. (2019), we test vanilla VAEs and assess generative performance on CIFAR-10 (Krizhevsky, 2009), CIFAR-100 (Krizhevsky, 2009), and CelebA (Liu et al., 2015) datasets. We compare our HCNN-VAE against the Euclidean and two hybrid models. Following prior works, the hybrid models only include a latent hyperbolic distribution and no hyperbolic layers. Specifically, we employ the wrapped normal distributions in the Lorentz model (Nagano et al., 2019) and the Poincaré ball (Mathieu et al., 2019), respectively. Main results The results in Table 3 show that our HCNN-VAE outperforms all baselines. Likewise, the hybrid models improve performance over the Euclidean model, indicating that learning the latent embeddings in hyperbolic spaces is beneficial. This is likely due to the higher representation capacity of the hyperbolic space, which is crucial in low dimensional settings. However, our HCNN is better at leveraging the advantages of hyperbolic geometry due to its fully hyperbolic architecture. These results suggest that our method is a promising approach for generation and for modeling latent structures in image data. Table 3: Reconstruction and generation FID of manifold VAEs across five runs (lower is better). \| \| CIFAR-10 \| \| CIFAR-100 \| \| CelebA \| \|------------------\|----------\|----------\|-----------\|----------\|----------\| \| \| Rec. FID \| Gen. FID \| Rec. FID \| Gen. FID \| Rec. FID \| \| Euclidean \| 61.21±0.72 \| 92.40±0.80 \| 63.81±0.47 \| 103.54±0.84 \| 54.80±0.29 \| \| Hybrid Poincaré (Mathieu et al., 2019) \| 59.85±0.50 \| 90.13±0.77 \| 62.64±0.43 \| 98.19±0.57 \| 54.62±0.61 \| \| Hybrid Lorentz (Nagano et al., 2019) \| 59.29±0.47 \| 90.91±0.84 \| 62.14±0.35 \| 98.34±0.62 \| 54.64±0.34 \| \| HCNN Lorentz (Ours) \| 57.78±0.56 \| 89.20±0.85 \| 61.44±0.64 \| 100.27±0.84 \| 54.17±0.66 \| Figure 4: Embeddings of MNIST dataset in 2D latent space of VAEs (with gen. FID). Colors represent golden labels and Lorentz embeddings are projected onto the Poincaré ball for better visualization. Analysis of latent embeddings The latent embedding space is a crucial component of VAEs as it influences how the data’s features are encoded and used for generating the output. We visually analyze the distribution of latent embeddings inferred by the VAEs. For this, the models are retrained on the MNIST (Lecun et al., 1998) dataset with an embedding dimension $d_E = 2$. Then, the images of the training dataset are passed through the encoder and visualized as shown in Figure 4. We observe the formation of differently shaped clusters that correlate with the ground truth labels. While the embeddings of the Euclidean and hybrid models form many clusters that direct towards the origin, the HCNN-VAE obtains rather curved clusters that maintain a similar distance from the origin. The structures within the HCNN’s latent space can be interpreted as hierarchies where the distance to the origin represents hierarchical levels. As these structures cannot be found for the hybrid model, our results suggest that hybrid HNNs using only a single hyperbolic layer have little impact on the model’s Euclidean characteristics. Conversely, our fully hyperbolic architecture significantly impacts how features are represented and learned, directing the model toward tree-like structures. 6 Conclusion In this work, we proposed HCNN, a generalization of the convolutional neural network that learns latent feature representations in hyperbolic spaces. To this end, we formalized the necessary modules in the Lorentz model, deriving novel formulations of the convolutional layer, batch normalization, and multinomial logistic regression. We empirically demonstrated that ResNet and VAE models based on our hyperbolic framework achieve better performance on standard vision tasks than Euclidean and hybrid decoder baselines, especially in adversarial and lower dimensional settings. Additionally, we showed that using the Lorentz model in HNNs leads to better stability and performance than the Poincaré ball. However, hyperbolic CNNs are still in their early stages and introduce mathematical complexity and computational overhead. For this, we explored HECNN models with the benefit of targeting only specific parts of the encoder, allowing for faster runtimes and larger models. Moreover, our framework currently relies on generalizations of neural network layers that were designed for Euclidean geometry and might not fully capture the unique properties of hyperbolic geometry. Further research is needed to fully understand the properties of HCNNs and address open questions such as optimization, scalability, and performance on other deep learning problems. We hope our work will inspire future research and development in this exciting and rapidly evolving field. ACKNOWLEDGMENTS This work was performed on the HoreKa supercomputer funded by the Ministry of Science, Research and the Arts Baden-Württemberg and by the Federal Ministry of Education and Research. Ahmad Bdeir and Kristian Schwethelm were funded by the European Union’s Horizon 2020 research and innovation programme under the SustInAfrica grant agreement No 861924. Kristian Schwethelm was also funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - project number 225197905. REFERENCES Mina Ghadimi Atigh, Julian Schoep, Erman Acar, Nanne van Noord, and Pascal Mettes. Hyperbolic image segmentation. 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Nu7dDaVF5a	The model is not scale/rotation/translation invariant. I think the main reason is the use of point position as input to the decoder, which means if the coordinates system is changed, the output of the decoder is also changed. Similarly, the surface normal or view directions should also be in the local coordinates system, otherwise the output will change if the scene was translated. I wonder how sensitive of the current model to random rotation/translation.	3D Reconstruction with Generalizable Neural Fields Using Scene Priors Yang Fu† Shalini De Mello Xueting Li Amey Kulkarni Jan Kautz Xiaolong Wang Sifei Liu 1University of California, San Diego 2NVIDIA Abstract High-fidelity 3D scene reconstruction has been substantially advanced by recent progress in neural fields. However, most existing methods train a separate network from scratch for each individual scene. This is not scalable, inefficient, and unable to yield good results given limited views. While learning-based multi-view stereo methods alleviate this issue to some extent, their multi-view setting makes it less flexible to scale up and to broad applications. Instead, we introduce training generalizable Neural Fields incorporating scene Priors (NFPs). The NFP network maps any single-view RGB-D image into signed distance and radiance values. A complete scene can be reconstructed by merging individual frames in the volumetric space WITHOUT a fusion module, which provides better flexibility. The scene priors can be trained on large-scale datasets, allowing for fast adaptation to the reconstruction of a new scene with fewer views. NFP not only demonstrates SOTA scene reconstruction performance and efficiency, but it also supports single-image novel-view synthesis, which is underexplored in neural fields. More qualitative results are available at: https://oasisyang.github.io/neural-prior. 1 Introduction Reconstructing a large indoor scene has been a long-standing problem in computer vision. A common approach is to use the Truncated Signed Distance Function (TSDF) (Zhou et al., 2018; Dai et al., 2017b) with a depth sensor on personal devices. However, the discretized representation with TSDF limits its ability to model fine-grained details, e.g., thin surfaces in the scene. Recently, a continuous representation using neural fields and differentiable volume rendering (Guo et al., 2022; Yu et al., 2022; Azinović et al., 2022; Wang et al., 2022b; Li et al., 2022) has achieved impressive and detailed 3D scene reconstruction. Although these results are encouraging, all of them require training a distinct network for every scene, leading to extended training durations with the demand of a substantial number of input views. To tackle these limitations, several works learn a generalizable neural network so that the representation can be shared among different scenes (Wang et al., 2021b; Zhang et al., 2022; Chen et al., 2021; Long et al., 2022; Xu et al., 2022). While these efforts scale up training on large-scale scene datasets, introduce generalizable intermediate scene representation, and significantly cut down inference time, they all rely on intricate fusion networks to handle multi-view input images at each iteration. This adds complexity to the training process and limits flexibility in data preprocessing. In this paper, we propose to perform 3D reconstruction by learning generalizable Neural Fields using scene Priors (NFPs). Such priors are largely built upon depth-map inputs (given posed RGB-D images). By leveraging the priors, our NFPs network allows for a simple and flexible design with single-view inputs during training, and it can efficiently adapt to each novel scene using fewer input views. Specifically, full scene reconstruction is achieved by directly merging the posed multi-view frames and their corresponding fields from NFPs, without the need for learnable fusion blocks. A direct way to generalize per-scene Nerf optimization is to encode each single-view input image into an intermediate representation in the volumetric space. Yet, co-learning the encoder and the † This work was done while Yang Fu was a research intern at NVIDIA. Figure 1: We propose Neural Fields scene Prior (NFP) to enable fast reconstruction of geometry and texture of indoor scenes. Our method first (a) learns a generalizable network as a scene prior that obtains a coarse scene reconstruction in a feed-forward manner. Next, we directly fuse the per-view results and (b) perform per-scene optimization in a more accurate and efficient way leading to high-quality surface reconstruction and realistic texture reconstruction. NeRF presents significant challenges. Given that a single-view image captures only a thin segment of a surface, it becomes considerably harder to discern the geometry compared to understanding the texture. Thus, to train NFPs, we introduce a two-stage paradigm: (i) We train a geometric reconstruction network to map depth images to local SDFs; (ii) We adopt this pre-trained network as a geometric prior to support the training of a separate color reconstruction network, as a texture prior, in which the radiance function can be easily learned with volumetric rendering (Wang et al., 2021a; Yariv et al., 2021), given the SDF prediction. Dense voxel grids are a popular choice in many NeRF-based rendering techniques (Yen-Chen et al., 2020; Chen et al., 2021; Liu et al., 2020; Huang et al., 2021; Takikawa et al., 2021; Sun et al., 2022b; Wang et al., 2022b). However, for the single-view input context, they fall short for two main reasons. First, the single-view image inherently captures just a thin and confined segment of surfaces, filling only a minuscule fraction of the entire voxel space. Second, dense voxel grids employ uniform sampling, neglecting surface priors like available depth information. Instead, we resort to a surface representation: we build a set of projected points in the 3D space as keypoint, from where a continuous surface can be decoded. The keypoint representation spans a compact 2D surface representation, allowing dense sampling close to the surface, which significantly enhances scalability. NFPs can easily facilitate further fine-tuning on large-scale indoor scenes. Given the pretrained geometry and texture network as the scene prior, the single-scene reconstruction can be performed by optimizing the aggregated surface representation and the decoders. With coarse reconstruction from the generalized network and highly compact surface representation, our approach achieves competitive scene reconstruction and novel view synthesis performance with substantially fewer views and faster convergence speed. In summary, our contributions include: • We propose NFPs, a generalizable scene prior that enables fast, large-scale scene reconstruction. • NFPs facilitate (a) single-view, across-scene input, (b) direct fusion of local frames, and (c) efficient per-scene fine-tuning. • We introduce a continuous surface representation, taking advantage of the depth input and avoiding redundancy in the uniform sampling of a volume. • With the limited number of views, we demonstrate competitive performance on both the scene reconstruction and novel view synthesis tasks, with substantially superior efficiency than existing approaches. 2 RELATED WORK Reconstructing and rendering large-scale indoor scenes is crucial for various applications. Depth sensors, on the other hand, are becoming increasingly common in commercial devices, such as Kinect (Zhang, 2012; Smisek et al., 2013), iPhone LiDAR (Nowacki & Woda, 2019), etc. Leveraging depth information in implicit neural representations is trending. We discuss both these topics in detail, in the following. Multi-view scene reconstruction. Reconstructing 3D scenes from images was dominated by multi-view stereo (MVS) (Schönberger et al., 2016; Schönberger & Frahm, 2016), which often follows the single-view depth estimation (e.g., via feature matching) and depth fusion process (Newcombe et al., 2011; Dai et al., 2017b; Merrell et al., 2007). Recent learning-based MVS methods (Cheng et al., 2020; Düzceker et al., 2020; Huang et al., 2018; Luo et al., 2019) substantially outperform the conventional pipelines. For instance, Yao et al. (2018); Luo et al. (2019) build the cost-volume based on 2D image features and use 3D CNNs for better depth estimation. Another line of works (Sun et al., 2021; Bi et al., 2017) fuse multi-view depth and reconstruct surface meshes using techniques such as TSDF fusion. Instead of fusing the depth, Wei et al. (2021), Wang et al. (2021b), Zhang et al. (2022), and Xu et al. (2022) directly aggregate multi-view inputs into a radiance field for coherent reconstruction. The multi-view setting enables learning generalizable implicit representation, however, their scalability is constrained as they always require multi-view RGB/RGB-D data during training. Our approach, for the first time, learns generalizable scene priors from single-view images with substantially improved scalability. Neural Implicit Scene Representation. A growing number of approaches (Yariv et al., 2020; Wang et al., 2021a; Yariv et al., 2021; Oechsle et al., 2021; Niemeyer et al., 2020; Sun et al., 2022a) represent a scene by implicit neural representations. Although these methods achieve impressive reconstruction of objects and scenes with small-scale and rich textures, they hardly faithfully reconstruct large-scale scenes due to the shape-radiance ambiguity suggested in (Zhang et al., 2020; Wei et al., 2021). To address this issue, Guo et al. (2022) and Yu et al. (2022) attempt to build the NeRF upon a given geometric prior, i.e., sparse depth maps and pretrained depth estimation networks. However, these methods take a long time to optimize on an individual scene. As mentioned previously, generalizable NeRF representations with multi-view feature aggregation are studied (Chen et al., 2021; Wang et al., 2021b; Zhang et al., 2022; Johari et al., 2022; Xu et al., 2022). However, they still focus on reconstructing the scene’s appearance, e.g., for novel view synthesis, but cannot guarantee high-quality surface reconstruction. Depth-supervised reconstruction and rendering. With the availability of advanced depth sensors, many approaches seek depth-enhanced supervision of NeRF (Azinović et al., 2022; Li et al., 2022; Zhu et al., 2022; Sucar et al., 2021; Yu et al., 2022; Williams et al., 2022; Xu et al., 2022; Deng et al., 2022) since depth information is more accessible. For instance, Azinović et al. (2022) enables detailed reconstruction of large indoor scenes by comparing the rendered and input RGB-D images. Unlike most methods that use depth as supervision, Xu et al. (2022), Williams et al. (2022) and Dong et al. (2023) build the neural field conditioned on the geometric prior. For example, Point-NeRF pretrains a monocular depth estimation network and generates a point cloud by lifting the depth prediction. Compared to ours, their geometric prior is less integrated into the main reconstruction stream since it is separately learned and detached. Furthermore, these methods only consider performing novel view synthesis (Xu et al., 2022; Deng et al., 2022), where the geometry is not optimized, or perform pure geometric (Yu et al., 2022; Li reconstruction. In contrast, our approach makes the scene prior and the per-scene optimization a unified model that enables more faithful and efficient reconstruction for both color and geometry. 3 METHOD Given a sequence of RGB-D images and their corresponding camera poses, our goal is to perform fast and high-quality scene reconstruction. To this end, we learn a generalizable neural scene prior, which encodes an RGB image and its depth map as continuous neural fields in 3D space and decodes them into signed distance and radiance values. As illustrated in Fig. 2, we first extract generalizable surface features from geometry and texture encoders (Sec. 3.1). Then, pixels with depth values are back-projected to the 3D space as keypoints, from which continuous fields can be built with the proposed surface representation (Sec. 3.2). Motivated by previous works (Wang et al., 2021a; Yariv et al., 2021), we utilize two separate MLPs to decode the geometry and texture representations, which are further rendered into RGB and depth values (Sec. 3.3). To obtain high-quality surface reconstruction, we further propose to optimize the neural representation on top of the learned geometric and texture prior for a specific scene (Sec. 3.4). 3.1 Constructing Surface Feature Given an RGB-D image $\{I, D\}$, we first project the depth map into 3D point clouds in the world coordinate system using its camera pose $\{R, t\}$ and intrinsic matrix $K$. We sub-sample $M$ points via Farthest Point Sampling (FPS), denoted as $\{p_m\}, m \in [0, M - 1]$, which are used as keypoints representing the discrete form of surfaces. We extract generalizable point-wise geometry and texture features, as described below, which are further splatted onto these keypoints. Both encoders are updated when training the NFP. Geometry encoder. For each surface point, we apply the K-nearest neighbor (KNN) algorithm to find $K - 1$ points and construct a local region with $K$ points. Thus, we obtain a collection of $M$ local regions, $\{p_m, \{p_k\}_{k \in \Psi_m}\}, \forall m \in [0, M - 1]$, where $\Psi_m$ is the neighbor index set of point $p_m$ and $\|\Psi_m\| = K - 1$. Then, we utilize a stack of PointConv (Wu et al., 2019) layers to extract the geometry feature from each local region $f_{geo}^m = \text{PointConv}(\{p_m, \{p_k\}_{k \in \Psi_m}\})$. Texture encoder. In addition, we extract RGB features for the keypoints via a 2D convolutional neural network. In particular, we feed an RGB image $I$ into an UNet (Ronneberger et al., 2015) with ResNet34 (He et al., 2016) as the backbone, which outputs a dense feature map. Then, we splat the pixel-wise features $f_{tex}^m$ onto the keypoints, according to the projection location of the surface point $p_m$ from the image plane. Thus, each surface point is represented by both a geometry feature and a texture feature, denoted by $f(p_m) = [f_{geo}(p_m), f_{tex}(p_m)]$. 3.2 Continuous Surface Implicit Representation Given the lifted keypoints and their projected geometry and texture features, in this section, we introduce how to construct continuous implicit fields conditioned on such discrete representations. We follow a spatial interpolation strategy: for any query point $x$ (e.g., in a typical volume rendering process, it can be a sampled point along any ray), we first find the $K$ nearest surface points $\{p_v\}_{v \in V}$, where $V$ is a set of indices of the neighboring surface points. Then, the query point’s feature can be obtained via aggregation of its neighboring surface points. In particular, we apply distance-based spatial interpolation as \[ f(x) = \frac{\sum_{v \in V} \omega_v f(p_v)}{\sum_{v \in V} \omega_v}; \quad \omega_v = \exp(-\|\|x - p_v\|\|), \] where $f(x)$ represents either the geometry $f_{geo}(x)$ or the texture $f_{tex}(x)$ feature, and $p_v$ is the position of the $v$-th neighbouring keypoint. With distance-based spatial interpolation, we establish continuous implicit fields for any point from the discrete keypoints. The continuous representation suffers from two drawbacks: First, when a point is far away from the surface, $f(x)$ is no longer a valid representation, but will still contribute to decoding and rendering. Second, the distance $ \omega_v $ is agnostic to the tangent direction and hence is likely to blur the boundaries. To mitigate the first problem, we incorporate an additional MLP layer that takes into account both the original surface feature $ f(p_v) $ and its relative distance to the query point $ x - p_v $, and outputs a distance-aware surface feature $ f(p_v^x) = \text{MLP}(f(p_v), x - p_v) $. Subsequently, this refined surface feature $ f(p_v^x) $ replaces the original surface feature in Eq. 1 to obtain the feature of query point $ x $. In addition, we ensure that the sampled points lie near the surface via importance sampling. We resolve the second issue via providing the predicted normal to the decoders as an input. We refer to Sec. 3.3 and 3.4 for details. ### 3.3 Generalizable Neural Scene Prior To reconstruct both geometry and texture, i.e., a textured mesh, a direct way is to decode the geometry and texture surface representation (Sec. 3.2) into signed distance and radiance values, render them into RGB and depth pixels (Guo et al., 2022; Yu et al., 2022), and then supervise them by the ground-truth RGB-D images. Unlike the multi-view setting that covers a significant portion of the volumetric space, the single-view input only encompasses a small fraction of it. From our experiments, we found that the joint training approach struggles to generate accurate geometry. Hence, we first learn a geometric network that maps any depth input to its corresponding SDF (Sec. 3.3.1). Once a coarse surface is established, learning the radiance function initialized by it becomes much easier – we pose it in the second stage where a generalizable texture network is introduced similarly (Sec. 3.3.2). #### 3.3.1 Generalizable Geometric Prior We represent scene geometry as a signed distance function, where in our case, it is conditioned on the geometric surface representation $ f_{\text{geo}}(x) $ to allow for generalization ability across different scenes. Specifically, along each back-projected ray with camera center $ o $ and ray direction $ r $, we sample $ N $ points as $ x_i = o + d_i r, \forall i \in [0, N - 1] $. For each sampled points $ x_i $, its geometry feature $ f_{\text{geo}}(x_i) $ can be computed by equation 1. Then, the geometry decoder $ \phi_G $, taking the point position and its geometry feature as inputs, maps each sampled point to a signed distance, which is defined as $ s(x_i) = \phi_G(f_{\text{geo}}(x_i), x_i) $. Note that we also apply positional encoding $ \gamma(\cdot) $ to the point position as suggested in Mildenhall et al. (2020). We omit it for brevity. Following the formulation of NeuS (Wang et al., 2021a), the estimated depth value $ \hat{d} $ is the expected values of sampled depth $ d_i $ along the ray: \[ \hat{d} = \sum_{i=0}^{N-1} T_i \alpha_i d_i; \quad T_i = \prod_{j=1}^{i-1} (1 - \alpha_j) \] \[ \alpha_i = \max \left( \frac{\sigma_s(s(x_i)) - \sigma_s(s(x_{i+1}))}{\sigma_s(s(x_i))}, 0 \right), \] where $ T_i $ represents the accumulated transmittance at point $ x_i $, $ \alpha_i $ is the opacity value and $ \sigma_s $ is a Sigmoid function modulated by a learnable parameter $ s $. Geometry objectives. To optimize the generalizable geometric representation, we apply a pixel-wise rendering loss on the depth map, \[ L_{\text{depth}} = \|\hat{d} - D(x, y)\|. \] Inspired by (Azinović et al., 2022; Li et al., 2022), we approximate ground-truth SDF based on the distance to observed depth values along the ray direction, $ b(x_i) = D(x, y) - d_i $. Thus, for points that fall in the near-surface region ($\|b(x_i)\| \leq \tau, \tau $ is a truncation threshold), we apply the following approximated SDF loss \[ L_{\text{near}} = \|s(x_i) - b(x_i)\| \] We also adopt a free-space loss (Ortiz et al., 2022) to penalize the negative and large positive predictions. \[ L_{\text{free}} = \max \left( 0, e^{-cs(x_i)} - 1, s(x_i) - b(x_i) \right), \] where $\epsilon$ is the penalty factor. Then, our approximated SDF loss is $$L_{\text{sdf}} = \begin{cases} L_{\text{near}} & \text{if } b(x_i) \leq \|\tau\| \\ L_{\text{free}} & \text{otherwise} \end{cases}$$ The approximated SDF values provide us with more explicit and direct supervision than the rendering depth loss (Eq. equation 3). Surface regularization. To avoid artifacts and invalid predictions, we further use the Eikonal regularization term (Yariv et al., 2021; Ortiz et al., 2022; Wang et al., 2021a), which aims to encourage valid SDF values via the following, $$L_{\text{eik}} = \|\|\nabla_{x_i}s(x_i) - 1\|\|_2^2,$$ where $\nabla_{x_i}s(x_i)$ is the gradient of predicted SDF w.r.t. the sampled point $x_i$. Therefore, we update the geometry encoder and decoder with the generalizable geometry loss as follows, $$L_{\text{geo}} = \lambda_{\text{depth}} L_{\text{depth}} + \lambda_{\text{sdf}} L_{\text{sdf}} + \lambda_{\text{eik}} L_{\text{eik}}$$ ### 3.3.2 Generalizable Texture Prior We build the 2nd stage – the generalizable texture network following the pretrained geometry network, as presented in Sec. 3.3.1, which offers the SDF’s prediction as an initialization. Specifically, we learn pixel-wise RGB features, as described in Sec. 3.1, and project them onto the corresponding keypoints. Following the spatial interpolation method in Sec. 3.2, we query the texture feature of any sampled point in 3D space. As aforementioned, the spatial interpolation in Eq. equation 1 is not aware of the surface tangent directions. For instance, a point at the intersection of two perpendicular planes will be interpolated with keypoints coming from both planes. Thus, representations at the boundary regions can be blurred. To deal with it, we further concatenate the surface normal $\nabla_{x_i}s(x_i)$ predicted in the first stage with the input to compensate for the missing information. With a separate texture decoder $\phi_{\text{tex}}$, the color of point $x_i$ is estimated, conditioned on the texture feature $f_{\text{tex}}(x_i)$ and the surface normal $\nabla_{x_i}s(x_i)$, $$c(x_i) = \phi_{\text{tex}}(f_{\text{tex}}(x_i), r, \nabla_{x_i}s(x_i)),$$ where $r$ is the ray direction. Here we omit the positional encoding of the point’s position and ray direction for conciseness. Therefore, the predicted pixel color can be expressed as $\hat{c} = \sum_{i=1}^{N} T_i \alpha_i c_i$, where $T_i$ and $\alpha_i$ are defined same as Eq. equation 2. We supervise the network by minimizing the L2 loss between the rendered pixel RGB values and their ground truth values $$L_{\text{rgb}} = \|\|\hat{c} - I(x,y)\|\|_2^2.$$ Meanwhile, we jointly learn the geometry network including the PointConv encoder and geometry decoder introduced in Sec. 3.2, via the same $L_{\text{geo}}$. Thus, the total loss function for generalizable texture representation learning is $$L_{\text{tex}} = \lambda_{\text{depth}} L_{\text{depth}} + \lambda_{\text{sdf}} L_{\text{sdf}} + \lambda_{\text{eik}} L_{\text{eik}} + \lambda_{\text{rgb}} L_{\text{rgb}}.$$ During volumetric rendering, to restrict the sampled points from being concentrated on the surface, we perform importance sampling based on: (i) the predicted surface as presented in Wang et al. (2021a), and (ii) the input depth map. More details are in the supplementary material. ### 3.4 Prior-guided Per-scene Optimization To facilitate large-scale, high-quality scene reconstruction, we can further finetune the pretrained generalizable geometric and texture prior to individual scenes, with multi-view frames. Specifically, we first directly fuse the geometry and texture feature of multi-view frames via the scene prior networks. No further learnable modules are required, in contrast, to (Chen et al., 2021; Zhang et al., 2022; Li et al., 2022). Then, we design a prior-guided pruning and sampling module, which lets optimization happen near surfaces. In particular, we initialize the grid in the volumetric space via learn NSP and estimate the SDF value of each grid by its corresponding feature, and remove the grids whose SDF values are larger than a threshold. We note that the generalizable scene prior can be combined with various optimization strategies (Xu et al., 2022; Yu et al., 2022; Wang et al., 2022b). More details can be found in the supplementary materials. During the finetuning, we update the scene-prior feature, and the weights of the MLP decoders to fit the captured images for a specific scene. Besides the objective functions described in Eq. equation 11, we also introduce the smoothness regularization term to minimize the difference between the gradients of nearby points $$L_{\text{smooth}} = \|\|\nabla_x s(x_i) - \nabla_{x_i + \sigma} s(x_i + \sigma)\|\|_2,$$ where $\sigma$ is a small perturbation value around point $x_i$. Thus, the total loss function for per-scene optimization is $$L_{\text{scene}} = \lambda_{\text{depth}} L_{\text{depth}} + \lambda_{\text{sdf}} L_{\text{sdf}} + \lambda_{\text{eik}} L_{\text{eik}} + \lambda_{\text{rgb}} L_{\text{rgb}} + \lambda_{\text{smooth}} L_{\text{smooth}}.$$ 4 EXPERIMENTS In this work, we introduce a generalizable network that can be applied to both surface reconstruction and novel view synthesis from RGB-D images in an offline manner. To our best knowledge, there is no prior work that aims for both two tasks. To make fair comparisons, we compare our work with the state-of-the-art (STOA) approaches of each task, respectively. 4.1 BASELINES, DATASETS AND METRICS Baselines. To evaluate surface reconstruction, we consider the following two groups of methods: First, we compared our method with RGB-based neural implicit surface reconstruction approaches: ManhattanSDF (Guo et al., 2022) and MonoSDF (Yu et al., 2022) which involve an additional network to extract the geometric prior during training. Second, we consider several RGB-D surface reconstruction approaches that share similar settings with ours: Neural-RGBD (Azinović et al., 2022) and Go-surf (Wang et al., 2022b). In addition, to have a fair comparison, we finetune ManhattanSDF and MonoSDF with ground-truth depth maps as two additional baselines and denoted as ManhattanSDF* and MonoSDF. We follow the setting in (Guo et al., 2022; Azinović et al., 2022) and evaluate the quality of the mesh reconstruction in different scenes. We note that all the above approaches perform per-scene optimization. To evaluate the performance in novel view synthesis, we compare our method with the latest NeRF-based methods in novel view synthesis, including NeRF (Mildenhall et al., 2020), NSVF (Liu et al., 2020), NerfingMVS (Wei et al., 2021), IBRNet (Wang et al., 2021b) and NeRFusion Zhang et al. (2022). As most of existing works are only optimized with RGB data, we further evaluate the Go-surf for novel view synthesis from RGB-D images as another baseline. We adopt the evaluation setting in NerfingMVS, where we evaluate our method on 8 scenes, and for each scene, we pick 40 images covering a local region and hold out 1/8 of these as the test set for novel view synthesis. Datasets. We mainly perform experiments on ScanNetV2 (Dai et al., 2017a) for both surface reconstruction and novel view synthesis tasks. Specifically, we first train the generalizable neural scene prior on the ScanNetV2 training set and then evaluate its performance in two testing splits proposed by Guo et al. (2022) and Wei et al. (2021) for surface reconstruction and novel view synthesis, respectively. The GT of ScanNetV2, produced by BundleFusion Dai et al. (2017b), is \| Method \| depth \| opt. (min) \| Acc↓ \| Comp↓ \| Prec↑ \| Recall↑ \| F-score↑ \| \|-------------------------\|-------\|------------\|------\|-------\|-------\|---------\|----------\| \| ManhattanSDF (Guo et al., 2022) \| SfM \| 640 \| 0.072 \| 0.068 \| 0.621 \| 0.586 \| 0.602 \| \| \| network \| 720 \| 0.039 \| 0.044 \| 0.775 \| 0.722 \| 0.747 \| \| MonoSDF (Yu et al., 2022) \| network \| 480 \| 0.051 \| 0.048 \| 0.720 \| 0.674 \| 0.696 \| \| NeuRIS (Wang et al., 2022a) \| network \| 30 \| 0.042 \| 0.056 \| 0.751 \| 0.678 \| 0.710 \| \| FastMono (Dong et al., 2023) \| network \| 30 \| 0.038 \| 0.044 \| 0.786 \| 0.727 \| 0.755 \| \| HelixSurf (Liang et al., 2023) \| network \| 30 \| \| \| \| \| \| \| ManhattanSDF (Guo et al., 2022) \| GT. \| 640 \| 0.027 \| 0.032 \| 0.915 \| 0.883 \| 0.907 \| \| MonoSDF* (Yu et al., 2022) \| GT. \| 720 \| 0.033 \| 0.026 \| 0.942 \| 0.912 \| 0.926 \| \| Neural-RGBD (Azinović et al., 2022) \| GT. \| 240 \| 0.055 \| 0.022 \| 0.932 \| 0.918 \| 0.925 \| \| Go-surf (Wang et al., 2022b) \| GT. \| 35 \| 0.052 \| 0.018 \| 0.946 \| 0.956 \| 0.950 \| \| Ours-prior (w/o per-scene opt.) \| – \| – \| 0.084 \| 0.057 \| 0.695 \| 0.764 \| 0.737 \| \| Ours (w per-scene opt.) \| GT. \| 15 \| 0.049 \| 0.017 \| 0.947 \| 0.962 \| 0.954 \| Table 1: Quantitative comparisons for mesh reconstruction on ScanNet. We compare with a number of baselines. “∗” is our re-implementation with dense ground-truth depth map. “opt.” stands for the optimization time for per-scene fine-tuning. \| Method \| #frame \| Acc ↓ \| Comp ↓ \| C-ℓ₁ ↓ \| NC ↑ \| F-score ↑ \| \|-------------------------\|--------\|-------\|--------\|--------\|------\|-----------\| \| BundleFusion (Dai et al., 2017b) \| 1,000 \| 0.0191 \| 0.0581 \| 0.0386 \| 0.9027 \| 0.8439 \| \| COLMAP (Schönberger et al., 2016) \| 1,000 \| 0.0271 \| 0.0322 \| 0.0296 \| 0.9134 \| 0.8744 \| \| ConvOccNets (Peng et al., 2020) \| 1,000 \| 0.0498 \| 0.0524 \| 0.0511 \| 0.8607 \| 0.6822 \| \| SIREN (Sitzmann et al., 2020) \| 1,000 \| 0.0229 \| 0.0412 \| 0.0320 \| 0.9049 \| 0.8515 \| \| Neural RGBD (Azinović et al., 2022) \| 1,000 \| 0.0151 \| 0.0197 \| 0.0174 \| 0.9316 \| 0.9635 \| \| Go-surf (Wang et al., 2022b) \| 1,000 \| 0.0158 \| 0.0195 \| 0.0177 \| 0.9317 \| 0.9591 \| \| Ours \| 1,000 \| 0.0172 \| 0.0192 \| 0.0177 \| 0.9311 \| 0.9529 \| \| Go-surf (Wang et al., 2022b) \| 30 \| 0.0246 \| 0.0442 \| 0.0336 \| 0.9117 \| 0.9042 \| \| Ours \| 30 \| 0.0177 \| 0.0292 \| 0.0234 \| 0.9207 \| 0.9311 \| Table 2: Quantitative evaluation of the reconstruction quality on 10 synthetic scenes. Our method show competitive results when being reconstructed using only 30 frames used per room, in the lower part of the table. known to be noisy, making accurate evaluations against it challenging. To further validate our method, we also conduct experiments on 10 synthetic scenes proposed by Azinović et al. (2022). Evaluation Metrics. For 3D reconstruction, we evaluate our method in terms of mesh reconstruction quality used in Guo et al. (2022). Meanwhile, we measure the PSNR, SSIM, and LPIPS for novel view synthesis quality. 4.2 Comparisons with the state-of-the-art methods Surface reconstruction. Table 1 provides a quantitative comparison of our methods against STOA approaches for surface reconstruction (Guo et al., 2022; Yu et al., 2022; Wang et al., 2022a; Liang et al., 2023). Within our methods, the feed-forward NFPs are denoted as Ours-prior, while the per-scene optimized networks are labeled as Ours. We list the RGB- and RGB-D-based approaches as in the top and the middle rows, whereas ours are placed in the bottom section. While we include ManhattanSDF (Guo et al., 2022) and MonoSDF (Yu et al., 2022) that are supervised by predicted or sparse depth information as in the top row, to ensure fair comparisons, we re-implement them by replacing the original supervision with ground-truth depth, as in the middle row (denoted by ‘∗’). Generally, using ground-truth depths can always enhance the reconstruction performance. Comparison with NFPs on ScanNet. In contrast to all the other approaches that all require time-consuming per-scene optimization, the NFPs can extract the geometry structure through a single forward pass. The results in Table 1 demonstrate that, even without per-scene optimization, the NFPs network not only achieves performance on par with RGB-based approaches but also operates hundreds of times faster. Note in contrast to all the other approaches in Table 1 that use around 400 frames to optimize the scene-specific neural fields, Ours-prior only takes around 40 frames per scene as inputs to achieve comparable mesh reconstruction results without per-scene optimization. Comparison with optimized NFPs on ScanNet. We further perform per-scene optimization on top of the NFPs network. Compared with methods using additional supervision or ground truth depth maps, our method demonstrates more accurate results on the majority of the metrics. More importantly, our method is either much faster, compared with the SOTA approaches. Some qualitative results are shown in Fig. 3 and more results can be found in the supplementary materials. Comparison on synthetic scenes. Table 2 compares our approach with the most recent works on neural surface reconstruction from RGB-D images. The results demonstrate that our method achieves comparable performance with most existing works, even when optimizing with a limited number of frames, such as 1,000 vs 30. Results on novel view synthesis. To validate the learned radiance representation, we further conduct experiments on novel view synthesis. The quantitative results and qualitative results are shown in Table 3 and Fig. 4. Table 3 shows that the proposed method achieves comparable if not better results compared to SOTA novel view synthesis methods (Wang et al., 2021b; Zhang et al., 2022; Liu et al., 2020). We note that our method outperforms Go-surf in this instance, even when both methods achieve comparable geometric reconstruction performance. This suggests that our learned prior representation offers distinct advantages for novel view synthesis. In addition, from Fig. 4, both NerfingMVS (Wei et al., 2021) and Go-surf (Wang et al., 2022b) fail on scenes with complex geometry and large camera motion. The generalized representation enables the volumetric rendering to focus on more informative regions during optimization and improves its performance for rendering RGB images of novel views. ### 4.3 Ablation Studies We further perform the ablation studies to evaluate the effectiveness and the efficiency of the neural prior network. Effectiveness of generalized representation. Table 4 shows the results with and without the generalized representation. For the model without generalized representation, we randomly initialize the parameters of feature grids and decoders while keeping the other components unchanged. We observe that the model integrated with geometry prior and/or color prior can consistently improve the performance on 3D reconstruction and novel view synthesis. Fast optimization. Our approach can achieve high-quality reconstruction at approximately 1.5K iterations within 15 minutes. As illustrated in Fig. 5, our method achieves a high F-score at the very early training stage, while Manhattan SDF* (Guo et al., 2022) and MonoSDF* (Yu et al., 2022) take much more iterations to reach a similar performance. ### 5 Conclusion In this work, we present a generalizable scene prior that enables fast, large-scale scene reconstruction of geometry and texture. Our model follows a single-view RGB-D input setting and allows non-learnable direct fusion of images. We design a two-stage paradigm to learn the generalizable geometric and texture networks. Large-scale, high-fidelity scene reconstruction can be obtained with efficient fine-tuning on the pretrained scene priors, even with limited views. We demonstrate that our approach can achieve state-of-the-art quality of indoor scene reconstruction with fine geometric details and realistic texture. \| Method \| PSNR ↑ \| SSIM ↑ \| LPIPS ↓ \| \|-------------------------\|--------\|--------\|---------\| \| NeRF (Mildenhall et al., 2020) \| 24.04 \| 0.860 \| 0.334 \| \| NSVF (Liu et al., 2020) \| 26.01 \| 0.881 \| – \| \| NerFingMVS (Wei et al., 2021) \| 26.37 \| 0.903 \| 0.245 \| \| IBRNet (Wang et al., 2021b) \| 25.14 \| 0.871 \| 0.266 \| \| NeRFusion (Zhang et al., 2022) \| 26.49 \| 0.915 \| 0.209 \| \| Go-surf (Wang et al., 2022b) \| 25.47 \| 0.894 \| 0.420 \| \| Ours \| 26.88 \| 0.909 \| 0.244 \| Table 3: Quantitative comparisons for novel view synthesis on ScanNet. The best two results of different metrics are highlighted. \| Geo. prior \| Acc ↓ \| Comp ↓ \| F-score ↑ \| \|------------\|-------\|--------\|-----------\| \| \| 0.079 \| 0.031 \| 0.851 \| \| ✓ \| 0.046 \| 0.030 \| 0.862 \| \| Color prior \| \| \| \| \| PSNR ↑ \| 25.87 \| 0.899 \| 0.415 \| \| ✓ \| 26.88 \| 0.909 \| 0.246 \| Table 4: Ablation studies on geometric and texture prior. We report both mesh reconstruction metrics and novel view synthesis metrics. Figure 5: Ablation studies on the number of training iterations for per-scene optimization. Acknowledgement This work was supported, in part, by NSF CAREER Award IIS-2240014, the Qualcomm Innovation Fellowship, Amazon Research Award. REFERENCES Dejan Azinović, Ricardo Martin-Brualla, Dan B Goldman, Matthias Nießner, and Justus Thies. Neural rgb-d surface reconstruction. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 6290–6301, 2022. Sai Bi, Nima Khademi Kalantari, and Ravi Ramamoorthi. 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WXXuORQwbQ	In Figure 6 and Table 4. It seems the model gets worse when using 50 or 10 masks which is strange.Why would using 3 masks be better than 10 or 50 masks? Why would it be even better than using the full dense tensor?	Sparse Mask Representation for Human-Scene Interaction Anonymous authors Paper under double-blind review Abstract Human-scene interaction is an active research topic with several applications in robotics, virtual experiences, gaming, surveillance, and healthcare. Despite efforts to improve the network architectures to achieve better results or optimize models for faster inference, a crucial aspect of input dimensionality has been somewhat overlooked. This paper introduces Sparse Mask Representation, a simple yet effective approach to enhance the inference speed of human-scene interaction models and improve the model’s effectiveness by exploring the sparsity of high-dimensional inputs. Specifically, our method utilizes sparse masks to convert high-dimensional inputs into sparse tensors in a compressed COO format. Our approach not only effectively streamlines computational speed but also eliminates non-useful input information, thereby enhancing overall model performance. We conducted rigorous experiments across three datasets, with a specific emphasis on tasks related to contact prediction and scene synthesis. The results underscore the substantial enhancements realized by our proposed method in terms of accuracy and inference time, surpassing existing state-of-the-art approaches. 1 Introduction Human-scene interaction explores how humans perceive, navigate, and engage with the environment around them (Hassan et al., 2021). Recently, there has been significant attention on learning the dynamics between humans and the environment (Li et al., 2019; Zhang et al., 2022; Yi et al., 2023). To enhance the modeling and understanding of human pose within diverse environments, researchers have investigated several topics such as human-scene interactions (Hassan et al., 2021; Luo et al., 2023), human-scene synthesis (Zhao et al., 2022b; Shen et al., 2023; Blinn et al., 2021), or human pose contact prediction (Zheng et al., 2022; Huang et al., 2022). Gaining a comprehensive understanding of human posture and interactions with the environment is crucial for various downstream applications (Ye et al., 2022) such as human-robot interaction (Romero et al., 2017; Yi et al., 2022a), realistic virtual experiences (Arsalan Soltani et al., 2017; Zhao et al., 2022b), game animations (Habermann et al., 2021), intuitive interfaces (Zou et al., 2018), advanced surveillance systems (Benfold and Reid, 2009), and healthcare applications (Meng et al., 2023). In the domain of human-scene interaction, numerous approaches concentrate on generating high-quality scenes based on human contacts and interactions (Hassan et al., 2021; Wang et al., 2022b; Jiang et al., 2022a; Zheng et al., 2022; Yi et al., 2022a; Ye et al., 2022; Wang et al., 2022a; Yi et al., 2023). While the development of complex networks capable of handling the intricacies of scene generation tasks is essential, it also poses challenges in terms of inference speed (Lee et al., 2023) and effectively process the data (Bautista et al., 2022). Yet, many works have acknowledged this problem and thus focused on lightweight architectures, model pruning, or quantization to improve model accuracy and enhance inference speed (Riegler et al., 2017; Tatarchenko et al., 2017; Zhang et al., 2022; Schwarz and Behnke, 2020). However, despite recent developments, current methods still struggle to process complex input structures such as 3D human poses, complex temporal dynamics, or realistic human-scenes interaction. In this paper, unlike previous methods that primarily focus on designing lightweight models, quantization, model pruning, or diffusion models to enhance human-scene interaction (Hassan et al., 2019; Liu et al., 2022; Jiang et al., 2022b), we propose a solution that focuses on effectively representing the input data. We are motivated by the fact that the input data for human-scene interaction are... complex but sparse data structures while having an effective way to represent the input has shown significant improvement in terms of both accuracy and inference speed in other tasks such as affordance learning (Morais et al., 2021) or NeRF-based scene generation (Zhao et al., 2022a; Niemeyer et al., 2022). In particular, we propose Sparse Mask Representation (SMR), a simple, yet effective method for human-scene interaction. Unlike other solutions, our simple method utilizes a set of sparse masks to effectively select important information from the input (Figure 1). The sparse marks are then integrated into the human-scene deep backbone by replacing traditional tensor operations with sparse operations. By utilizing sparse operations, our method significantly reduces the computational cost. Intensive experiments show that our method outperforms recent works in contact prediction and scene synthesis tasks while achieving much faster inference speed. Our key contributions are as follows: - We introduce sparse mask representation, a simple yet effective method for representing high-dimensional human-scene interaction data. - We apply our method to different downstream human-scene interaction tasks and demonstrate its effectiveness in terms of accuracy and inference speed. 2 RELATED WORK Human-scene Interaction. The human body plays a significant role in facilitating physical interactions (Romero et al., 2017) and in comprehending the contact between humans and their environmental scenes (Li et al., 2019). With advancements in human modeling techniques such as SMPL (Loper et al., 2015), SMPL-X (Pavlakos et al., 2019), MANO (Romero et al., 2017), and FLAME (Li et al., 2017), researchers explore new methods to integrate human skeletons into scenes. For instance, Wang et al. (2017) propose learning affordances from videos to position skeletons in static images. Li et al. (2019) introduce a generative model of 3D poses for predicting plausible human poses within scenes. Several works also focus on collecting or generating data that involve human-scene interactions. Puig et al. (2018) provide a simulated 3D environment where humanoid agents can interact with 3D objects. BEHAVE (Bhatnagar et al., 2022) provides a dataset of real full-body human parameters using the SMPL model while interacting and manipulating objects in 3D with contact points. Based on these datasets, various approaches have been introduced to learn human-scene interaction through scene population (Hassan et al., 2021; Wang et al., 2022b; Jiang et al., 2022a), understand affordances in 3D indoor environments (Li et al., 2019; Kulal et al., 2023; Luo et al., 2023), capture hand and body motions together (Romero et al., 2017; Pavlakos et al., 2019), generate 3D people in scenes (Nie et al., 2022; Wang et al., 2022c), synthesize scene from human motion with diffusion models (Zheng et al., 2022; Yi et al., 2022a; Ye et al., 2022; Wang et al., 2022a; Yi et al., 2023), or track human-object interactions (Blinn et al., 2021; Yi et al., 2022b; Xie et al., 2023). These works contribute to advancing the understanding of human-object interactions, 3D scene generation, and human pose estimation in diverse real-world scenarios (Zhang et al., 2020a; Wang et al., 2022a). Lightweight Architecture. Lightweight methods focus on efficient neural network designs for faster inference and low power consumption in resource-limited scenarios. Network pruning, a prominent approach to achieve this, has been exemplified by (Han et al., 2015; Chakraborty et al., 2018), to eliminate redundancy in large deep networks. Kahatapitiya and Rodrigo (2021) explore the separation of redundancy and represent it using a smaller parameter set. Quantization techniques (Liu et al., 2018; 2019) leverage lower-bit weight value representations to minimize memory use. Knowledge distillation (Hinton et al., 2015) has emerged as a technique to train lightweight student networks that mimic the behavior of more complex teacher networks. Finally, neural architecture search methods (Guo et al., 2020; Yang et al., 2020; Zoph and Le, 2017; Pham et al., 2018) automatically discover architectures that balance compactness and performance. Lightweight architectures are considered in the context of human-scene interaction to address the complexity of trajectory prediction (Liu et al., 2022; Katariya et al., 2022) or dynamic scene generation (Su et al., 2022; Arad Hudson and Zitnick, 2021). While lightweight networks are appealing, limitations such as reducing modeling capacity and compromising accuracy performance in complex tasks are noteworthy (Cheng et al., 2018). Additionally, they are more prone to overfitting and may struggle to maintain fine-grained information and generalizability (Gupta et al., 2015). Sparse Coding. In addition to lightweight architecture, another direction significantly enhances inference speed is sparse coding. Unlike focusing on architecture design, these methods concentrate on input utilization during learning and inference (Liu et al., 2015). Sparse coding approaches do not modify architectures, instead, they target input format (Choy et al., 2019) and kernel design (Liu et al., 2015; Gray et al., 2017). Specifically, Graham et al. (2018) address inefficiencies in dense convolutional networks by introducing specialized sparse convolutional operations for spatially sparse data and developing submanifold sparse convolution. Chen (2018) directly calculate convolution with a sparse kernel, customize dataflow and memory access instead of converting to matrix multiplication. Graham et al. (2018) develop an implementation of sparse convolution for high-dimensional, sparse input data. Recently, Sylos Labini et al. (2022) present a 1-dimensional blocking algorithm for accelerating sparse matrix multiplication that constructs dense blocks from sparse matrices, providing theoretical guarantees on density. Although sparse coding works have demonstrated effectiveness in various tasks, they have not been widely applied in human-scene interactions yet, primarily due to limitations in dealing with temporal, contextual dependencies, or the dynamic evolution of interactions over time (Ren et al., 2018). By effectively handling redundant information from inputs, our strategy overcomes these limitations and opens up new possibilities for enhancing real-time interaction prediction and optimizing the efficiency of associated downstream tasks. ### 3 Sparse Mask Representation for Human-Scene Interaction #### 3.1 Motivation Sparse kernels have gained significant popularity in the development of efficient models (Choy et al., 2019; Gray et al., 2017; Graham et al., 2018). However, when compressing models through parameter-space sparsity, the networks still operate on dense tensors, and all intermediate activations within these networks are also dense tensors. This leads to redundancy in the data space during the establishment of computational matrices. Consequently, the full potential of sparse kernels is not maximized. To address this issue, we have a shift in focus towards spatially sparse tensor data, with particular emphasis on sparse high-dimensional 3D inputs and convolution on the surface of 3D objects/humans. In this way, we allow a more efficient utilization of computational resources. By leveraging sparsity in the input, computations between the kernel and input only occur on existing data points, significantly reducing the computational workload based on the input’s sparsity. To achieve sparsity in the input, a binary sparse mask is employed to identify which data points would be utilized for the learning process, ensuring the effective utilization of computational resources and enhancing the overall efficiency of the network. In practice, we observe that increasing the sparsity of the sparse mask results in the loss of input data information and affects the model’s performance. Therefore, we utilize multiple sparse masks to generate multiple sparse inputs. As the sparse masks remain unchanged during the learning process, our objective is to assess the contribution of each mask to the task. This assessment allows us to maintain the sparsity of the mask while discarding the masks that do not significantly contribute to the final results, thereby reducing the inference speed and improving model accuracy. Our sparse masks then can be integrated into traditional networks to perform human-scene interaction tasks. Figure 2 shows an overview of our method. Figure 2: An overview of our method. The red cells denote the non-zero kernel weights and mask values, blue cells denote the coordinate values, green cells denote the non-zero contact values and white cells denote zero values. 3.2 Sparse Mask Representation Human-Scene Representation. We follow Hassan et al. (2021) to represent the human-scene interaction. In particular, the human-scene input tensor $I$ is defined as $I = (V, F)$, where $V \in \mathbb{R}^{N_v \times 3}$ is body vertices and $F \in \mathbb{R}^{N_v \times N_c}$ is the contact label of the vertices. $N_v$ is the number of vertices, and $N_c$ is the number of labels. Sparse Mask. Our goal is to convert the human-scene input tensor $I$ into a sparse tensor $I' \in \mathbb{R}^{N_v \times N_S}$ for a more efficient representation ($N_S = N_c + 3$). We define a sparse mask $M \in \mathbb{R}^{N_v \times N_S}$ and calculate $I' = M \circ I$, where $\circ$ denotes element-wise multiplication. Each element in the sparse mask $M$ is sampled from a binomial distribution. The sparsity of $M$ is controlled via a sparsity ratio parameter which indicates the non-zero value ratio of the mask. Intuitively, the sparse mask $M$ is a matrix with only 0 or 1 values to mask out the unnecessary information from the input. In practice, applying only a single high-sparsity mask $M$ to the input causes significant information loss hence heavily affecting the effectiveness of the model. To overcome this limitation, we apply $K$ multiple sparse masks $\{M_1, M_2, ..., M_K\}$ to the input with the expectation that each sparse mask $M_k$ would learn different important information from the input. We note that each sparse mask $M_k$ is applied independently to the input to obtain the sparse tensor $I'_k$, and $K$ is the hyper-parameter that indicates how many sparse masks we use during training. Sparse Mask Representation. After applying the sparse mask $M_k$ to the input tensor $I$, we obtain a sparse tensor $I'_k = M_k \circ I$ which has a high proportion of zero values. Consequently, the conventional dense representation is inefficient for representing the sparse tensor $I'_k$ during the learning process. Additionally, effectively storing only non-zero values in the sparse tensor facilitates computation (Tew, 2016). To efficiently represent the sparse mask, we find out that the COO format introduced by Chou et al. (2018) is best fitted since this format is based on the coordinates of non-zero values, and is efficient for neighborhood queries. This representation includes a coordinate matrix $C'_k \in \mathbb{R}^{N'_k \times 2}$ and an associated feature matrix $S'_k \in \mathbb{R}^{N'_k \times N_S}$ where $N'_k$ denotes the number of non-zero values in $I'_k$. The COO format not only saves memory by removing zero-values from the sparse tensor but also streamlines the computation process for $I'_k$. The sparse tensor $I'_k$ is represented as $I'_k = (C'_k \| S'_k)$, where $C'_k$ and $S'_k$ are defined as: $$ C'_k = \begin{bmatrix} b_1 & x_1 \\ \vdots & \vdots \\ b_{N'_k} & x_{N'_k} \end{bmatrix}, \quad S'_k = \begin{bmatrix} s^T_1 \\ \vdots \\ s^T_{N'_k} \end{bmatrix} $$ (1) where $(b_i, x_i)$ is the frame index and coordinate of $i$-th feature $s_i \in \mathbb{R}^{N_S}$. Sparse Mask Selection. Although using a list of sparse masks preserves the model’s performance compared to using a single mask, it leads to the fact that some sparse masks capture duplicate information or unnecessary features in the input which may have a negative effect on the results or slow down the inference. To resolve this problem, we define the learnable mask score $\alpha \in \mathbb{R}^K$ to indicate the importance of each sparse mask. This mask score is calculated based on the contribution of each mask to the final results and the similarity between corresponding masks as follows: $$\alpha_{(t+1,k)} = \alpha_{(t,k)} + \frac{1}{K-1} \sum_{i \neq k, 1 \leq i \leq K} \left( 1 - \frac{\\|O_{(t,i)}^\top O_{(t,k)}\\|^2_F}{\\|O_{(t,k)}^\top O_{(t,k)}\\|_F \\|O_{(t,i)}^\top O_{(t,i)}\\|_F} \right)$$ where $\\|\cdot\\|_F$ is the Frobenius norm; $t$ corresponds to iteration during learning; $O_k$ is the output tensor corresponding to mask $M_k$. Our goal is to compare the differences in distribution between features outputted from different sparse masks to identify which masks mostly produce the same outputs and then discard the redundant ones during the inference process. We note that during training, we utilize $K$ sparse masks and calculate the associated mask scores, while during testing, we select $\kappa$ masks ($\kappa \ll K$) based on the mask score $\alpha$ to use only the useful masks. Using Sparse Mask Representation in Deep Layers. To employ our sparse mask representation in different network layers during training, we simply replace the conventional matrix operations with sparse matrix operations, utilizing input from our sparse mask. This strategy can be applied across different layers, including convolution, batch normalization, pooling, and more, using the COO format (Chou et al., 2018; Choy, 2020), all without necessitating changes to the network architecture. More details on sparse mask implementation can be found in our Appendix B. 3.3 Sparse Network for Human-Scene Interaction Contact Prediction. We train the conditional Variational Autoencoder (cVAE) model, as implemented in POSA (Hassan et al., 2021) for contact prediction. As the input is in the form of a sparse tensor, we replace each layer in Hassan et al. (2021) with a corresponding sparse layer to produce the sparse tensor. This sparse tensor is then passed as the input to the subsequent layer of the network. Note that we only change the original tensor to our sparse tensor, while keeping the whole network unchanged. The Appendix D shows a detailed comparison between our model and POSA. Scene Synthesis. After predicting the contact labels of body vertices in each frame by integrating our sparse tensor into the cVAE model Hassan et al. (2021), we perform the scene synthesis task as a downstream task. We follow the approach outlined by Ye et al. (2022) to conduct the experiment. In particular, we generate objects that make contact with the human body based on the predicted contact points mentioned earlier. The objects that are successfully generated should not penetrate the human body and should align well with the human’s intention. 4 Experiments 4.1 Contact Prediction Datasets. We use the PROXD (Hassan et al., 2019), GIMO (Zheng et al., 2022), and BEHAVE (Bhatnagar et al., 2022) datasets for contact prediction. In all datasets, the human body is modeled using SMPL-X format (Pavlakos et al., 2019). In the PROXD dataset, the contact labels are obtained from PROX-E dataset (Zhang et al., 2020b). Evaluation Metrics. As in (Ye et al., 2022), the Reconstruction Accuracy and Consistency Score are used for comparing the effectiveness of different methods. We also compare the inference time (second per sample) of all methods on the same NVIDIA Tesla V100 GPU. Baselines. We compare our SMR method with recent works, including POSA (Hassan et al., 2021), ContactFormer (Ye et al., 2022), multi-layer perceptron predictor or bidirectional LSTM (Greff et al., 2016), MIME (Yi et al., 2023), PIAL-Net (Luo et al., 2023), and HOT (Chen et al., 2023). We train our SMR using $K = 10$ masks and keep only $\kappa = 3$ masks with the highest values of mask score $\alpha$ during inference. More implementation details are in Appendix C. \| Methods \| PROXD \| GIMO \| BEHAVE \| Inference Speed (s/sample) \| \|------------------\|-------\|------\|--------\|----------------------------\| \| \| Recons. Acc. (%) \| Consistency Score \| Recons. Acc. (%) \| Consistency Score \| Recons. Acc. (%) \| Consistency Score \| \| MLP Predictor \| 90.84 (+2.85) \| 0.892 (+0.089) \| 80.7 (+1.4) \| 0.801 (+0.142) \| 82.5 (+11.3) \| 0.724 (+0.149) \| 0.11 (<× 12.2) \| \| LSTM Predictor \| 90.91 (+2.78) \| 0.921 (+0.06) \| 83.2 (+8.9) \| 0.814 (+0.129) \| 80.8 (+13.0) \| 0.766 (+0.107) \| 0.17 (<× 18.9) \| \| POSA \| 91.12 (+2.57) \| 0.882 (+0.099) \| 89.9 (+2.2) \| 0.909 (+0.034) \| 89.7 (+4.1) \| 0.854 (+0.019) \| 0.28 (<× 31.1) \| \| ContactFormer \| 91.27 (+2.42) \| 0.952 (+0.029) \| 90.7 (+1.4) \| 0.912 (+0.031) \| 91.1 (+2.7) \| 0.845 (+0.028) \| 0.20 (<× 22.2) \| \| MIME \| 90.97 (+2.72) \| 0.902 (+0.079) \| 89.9 (+2.2) \| 0.911 (+0.032) \| 90.2 (+3.6) \| 0.854 (+0.019) \| 0.54 (<× 60.0) \| \| PIAL-Net \| 92.04 (+1.65) \| 0.953 (+0.028) \| 91.1 (+1.0) \| 0.934 (+0.009) \| 89.9 (+3.9) \| 0.864 (+0.009) \| 2.97 (<× 330.0) \| \| HOT \| 90.9 (+2.79) \| 0.966 (+0.015) \| 90.3 (+1.8) \| 0.900 (+0.043) \| 91.7 (+2.1) \| 0.821 (+0.052) \| 1.12 (<× 124.4) \| \| SMR (Ours) \| 93.69 \| 0.981 \| 92.1 \| 0.943 \| 93.8 \| 0.873 \| 0.009 \| Table 1: Contact prediction results. Figure 3: Contact prediction visualization between different methods. We can see that LSTM (b) and POSA (c) show the mismatch between the Floor and the Couch; ContactFormer (d) and HOT (f) cannot differentiate between Couch and Bed, while our method shows reasonable predictions. Results. Table 1 shows the comparison between our method and other baselines. This table indicates that our model surpasses all other baselines by a large margin with a reconstruction accuracy of 93.69%, and a consistency score of 0.981 on PROXD dataset. Furthermore, our inference speed is 0.009 second/sample, which is approximately 12 times faster than the runner-up. Visualization. Figure 3 shows the qualitative comparison of contact prediction results with different methods. It is notable that our method stands out by achieving accurate contact predictions in both the contact labels and contact location compared to other methods. ### 4.2 Scene Synthesis Datasets. In the human-scene synthesis task, we use the PROXD (Hassan et al., 2019) and GIMO (Zheng et al., 2022) datasets for conducting experiments as in recent works. Note that BEHAVE (Bhatnagar et al., 2022) dataset cannot be used in the scene synthesis task since this dataset only has contacts with independent objects, not ones synchronized in a scene. Baselines. We compare our method with recent baselines on the scene synthesis domain, including ContactICP (Besl and McKay, 1992), PosePrior (Moreno-Noguer et al., 2008), SUMMON (Ye et al., 2022), MIME (Yi et al., 2023), and SceneDiffuser (Huang et al., 2023). Our SMR is trained using $K = 10$ masks, then 3 masks with the highest values of mask score $\alpha$ are kept during inference. Evaluation Protocol. We use the non-collision score proposed in Zhang et al. (2020b) as a metric for the scene synthesis task. Furthermore, we perform a user study to compare different methods. Results. Table 2 and Figure 4 provide a comprehensive comparison between scene synthesis results. ContactICP, although exhibiting relatively lower non-collision values, represents an initial approach in this task. Pose Priors (Moreno-Noguer et al., 2008) demonstrates improvements by incorporating pose information, resulting in enhanced reconstruction accuracy. Recent works such as SUMMON (Ye et al., 2022), MIME (Yi et al., 2023), and SceneDiffuser (Huang et al., 2023) show significant advancements, outperforming PosePriors, and achieving notably higher scores on both datasets. However, our method surpasses all other techniques with a recognizable margin, demonstrating a clear improvement in the scene synthesis task. Table 2: Scene synthesis results. The non-collision score is reported on the PROXD dataset and GIMO dataset. \| Methods \| PROXD \| GIMO \| \|---------------\|-----------\|-----------\| \| ContactICP \| 0.654 (+0.282) \| 0.820 (+0.131) \| \| PosePriors \| 0.703 (+0.233) \| 0.798 (+0.171) \| \| SUMMON \| 0.851 (+0.085) \| 0.951 (+0.018) \| \| MIME \| 0.897 (+0.039) \| 0.938 (+0.031) \| \| SceneDiffuser \| 0.914 (+0.022) \| 0.942 (+0.027) \| \| SMR (Ours) \| 0.936 \| 0.969 \| Table 3: Comparison between different sparse representation methods on PROXD dataset in the contact prediction task. \| Methods \| Reconstruction Accuracy (%) \| Consistency Score \| Inference Speed (s/sample) \| \|---------------\|-----------------------------\|-------------------\|-----------------------------\| \| POSA \| 91.12 (+2.57) \| 0.882 (+0.099) \| 0.28 (+× 31.1) \| \| ME \| 83.61 (+10.08) \| 0.797 (+0.184) \| 0.008 (+× 1.13) \| \| EsCoin \| 69.78 (+23.91) \| 0.721 (+0.260) \| 0.17 (+× 1.89) \| \| pSConv \| 90.24 (+3.45) \| 0.825 (+0.156) \| 0.084 (+× 9.33) \| \| 1-D Blocking \| 88.77 (+4.92) \| 0.912 (+0.069) \| 0.15 (+× 16.7) \| \| SMR (Ours) \| 93.69 \| 0.981 \| 0.009 \| Figure 4: Scene synthesis visualization between different methods. Our method stands out by efficiently utilizing predicted contacts to produce more reasonable and comprehensive scenes. User Study. We conduct a user study with 40 participants from various backgrounds. In this study, participants are presented with a choice between our proposed SMR and current state-of-the-art models, displayed side by side. Both sets of samples are generated using the PROXD test set. This process is repeated five times for each model and the user scores are from 1 to 5. There are two judgment criteria: (i) “Naturalness” identifies if the position and orientation of facilities are generated properly in the scene and matched with the human poses or not, and (ii) “Non-Collision” shows if the generated object collides with human motions. The results in Figure 5 show that, in most instances, our method is the preferred choice over the compared models. More qualitative results can be found in our Demonstration Video. 4.3 Comparison with Other Sparse Representation Methods Baselines. We compare the effectiveness of the proposed method with four other sparse representation works: ME (Choy et al., 2019), EsCoin (Chen, 2018), pSConv (Kundu et al., 2019), and 1-D Blocking (Jin et al., 2014). Implementation. We use the baseline POSA (Hassan et al., 2021) as the network for contact prediction and report the results in terms of both accuracy (Reconstruction Accuracy and Consistency Score) and inference speed (second/sample). Figure 5: The user evaluation of our method, ground-truth (GT), and other baselines. Figure 6: Effectiveness of models with different sparsity ratios and the number of masks. \| Test Cases \| #Avg. Vertices with contacts ↓ \| #Avg. Vertices need to predict ↓ \| Correct Vertices prediction (%) ↑ \| Reconstruction Accuracy (%) ↑ \| Consistency Score ↑ \| Inference Speed (s/sample) ↓ \| \|---------------------\|---------------------------------\|----------------------------------\|-----------------------------------\|-------------------------------\|----------------------\|-----------------------------\| \| Original Input \| 121 \| 655 \| 90.31 \| 91.12 \| 0.882 \| 0.28 \| \| Keep all 50 masks \| 107 (↓ × 1.13) \| 603 (↓ × 1.08) \| 88.73 (+ 1.58) \| 89.46 (- 1.66) \| 0.935 (+ 0.053) \| 0.451 (+ × 1.61) \| \| Keep only 01 mask \| 12 (↓ × 10.08) \| 66 (↓ × 9.92) \| 54.67 (- 35.64) \| 83.61 (- 7.51) \| 0.763 (- 0.119) \| 0.008 (↓ × 35.0) \| \| Keep only 03 masks \| 41 (↓ × 2.95) \| 66 (↓ × 9.92) \| 95.65 (+ 5.34) \| 93.69 (+ 2.57) \| 0.981 (+ 0.099) \| 0.009 (↓ × 31.1) \| \| Keep only 10 masks \| 48 (↓ × 2.52) \| 72 (↓ × 9.1) \| 92.07 (+ 1.76) \| 90.80 (- 0.32) \| 0.989 (+ 0.107) \| 0.143 (↓ × 1.96) \| Table 4: Redundant information analysis by selecting the masks based on mask score $\alpha$. POSA network (Hassan et al., 2021) is used as the backbone. Results. Table 3 presents the performance of different sparse representation methods. We can see that our method achieves the highest accuracy compared to all the other sparse coding baselines. In terms of inference speed, our method is only slower than ME (Choy et al., 2019) (0.009 second/sample vs. 0.008 second/sample) while our accuracy is 10.08% higher than ME. 4.4 Sparse Mask Analysis Sparsity ratio and the number of sparse masks. Figure 6 illustrates the correlation between reconstruction accuracy and inference speed of our method under different values of sparsity ratio and the number of sparse masks $K$. We note that $K = 50$ masks are used during training. During inference, we consequently only select $\kappa$ masks based on the value of mask score $\alpha$. We can see that using $\kappa = 1$ mask leads to faster model performance, however, this also significantly reduces accuracy due to the loss of input information. In contrast, employing multiple sparse masks helps retain essential information and improves the overall model performance. Overall, the experiment in Figure 6 shows that using $\kappa = 3$ masks with 90% sparsity ratio during the inference brings the balance of the accuracy and inference speed. How do sparse masks help reduce input information? Our sparse masks work as a filter to reduce non-useful information in the human-scene input data. In particular, the sparse mask reduces the vertices in human-scene representation and hence, influences inference speed and accuracy. Table 4 illustrates how sparse masks help reduce non-useful input information. The “Original Input” uses all vertices as input; the “Keep only 01 mask” setup uses only $\kappa = 1$ mask during inference. Similarly, we set up our method with 3 masks, 10 masks, and all 50 masks for the inference process, respectively. Note that, our method is trained with $K = 50$ sparse masks, each with a 90% sparsity ratio. The masks are kept based on the mask score $\alpha$ illustrated in Section 3.2. As shown in Table 4, the model using just 1 sparse mask reduces vertex processing requirements by 90%, significantly enhancing inference speed but causing a 7.51% accuracy drop compared to the Original Input setup. With 50 masks, our SMR approach maintains accuracy but increases inference time since too many masks are used. Using the mask score $\alpha$, we can remove non-useful masks and retain only 10 or even 3 informative masks during inference. We see that using only 3 masks during inference helps reduce the verticle input while increasing the accuracy and reducing the inference speed. ![Figure 7](image) Sparse Masks Selection. Figure 7 presents the similarity between features of POSA baseline (Hassan et al., 2021) and features of our SMR model when we keep 1, 3, 5, and all 10 sparse masks during the inference. We train the SMR model with $K = 10$ sparse masks, each mask has a sparsity ratio of 90% in this experiment. The mask score $\alpha$ is used to rank and choose useful masks during inference. To compare feature similarity maps, we pass test samples of PROXD dataset (Hassan et al., 2019) to both POSA and our SMR model with the corresponding number of masks. Then, we extract the features from each layer and use the Euclidean distance to compute similarity. While features extracted from the POSA Network remain unchanged in all setups, features of our SMR change when the number of sparse masks is changed. We can see that in Figure 7(a), using only 1 mask with the highest mask score $\alpha$ only maintains feature similarity at abstract layers and the dissimilarity significantly increases in later layers (lightens in early layers and darkens in latter ones). Using 3 masks (Figure 7(b)) or 5 masks (Figure 7(c)) shows good feature similarity within corresponding masks (most features show high similarity in their corresponding layers). This behavior shows that the representations extracted from each layer in our model are distinctive, highlighting how our proposed method handles redundant information compared with all features from the setup that does not use the mask score $\alpha$ to select the useful masks (Figure 7(d)). ## 5 DISCUSSION We have presented sparse mask representation, a simple yet efficient approach for representing complex human-scene interaction data. Our goal is to expedite the inference process and enhance network performance by reducing redundant information. We have employed our method across various downstream tasks, such as contact prediction and scene synthesis, demonstrating its effectiveness in terms of both accuracy and inference speed. Although our method shows potential to improve the human-scene interaction task, it does have limitations. First, since our method involves processing the input data using multiple masks, the training time of our model is typically longer than that of the baseline network due to the large number of random masks being used. We note that our strategy currently sacrifices the training time for the inference time. Second, it is challenging to apply our method to recent diffusion works for scene synthesis as the network for diffusion models is relatively simple and the training of diffusion models involves adding noise, which is not compatible with our strategy for effectively representing the input data. Finally, choosing the right sparsity pattern or sparsity ratio can impact the quality of the representation, which requires parameter tuning for a specific task. Failure to make an appropriate selection regarding the number of sparse masks or the sparsity ratio may result in the presence of redundant sparse masks or information loss in inputs, and vice versa. Both cases can lead to inaccurate contact predictions, a primary cause of failure in the synthesis of the existence, position, and orientation of objects in the generated scene. Please refer to Section F in our Appendix for a more comprehensive examination of instances where these failures occur. There are several avenues for future research from our work. First, developing methods that dynamically adjust the sparsity ratio during inference based on real-time context could improve the flexibility of our approach. 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6vtGG0WMne	At 500:1 imbalance ratios, multiple benchmark methods show numbers decimated to zero. A previous study, MMM, by Mirza et al. '21 appears to suggest that even at such a ratio, classical resampling and even baselines report an above zero performance. Could you reason about the disparity?	REGULATING IMBALANCED DEEP MODELS WITH USER-SPECIFIED METRICS Anonymous authors Paper under double-blind review ABSTRACT Deep learning models implemented in real-world applications still face challenges from imbalanced data. Existing methods address the imbalance problem by balancing the models between the minority class and the majority class. However, practical applications may require an imbalanced optimization strategy that selectively unbalances the models and makes them more suitable for the applications than the balanced models. In this work, we first give a formal definition to accurately quantify the degree of imbalance of a model. Then, we propose a bias adjustment method that can efficiently optimize the model to a specified imbalance state according to application metrics or requirements so that this method has wide applicability. Finally, we introduce a training strategy that is advantageous to select the optimal representation parameters of the model during traditional training process. Extensive experiments verify the effectiveness and efficiency of our method, and compared with state-of-the-art algorithms, our method has significant improvement in different metrics including accuracy, F1 value and G-means. 1 INTRODUCTION Deep learning has achieved enormous success in various fields, but it also faces a challenge due to imbalanced data. In fact, the datasets for many applications are imbalanced, where the majority class dominates most of the data while the minority class has few samples. The ratio of their sample sizes may be many orders of magnitude. A deep network model trained on such an imbalanced dataset will be seriously biased towards the majority class, resulting in misclassification of the minority class samples. This class-imbalance problem appears in many applications, such as sentiment classification (Wang et al., 2021), Twitter spam detection (Li & Liu, 2018), object detection (Oksuz et al., 2020) and medical science (Khushi et al., 2021). There are many works to solve the imbalance problems in deep learning, which design different optimization objectives to balance the models to improve the performance of the minority class. In this paper, we call the above objectives as balanced optimization objectives. These works focus on a common imbalance scenario where the training data is unbalanced due to manual sampling errors or the scarcity of sampled objects but the actual test set is balanced, e.g. the long-tailed problem in image classification (Tan et al., 2020). This balance effect is shown on data(1) of Figure 1. The circles and pentagrams represent the samples of majority and minority classes, respectively. The lines represent the boundary of the models. The samples above and below the lines are predicted into the majority class and the minority class respectively. In Figure 1(1), the green line represents the best boundary line. It means that the model trained on the unbalanced dataset can achieve the best accuracy on a balanced dataset. These research works mainly include re-sampling (Chawla et al., 2002; Liang et al., 2022), class-level or instance-level re-weighting (Lin et al., 2017; Liu et al., 2021), and two-stage methods (Wahab et al., 2017; Guo et al., 2022). Re-sampling uses down/up-sampling to obtain a balanced dataset and optimize a model on this dataset (Drummond et al., 2003; Barandela et al., 2004). Recently, re-weighting methods learn instance-level weight values with a balanced dataset, so that the models guided by these weights can achieve optimal performance on balanced test datasets (Ren et al., 2018; Hu et al., 2019; Liu et al., 2021; Guo et al., 2022). In two-stage methods, the models are also corrected by class-balanced optimization strategies at the second stage (Kang et al., 2019). Figure 1: On data(1), the green line equally divides the circles and pentagrams and has the best accuracy. On data(2), G-means considers the recall of the minority and majority class, so the blue line is the best due to recognizing all pentagrams without significantly reducing the majority class recall. On data(3), F1 value takes into account the recall and precision of the minority class, so orange line is the best due to improving the recall of the pentagrams while not overly misclassifying the circles. However, in many real-world applications, the online test data is also unbalanced like the training data and these applications have a certain preference among classes. We roughly divide them into 2 cases. In the first case, the minority class is more important than the majority class. For example, in financial fraud detection (Priscilla & Prabha [2020], Warghade et al. [2020]), fraudulent customers (i.e. minority class) are much more valued than normal customers (i.e. majority class), and the detection system is unwilling to omit any customer who may be fraudulent. Data(2) in Figure 1 shows this situation. The blue line as the best boundary identifies all fraudulent customers (pentagrams) and G-means is the metric for fraud detection (Sisodia et al. [2017]). Similar situations appear in disease detection (Cui et al. [2020]), information security (Shu et al. [2022]), crime prediction (Hossain et al. [2020]), etc. In the second case, the majority class is more critical. For example, in customer complaints recognition, the system pays more attention to major incidents (i.e. majority class) but minor incidents (i.e. minority class) cannot be ignored either. Thus, while improving the recall of the minor incidents, the system does not want to excessively misclassify the complaints of major incidents. In Figure 1, the orange line is the optimal boundary and corresponds to the best F1 value for customer complaints data (Tang et al. [2021]). In summary, the Figure 1 demonstrates that the metrics for different applications require models to be biased between minority and majority classes, i.e. this green line as a balanced boundary is not the best for F1 value and G-means. However, existing methods aimed at obtaining balanced models and cannot generalize well to other applications. Although re-weighting methods can tune hyperparameters (i.e. class weights) to control the degree of model imbalance, the large range of values leads to serious time-consuming for searching optimal values. How to make the models efficiently achieve the appropriate imbalance correction becomes a new challenge. In this paper, we propose a new optimization method that can efficiently adjust the models to specified imbalance states according to application metrics or requirements. This method compensates for the lack of explicitly regulating the model imbalance in existing works, so that the method can be broadly applicable to many scenarios that require a variable degree of model imbalance. Specifically, we first use a class probability distribution to formally define model imbalance state (MIS) which can describe and quantify the imbalance of a model. Then, we propose bias adjustment (BA) method that optimizes the bias of the last layer of a model to make the deformed model reach the optimal MIS in applications. BA is an efficient method because it participates in a simple calculation in the last layer of the model and optimizes a few parameters with the same number of labels. In addition, BA has wide applicability since users can give imbalance metrics to determine the target MIS for BA optimization. Finally, we introduce an overall training strategy which uses the BA method to correct the biased model in every epoch of the traditional training process. An advantage is that the strategy facilitates the discovery of optimal model parameters for representation learning. In brief, we summarize four main contributions as follows. (1) We give a formal definition of model imbalance state (MIS) so that the imbalance of a model can be precisely quantified. (2) We propose a bias adjustment (BA) method that can efficiently correct the imbalanced models and broadly adapt to different applications based on user-specified metrics. (3) We introduce a training strategy to discover the optimal representation parameters during imbalanced learning. (4) We perform extensive experiments to verify the effectiveness and efficiency of our method. 2 BACKGROUND Imbalanced Classification. Let $X = ((x_i, y_i))_{i=1}^{N}$ be a training set, where $x_i$ is the i-th sample, $y_i$ is the corresponding label and $N$ is the number of samples. In K-category classification, the label $y$ has $K$ possible values which are denoted as $C_1, ..., C_K$. Let $n_1, ..., n_K$ be the number of training samples for the class $C_1, ..., C_K$ respectively. A deep learning model $\Phi$ can be viewed as a mapping function from input $x$ to target $\hat{y}$, that is, $\hat{y} = \Phi(x)$, and $\hat{y}$ is also the prediction for input $x$. The learning goal is to reduce the difference between the prediction $\hat{y}$ and the real label $y$. We use cross-entropy loss function $l(y, \hat{y})$ to measure this difference. Thus, the training objective is to minimize the sum of the losses over the entire training set $X$, formally, the overall loss $L_{CE} = \min \sum_{i=1}^{N} l(y_i, \Phi(x_i))$. However, if $\exists i, j \ n_i \ll n_j$, then, the training set is imbalanced. The model learned from $L_{CE}$ will be seriously biased toward the majority class, that is, most samples of the minority class $C_i$ may be misclassified into the majority class $C_j$. Decoupling Models. Recently, researchers have proposed to solve the imbalanced classification by decoupling the models. The deep learning models consist of complex computational structures, though it can be simply divided into two parts, that is, the backbone and classifier modules. The backbone module is used to obtain the feature representation of the input, such as a BERT model (Devlin et al., 2018) can extract the representation of the text. We denote $z = f(x; \theta)$ as the representation of the input $x$, where $f$ is the function of the backbone module and $\theta$ is the parameter. The classifier module refers to the last layer of a deep learning model, which takes the representation $z$ as input and outputs the label probability. Generally, the last layer is a linear classifier and we denote $W = \{w_i\}_{i=1}^{K}$ and $b = \{b_i\}_{i=1}^{K}$ are the linear weight matrix and bias respectively, where $w_i \in R^d$ and $b_i$ are the weight and bias corresponding to label $C_i$. The probability of $C_i$ is calculated by softmax: $$\hat{y}_i = \frac{\exp(w_i^T z + b_i)}{\sum_{k=1}^{K} \exp(w_k^T z + b_k)}$$ The researchers found that the classifier module was mainly affected by the imbalanced data, rather than the backbone module. Therefore, many works are devoted to adjusting the classifier module to tackle the imbalanced classification (Kang et al., 2019). Classifier from Probability Theory. From the perspective of probability theory, the classifier is the conditional probability $p(y\|z)$ that is the probability distribution of labels under the given representation $z$, and the label corresponding to the maximum probability is the predicted result. According to Bayesian formula, the probability of label $C_i$ is $$p(y = C_i\|z) = \frac{p(z\|C_i)p(C_i)}{\sum_{i=k}^{K} p(z\|C_k)p(C_k)} = \frac{\exp(w_i^T z + b'_i + \ln p(C_i))}{\sum_{k=1}^{K} \exp(w_k^T z + b'_k + \ln p(C_k))}$$ where assuming $p(z\|C_i)$ is the member of exponential family distributions and is further restricted to be a linear function of $z$ in the exponent (Bishop & Nasrabadi, 2006), and $w_i$ and $b'_i$ are the linear parameters of $p(z\|C_i)$. Comparing the forms of Eq.(1) and Eq.(2), it can be found that $$b_i = b'_i + \ln p(C_i)$$ It suggests that the estimate of the class probability $p(C_i)$ is included in the bias of the last layer. 3 ALGORITHM In this section, we first give the definition of model imbalance state (MIS) and then propose the bias adjustment method that can efficiently correct the imbalance. Thereafter, we introduce a training strategy for the imbalanced classification. 3.1 MODEL IMBALANCE MEASURE Recently, many works focus on building a class-balanced model, but the balance between minority and majority classes is not always optimal. In fact, the optimal model in different applications has varied degrees of imbalance among classes. To measure the degree of imbalance of a model, we introduce the concept of model imbalance state (MIS) which is denoted as $P \in R^K$ and $P_i$ records the model prediction probability of label $C_i$. If the value of $P_i$ is large, the model will be biased towards label $C_i$ and become imbalanced. We can estimate $P_i$ from the training set $X$. Formally, given a model $\Phi$ and a dataset $X$, $P_i$ can be obtained by $$p(C_i\|\Phi) = \int p(C_i\|x,\Phi)p(x)dx = E(p(C_i\|x,\Phi)) \approx \frac{1}{N} \sum_{x \in X} p(C_i\|x,\Phi)$$ Eq.(4) shows $P_i$ can be estimated as the average prediction probability of label $C_i$ on $X$. ### 3.2 Bias Adjustment When the model is trained in a traditional way on the imbalanced dataset, the predicted probability of the majority class in MIS is much greater than that of the minority class. The main idea of this work is to correct the model by adjusting the MIS, such as increasing the minority class prediction probability. Additionally, inspired by Eq.(3), bias contains the estimation of class probability, thus we propose a bias adjustment method (BA) that only adjusts the bias to change the MIS. Specifically, given an expected class probability distribution $r$, BA adjusts the bias $b$ to make the model imbalance state $P$ close to $r$ and uses KL divergence to build this objective $L_{bal}$ as follows $$L_{bal} = -\sum_{i=1}^{K} r_i \ln \left( \frac{P_i(b)}{r_i} \right) \Leftrightarrow -\sum_{i=1}^{K} r_i \ln (P_i(b))$$ where the right side removes the constant term $\sum_{i=1}^{K} r_i \ln (r_i)$ that is independent of $b$. In practice, the class probability distribution $r$ is generally unknown in applications. BA uses a search strategy to find the optimal $r^$ which is based on imbalance metrics on a validation set, i.e. F1 value, G-means. The details of this search strategy and bias optimization are as follows. Search Strategy.* This work mainly discusses the binary classification, so only the minority class probability $r_1$ needs to be adjusted and the majority class probability $r_2$ can be determined by $r_2 = 1 - r_1$. In order to perform an efficient search, BA finds $r_1^$ from $(0, 1)$ step by step with the precision of 10 powers. BA first finds the best value $a_1 10^{-1}$ where $a_1$ from $\{1, ..., 9\}$, and then finds the best value $a_1 10^{-1} + a_2 10^{-2}$ where $a_2$ from $\{-9, ..., 9\}$, and similarly finds $a_1 10^{-1} + a_2 10^{-2} + a_3 10^{-3}$ and $a_3 \in \{-9, ..., 9\}$. Generally, the precision taken to $10^{-2}$ or $10^{-3}$ is enough. Bias Optimization.* BA uses gradient descent to calculate the optimal $b^$ for the objective $L_{bal}$. In fact, this is a simple optimization because $b$ participates in the calculation at the last layer of the model and has only $K$ parameters. Therefore, BA can first store the calculation results in the model feed-forward and adopt the entire training data set as a batch, which can calculate the optimal $b^$ efficiently and accurately. ### 3.3 A Training Strategy for Imbalanced Classification We introduce a new training strategy that alters the MIS in the traditional training process. This strategy does not need to change the traditional training method and still uses the $L_{CE}$ to update the model parameters. However, after each epoch in the training, the strategy will try to use the BA to correct the bias and validate the model to retain the best model. The motivation is that, the traditional training way on the imbalanced dataset may only slightly affect the backbone module but seriously shift the classifier. Therefore, after training in the traditional way, the strategy keeps the backbone module constant and only adjusts the bias of the classifier. The experimental results indicate that the effect of the bias adjustment is excellent in the binary classification task. Discussion. The training strategy has two advantages for imbalanced classification. First, the strategy is beneficial to discover the best backbone module parameters because it can adjust the imbalance and validate the model at each epoch of training. In contrast, the two-stage methods train a stable model on an imbalanced dataset in the first stage and then balance the classifier in the second stage. It is difficult to ensure that the parameters of backbone module are optimal. Although the two-stage methods can also adjust the classifier at each epoch, the optimization of the classifier is more time-consuming than BA. Second, our strategy enables the model to meet different application requirements. Because the strategy corrects the imbalance based on user-specified metrics, in Table 1: Statistics of three datasets \| Data sets \| Classes \| Training Samples \| Testing Samples \| \|-----------\|---------\|------------------\|----------------\| \| CIFAR-10 \| 2 \| $2 \times 5000$ \| $2 \times 1000$ \| \| SST-2 \| 2 \| $2 \times 5000$ \| $2 \times 5000$ \| \| AG \| 2 \| $2 \times 20000$ \| $2 \times 20000$ \| other words, the degree of model imbalance can be determined by the requirement of application. However, traditional imbalanced approaches either obtain class-balanced models that may not be applicable, or suffer from expensive hyperparameter tuning to suit the needs of the application. 4 EXPERIMENTS 4.1 EXPERIMENTAL SETUP Datasets and Evaluation. We use CIFAR-10 (Schneider et al., 2019) for image classification and adopt SST-2 sentiment analysis data (Socher et al., 2013) and AG news data (Zhang et al., 2015) for text classification. Specifically, we select the class 0 and 1 from CIFAR-10 and the class “World” and “Sci/Tech” from AG to form binary classification datasets. The statistics of the datasets are shown in Table 1. Further, we construct the imbalanced datasets from these three datasets. We set the example ratios of majority class to minority class 10:1, 50:1, 100:1 and 500:1 for the CIFAR-10 and SST-2 datasets, and we set the extremely imbalance ratio of 1000:1 for the AG dataset. In addition, we use 3 metrics to evaluate the results, which are accuracy rate, F1 value of the minority class and G-means. The F1 value comprehensively measures the precision and recall of the minority class, and G-means calculates the geometric mean of the recall of the majority and minority classes (Du et al., 2017). It is noted that the accuracy is evaluated on a balanced test set, and F1 and G-means are calculated on a test set with an imbalance ratio equal to the training set. Comparison Methods. We compare our method with six approaches: (1) Baseline, the model is directly trained on an imbalanced training set with cross-entropy loss. (2) Proportion, an empirical class weighting method that weights examples by inverse class frequency. (3) Auto-Weighting, the method proposed by Hu et al. (Hu et al., 2019) can learn data weights from a small validation set. (4) cRT (Kang et al., 2019), a two-stage method that re-trains the classifier with class-balanced sampling at the second stage. (5) LWS (Kang et al., 2019), a two-stage method that learns the scaling factors for the classifier at the second stage. (6) POT (Guo et al., 2022), is the SOTA approach that considers automatic weighting and the two-stage method. The experimental details are described in Section A. 4.2 RESULTS OF DIFFERENT METRICS ON TEXT CLASSIFICATION AND IMAGE CLASSIFICATION Results on Accuracy. The accuracy results on SST-2 and CIFAR-10 are shown in Table 2. There are mainly the following three observations. (1) Our method achieves the best accuracy at different imbalance ratios on datasets SST-2 and CIFAR-10, which shows that just adjusting the bias of the classifier can greatly improve the distorted model due to the imbalanced data. It also implies that the impact of imbalanced learning on the backbone module and the classifier weight parameter may not be severe. (2) When the dataset is more imbalanced, our method is more advantageous than other methods. For example, on SST-2, our method outperforms proportion by only 0.19 accuracy points at 10:1 but 7 points at 500:1, and on CIFAR-10, our method exceeds POT by only about 1 point at 10:1 but more than 3 points at 500:1. The possible reason is that when the number of minority class examples decreases, our method of adjusting the bias is less susceptible to overfitting than optimizing the entire model parameters or the classification module. (3) The performance of the two-stage methods POT and cRT on CIFAR-10 is stable and excellent at different imbalance ratios, but the results on SST-2 are lower than proportion. This may be because the representation learning of the model at first stage is of high quality on CIFAR-10, while it is not optimal on SST-2. Table 2: Results of accuracy on SST-2 and CIFAR-10 under different imbalance ratios \| Methods \| SST-2 \| \| \| \| CIFAR-10 \| \| \| \| \|---------------\|-------\|------\|------\|------\|----------\|------\|------\|------\| \| Imbalance Ratios \| 500:1 \| 100:1 \| 50:1 \| 10:1 \| 500:1 \| 100:1 \| 50:1 \| 10:1 \| \| Baseline \| 50.00 \| 58.61 \| 66.97 \| 82.46 \| 62.27 \| 78.25 \| 84.24 \| 96.17 \| \| Proportion \| 57.03 \| 79.00 \| 83.13 \| 87.77 \| 68.39 \| 81.65 \| 87.80 \| 96.93 \| \| Auto-Weighting\| 50.25 \| 61.16 \| 61.54 \| 81.96 \| 57.99 \| 76.81 \| 84.98 \| 96.04 \| \| LWS \| 50.13 \| 56.17 \| 59.82 \| 79.95 \| 61.95 \| 76.05 \| 77.44 \| 90.88 \| \| cRT \| 50.13 \| 55.97 \| 60.29 \| 78.51 \| 76.81 \| 88.18 \| 91.52 \| 97.43 \| \| POT \| 52.41 \| 63.78 \| 75.40 \| 81.95 \| 77.10 \| 89.11 \| 91.72 \| 96.42 \| \| Ours. \| 64.45 \| 80.31 \| 83.76 \| 87.96 \| 80.89 \| 91.72 \| 93.98 \| 98.04 \| Table 3: Results of F1 value on SST-2 and CIFAR-10 under different imbalance ratios \| Methods \| SST-2 \| \| \| \| CIFAR-10 \| \| \| \| \|---------------\|-------\|------\|------\|------\|----------\|------\|------\|------\| \| Imbalance Ratios \| 500:1 \| 100:1 \| 50:1 \| 10:1 \| 500:1 \| 100:1 \| 50:1 \| 10:1 \| \| Baseline \| 0.00 \| 10.18 \| 31.67 \| 65.43 \| 29.33 \| 73.23 \| 72.67 \| 95.74 \| \| Proportion \| 4.33 \| 17.54 \| 34.96 \| 65.17 \| 34.11 \| 72.45 \| 75.60 \| 96.20 \| \| Auto-Weighting\| 0.00 \| 8.62 \| 17.24 \| 54.49 \| 19.05 \| 74.02 \| 73.20 \| 95.75 \| \| LWS \| 0.00 \| 11.96 \| 22.17 \| 58.97 \| 5.12 \| 50.51 \| 52.11 \| 89.92 \| \| cRT \| 0.00 \| 12.25 \| 27.81 \| 59.69 \| 29.96 \| 71.30 \| 71.93 \| 95.31 \| \| POT \| 3.84 \| 7.72 \| 15.05 \| 52.80 \| 2.40 \| 22.76 \| 43.66 \| 94.71 \| \| Ours. \| 12.15 \| 22.42 \| 41.36 \| 66.79 \| 44.64 \| 77.66 \| 76.89 \| 96.24 \| It illustrates that the well-trained backbone module parameters are crucial for two-stage methods, and further analysis is presented in section [4.3.2]. Results on F1 Value. The results of F1 value are shown in Table 3. There are two main conclusions. (1) Our method also achieves the best performance on F1 value in all cases. In particular, on SST-2, our method outperforms the second-best by nearly 8 F1 points at 500:1 and 6 F1 points at 50:1, and on CIFAR-10 with the ratio of 500:1, our method surpasses the second-best method by more than 10 points. It demonstrates that our method has the dominant performance on the F1-value metric, and it also suggests that modifying the model imbalance state (MIS) to fit the metric is effective. (2) The two-stage methods POT, cRT and LWS, and the auto-weighting method all hardly work on the F1-value metric. The F1 values of these methods are almost lower than baseline at different imbalance ratios on SST-2 and CIFAR-10. It indicates that learning with the objective on a balanced dataset or class-balanced sampling is not suitable for the F1-value metric. In other words, the MIS suitable for F1 value is more likely to have a small minority class probability, rather than a completely balanced minority class and majority class. Results on G-means. The results of G-means are shown in Table 4, and we can obtain the following two observations. (1) Similar to the results of the F1 value, our method has the best performance on G-means in all cases. Especially on SST-2 dataset, our method exceeds the second-best method by 13 G-means points at 500:1 and 10 points at 100:1. It again illustrates the superiority of our method and the importance of adjusting the MIS to fit the metrics. (2) The performance of the two-stage methods POT and cRT is better than the baseline on the G-means metric. Especially when Table 4: Results of G-means on SST-2 and CIFAR-10 under different imbalance ratios \| Methods \| SST-2 \| \| \| \| CIFAR-10 \| \| \| \| \|---------------\|-------\|------\|------\|------\|----------\|------\|------\|------\| \| Imbalance Ratios \| 500:1 \| 100:1 \| 50:1 \| 10:1 \| 500:1 \| 100:1 \| 50:1 \| 10:1 \| \| Baseline \| 0.00 \| 22.07 \| 53.63 \| 80.66 \| 34.13 \| 80.93 \| 80.94 \| 97.02 \| \| Proportion \| 0.00 \| 74.00 \| 83.82 \| 87.62 \| 62.31 \| 83.14 \| 87.26 \| 97.94 \| \| Auto-Weighting\| 0.00 \| 48.45 \| 59.85 \| 83.06 \| 28.26 \| 80.59 \| 81.03 \| 97.37 \| \| LWS \| 0.00 \| 33.10 \| 52.27 \| 77.66 \| 72.93 \| 74.92 \| 68.64 \| 90.98 \| \| cRT \| 0.00 \| 41.22 \| 52.69 \| 78.03 \| 93.43 \| 94.55 \| 94.77 \| 98.45 \| \| POT \| 59.80 \| 64.55 \| 71.30 \| 81.12 \| 90.65 \| 94.71 \| 94.53 \| 97.32 \| \| Ours. \| 73.68 \| 84.87 \| 85.41 \| 88.04 \| 93.51 \| 97.69 \| 97.93 \| 98.86 \| Table 5: Results of different metrics on AG with the imbalance ratio of 1000:1 \| Methods \| Accuracy \| F1 Value \| G-means \| \|-------------\|----------\|----------\|---------\| \| Baseline \| 52.02 \| 3.2 \| 6.32 \| \| Proportion \| 82.78 \| 27.22 \| 77.25 \| \| Auto-Weighting \| 82.24 \| 24.73 \| 76.48 \| \| LWS \| 70.65 \| 17.02 \| 49.35 \| \| cRT \| 83.67 \| 4.23 \| 79.22 \| \| POT \| 73.50 \| 8.46 \| 67.16 \| \| Ours. \| 86.71 \| 29.11 \| 86.02 \| Figure 2: Results of three metrics on different minority class probabilities (a) Results on SST-2 (b) Results on CIFAR-10 the imbalance is serious, the baseline method is almost invalid on G-means. It shows that the MIS suitable for G-means requires a greater probability of the minority class than the F1-value metric. Results on AG Dataset. Table 5 shows the results of accuracy, F1 value and G-means on the AG dataset with the extremely imbalanced ratio. We summarize the following two conclusions. First, we can see that our method achieves the best results on all metric, which further proves that the method of modifying MIS to adapt metrics is successful. Second, the experimental results show that our method is also effective in extremely imbalanced conditions. 4.3 Inside Analysis and Efficiency Comparison 4.3.1 Inside Analysis Optimal MIS on Different Metrics. To validate that different metrics correspond to different optimal MIS, we present the results of regulating the model imbalance to different MIS. Here, we express the MIS in terms of minority class probabilities. Figure 2 show the results of accuracy, F1 value and G-means on SST-2 and CIFAR-10. We can observe that the probabilities corresponding to the highest values are different among these metrics. Specifically, the probability of the highest F1 value is less than 0.1 on SST-2 and close to 0 on CIFAR-10, which is much smaller than that of G-means and accuracy. This is because the F1 value takes into account the precision and recall of the minority class. If the minority class probability is great, a large number of majority classes will be misclassified into the minority class, which will greatly reduce the precision of the minority class and result in a decrease in F1 value. Thus, a high F1 value may favor a small minority class probability. On the contrary, the highest G-means requires a great minority class probability. Because G-means considers the recall of both the minority class and majority class, and the great minority class probability significantly improves the recall of the minority class, thereby increasing the G-means. For the accuracy, the best probabilities is close to 0.5 on SST-2 and is 0.1 on CIFAR-10, which shows the optimal MIS may also vary on different datasets. In summary, the different imbalance metrics prefer different MIS, and the balanced optimization strategy may be not best, e.g. on the F1 value. It indicates that the work of regulating the model imbalance is necessary and the BA method is effective. Impact of Epochs on the Results. We show the BA correction results at each epoch during the traditional training process. The results of different metrics on SST-2 and CIFAR-10 are shown Figure 3: Results of three metrics at each epoch during training Table 6: Comparison to the two-stage methods with tuning the epochs on SST-2 \| Methods \| Accuracy \| F1 Value \| G-means \| time(h) \| \|------------------\|----------\|----------\|---------\|---------\| \| LWS(10th epoch) \| 50.13 \| 0.00 \| 0.00 \| 0.25 \| \| LWS(best epoch) \| 53.12 \| 6.04 \| 45.36 \| 2.34 \| \| cRT(10th epoch) \| 50.13 \| 0.00 \| 0.00 \| 0.25 \| \| cRT(best epoch) \| 58.80 \| 3.57 \| 66.66 \| 2.37 \| \| POT(10th epoch) \| 52.41 \| 3.84 \| 59.80 \| 0.46 \| \| POT(best epoch) \| 62.54 \| 10.30 \| 66.74 \| 4.44 \| \| Ours. \| 64.45\| 12.15\| 73.68\| 0.30\| As the training epoch increases, the results on these metrics are roughly stable but still vary. Especially, this variation is significant on the F1 value of CIFAR-10. It indicates that the number of training epochs may greatly affect the models and the quality of the representation parameters determines the performance. However, in actual implementation, due to the influence of imbalanced data, we cannot explicitly know the optimal representation parameters of the model on which epoch during training. Therefore, we encourage correcting and validating the model on each epoch to obtain the best representation parameters. ### 4.3.2 Efficiency Comparison Compared to the Two-stage Methods. We tune the best epoch for the two-stage methods and compare the performance and time consumption with our method. The results are shown in Table 6. We summarize the following three points. (1) The results of the two-stage methods at the best epoch are significantly improved compared to the 10th epoch, which illustrates the importance of the selection of the optimal model parameters. (2) The results of the two-stage methods at the best epoch are still lower than our method, which indicates the effectiveness of our method. (3) The time consumption of tuning the epochs for the two-stage methods is about 10 times that of our method, which shows the efficiency of our method. Compared to the class-level weighting Method. We set the minority class weight to 1 and we use grid search to find the optimal weight value from $(0, 10)$ for the majority class. We sequentially increase the number of weight values according to the exponential of 2, so we tested a total of $2^{11}-1$ and $2^{10}-1$ values for SST2-2 and CIFAR-10, and our method is tested once. This comparison is shown in Figure 4, where the triangles represent the results of our method. We summarize the three points. (1) The time cost of tuning the weights to get good results differs from our method by 2-3 orders of magnitude. For example, in Figure 4(a), the weighting method takes nearly 100 hours and the accuracy is close to 80%, while our method takes less than 1 hour to achieve an accuracy of more than 80%. It demonstrates the clear advantage of our method in terms of efficiency compared to the tuning the weights. (2) The weighting method by tuning a large number of weights is still almost lower than our method on all metrics, which further illustrates the effectiveness of our method. (3) In Figure 4(a), our method is slightly lower than the weighting method on F1 value, which shows that optimizing the entire model parameters can perform better than adjusting the bias. Figure 4: Comparison to the weighting method with tuning the weight values. These results are on the imbalance ratio of 100:1, and CW and Ours. represent class weighting and our method. 5 RELATED WORK Re-sampling. The re-sampling methods obtain the balanced deep models by re-balancing the training data distribution. These methods mainly include up-sampling method of increasing the minority class samples (Chawla et al., 2002; Shi et al., 2022) and down-sampling method of reducing the majority class samples (Drummond et al., 2003; Barandela et al., 2004; Liang et al., 2022) to achieve the balance of the number of samples between classes. Weighting Methods. There are very rich works on weighting samples for imbalance classification, and we summarize into two main groups, namely empirical weighting and automatic weighting. The empirical weighting methods set the manual weight values to the samples, such as inverse class frequency (Wang et al., 2017), inverse square root of class frequency (Mikolov et al., 2013; Mahajan et al., 2018), calculating weight values based on the effective number of examples (Cui et al., 2019), hard example mining (Dong et al., 2017; Shrivastava et al., 2016) and Focal loss (Lin et al., 2017). The automatic weighting methods obtain adaptive weights through learning mechanisms. Ren et al. (Ren et al., 2018) and Hu et al. (Hu et al., 2019) proposed to learn the example weights by a meta-learning paradigm. Similar methods also include the work of Liu et al. (Liu et al., 2021), Meta-weight-net (Shu et al., 2019) and Meta-class-weight (Jamal et al., 2020). Recently, Guo et al. (Guo et al., 2022) proposed an automatic weighting method based on optimal transport (OT). However, these automatic weighting methods are all based on a balanced validation set to learn the sample weights. Two-stage Methods. The two-stage methods focus on representation learning at first stage and re-balance the classifier at second stage, such as OLTR (Liu et al., 2019), LDAM (Cao et al., 2019) and cRT (Kang et al., 2019). Experiments prove the effectiveness of these training strategies for addressing the imbalance problem (Li et al., 2021; Zhong et al., 2021). The above methods aim to obtain balanced deep models by re-balancing data, learning weights on a balanced data set or balanced optimization strategies. There is a key difference between our work and theirs. We believe that a balanced model may be not beneficial for practical applications, thus, we propose to regulate the degree of model imbalance to promote applicability. 6 CONCLUSION To solve the imbalance problem of different applications, we propose a new optimization strategy that can efficiently regulate the imbalanced deep models based on the user-specified metrics. Thus, this strategy can be widely applied to different scenarios. Specifically, we define model imbalance state and propose the BA method that can efficiently correct the distorted model, and finally we introduce the overall training framework. The experimental evaluation shows that our algorithm can achieve significant improvement compared with the SOTA method in terms of efficiency and effectiveness. 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Va4t6R8cGG	How fast is the proposed method compared to other methods? I understand that there are some GFLOPs comparisons in the supplementary, but it is difficult to compare the methods due to the presence of other parts (such as LTC or person detector). Could we see a speed comparison instead?	END-TO-END SPATIO-TEMPORAL ACTION LOCALISATION WITH VIDEO TRANSFORMERS Anonymous authors Paper under double-blind review ABSTRACT The most performant spatio-temporal action localisation models use external person proposals and complex external memory banks. We propose a fully end-to-end, transformer based model that directly ingests an input video, and outputs tubelets – a sequence of bounding boxes and the action classes at each frame. Our flexible model can be trained with either sparse bounding-box supervision on individual frames, or full tubelet annotations: In both cases, it predicts coherent tubelets as the output. Moreover, our end-to-end model requires no additional pre-processing in the form of proposals, or post-processing in terms of non-maximal suppression. We perform extensive ablation experiments, and significantly advance the state-of-the-art results on four different spatio-temporal action localisation benchmarks with both sparse keyframes and full tubelet annotations. 1 INTRODUCTION Spatio-temporal action localisation is an important problem with applications in advanced video search engines, robotics and security among others. It is typically formulated in one of two ways: Firstly, predicting the bounding boxes and actions performed by an actor at a single keyframe given neighbouring frames as spatio-temporal context (Gu et al., 2018; Li et al., 2020a). Or alternatively, predicting a sequence of bounding boxes and actions (i.e. “tubes”), for each actor at each frame in the video (Soomro et al., 2012; Jhuang et al., 2013). The most performant models (Pan et al., 2021; Arnab et al., 2022; Wu et al., 2022; Feichtenhofer et al., 2019), particularly for the first, keyframe-based formulation of the problem, employ a two-stage pipeline inspired by the Fast-RCNN object detector (Girshick, 2015): They first run a separate person detector to obtain proposals. Features from these proposals are then aggregated and classified according to the actions of interest. These models have also been supplemented with memory banks containing long-term contextual information from other frames (Wu et al., 2019; 2022; Pan et al., 2021; Tang et al., 2020), and/or detections of other potentially relevant objects (Tang et al., 2020; Arnab et al., 2021b) to capture additional scene context, achieving state-of-the-art results. And whilst proposal-free algorithms, which do not require external person detectors, have been developed for detecting both at the keyframe-level (Köpiklü et al., 2019; Chen et al., 2021; Sun et al., 2018) and tubelet-level (Kalogeiton et al., 2017; Zhao et al., 2022b), their performance has typically lagged behind their proposal-based counterparts. Here, we show for the first time that an end-to-end trainable spatio-temporal model outperforms a two-stage approach. As shown in Fig. 1, we propose our Spatio-Temporal Action Transformer (STAR) that consists of a transformer architecture, and is based on the DETR (Carion et al., 2020) detection model. Our model is “end-to-end” in that it does not require pre-processing in the form of proposals, nor post-processing in the form of non-maximal suppression (NMS) in contrast to the majority of prior work. The initial stage of the model is a vision encoder. This is followed by a decoder that processes learned latent queries, which represent each actor in the video, into output tubelets – a sequence of bounding boxes and action classes at each time step of the input video clip. Our model is versatile in that we can train it with either fully-labeled tube annotations, or with sparse keyframe annotations (when only a limited number of keyframes are labelled). In the latter case, our network still predicts tubelets, and learns to associate detections of an actor, from one frame to the next, without explicit supervision. This behaviour is facilitated by our formulation of factorised queries, decoder architecture and tubelet matching in the loss which all contain temporal inductive biases. Figure 1: We propose an end-to-end Spatio-Temporal Action Recognition model named STAR. Our model is end-to-end in that it does not require any external region proposals to predict tubelets – sequences of bounding boxes associated with a given person in every frame and their corresponding action classes. Our model can be trained with either sparse box annotations on selected keyframes, or full tubelet supervision. We conduct thorough ablation studies of these modelling choices, confirming the benefit of temporal inductive biases in our model design. Informed by these experiments, we achieve state-of-the-art on both keyframe-based action localisation datasets like AVA (Gu et al., 2018) and AVA-Kinetics (Li et al., 2020a), and also tubelet-based datasets like UCF101-24 (Soomro et al., 2012) and JHMDB (Jhuang et al., 2013). In particular, we achieve a Frame mAP of 45.1 on AVA-Kinetics, outperforming the best previous results achieved by a massive video foundation model (Wang et al., 2023). In addition our Video AP50 on UCF101-24 surpasses prior work (Zhao et al., 2022b) by 13.2 points. Moreover, our state-of-the-art results are achieved with a single forward-pass through the model, using only a video clip as input, and without any separate external person detectors providing proposals (Wu et al., 2022; Wang et al., 2022; 2023), complex memory banks (Wu et al., 2022; Zhao et al., 2022b; Pan et al., 2021), or additional object detectors (Tang et al., 2020; Arnab et al., 2021b), as used by the prior state-of-the-art. Furthermore, we outperform these complex, prior two-stage models whilst also having additional functionality in that our model predicts tubelets, that is, temporally consistent bounding boxes at each frame of the input video clip. 2 RELATED WORK Models for spatio-temporal action localisation have typically built upon advances in object detectors for images. The most performant methods (Pan et al., 2021; Wu et al., 2022; Tang et al., 2020; Arnab et al., 2022) are based on ‘two-stage’ detectors like Fast-RCNN (Girshick, 2015). These models use external, pre-computed person detections, and use them to ROI-pool features which are then classified into action classes. Although these models are cumbersome in that they require an additional model and backbone to first detect people, and therefore additional detection training data as well, they are currently the leading approaches on datasets such as AVA (Gu et al., 2018). Such models using external proposals are also particularly suited to datasets such as AVA as each person is exhaustively labelled as performing an action, and therefore there are fewer false-positives from using action-agnostic person detections compared to datasets such as UCF101 (Soomro et al., 2012). The accuracy of these two-stage models has further been improved by incorporating more contextual information using feature banks extracted from additional frames in the video (Wu et al., 2022; Pan et al., 2021; Tang et al., 2020; Wu et al., 2019) or by using detections of additional objects in the scene (Arnab et al., 2021b; Baradel et al., 2018; Wang & Gupta, 2018). Both of these cases entail significant extra computation and complexity to train additional auxiliary models, and to precompute features from them that are then used during training and inference of the localisation model. Our proposed method, in contrast, is end-to-end in that it directly produces detections without any additional inputs besides a video clip. Moreover, it outperforms these prior works without resorting to external proposals or memory banks, showing that a transformer backbone is sufficient to capture long-range dependencies in the input video. In addition, unlike previous two-stage methods, our method directly predicts tubelets: a sequence of bounding boxes and actions for each frame of the input video, and can do so even when we do not have full tubelet annotations available. A number of proposal-free action localisation models have also been developed (Köpüklü et al., 2019; Chen et al., 2021; Sun et al., 2018; Girdhar et al., 2019; Kalogeiton et al., 2017; Zhao et al., 2022b). These methods are based upon alternative object detection architectures such as SSD (Liu et al., 2016), CentreNet (Zhou et al., 2019), YOLO (Redmon et al., 2016), DETR (Carion et al., 2020) and Sparse-RCNN (Sun et al., 2021). However, in contrast to our approach, they have been outperformed by their proposal-based counterparts. Moreover, some of these methods (Köpüklü et al., 2019; Girdhar et al., 2019; Sun et al., 2018) also consist of separate network backbones for Figure 2: Our model processes a fixed-length video clip, and for each frame, outputs tubelets (i.e. linked bounding boxes with associated action class probabilities). It consists of vision encoder which outputs a video representation, $ x \in \mathbb{R}^{T \times h \times w \times d} $. The video representation, along with learned queries, $ q $ (which are factorised into spatial $ q^s $ and temporal components $ q^t $) are decoded into tubelets by a decoder of $ L $ layers followed by shallow box and class prediction heads. learning video feature representations and proposals for a keyframe, and are thus effectively two networks trained jointly, and cannot predict tubelets either. Among prior works that do not use external proposals, and also directly predict tubelets (Kalogeiton et al., 2017; Li et al., 2020b; Song et al., 2019; Li et al., 2018; Singh et al., 2017), our work is most similar to TubeR (Zhao et al., 2022b) given that our model is also based on DETR. Our model, however, is designed with additional temporal inductive biases which improves accuracy (without using external memory banks precomputed offline like Zhao et al. (2022b). And moreover, unlike TubeR, we also demonstrate how our model can predict tubelets (i.e. predictions at every frame of the input video), even when we only have sparse keyframe supervision (i.e. ground truth annotation for a limited number of frames) available. Finally, we note that DETR has also been extended as a proposal-free method to addressing other localisation tasks in video such as temporal localisation (Liu et al., 2022; Zhang et al., 2021; Nawhal & Mori, 2021), instance segmentation (Wang et al., 2021) and moment retrieval (Lei et al., 2021). 3 Spatio-Temporal Action Transformer Our proposed model ingests a sequence of video frames, and directly predicts tubelets (a sequence of bounding boxes and action labels). No external person detections (Pan et al., 2021; Wang et al., 2023; Tong et al., 2022), or memory banks (Zhao et al., 2022b; Wu et al., 2022), are needed. As summarised in Fig. 2, our model consists of a vision encoder (Sec. 3.1), followed by a decoder which processes learned query tokens into output tubelets (Sec. 3.2). We incorporate temporal inductive biases into our decoder to improve accuracy and tubelet prediction with weaker supervision. Our model is inspired by the DETR architecture (Carion et al., 2020) for object detection in images, and is also trained with a set-based loss and Hungarian matching. We detail our loss, and how we can train with either sparse keyframe supervision or full tubelet supervision, in Sec. 3.3. 3.1 Vision Encoder The vision backbone processes an input video, $ X \in \mathbb{R}^{T \times H \times W \times 3} $ to produce a feature representation of the input video $ x \in \mathbb{R}^{t \times h \times w \times d} $. Here, $ T, H $ and $ W $ are the original temporal-, height- and width-dimensions of the input video respectively, whilst $ t, h $ and $ w $ are the spatio-temporal dimensions of their feature representation, and $ d $ its latent dimension. When using a transformer backbone, these spatio-temporal dimensions depend on the patch size when tokenising the input, and when using a convolutional backbone, they depend on the overall stride. To retain spatio-temporal information, we remove the spatial- and temporal-aggregation steps at the end of the original backbone. And if the temporal patch size (or stride) is larger than 1, we bilinearly upsample the final feature map along the temporal axis to maintain the original temporal resolution. 3.2 Tubelet Decoder Our decoder processes the visual features, $ x \in \mathbb{R}^{T \times h \times w \times c} $, along with learned queries, $ q \in \mathbb{R}^{T \times S \times d} $, to output tubelets, $ y = (b, a) $ which are a sequence of bounding boxes, $ b \in \mathbb{R}^{T \times S \times 4} $ and corresponding actions, $ a \in \mathbb{R}^{T \times S \times C} $. Here, $ S $ denotes the maximum number of bounding boxes per frame (padded with “background” as necessary) and $C$ denotes the number of output classes. The idea of decoding learned queries into output detections using the transformer decoder architecture of Vaswani et al. (2017) was used in DETR (Carion et al., 2020). In summary, the decoder of (Carion et al., 2020; Vaswani et al., 2017) consists of $L$ layers, each performing a series of self-attention operations on the queries, and cross-attention between the queries and encoder outputs. We modify the queries, self-attention and cross-attention operations for our spatio-temporal localisation scenario, as shown in Fig. 2 and 3 to include additional temporal inductive biases, and to improve accuracy as detailed below. Queries Queries, $q$, in DETR, are decoded using the encoded visual features, $x$, into bounding box predictions, and are analogous to the “anchors” used in other detection architectures such as Faster-RCNN (Ren et al., 2015). The most straightforward way to define queries is to randomly initialise $q \in \mathbb{R}^{T \times S \times d}$, where there are $S$ bounding boxes at each of the $T$ input frames in the video clip. However, we find it is more effective to factorise the queries into separate learned spatial, $q^s \in \mathbb{R}^{S \times d}$, and temporal, $q^t \in \mathbb{R}^{T \times d}$ parameters. To obtain the final tubelet queries, we simply repeat the spatial queries across all frames, and add them to their corresponding temporal embedding at each location, as shown in Fig. 2. More concretely $q_{ij} = q^t_i + q^s_j$ where $i$ and $j$ denote the temporal and spatial indices respectively. The factorised query representation means that the same spatial embedding is used across all frames. Intuitively, this encourages the $i^{th}$ spatial query embedding, $q^s_i$, to bind to the same location across different frames of the video, and since objects typically have small displacements from frame to frame, may help to associate bounding boxes within a tubelet together. We verify this intuition empirically in the experimental section. Decoder layer The decoder layer in the original transformer (Vaswani et al., 2017) consists of self-attention on the queries, $q$, followed by cross-attention between the queries and the outputs of the encoder, $x$, and then a multilayer perceptron (MLP) layer (Hendrycks & Gimpel, 2016): $$u^\ell = \text{MHSA}(q^\ell) + q^\ell, \quad v^\ell = \text{CA}(u^\ell, x) + u^\ell, \quad z^\ell = \text{MLP}(v^\ell) + v^\ell,$$ where $z^\ell$ is the output of the $\ell^{th}$ decoder layer, $u$ and $v$ are intermediate variables, MHSA denotes multi-headed self-attention and CA denotes cross-attention. Note that the inputs to the MLP, self- and cross-attention operations are layer-normalised (Ba et al., 2016), which we omit here for clarity. In our model, we factorise the self- and cross-attention layers across space and time respectively as shown in Fig. 3, to introduce a temporal locality inductive bias, and also to increase model efficiency. Concretely, when applying MHSA, we first compute the queries, keys and values, over which we attend twice: first independently at each time step with each frame, and then, independently along the time axis at each spatial location. Similarly, we modify the cross-attention operation so that only tubelet queries and backbone features from the same time index attend to each other. Localisation and classification heads We obtain the final predictions of the network, $y = (b, a)$, by applying a small feed-forward network to the outputs to the decoder, $z$, following DETR (Carion et al., 2020). The sequence of bounding boxes, $b$, is obtained with a 3-layer MLP, and is parameterised by the box center, width and height for each frame in the tubelet. A single-layer linear projection is used to obtain class logits, $a$. As we predict a fixed number of $S$ bounding boxes per frame, and $S$ is more than the maximum number of ground truth instances in the frame, we also include an additional class label, $\emptyset$, which represents the “background” class which tubelets with no action class can be assigned to. 3.3 Training objective Our model predicts bounding boxes and action classes at each frame of the input video. Many datasets, however, such as AVA (Gu et al., 2018), are only sparsely annotated at selected keyframes of the video. In order to leverage the available annotations, we compute our training loss, Eq. 2, only at the annotated frames of the video, after having matched the predictions to the ground truth: \[ L(y, \hat{y}) = \frac{1}{\|T\|} \sum_{t \in T} L_{\text{frame}}(y_t, \hat{y}_t), \] where $T$ is the set of labelled frames; $y$ and $\hat{y}$ denote the ground truth and predicted tubelets after matching. Following DETR (Carion et al., 2020), our training loss at each frame, $L_{\text{frame}}$, is a sum of an $L_1$ regression loss on bounding boxes, the generalised IoU loss (Rezatofighi et al., 2019) on bounding boxes, and a cross-entropy loss on action labels: \[ L_{\text{frame}}(b^t_i, \hat{b}^t_i, a^t_i, \hat{a}^t_i) = \sum_i L_{\text{box}}(b^t_i, \hat{b}^t_i) + L_{\text{iou}}(b^t_i, \hat{b}^t_i) + L_{\text{class}}(a^t_i, \hat{a}^t_i). \] Matching Set-based detection models such as DETR can make predictions in any order, which is why the predictions need to be matched to the ground truth before computing the training loss. The first form of matching that we consider is to independently perform bipartite matching at each frame to align the model’s predictions to the ground truth (or the $\emptyset$ background class) before computing the loss. In this case, we use the Hungarian algorithm (Kuhn, 1955) to obtain $T$ permutations of $S$ elements, $\hat{\pi}^t \in \Pi^t$, at each frame, where the permutation at the $t^{th}$ frame minimises the per-frame loss, \[ \hat{\pi}^t = \arg \min_{\pi \in \Pi^t} L_{\text{frame}}(y^t, \hat{y}_{\pi(t)}^t). \] An alternative is to perform tubelet matching, where all queries with the same spatial index, $q^s$, must match to the same ground truth annotation across all frames of the input video. Here the permutation is obtained over $S$ elements as \[ \hat{\pi} = \arg \min_{\pi \in \Pi} \frac{1}{\|T\|} \sum_{t \in T} L_{\text{frame}}(y^t, \hat{y}_{\pi(t)}^t). \] Intuitively, tubelet matching provides stronger supervision when we have full tubelet annotations available. Note that regardless of the type of matching that we perform, the loss computation and the overall model architecture remains the same. Note that we do not weight terms in Eq. 3, for both matching and loss calculation, for simplicity, and to avoid having additional hyperparameters, as also done by Minderer et al. (2022). 3.4 Discussion As our approach is based on DETR, it does not require external proposals nor non-maximal suppression for post-processing. The idea of using DETR for action localisation has also been explored by TubeR (Zhao et al., 2022b) and WOO (Chen et al., 2021). There are, however, a number of key differences: WOO does not detect tubelets at all, but only actions at the center keyframe. We also factorise our queries in the spatial and temporal dimensions (Sec. 3.2) to provide inductive biases urging spatio-temporal association. Moreover, we predict action classes separately for each time step in the tubelet, meaning that each of our queries binds to an actor in the video. TubeR, in contrast, parameterises queries such that they are each associated with separate actions (features are average-pooled over the tubelet, and then linearly classified into a single action class). This choice also means that TubeR requires an additional “action switch” head to predict when tubelets start and end, which we do not require as different time steps in a tubelet can have different action classes in our model. Furthermore, we show experimentally (Tab. 1) that TubeR’s parameterisation obtains lower accuracy. We also consider two types of matching in the loss computation (Sec. 3.3) unlike TubeR, with “tubelet matching” designed for predicting more temporally consistent tubelets. And in contrast to TubeR, we experimentally show how our decoder design allows our model to accurately predict tubelets even with weak, keyframe supervision. Finally, TubeR requires extra complexity in the form of a “short-term context module” (Zhao et al., 2022b) and the external memory bank of Wu et al. (2019) which is computed offline using a separate model to achieve strong results. As we show experimentally, we outperform TubeR without any additional modules, meaning that our model does indeed produce tubelets in an end-to-end manner. 4 EXPERIMENTAL EVALUATION 4.1 EXPERIMENTAL SET-UP Datasets We evaluate on four spatio-temporal action localisation benchmarks. AVA and AVA-Kinetics contain sparse annotations at each keyframe, whereas UCF101-24 and JHMDB51-21 contain full tubelet annotations. AVA (Gu et al., 2018) consists of 430, 15-minute video clips from movies. Keyframes are annotated at every second in the video, with about 210,000 labelled frames in the training set, and 57,000 in the validation set. There are 80 atomic actions labelled for every actor in the clip, of which 60 are used for evaluation (Gu et al., 2018). Following standard practice, we report the Frame Average Precision (fAP) at an IoU threshold of 0.5 using the latest v2.2 annotations (Gu et al., 2018). AVA-Kinetics (Li et al., 2020a) is a superset of AVA, and adds detection annotations following the AVA protocol, to a subset of Kinetics 700 (Carreira et al., 2019) videos. Only a single keyframe in a 10-second Kinetics clip is labelled. In total, about 140,000 labelled keyframes are added to the training set, and 32,000 to the validation sets of AVA. Once again, we follow standard practice in reporting the Frame AP at an IoU threshold of 0.5. UCF101-24 (Soomro et al., 2012) is a subset of UCF101, and annotates 24 action classes with full spatio-temporal tubes in 3,207 untrimmed videos. Note that actions are not labelled exhaustively as in AVA, and there may be people present in the video who are not performing any labelled action. Following standard practice, we use the corrected annotations of Singh et al. (2017). We report both the Frame AP, which evaluates the predictions at each frame independently, and also the Video AP, which uses a 3D, spatio-temporal IoU to match predictions to targets. Since UCF101-24 videos are up to 900 frames long (median length of 164 frames), and our network typically processes $T = 32$ frames at a time, we link together tubelet predictions from our network into full-video-tubes using the same causal linking algorithm as (Kalogeiton et al., 2017; Li et al., 2020b) for fair comparison. JHMDB51-21 (Jhuang et al., 2013) also contains full tube annotations in 928 trimmed videos. However, as the videos are shorter and at most 40 frames, we can process the entire clip with our network, and do not need to perform any linking. Implementation details For our vision encoder backbone, we consider both transformer-based (ViViT Factorised Encoder (Arnab et al., 2021a)), and convolutional (CSN (Tran et al., 2019)) backbones. For ViViT, we use the “Base” and “Large” model sizes (Dosovitskiy et al., 2021), which are typically first pretrained on image datasets like ImageNet-21K (Deng et al., 2009) and then finetuned on video datasets like Kinetics (Kay et al., 2017). We also use CSN-152 pretrained on Instagram65M (Mahajan et al., 2018) and then Kinetics following Zhao et al. (2022b). Our models process $T = 32$ frames unless otherwise specified, with $S = 64$ spatial queries per frame and latent decoder dimensionality of $d = 2048$. Exhaustive implementation details and training hyperparameters are included in the supplement. We will also release all code and models upon acceptance. 4.2 ABLATION STUDIES We analyse the design choices in our model by conducting experiments on both AVA (with sparse per-frame supervision) and on UCF101-24 (where we can evaluate the quality of our predicted tubelets). Unless otherwise stated, our backbone is ViViT-Base pretrained on Kinetics 400, and the frame resolution is 160 pixels (160p) on the smaller side. Comparison of detection architectures Table 1 compares our model, where each query represents a person, and all of their actions (Sec. 3.2) to the approach of TubeR (Zhao et al., 2022b) (Sec. 3.4), where there is a separate query for each action being performed. We observe that our parameterisation has a substantial impact, with our method outperforming binding to actions by 3.1 points with a ViViT backbone, and 2.1 points with a CSN backbone on the AVA dataset, therefore motivating the design of our decoder. Appendix C shows that this trend is consistent on UCF101-24 and JHMDB too. Another architectural baseline that we can compare to is that of a two-stage Fast-RCNN model using external person detections from Wu et al. (2019), as used by (Wu et al., 2022; Feichtenhofer et al., 2019; Fan et al., 2021; Arnab et al., 2022). This baseline using the same ViViT-B backbone achieved Table 1: Comparison of detection architectures on AVA controlling for the same resolution (160p) and training settings. Binding each query to a person, rather than to an action (as done in TubeR (Zhao et al., 2022b)) yields solid improvements. We report the mean AP for both ViViT-B and CSN-152 backbones. \| Query binds to action \| ViViT-B \| CSN-152 \| \|----------------------\|---------\|---------\| \| Ours, query binds to person \| 23.6 \| 25.7 \| \| \| 26.7 \| 27.8 \| Table 2: Comparison of independent and factorised queries on the AVA and UCF101-24 datasets. Factorised queries are particularly beneficial for predicting tubelets, as shown by the VideoAP on UCF101-24 which has full tube annotations. Both models use tubelet matching in the loss. \| Query \| AVA \| UCF101-24 \| \|----------------\|-----------\|-----------\| \| \| fAP \| vAP \| vAP20 \| vAP50 \| vAP50:95 \| \| Independent \| 25.2 \| 85.6 \| 86.3 \| 59.5 \| 28.9 \| \| Factorised \| 26.3 \| 86.5 \| 87.4 \| 63.4 \| 29.8 \| Table 3: Comparison of independent and tubelet matching for computing the loss on AVA and UCF101-24. Tubelet matching helps for tube-level evaluation metrics like the Video AP (vAP). Note that tubelet matching is actually still possible on AVA as the annotations are at 1fps with actor identities. \| Query \| AVA \| UCF101-24 \| \|----------------\|-----------\|-----------\| \| \| fAP \| vAP \| vAP20 \| vAP50 \| vAP50:95 \| \| Per-frame matching \| 26.7 \| 88.2 \| 85.7 \| 63.5 \| 29.4 \| \| Tubelet matching \| 26.3 \| 86.5 \| 87.4 \| 63.4 \| 29.8 \| a mean AP of 25.2, which is still 1.5 points below our model, emphasising the promise of our end-to-end approach. Note that the proposals of Wu et al. (2019) obtain an AP50 of 93.9 for person detection on the AVA validation set. They were obtained by first pretraining a Faster-RCNN (Ren et al., 2015) detector on COCO keypoints, and then finetuning on the person boxes from the AVA training set, using a resolution of 1333 on the longer side. Our model is end-to-end, and does not require any external proposals generated by a separate model at all. Comparison to TubeR The second row of Tab. 1 using a CSN-152 backbone corresponds to our reimplemention of TubeR. By keeping all other training hyperparameters constant, we observe that our query binding provides an improvement of 2.1 mAP points in a fair comparison. Note that we could not use the public TubeR code (Zhao et al., 2022a), as it does not reproduce the paper’s results: A higher resolution 256p model achieved only 20 mAP when trained with the public code, whilst it is reported to achieve 31.1. Exhaustive details on our attempts to reproduce TubeR with the authors’ public code is in Appendix B. Query parameterisation Table 2 compares our independent and factorised query methods (Sec. 3.2) on AVA and UCF101-24. We observe that factorised queries consistently provide improvements on both the Frame AP and the Video AP across both datasets. As hypothesised in Sec. 3.2, we believe that this is due to the inductive bias present in this parameterisation. Note that we can only measure the Video AP on UCF101-24 as it has tubes labelled. We also show in Appendix C that these observations are consistent on the JHMDB dataset too. Matching for loss calculation As described in Sec. 3.3, when matching the predictions to the ground truth for loss computation, we can either independently match the outputs at each frame to the ground truths at each frame, or, we can match entire predicted tubelets to the ground truth tubelets. Table 3 shows that tubelet matching does indeed improve the quality of the predicted tubelets, as shown by the Video AP on UCF101-24. However, this comes at the cost of the quality of per-frame predictions, i.e. Frame AP (fAP). This suggests that tubelet matching improves the association of bounding boxes predicted at different frames (hence higher Video AP), but may also impair the quality of the bounding boxes predicted at each frame (Frame AP). Note that it is technically possible for us to also perform tubelet matching on AVA, since AVA is annotated at 1fps with actor identities, and our model is input 32 frames at 12.5fps (therefore 2.56 seconds of temporal context) meaning that we have sparse tubelets with 2 or 3 annotated frames. As tubelet matching helps with the overall Video AP, we use it for subsequent experiments on UCF101-24 and JHMDB51-21. For AVA, we use per-frame matching as the standard evaluation metric is the Frame AP, and annotations are sparse at 1fps. Weakly-supervised tubelet detection Our model can predict tubelets even when the ground truth annotations are sparse and only labelled at certain frames (such as the AVA dataset). We quantita- Table 5: Effect of decoder depth on performance on the AVA dataset. Performance saturates at $L = 6$ layers. \| Layers ($L$) \| 0 \| 1 \| 3 \| 6 \| 9 \| \|-------------\|-----\|-----\|-----\|-----\|-----\| \| mAP ↑ \| 23.4\| 24.6\| 26.2\| 26.5\| 26.7\| Table 6: Effect of the type of attention used in the decoder on AVA. Factorised attention is both more accurate and efficient (almost half of the GFLOPs per decoder layer). \| Decoder attention \| mAP \| GFLOPs \| \|-------------------\|-----\|--------\| \| Full \| 26.4\| 10.5 \| \| Factorised \| 26.7\| 5.3 \| tively measure this ability of our model on UCF101-24 which has full tube annotations. We do so by subsampling labels from the training set, and evaluating the full tubes on the validation set. As shown in Tab. 4, we still obtain meaningful tube predictions, with a Video AP20 of 77.1, when using only a single frame of annotation from each UCF video clip. When retaining 1 frame of supervision for every 24 labelled frames (which is roughly 1fps and corresponds to the AVA dataset’s annotations), we observe minimal deterioration with respect to the fully supervised model (all Video AP metrics are within 0.7 points). Retaining 1 frame of annotation for every 12 consecutive labelled frames also performs similarly to using all frames in the video clip. These results suggest that due to the redundancy in the data (motion between frames is often limited), and the inductive bias of our model, we do not require each frame in the tube to be labelled in order to predict accurate tubelets. Decoder design Tables 5 and 6 analyse the effect of the decoder depth and the type of attention in the decoder (described in Sec. 3.2). As seen in Tab. 5, detection accuracy on AVA increases with the number of decoder layers, plateauing at around 6 layers. It is possible to use no decoder layers too: In this case, instead of learning queries $q$ (Sec. 3.2), we simply interpret the outputs of the vision encoder (Sec. 3.1), $x$, as our queries and apply the localisation and classification heads directly upon them. Using decoder layers, however, can provide a performance increase of up to 3.3 mAP points (14% relative), emphasising their utility. Table 6 shows that factorised attention in the decoder is more accurate than standard, “full” attention between all queries and visual features. Moreover, it is more efficient too, using almost half of the GFLOPs at each decoder layer. Additional analysis We further analyse the effect of frame resolution, and pretraining in Appendix C. As expected, we find that higher resolutions, and larger scale pretraining, using CLIP (Radford et al., 2021), improves accuracy. We make use of these observations in our following state-of-the-art comparisons. The supplement also visualises our predicted tubelets. ### 4.3 Comparison to State-of-the-Art We compare our model to the state-of-the-art on datasets with both sparsely annotated keyframes (AVA and AVA-Kinetics), and full tubes (UCF101-24 and JHMDB). AVA and AVA-Kinetics Table 7 compares to prior work on AVA and AVA-Kinetics. The best previous methods relied on external proposals (Wang et al., 2022; Tong et al., 2022; Arnab et al., 2022) and external memory banks (Pan et al., 2021; Wu et al., 2022) which we outperform. There are fewer end-to-end approaches, and we outperform these by an even larger margin. Note that though TubeR (Zhao et al., 2022b) is a proposal-free approach, their best results are actually obtained with the external memory of Wu et al. (2019). Consequently, we have reported the end-to-end, and external-memory versions of TubeR (“TubeR + LTC”) separately in Tab. 7. Furthermore, as detailed in Appendix B, the TubeR public code also shows additional object detection pretraining on COCO that is not used by any other prior work. Observe that we outperform TubeR using the same CSN-152 backbone, and then improve further using larger transformer backbones. We achieve greater relative improvements on AVA-Kinetics, showing that our end-to-end approach can leverage larger datasets more effectively. To our knowledge, we surpass the best previous results on AVA-Kinetics, achieving a Frame AP of 45.1. Notably, we outperform InternVideo (Wang et al., 2022) and VideoMAE-v2 (Wang et al., 2023), which are two recent video foundational models using more powerful backbones and larger, proprietary, web-scale video datasets. Note that InternVideo consists of two different encoders, one of which is also initialised from CLIP (Radford et al., 2021). And like Wang et al. (2022), we achieve our best AVA results by training a model on AVA-Kinetics, and then evaluating it only on the AVA validation set. Moreover, note that we do not perform any Table 7: Comparison to the state-of-the-art (reported with mean Average Precision; mAP) on AVA v2.2 and AVA-Kinetics (AVA-K). Methods using external proposals are also trained on additional object detection and human pose data. Unless otherwise stated, separate models are trained for AVA and AVA-Kinetics. * denotes the model was trained on AVA-Kinetics and evaluated on AVA. “Res.” denotes the frame resolution of the shorter side. Web-scale foundational models are denoted in grey. \| Pretraining \| Views \| AVA \| AVA-K \| Res. \| Backbone \| End-to-end \| \|-------------\|-------\|-----\|-------\|------\|----------\|------------\| \| MViT-B (Pan et al., 2021) \| K400 \| 1 \| 27.3 \| – \| – \| MViT \| x \| \| Unified (Arnab et al., 2021b) \| K400 \| 6 \| 27.7 \| – \| 320 \| SlowFast \| x \| \| AIA (Tang et al., 2020) \| K700 \| 18 \| 32.3 \| – \| 320 \| SlowFast \| x \| \| ACAR (Pan et al., 2021) \| K700 \| 6 \| 33.3 \| 36.4 \| 320 \| SlowFast \| x \| \| TubeR (Zhao et al., 2022b) with LTC \| IG65M→K400, COCO \| 2 \| 33.6 \| – \| 256 \| CSN-152 \| x \| \| MeMViT (Wu et al., 2022) \| K700 \| – \| 34.4 \| – \| 312 \| MViT v2 \| x \| \| Co-finetuning (Arnab et al., 2022) \| IN21K→K700, MiT SSv2 \| 1 \| 32.8 \| 33.1 \| 320 \| ViViT/L \| x \| \| VideoMAE (Tong et al., 2022) \| JFT WTS→K700, MiT SSv2 \| 1 \| 36.1 \| 36.2 \| 320 \| ViViT/L \| x \| \| InternVideo* (Wang et al., 2022) \| SSL K700 → Sup. K700. \| – \| 39.3 \| – \| – \| ViViT/L \| x \| \| VideoMAE v2 (Wang et al., 2023) \| 7 different datasets \| – \| 41.0 \| 42.5 \| – \| Uniformer v2 \| x \| \| Action Transformer (Li et al., 2020a) \| K400 \| 1 \| 23.0 \| 400 \| I3D \| ✓ \| \| WOO (Chen et al., 2021) \| K600 \| 1 \| 28.3 \| – \| 320 \| SlowFast \| ✓ \| \| TubeR (Zhao et al., 2022b) \| IG65M→K400, COCO \| 1 \| 31.1 \| – \| 256 \| CSN-152 \| ✓ \| \| STAR(CSN-152 (ours) \| IG65M→K400 \| 1 \| 31.4 \| 35.8 \| 256 \| CSN-152 \| ✓ \| \| STAR/B (ours) \| IN21K→K400 \| 1 \| 30.0 \| 36.6 \| 320 \| ViViT/B \| ✓ \| \| STAR/L (ours) \| CLIP→K700 \| 1 \| 33.9 \| 39.1 \| 320 \| ViViT/B \| ✓ \| \| STAR/L (ours)* \| CLIP→K700 \| 1 \| 39.2 \| 44.5 \| 320 \| ViViT/L \| ✓ \| \| STAR/L (ours) \| CLIP→K700 \| 1 \| 42.5 \| 45.1 \| 420 \| ViViT/L \| ✓ \| Table 8: State-of-the-art comparison on datasets with tubelet annotations, UCF101 and JHMDB. \| Pretraining \| FAP \| vAP20 \| vAP50 \| vAP50:95 \| FAP \| vAP20 \| vAP50 \| Backbone \| \|-------------\|-----\|-------\|-------\|-----------\|-----\|-------\|-------\|----------\| \| ACT (Kalogeiton et al., 2017) \| IN1K \| 67.1 \| 77.2 \| 51.4 \| 25.0 \| 65.7 \| 74.2 \| 73.7 \| VGG \| \| MOC (Li et al., 2020b) \| IN1K → COCO \| 78.0 \| 82.8 \| 53.8 \| 28.3 \| 70.8 \| 77.3 \| 77.2 \| DLA34 \| \| Unified (Arnab et al., 2021b) \| K600 \| 79.3 \| – \| – \| – \| – \| – \| SlowFast \| \| WOO (Chen et al., 2021) \| K600 \| – \| – \| – \| 80.5 \| – \| – \| SlowFast \| \| TubeR (Zhao et al., 2022b) \| IG65M→K400 \| 83.2 \| 83.3 \| 58.4 \| 28.9 \| – \| 87.4 \| 82.3 \| CSN-152 \| \| TubeR with flow (Zhao et al., 2022b) \| K400 \| 81.3 \| 85.3 \| 60.2 \| 29.7 \| – \| 81.8 \| 80.7 \| I3D \| \| STAR(CSN-152 (ours) \| IG65M→K400 \| 86.7 \| 87.0 \| 65.4 \| 30.6 \| 93.5 \| 96.3 \| 95.4 \| CSN-152 \| \| STAR/B (ours) \| IN21K→K400 \| 87.3 \| 88.2 \| 68.6 \| 31.7 \| 86.9 \| 89.5 \| 88.2 \| ViViT/B \| \| STAR/L (ours) \| CLIP→K700 \| 90.3 \| 89.8 \| 73.4 \| 35.8 \| 92.1 \| 93.1 \| 92.6 \| ViViT/L \| Test-time augmentation, in contrast to previous works that ensemble results over multiple resolutions and/or left/right flips as denoted by the “Views” column. UCF101-24 Table 8 shows that we outperform prior work on UCF101-24, both in terms of frame-level (Frame AP), and tube-level metrics (Video AP). We achieve state-of-the-art results with a ViViT-Base backbone, and improve further by scaling up to ViViT-Large, consistent with our results on AVA (Tab. 7). Moreover, note how we substantially outperform TubeR (Zhao et al., 2022b) using the same CSN-152 backbone. To our knowledge, we outperform the best previous reported Video AP50 number by 13.2 points. Note that as UCF videos are up to 900 frames, and as our network processes $T = 32$ frames, we follow prior works and link together tubelets using the same causal algorithm as (Kalogeiton et al., 2017; Li et al., 2020b; Arnab et al., 2021b) for fair comparison. JHMDB51-21** Table 8 also shows that we surpass the state-of-the-art on JHMDB. Once again, we outperform TubeR (Zhao et al., 2022b) with the same CSN-152 backbone. The CSN-152 backbone outperforms ViViT in this case, possibly because this is the smallest dataset and larger backbones can overfit more easily. The videos in this dataset are trimmed (meaning that labelled actions are being performed on each frame), and also shorter. Therefore, the Video AP is not as strict as it is on UCF101-24. Additionally, as the input videos are a maximum of 40 frames, we set $T = 40$ in our model so that we process the entire clip at once without needing to link tubelets. ## 5 Conclusion and Future Work We have presented STAR, an end-to-end spatio-temporal action localisation model that can output tubelets, when either sparse keyframe, or full tubelet annotation is available. Our approach achieves state-of-the-art results on four action localisation datasets for both frame-level and tubelet-level predictions (in particular, we obtain 45.1% mAP on the challenging AVA-Kinetics dataset), outperforming complex methods that use external proposals and memory banks. Future work is to extend our method beyond fixed action classes to open vocabularies. REFERENCES Anurag Arnab, Mostafa Dehghani, Georg Heigold, Chen Sun, Mario Lučić, and Cordelia Schmid. ViViT: A Video Vision Transformer. In ICCV, 2021a. Anurag Arnab, Chen Sun, and Cordelia Schmid. Unified graph structured models for video understanding. In ICCV, 2021b. Anurag Arnab, Xuehan Xiong, Alexey Gritsenko, Rob Romijnders, Josip Djolonga, Mostafa Dehghani, Chen Sun, Mario Lučić, and Cordelia Schmid. Beyond transfer learning: Co-finetuning for action localisation. In arXiv preprint arXiv:2207.03807, 2022. Jimmy Lei Ba, Jamie Ryan Kiros, and Geoffrey E Hinton. Layer normalization. In arXiv preprint arXiv:1607.06450, 2016. Fabien Baradel, Natalia Neverova, Christian Wolf, Julien Mille, and Greg Mori. Object level visual reasoning in videos. In ECCV, 2018. Nicolas Carion, Francisco Massa, Gabriel Synnaeve, Nicolas Usunier, Alexander Kirillov, and Sergey Zagoruyko. 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Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai, Thomas Unterthiner, Mostafa Dehghani, Matthias Minderer, Georg Heigold, Sylvain Gelly, et al. An image is worth 16x16 words: Transformers for image recognition at scale. In ICLR, 2021. Haoqi Fan, Bo Xiong, Karttikeya Mangalam, Yanghao Li, Zhicheng Yan, Jitendra Malik, and Christoph Feichtenhofer. Multiscale vision transformers. In ICCV, 2021. Christoph Feichtenhofer, Haoqi Fan, Jitendra Malik, and Kaiming He. Slowfast networks for video recognition. In ICCV, 2019. Golnaz Ghiasi, Yin Cui, Aravind Srinivas, Rui Qian, Tsung-Yi Lin, Ekin D Cubuk, Quoc V Le, and Barret Zoph. Simple copy-paste is a strong data augmentation method for instance segmentation. In CVPR, 2021. Rohit Girdhar, Joao Carreira, Carl Doersch, and Andrew Zisserman. Video action transformer network. In CVPR, 2019. Ross Girshick. Fast r-cnn. In ICCV, 2015. Chunhui Gu, Chen Sun, David A Ross, Carl Vondrick, Caroline Pantofaru, Yeqing Li, Sudheendra Vijayanarasimhan, George Toderici, Susanna Ricco, Rahul Sukthankar, et al. Ava: A video dataset of spatio-temporally localized atomic visual actions. In CVPR, 2018. Dan Hendrycks and Kevin Gimpel. Gaussian error linear units (gelus). In arXiv preprint arXiv:1606.08415, 2016.
eO6lXIWyxn	There are so many metrics used for the evaluation (FID, CLIP-S, and OCR). Which one is the most appropriate to evaluate the overall performance? Or is there any way to combine all of them as a final metric?	SUBMISSION HAS BEEN WITHDRAWN Anonymous authors Paper under double-blind review ABSTRACT We thank the reviewers for their valuable comments. After careful consideration, we think our paper is inappropriate for ICLR and decided to withdraw our paper.
4pW8NL1UwH	Equation 3 seems to model generations as P_{\pi_{\theta}} (y^{i}_{j,k} \| x^{i}) , but for autoregressive models that the authors study, the probability should be modelled as P_{\pi_{theta}}(y^{i}_{j,k} \| x^{i}, y^{i}_{j, <k}), which in turn renders it intractable to compute	LIRE: LISTWISE REWARD ENHANCEMENT FOR PREFERENCE ALIGNMENT Anonymous authors Paper under double-blind review ABSTRACT Recently, tremendous strides have been made in the domain of Natural Language Generation (NLG) due to the vast advances in Large Language Models (LLMs). However, often trained on large-scale unsupervised data, LLMs may generate toxic or unhelpful content for lack of human supervision. Leveraging reinforcement learning with human feedback (RLHF) turns out a good remedy for this problem and has been prevalent among researchers. However, RLHF is notoriously unstable and hyperparameter-sensitive, which hinders an all-compassing and sustainable LLM system. For the above reason, we propose a new approach: LIRE, which stands for Listwise Reward Enhancement for Preference Alignment, to optimize rewards through a listwise paradigm. We directly incorporate the rewards of multiple candidates into the listwise loss and optimize against it in a compact and effective framework, without explicit modeling of the Bradley-Terry model. Furthermore, we propose a self-enhancement algorithm to progressively optimize the reward through iterative training. Our work also entails extensive experiments to demonstrate the stability and consistency of the model performance without heavy hyperparameter tuning, while still surpassing the state-of-the-art methods in preference alignment tasks. 1 INTRODUCTION While a growing plethora of large language models (LLMs) have exhibited incredible performance in a broadening scope of tasks and applications such as summarization, machine translation, and dialog generation Nakano et al. (2021); Stiennon et al. (2020); Brown et al. (2020); Zhao et al. (2023a), they can still output contents that are harmful, biased or simply do not agree with standard human perception Mathur et al. (2020); Fernandes et al. (2023). This is an inherent problem existing in the extensive data sources during model training Ouyang et al. (2022); Bai et al. (2022); Song et al. (2023), and can be alleviated by incorporating certain restrictions or limitations to align the output generation towards human desires and specifications Ngo (2022); Kenton et al. (2021). Existing methods focus on employing reinforcement learning from human feedback (RLHF) to fine-tune the pre-trained LLMs Christiano et al. (2017); Stiennon et al. (2020); Ouyang et al. (2022); Xue et al. (2023), which concept was originally introduced in the field of robotics and Atari games Christiano et al. (2017); Ibarz et al. (2018). RLHF in LLM introduces a paradigm that involves leveraging supervised fine-tuning (SFT) on the initial models, fitting the reward model to human preferences, and then using Reinforcement Learning (RL) algorithms such as Proximal Policy Optimization (PPO) Schulman et al. (2017) to optimize a policy that doesn’t drift overly far from the original model Rafailov et al. (2023). Such methods successfully incorporate human preferences into data training and achieve satisfying results to a large extent. However, PPO is trained in a pointwise manner and optimizes at every single step based on the rewards, penalizing fragments within a segment equally and disregarding the truly informative parts. Alternatively, pairwise ranking leverages a comparison between a positive and a negative sample to incorporate context information. Methods such as DPO Rafailov et al. (2023), PRO Song et al. (2023), and RRRF Yuan et al. (2023) all leverage a pairwise comparison model to optimize the rewards. Nevertheless, the performance of pairwise ranking is heavily dependent on the quality of the sample pairs, and trivial negatives may yield suboptimal results. Moreover, if given a large candidate pool, performing pairwise comparisons among multiple samples entails a significant computation complexity. For the above reasons, we propose a listwise optimization approach: Listwise Reward Enhancement for Preference Alignment (LIRE). Instead of employing the Bradeley-Terry model Bradley & Terry (1952) or Plackett-Luce models Plackett (1975) to rank the candidates, we take a listwise approach by modeling the response probability distribution under the general policy gradient framework, with reward scores implicitly weighing samples differently during loss calculation. Essentially, LIRE does not rely on an ordinal ranking, instead, the ranking information is implicitly given by the reward scores. This is different from the top-k probability defined in ListNet Cao et al. (2007), which gives a permutation probability distribution that relies on the position of a response in the permutation. LIRE considers multiple responses simultaneously at each iteration and is therefore free from hard mining techniques to eliminate the influence of trivial negatives. We give the training pipeline of the proposed LIRE in Figure 1. The overarching concept is as follows: we first construct the candidate pool by gathering responses $A$ for queries $Q$ from different initial policies $\pi_{\theta_{init}}$. A popular approach to gathering data is to utilize LLM generations with various decoding strategies. Note that human preference data is also a kind of sampling data and constitutes our reservoir of candidates. After the responses are gathered, we have the environment to provide rewards $R$ and then leverage a listwise optimization approach. The updated model $\pi_\theta$ is re-initialized as the sampling policy and generates fresh responses that substitute the prior ones within the candidate pool. Through iterative training, the model progressively enhances the ability for preference alignment. Extensive experiments of the state-of-the-art methods are fairly conducted on multiple benchmarks of dialogue generation and summarization tasks. The results show that the proposed LIRE achieves superior and consistent performance in all the experiments, exhibiting more noticeable gains as we increase the size of the candidate pool. ![Figure 1](image-url) Figure 1. Training pipeline of the proposed LIRE framework. The candidate pool is initially constructed by gathering responses $A$ with different policies $\pi_{\theta_{init}}$ and rewards $R$ from the environment (Reward Model) before they are optimized in a listwise manner. The updated model $\pi_\theta$ is then re-initialized as the sampling policy and generates fresh responses that substitute the prior ones within the candidate pool. Through iterative training, the model progressively enhances the ability for preference alignment. ### 2 RELATED WORK Leveraging human feedback to improve model generation ability toward human desire renders it imperative given the quickly growing family of LLMs. Directly leveraging human feedback to optimize models generally requires an “optimizable” formulation of the feedback Fernandes et al. (2023). However, it is expensive and impractical to generate sufficient human feedback for LLM training in general cases, whether numerical, ranking-based, or even natural language-based. As an alternative, one line of work relies on models to produce feedback that approximates human perception Stiennon et al. (2020); Ouyang et al. (2022); Askell et al. (2021). Given enough feedback (preference data), RLHF has been extensively employed to optimize an LLM with various training objectives using a unified approach. SFT is an alternative approach that involves maximizing the likelihood of the top-1 candidate directly Zhou et al. (2023); Thoppilan et al. (2022). Both methods can be used in tandem as demonstrated in Ouyang et al. (2022), where InstructGPT is proposed to steer model generation better towards human instruction and desire. In the typical setting of RLHF, the model is first fine-tuned with the preference datasets, followed by a reward modeling procedure that gives scores to model output. Finally, RL policies are utilized to maximize the overall reward. This is an online procedure that requires multiple sampling from the updated policy and scoring during training, thus suffering complex training and high computation costs Gulcehre et al. (2023). Many methods have aimed to improve efficiency as well as performance for preference alignment over online RL policies such as PPO. DPO Rafailov et al. (2023) reformulates the constrained reward maximization problem as a direct policy optimization problem by correctly classifying the preference data, which proves to be performant and computationally lightweight. SLiC-HF Zhao et al. (2023b) utilizes the rank calibration loss and cross-entropy regularization loss to learn pairwise human feedback. Other approaches employ ranking-based methods to align preferences, which naturally extend beyond binary-format preference data. RRHF Yuan et al. (2023) learns to align scores of sampled responses with human preferences through pairwise ranking loss among multiple responses. PRO Song et al. (2023) iteratively contrasts the likelihood of the best response against the remaining responses on a rolling basis, using an extended pairwise Bradley-Terry comparison model. These methods consider not only the positive-labeled responses, as in the typical SFT loss, but also negative samples. Another line of work directly utilizes reward scores from reward models for filtering purposes to improve model generation. ReST Gulcehre et al. (2023) introduces two loops and frames the alignment problem as a growing batch RL problem. The outer loop is a Grow step that iteratively augments the training dataset, and the inner loop is an Improve step that involves filtering the generated data and fine-tuning a model on the filtered dataset with offline RL algorithms. Concurrent to this work, RAFT Dong et al. (2023) subsequently selects the $1/k$ percent of samples with the highest reward as the training samples and then fine-tune the model on this filtered dataset. While the above methods all bring improvement to better aligning model output with human preferences, we believe more research and effort should be devoted to this research topic. To the best of our knowledge, reward scores so far have not been explicitly integrated into the training objective, mainly limited to a filter function at most for data selection in offline settings such as in Dong et al. (2023); Gulcehre et al. (2023). Besides, the idea of listwise optimization has not yet been fully studied in this domain. In this paper, we introduce a framework that directly optimizes the expectation of rewards in a listwise fashion, and makes the model more “steerable”. 3 PRELIMINARIES In this section, we illustrate the motivation for the LIRE framework and the related preliminaries. To start with, we give the optimization objective in the common RLHF settings Ouyang et al. (2022); Stienmon et al. (2020); Ziegler et al. (2019): $$\max_{\pi_\theta} \mathbb{E}_{x \sim D, y \sim \pi_\theta(y\|x)} \left( r_\phi(x, y) \right) - \beta \mathbb{D}_{KL} \left( \pi_\theta(y\|x) \|\| \pi_{ref}(y\|x) \right),$$ where $r_\phi$ is the well-trained reward function, and $\pi_{ref}$ and $\pi_\theta$ are the reference policy and the LM policy, respectively. Rafailov et al. (2023) gives the optimal policy of the above KL-constrained objective and further derives this optimal policy under the famous Bradley-Terry model to model the preference. These methods directly or implicitly stem from Equation 1 and are thus always heavily dependent on the KL constraint. In view of the above reasons, we move one step back and start with the original policy gradient methods in RL. The general and coarser expression for the optimization objective in RLHF can be formulated as: $$J(\theta) = \mathbb{E}_{x \sim D, y \sim \pi_\theta(y\|x)} R(x, y) = \sum_{y, x} P_{\pi_\theta}(y\|x) R(x, y),$$ where $P_{\pi_\theta}$ is the probability distribution of the trajectory under some policy $\pi_\theta$, and $R(x, y)$ is the reward model that provides reward signals during training. The ultimate goal of policy gradient methods is to maximize the rewards of the trajectories under the policy $\pi_\theta$. Since this is an on-policy process, the training data has to be sampled iteratively as policy $\pi_\theta$ updates. PPO is a popular method that turns this on-policy learning into an off-policy process, by resorting to importance sampling as well as the KL penalty to approximate the true distribution of the unknown $P_{\pi_\theta}(y\|x)$ Schulman et al. (2017). In this paper, we propose an alternative to approximate $P_{\pi_\theta}(y\|x)$ with sampled responses and $R(x, y)$ with the reward scores. Specifically, our method initially models the probability distribution with the generated responses from LLMs and scores the responses using well-trained reward models. Subsequently, it optimizes the expectation of the final rewards in a listwise manner. 4 METHODOLOGY 4.1 LIRE: LISTWISE REWARD ENHANCEMENT FOR PREFERENCE ALIGNMENT In this section, we reformulate the preference alignment problem and introduce a listwise softmax loss in our LIRE framework. As illustrated in Figure 1, our framework comprises two main components: offline data generation and online model training. In the offline phase, we assume a set of queries $ Q = \{x^{(1)}, x^{(2)}, \ldots, x^{(N)}\} $ is given, and each query is associated with a list of offline responses $ A^{(i)} = \{y_1^{(i)}, \ldots, y_m^{(i)}\}, i \in \{1, \ldots, N\} $. Furthermore, each response $ y_j^{(i)} $ for query $ x^{(i)} $ is paired with a score $ R(x^{(i)}, y_j^{(i)}) $ by some reward model RM. During training, we aim to learn a language model parameterized by $ \theta $, which generates responses with better alignment with human preferences. First, we define a set of token prediction probabilities conditioned on $ x^{(i)} $ as $ P_{\pi_\theta}(y_{j,k}^{(i)}\|x^{(i)}) \in \mathbb{R}^{L \times V} $, where $ L $ is the sequence length and $ V $ the vocabulary size. The probability of the sentence $ y_j^{(i)} $ with $ K $ tokens are formulated as: \[ \pi_\theta(y_j^{(i)}\|x^{(i)}) = \prod_{k=1}^{K} P_{\pi_\theta}(y_{j,k}^{(i)}\|x^{(i)}, y_{j,<k}). \] Next, the probability of the response distribution against response set $ A^{(i)} $ is calculated as: \[ P_{\pi_\theta}(y^{(i)}\|x^{(i)}, A^{(i)}) = \frac{\exp\left(\frac{1}{T} \log \pi_\theta(y^{(i)}\|x^{(i)})\right)}{\sum_{j=1}^{m} \exp\left(\frac{1}{T} \log \pi_\theta(y_j^{(i)}\|x^{(i)})\right)}, \] where $ T $ is a temperature parameter to control the smoothness of the probability distribution. So far we have given an approximation of the $ P_{\pi_\theta} $ in Equation (2), we next derive the listwise loss of our LIRE objective. The general idea is that the quantized scores provide more specific and direct guidance to the model during training, compared to solely based on cardinal ranking numbers. Formally, the loss is calculated as: \[ J(\theta) = -\sum_{i=1}^{N} \mathbb{E}_{y^{(i)} \sim \pi_\theta(y^{(i)}\|x^{(i)})} R(x^{(i)}, y^{(i)}) \] \[ = -\sum_{i=1}^{N} \sum_{j=1}^{m} P_{\pi_\theta}(y_j^{(i)}\|x^{(i)}, A^{(i)}) R(x^{(i)}, y_j^{(i)}). \] In practice, we apply softmax to the reward scores of a single query $ R(x^{(i)}, y^{(i)}) $ due to its property of translation invariance. By doing so we mitigate the influence of different reward scales and maintain stable training parameter settings. To this end, we successfully derived the listwise loss of our LIRE objective. The sophisticated modeling of pairwise comparison among multiple responses has been safely circumvented and the objective in Equation (5) nicely resonates with our initial goal in Equation (2). To develop a general perception of what the model actually learns through the process, we next illustrate the derivative of $ J(\theta) $ with regard to model parameters $ \theta $. We also give a detailed derivation process in Appendix A.1. \[ \nabla J(\theta) = -\frac{1}{T} \sum_{i=1}^{N} \mathbb{E}_{y^{(i)} \sim \pi_\theta(y^{(i)}\|x^{(i)})} \left[ \nabla P_{\pi_\theta}(y^{(i)}\|x^{(i)}, A^{(i)}) \right] \] \[ \times \left( R(x^{(i)}, y^{(i)}) - \mathbb{E}_{y' \sim \pi_\theta(y'\|x^{(i)})} R(x^{(i)}, y'^{(i)}) \right). \] It shows that $ \nabla P_{\pi_\theta}(y^{(i)}\|x^{(i)}, A^{(i)}) $ is the normalized gradient of model predictions, multiplied by a demeaned reward score. These demeaned rewards act as a weighting mechanism that encourages responses with higher scores while depressing those with lower rewards. Relation with pairwise losses and DPO. When the number of candidate responses descends to 2, this listwise loss degenerates into a pairwise loss. Specifically, we rewrite Equation (6) into a pairwise formulation under 2 responses (omitting $A^{(i)}$ for clarity): $$\nabla J_{\text{LIRE-2}}(\theta) = -\frac{1}{T} \sum_{i=1}^{N} \left[ P_1 \times \nabla P_{\pi_\theta}(y_1^{(i)}\|x^{(i)}) + P_2 \times \nabla P_{\pi_\theta}(y_2^{(i)}\|x^{(i)}) \right],$$ where $P_j = \frac{P_{\pi_\theta}(y_j^{(i)}\|x^{(i)})^{\frac{1}{T}}}{\sum_m P_{\pi_\theta}(y_m^{(i)}\|x^{(i)})^{\frac{1}{T}}} \times \delta R(x^{(i)}, y_j^{(i)})$, and $\delta R(x^{(i)}, y_j^{(i)})$ is the corresponding demeaned reward scores, $j \in \{1, 2\}$, $m = 2$. Referring to our previous definition format, we reorganized the gradient of the DPO objective in the following: $$\nabla J_{\text{DPO}}(\pi_\theta; \pi_{\text{ref}}) = -\beta \sum_{i=1}^{N} \left[ r \times \nabla \log \pi_\theta(y_1^{(i)}\|x^{(i)}) + (1-r) \times \nabla \log \pi_\theta(y_2^{(i)}\|x^{(i)}) \right],$$ with $r$ defined by the policy $\pi_\theta$ and reference model $\pi_{\text{ref}}$. Interestingly, these two objectives resemble in that they can both be viewed as the weighted sum of gradients of two responses, with higher weights for preferred responses and lower weights for rejected ones. The difference is that in our LIRE, $P_j$ is determined by offline rewards together with the model predictions. In DPO, $r$ is determined by the differences in the rewards of two responses. Interestingly, we can further substitute $\nabla P_{\pi_\theta}(y_j^{(i)}\|x^{(i)})$ with $\nabla \log \pi_\theta(y_j^{(i)}\|x^{(i)})$ through some algebra and align the derivative objectives. Subsequently, our objective in Equation (7) takes the form: $$\nabla J_{\text{LIRE-2}}(\theta) = -\frac{1}{T^2} \sum_{i=1}^{N} \left[ \tilde{P}_1 \times \nabla \log \pi_\theta(y_1^{(i)}\|x^{(i)}) + \tilde{P}_2 \times \nabla \log \pi_\theta(y_2^{(i)}\|x^{(i)}) \right],$$ where $\tilde{P}_j = \frac{P_{\pi_\theta}(y_j^{(i)}\|x^{(i)})^{\frac{1}{T}}}{\sum_m P_{\pi_\theta}(y_m^{(i)}\|x^{(i)})^{\frac{1}{T}}} \times \delta R(x^{(i)}, y_j^{(i)})$. This way, the relation between LIRE and DPO becomes clearer. Please refer to Appendix A.2 for detailed derivation. 4.2 THE SELF-ENHANCEMENT ALGORITHM Algorithm 1: The self-enhancement strategy for reward maximization during progressive sampling and consecutive training process. An Evolve step is defined as a data generation procedure with policy $\pi_\theta$, followed by subsequent Iterate steps of policy training with regard to objective $J(\theta)$. Input: Input queries $x$, training objective $J(\theta)$, reward model RM, number of samples per query $m$, Language Model with initial policy $\pi_{\theta_{\text{init}}}$, Evolve steps $E$, Iterate steps $I$. 1. for $e = 1$ to $E$ do 2. Generate dataset $D_e$: for each query $x^{(i)}$, sample $m$ responses $A^{(i)} \sim \pi_\theta(y\|x^{(i)})$. 3. Score $D_e$ with the reward model RM. 4. for $i = 1$ to $I$ do 5. Update $\pi_\theta$ on data $D_e$ with the objective $J(\theta)$. 6. end 7. end Output: The learned policy $\pi_\theta$. To further boost the performance, we propose Algorithm 1 to conduct iterative data sampling and incremental policy updates. This iterative strategy is also adopted in works Gulcehre et al. (2023); Dong et al. (2023) and proves to be effective. The whole training outline are divided into two phases: Data Sampling (Evolve) and Policy Training (Iterate). We start by sampling responses from some policy $\pi_{\theta_{\text{init}}}$, and this can be pretrained LLMs or human preference, then we score the responses with some reward model RM. Afterwards, we initialize the target policy $\pi_\theta$ as the pretrained LLM and start to optimize the objective $J(\theta)$ in Equation (5). The current model again samples completions to construct a new candidate pool. One approach is to only keep new candidates with higher reward scores and discard those degraded ones, this way we can better ensure the policy is updated on a higher-quality dataset and prevent policy diverging. Specifically, $E = 1$ suggests we sample responses only once and then conduct training, without iterative sampling afterwards. \| Test Data \| Eval Metric \| ø \| PPO \| DPO \| PRO \| RRHF \| LIRE \| \|-----------\|-------------\|---\|-----\|-----\|-----\|------\|------\| \| HH Test \| PPL \| 10.98 \| 11.81 \| 16.04 \| 16.63 \| 14.66 \| 12.15 \| \| \| RM \| -0.93 \| -0.96 \| -0.87 \| -1.02 \| -0.96 \| -0.85 \| Table 2. Comparison of LIRE and other methods on Anthropic HH Dataset. ø refers to zero-shot results of Alpaca-7B. The best and second best results are marked with Bold and underlined format. 5 EXPERIMENTS 5.1 DATASETS For performance comparison, we mainly focus on dialogue generation and summarization tasks. For dialogue, we use Anthropic’s Helpful and Harmless (HH) dataset. Moreover, in order for a more diverse candidate pool, we sample responses with LLM completions due to their impressive language generation abilities. We follow Yuan et al. (2023) to sample responses from Alpaca-7B Taori et al. (2023) using diverse beam search. All the responses of a single query are scored by reward model RM. For summarization, we use the TL;DR Summarization dataset from Stiennon et al. (2020) and score the resulting responses by RM-SUM. 5.2 COMPARISON METHODS To demonstrate the ability of the proposed LIRE, we conduct an exhaustive investigation into the state-of-the-art methods on human preference alignment tasks. PPO is implemented according to the official code from trlx. DPO Rafailov et al. (2023) optimizes the constrained reward maximization problem in PPO using a single stage of policy training, so it is essentially easier to train and achieves better performance than PPO. PRO Song et al. (2023) and RRHF Yuan et al. (2023) are two preference ranking methods that both support multiple-response ranking. We follow the default configuration settings introduced in the official codes for each method and Lora Hu et al. (2021) is applied for the concern of computation and memory limitation. We implement these methods on Alpaca-7B as the base model. More implementation details can be found in Appendix A.4. 5.3 COMPARE AGAINST THE STATE-OF-THE-ARTS Firstly we conduct a thorough assessment of the methods introduced in Section 5.2 on the Human Preference HH dataset. The automatic evaluation is directed on HH test. We leverage Perplexity (PPL) using gpt2-medium and reward model RM. Since the reward score is our optimization target, we focus more on the analysis of this evaluation indicator. As shown in Table 2, when trained with the HH dataset, LIRE achieves the best performance with regard to the average reward score, with DPO attaining the second-best reward score at the sacrifice of a much lower PPL. As for PPO, it achieves a smaller PPL, very close to the zero-shot results. Our hypothesis is that models trained in a pointwise manner focus more on a single data sample, thus giving more coherent and certain predictions based on the preceding context. Besides, Table 1 gives human evaluation on a subset of Anthropic-HH. The first row gives win rates for human-written (HW) responses versus different methods, and the second row stands for direct comparison between LIRE versus other methods. Win rates greater than or equal to 50 are marked in orange. We also leverage the TL;DR summarization task to validate the proposed LIRE framework in Table 3. To avoid possible model hacking Skalse et al. (2022); Touvron et al. (2023) behavior or inflated reward scores due to overfitting, we additionally utilize another reward model RM-SUM* to evaluate the methods. Note that RM-SUM* and RM-SUM are two different training versions of the same model, and should have similar judgments toward the model responses. We employ RM-SUM* to investigate how the models perform under a reward criterion, which is not identical \| Test Data \| Eval Metric \| Ø \| PPO \| DPO \| PRO \| RRHF \| LIRE \| \|-----------\|-------------\|-----\|-----\|-----\|-----\|------\|------\| \| \| Rouge-L \| 0.096 \| 0.16 \| 0.29 \| 0.32 \| 0.20 \| 0.22 \| \| TL;DR \| RM-SUM \| -1.74 \| 1.16 \| 2.14 \| 1.49 \| 1.35 \| 2.76 \| \| \| RM-SUM* \| -0.31 \| 2.09 \| 1.89 \| 1.15 \| 0.82 \| 2.79 \| Table 3. TL;DR Summarization results of different methods. LIRE got the highest reward scores for both RM-SUM and RM-SUM, with DPO and PPO attaining the second-highest scores, respectively. Figure 2. Left: TL;DR Summarization win rate against human-written baselines. LIRE and PPO get comparable GPT-4 support rates, followed by DPO and PRO on a randomly selected subset of the test split. Right: Radar plot of the MT Bench. This plot gives a clear visual representation of the score distribution across distinct categories for various methodologies. LIRE exhibits the best scores in 6 out of 8 tasks and only slightly falls behind in Reasoning and Math. Apart from automatic evaluation metrics, we leverage GPT-4 to assess the quality of the summarizations since it is known to be greatly correlated with human judgments Liu et al. (2023); Song et al. (2023); Rafailov et al. (2023). We let GPT-4 judge whether the model responses or the human-written baselines are preferred on a subset of the test split. Figure 2 shows that LIRE and PPO achieve quite comparable GPT-4 votes, followed by DPO and PRO. We give real examples of model responses as well as reward scores in Appendix A.3 and evaluation prompts for GPT-4 in Appendix A.7 for further analysis. 5.4 DOES EXTRAPOLATION TO LARGER CANDIDATE POOL HELP? In this section, we explore if increasing the number of samples in our listwise optimization framework can bring a performance boost. For the dialogue task, we sample another 2 and 4 responses with Alpaca as stated in 5.1, resulting in HH-4 (4 responses) and HH-6 (6 responses). Besides, we adopt another dataset introduced by Yuan et al. (2023), which contains 5 candidate responses sampled by ChatGPT, text-davince-003, LLaMA Touvron et al. (2023) and Alpaca using Alpaca prompts Taori et al. (2023). All the responses are scored by ChatGPT on a scale of 10 and we call this dataset General-5. We use General-5 and a subset of it (General-2) to train the models and test on the MT-Bench introduced in Zheng et al. (2023), which contains 80 open-ended questions for evaluating chat assistants. For the summarization task, we directly leverage an Alpaca augmented TL;DR dataset introduced in Song et al. (2023), and we call this dataset TL;DR-3. We mainly compare PRO, RRHF, and LIRE since they are inherently compatible with multiple response comparison and do not require a reference model that adheres to the distribution of the preference data. Table 4 shows that when expanding the number of responses, all three methods witness different degrees of performance boost on the HH test set. Specifically, LIRE secures the largest reward score as well as the smallest PPL, and PRO and RRHF got analogous performance. We observe that expanding the candidate pool sizes brings more pronounced reward improvements for LIRE, which leverages a listwise optimization approach. For the other two methods that primarily leverage a pairwise approach, expanding from HH-4 to HH-6 results in comparatively smaller gains. Therefore, \| Methods \| HH-2 \| HH-4 \| HH-6 \| \|---------\|------\|------\|------\| \| \| RM \| PPL \| RM \| PPL \| RM \| PPL \| \| PRO \| 16.63\| -1.02\| 12.96\| -0.91\| 12.78\| -0.92\| \| RRHF \| 14.66\| -0.96\| 15.79\| -0.92\| 12.71\| -0.95\| \| LIRE \| 12.15\| -0.85\| 12.61\| -0.80\| 12.45\| -0.77\| Table 4. Influence of candidate pool Size for HH test set.* All three counterpart methods achieve an across-the-board enhancement in rewards when increasing the number of responses. \| Eval Metric \| TL;DR-3 \| General-2 \| General-5 \| \|-------------\|---------\|-----------\|-----------\| \| \| Rouge-L \| RM-SUM \| RM-SUM* \| ChatGPT \| ChatGPT \| \| PRO \| 0.33 \| 1.61 \| 1.05 \| 418 \| 405 \| \| RRHF \| 0.32 \| 2.83 \| 2.80 \| 399 \| 406 \| \| LIRE \| 0.23 \| 2.88 \| 3.00 \| 435 \| 467.5 \| Table 5. Performance of various methods evaluated on TL;DR-3 and General datasets. LIRE demonstrates consistent performance. We argue that an augment in the candidate pool during training exhibits a positive correlation with reward improvements in our LIRE framework. Likewise, compared with TL;DR, training with TL;DR-3 brings performance improvement across the methods. For the MT Bench, we see that using General-5 brings more evident benefits than using General-2 for LIRE. For PRO and RRHF the effect is minimal or even opposite. We conjecture that this is because General-2 includes higher-quality responses from ChatGPT and text-davinci-003. Except for the scores in Table 5, we also provide a Radar plot in Figure 2 that gives a clear visual representation of the score distribution across distinct categories for various methods. LIRE exhibits the best scores in 6 out of 8 tasks and only slightly falls behind in Reasoning and Math, striking a better balance across the tasks. Our hypothesis is that the flaw in the reward mechanism itself results in suboptimal performance in certain aspects such as math and reasoning. Generally, while adding model generations does bring out additional advantages, it is a diminishing return if we use a single model to do sampling and provide average-quality responses. Intuitively, higher-quality responses can provide more valuable information and direct the model to learn better preference representations, and diversity also matters because negatives are also important to help the model avoid less preferred patterns. ### 5.5 DO WE NEED TO INCORPORATE THE SFT LOSS? In this section, we explore the effect of integrating the supervised fine-tuning phase into the framework. SFT loss usually refers to the maximum likelihood loss on high-quality human-annotated data. Consequently, the loss is formulated as: $$L(\theta) = J(\theta) + \alpha L_{SFT}(\theta),$$ where $\alpha$ is a hyperparameter to control the weight of the SFT loss to the whole training objective. Specifically, $\alpha$ in Equation 10 should be a relatively small value to contribute a reasonable part to the final loss, otherwise, it will degrade the overall performance. We demonstrate the results on HH-4 in Table 6. Adding an SFT loss helps the model adhere to human preferences, which may introduce an extra reward boost within a limited range, with a suitable parameter of $\alpha$. In Appendix A.8 we explore another regularization technique by adding the KL divergence to preserve knowledge from the pretraining process. ### 5.6 DO MULTIPLE Evolve AND Iterate STEPS FURTHER BOOST PERFORMANCE? In this section, we explore the effects of multiple Evolve and Iterate steps in Algorithm 1. One better approach is to explicitly filter the newly generated candidates to only keep the higher-score responses. \| Iterate \| Evolve \| \|---------\|----------------\| \| I=1 \| E=1(HH) \| \| \| E=1(HH-4) \| \| \| E=2(HH-4)* \| \| \| E=3(HH-4)** \| \| I=2 \| -0.883 \| \| \| -0.977 \| \| \| -0.823 \| \| \| -0.759 \| \| I=3 \| -0.826 \| \| \| -0.779 \| \| \| -0.771 \| \| \| -0.756 \| \| \| -0.813 \| \| \| -0.774 \| \| \| -0.763 \| \| \| -0.731 \| Table 7. Reward score variations during multiple Evolve (E) and Iterate (I) steps. We observe a trend for growing rewards when we increase the steps for Evolve and Iterate. * represents the times of model resampling during training (illustrated as the “Re-initialize” arrow in Figure 1). This suggests that LIRE further boosts performance during iterative data generation and policy training. Figure 3. Left: Average reward scores when trained with different Evolve steps E and Iterate steps I. When trained with larger E and S, LIRE generally witness a reward gain. Right: RM score variation after LIRE enhancement. After LIRE training, most of the extreme cases of low scores are suppressed, which demonstrates the effectiveness of our proposed self-enhancement algorithm. as mentioned in Section 4.2, but here we just keep the human preference data in the candidate pool and replace model responses to avoid an utter distribution shift and maintain a consistent pool size. We also include an SFT loss during training. We experiment with different Evolve steps E and Iterate steps I. The details are listed in Table 7. Specifically, $E = 1(HH)$ means we only utilize the human preference data, without sampling from models. $E = 3(HH - 4)$, $I = 3$ means we sample 4 responses three times and train for 3 epochs in between. The general idea is depicted in Framework 1. We find that when increasing the number of data sampling steps, LIRE generally gives a reward gain. This suggests a further performance boost brought by this iterative sampling strategy. For a clear illustration, we plot the results of $(E = 1(HH), I = 3)$, $(E = 1(HH - 4), I = 3)$, $(E = 3(HH - 4), I = 1)$ when increasing training steps in Figure 3. Also, to understand the score changes brought by our framework from a micro perspective, we plot in Figure 3 the distribution of the reward scores before and after our LIRE enhancement. The result suggests that compared to zero-shot results of Alpaca, most of the extreme cases of low scores are suppressed (the dashed rectangular), thus improving the overall performance. However, we do observe that a fair amount of test samples have decreasing scores after policy training. We further explore this phenomenon with other comparing methods in Appendix A.9. 6 DISCUSSION In this paper, we propose LIRE, a listwise optimization scheme under the general policy gradient framework for preference alignment tasks. 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Glcsog6zOe	The false negative is misleading. In the appendix, the error explanation says that the environment reports that there is no keyboard. Are keyboards part of the possible objects? The observations in the prompt mention a computer but no keyboard.	TREE-PLANNER: EFFICIENT CLOSE-LOOP TASK PLANNING WITH LARGE LANGUAGE MODELS Mengkang Hu ♦ Yao Mu ♦ Xinmiao Yu ♥ Mingyu Ding♦ Shiguang Wu ◊ Wenqi Shao Qiguang Chen♥ Bin Wang ♦ Yu Qiao* Ping Luo* ♦ ABSTRACT This paper studies close-loop task planning, which refers to the process of generating a sequence of skills (a plan) to accomplish a specific goal while adapting the plan based on real-time observations. Recently, prompting Large Language Models (LLMs) to generate actions iteratively has become a prevalent paradigm due to its superior performance and user-friendliness. However, this paradigm is plagued by two inefficiencies: high token consumption and redundant error correction, both of which hinder its scalability for large-scale testing and applications. To address these issues, we propose TREE-PLANNER, which reframes task planning with LLMs into three distinct phases: plan sampling, action tree construction, and grounded deciding. TREE-PLANNER starts by using an LLM to sample a set of potential plans before execution, followed by the aggregation of them to form an action tree. Finally, the LLM performs a top-down decision-making process on the tree, taking into account real-time environmental information. Experiments show that TREE-PLANNER achieves state-of-the-art performance while maintaining high efficiency. By decomposing LLM queries into a single plan-sampling call and multiple grounded-deciding calls, a considerable part of the prompt are less likely to be repeatedly consumed. As a result, token consumption is reduced by 92.2% compared to the previously best-performing model. Additionally, by enabling backtracking on the action tree as needed, the correction process becomes more flexible, leading to a 40.5% decrease in error corrections. 1 INTRODUCTION Task planning is a significant topic in the field of robotics, where a system is tasked with crafting a sequence of mid-level actions (skills) that enable a robot to complete complex high-level tasks (Kaelbling & Lozano-Pérez, 2011). This involves a consideration of various factors, such as the capabilities of robots, the surrounding environment, and any constraints or uncertainties that might exist. An emerging trend within the field of task planning is using Large Language Models (LLMs) to generate actions directly (Huang et al., 2022a; Song et al., 2023), rather than searching within a pre-defined domain (Eysenbach et al., 2019; Xu et al., 2019). As shown in Figure 1, the commonly adopted paradigm for LLM-based planning can be summarized as follows: (i) prompt an LLM to generate one action at a time; (ii) execute the generated action and then append the obtained observation to the LLM; and (iii) generate the next action. We categorize such approaches as ITERATIVE-PLANNER, which enables the model to generate subsequent actions in an auto-regressive manner. Based on ITERATIVE-PLANNER, when errors occur during action execution, existing research endeavors either re-generate actions at the current timestep (Raman et al., 2022; Guo et al., 2023) or re-generate the entire plan from the initial timestep (Shinn et al., 2023), referred to as LOCAL REPLAN and GLOBAL REPLAN, respectively. Corresponding authors: Mingyu Ding and Ping Luo ({dingmyu, pluo.lhi}@gmail.com). ♦The University of Hong Kong. ♥Harbin Institute of Technology. ◊Noah’s Ark Laboratory. ♦Shanghai AI Laboratory. All methods above have the following two drawbacks: (i) Token Inefficiency: The expenses for a single LLM call increase proportionally with the number of tokens utilized, including both the prompt tokens and the generated tokens. However, in the scenario of task planning, the prompt tokens often consist of instructions, global information about the environment, in-context learning examples, and environmental observation (Vemprala et al., 2023) while the generated tokens predominantly represent a concise action. The discrepancy in the number of tokens between prompt tokens and generated tokens results in the issue of token inefficiency (Cheng et al., 2023). Moreover, due to the multi-step nature of a complex task (usually involving 5-20 steps), the prompt tokens incur repeated charges, leading to even higher costs. (ii) Correction Inefficiency: LOCAL REPLAN can be viewed as a trial-and-error approach implemented at the execution-failed time step, which makes it difficult for the model to detect errors that occurred several time steps earlier. While GLOBAL REPLAN can mitigate this problem by regenerating the entire plan, it may still come at the cost of increased time and token consumption. The token and correction inefficiencies inherent in ITERATIVE-PLANNER limit its applicability for large-scale inference or frequent use in everyday life. To address the issues above while maintaining high performance, we propose TREE-PLANNER as illustrated in Figure 2. In general, TREE-PLANNER divides the queries to an LLM into two parts: a single plan-sampling call and multiple grounded-deciding calls to reduce the repetitive computational cost for several components in prompt tokens. These two stages are bridged using a tree-like structure, which leads to more effective logical correction. More specifically, TREE-PLANNER first prompts the LLM to sample potential task plans with its inherent commonsense (Stage I). Subsequently, an action tree is constructed to aggregate the sampled plans (Stage II). Lastly, TREE-PLANNER instructs the LLM again in closed loops to reason on the action tree with the environmental observations (Stage III). In terms of token efficiency, TREE-PLANNER only charges once for global information about the environment and in-context examples in plan sampling. However, for ITERATIVE-PLANNER, this information must be charged at each time step. In terms of correction efficiency, the correction process based on the action tree can be seen as an intermediate between... LOCAL REPLAN and GLOBAL REPLAN. TREE-PLANNER not only reduces the likelihood of redundant decision-making at a specific time step through backtracking but also significantly reduces the time and tokens required to generate the entire plan from scratch. We demonstrate the effectiveness of TREE-PLANNER framework in VirtualHome (Puig et al., 2018), a simulated environment for complex household tasks. The experiments are conducted under two different settings: with correction and without correction. In with correction setting, the model is required to modify the plan when errors occur, while in without correction setting, the opposite is true. The main result shows that TREE-PLANNER achieves state-of-the-art results in both experimental settings, surpassing the best baseline models by 1.29% and 3.65% in terms of success rate, respectively. At the same time, TREE-PLANNER exhibits high efficiency. In terms of token efficiency, TREE-PLANNER reduces the token cost of ITERATIVE-PLANNER by 53.29%. Furthermore, when compared to LOCAL REPLAN and GLOBAL REPLAN under the with correction setting, TREE-PLANNER achieves even greater improvement with reductions of 74.36% and 92.24%, respectively. In terms of correction efficiency, TREE-PLANNER reduces the number of corrections by 37.99% and 40.52%, respectively. In further analysis, we formally verify the token efficiency of TREE-PLANNER and derive the critical value of the number of sampled plans required for the model to possess token efficiency. We also perform an ablation study on both plan sampling and grounded deciding, demonstrating the effectiveness of the individual components of TREE-PLANNER. Finally, we provide a manual error analysis of potential areas for improvement in the model. 2 PRELIMINARY Task and Motion Planning (TAMP) (Kaelbling & Lozano-Pérez, 2011) is the process of generating a sequence of actions and robot motions to achieve a desired goal in a given environment. As is shown in Figure 2, a high-level task description such as “Take nap” is decomposed into several mid-level actions. We assume the existence of a low-level controller that can execute these mid-level actions, which typically requires training using reinforcement learning (RL) methods or fine-tuning with expert data. Task planning can be categorized into closed-loop task planning and open-loop task planning. Open-loop task planning aims to decompose a high-level task description into a mid-level plan without any feedback from the environment. Closed-loop task planning, on the other hand, involves continuously adjusting planning strategies through perception and feedback mechanisms to adapt to environmental changes and uncertainties during execution. This paper focuses on closed-loop task planning, which is more suitable for task execution in dynamic and complex environments. Problem Setup We formulate the closed-loop task planning problem as a partially observable Markov decision processes (POMDPs) denoted by $ \langle S, O, A, T \rangle $, which is similar to Li et al. (2022a). $ S, O, A $ are sets of states, observations and actions respectively and $ T(s_{t+1}\|s_t, a_t) $ is a transition model. In a POMDP setting, the observation $ o_t $ represents a subset of the underlying state $ s_t $. Let $ g $ be the task, the optimal policy $ \pi(a_t\|g, h_t, o_t) $ must take into account not only the current observation $ o_t $, but also the entire history of actions $ h_t = \{a_1, \ldots, a_{t-1}\} $. 3 MODEL 3.1 PLAN SAMPLING Abstract specifications often restrict task planning. Take the “Take nap” task as an example, the robot needs to understand that napping can be done on a bed, and the bed is typically located in a bedroom. Many works hold the belief that LLMs trained on large-scale data encode commonsense knowledge about the real-world (Davison et al., 2019; Li et al., 2022b; Bian et al., 2023). Recently, several studies have investigated the integration of LLMs into task planning, which aims to address language ambiguities and provide robots with background knowledge (Huang et al., 2022a; Li et al., 2022a; Ahn et al., 2022). In contrast to these approaches, which typically use LLMs directly as policies, TREE-PLANNER prompts an LLM to generate prospective task plans before executing them in a specific environment. We consider this as a way to extract commonsense knowledge from LLM through sampling, which serves as prior knowledge for task planning. Let $ \rho_{ps} $ be the prompt for plan sampling, $ g $ be the task name, the process of plan sampling can be formalized as: $ \text{LLM}(\rho_{ps}, g) = c = \{c_1, c_2, \ldots, c_N\} $, where $ N $ is a hyper-parameter which determines the number of sampled plans. Each plan candidate $ c_i $ is a sequence of actions, i.e., $ c_i = \{a_{it} \| t = 1, \ldots, m(i)\} $. $ m(i) $ is the number of actions in plan $ i $ and $ a_{it} $ is the action of plan $ i $ at time step $ t $. The prompt consists of four parts: instruction, global information, initial observation, and in-context examples. The instruction provides the LLM with a clear and concise explanation of the process of task planning. The global information provides the LLM with background knowledge about the environment and available action space. The initial observation provides the LLM with an initial snapshot at the starting point of the task. The in-context examples are additional task plans that serve to indicate the format of the output plan and have also been proven to be helpful in enhancing performance (Brown et al., 2020). In Section 5.2, we provide a quantitative analysis of the upper-bound on plan sampling. ### 3.2 Action Tree Construction ![Figure 3: The process of constructing the action tree. Left: each path represents a sampled plan. Right: plans with the same prefix are aggregated together. Note that although certain paths have the same action ([Sleep]), they are not aggregated together due to inconsistent prefixes.] To select an optimal plan from potential plans, an obvious approach would be to execute and test each plan in the environment. However, this approach has two drawbacks: (i) It is time-consuming to execute multiple plans in the environment; (ii) Different plans may have overlapping parts, so repeating the execution of these overlapping parts in the environment is redundant. For example, in plan 1 and plan 2 shown in Figure 2, the first step in both plans is: “[Walk] <bedroom>”. Based on the previous analysis, we designed a structured representation that aggregates the sampled potential plans called Action Tree. As is shown in Figure 3, when two plans share a common prefix but differ in their actions at a specific time step, their shared prefix is aggregated into a single branch, while their differing actions form divergent paths. This process repeats iteratively until all sampled plans are organized into a complete tree structure. The motivation behind it is to convert the filtering of the plan level into a search at the action level, thereby reducing the execution time in the environment. An action tree with root node $ r $ can be formalized as $ T(c) = (V, E) $, where $ V $ and $ E $ represent the sets of nodes and edges respectively. Each node $ v $ is associated with an action $ a_v $ and a time step $ t_v $, i.e., $ v = (a_v, t_v) $. Each edge $ e $ represents a pair of adjacent actions in plan $ c_i $, i.e., $ E = \{e(v_1, v_2) \| v_1, v_2 \in V, v_1 = (a_{it}, t), v_2 = (a_{(t+1)}, t + 1)\} $. The root node $ r $ is not associated with any specific action, and its child nodes are the set of nodes obtained by aggregating the first action of each plan. The construction process of the action tree is presented in Algorithm 1. ### 3.3 Grounded Deciding During grounded deciding, an LLM functions as the policy $ \pi(a_t \| g, h_t, o_t) $. However, instead of sampling from the entire corpus of LLMs as the ITERATIVE-PLANNER, we limit the choices to a few child nodes of the current node at time $ t $ on the action tree. This process simulates the decision-making process of humans, who first propose several action options and then combine their current real-world observations to make decisions. Specifically, we provide an LLM with instruction, observation, and history (the previously executed actions) as prompts, and then the LLM chooses one from the child nodes of the current node. Furthermore, we also designed a corresponding error correction method. When a chosen action fails to execute in the environment, TREE-PLANNER (i) marks the nodes on the subtree rooted at the failed node as invalid nodes; (ii) traces back on the action tree to find the previous valid fork node with available valid child nodes. If all the child nodes of a particular node are invalid, then the fork node should also be marked as invalid. (iii) executes the inverse process of previously executed actions (e.g., the inverse of [SwitchOn] is [SwitchOff]) to recover the state of the agent; (iv) re-decides at the fork node. Error correction with grounded Algorithm 1: Action Tree Construction Input : c, r Output: r Function ConstructActionTree(c, r): forall ci ∈ c do n ← r; for t = 1 to m(i) do cn ← GetChildNode(n, ai_t); if cn is None then cn ← CreateChildNode(ai_t); AddChildNode(n, cn); end end n ← cn; end deciding is more effective than the commonly adopted methods presented in Section 1. This is because the action tree serves as an important prior to completing the current task. Therefore, when an error occurs at a node on the tree, it is possible to selectively backtrack on the action tree, thus alleviating repetitive decisions at a particular time step as in LOCAL REPLAN. Performing error correction on the action tree also relieves the need to return to the initial time step as in GLOBAL REPLAN, thereby reducing time and token consumption. The process described above is displayed in Figure 4. Quantitive analysis of the effectiveness of error correction is presented in Section 5.3. Figure 4: An overview of the process of grounded deciding. Left: When an error occurs, TREE-PLANNER tracks back and marks the nodes along the way as invalid. Afterward, TREE-PLANNER makes a new decision at the previous fork node. Right: After the action is successfully executed, TREE-PLANNER makes a decision at the current node, and then the agent moves on to the next level. 4 EXPERIMENTAL RESULTS 4.1 EXPERIMENTAL SETUP Environment. We conduct the experiments in the VirtualHome (VH) Environment (Puig et al., 2018), a simulation platform for household tasks. Each scene in every VH environment contains hundreds of objects. These objects may possess individual properties, and there may also be relationships between different objects. There are 28 different action types in VH, which are listed in Appendix A.1. The task-relevant goal conditions refer to a set of specific states of objects or predicates between objects. For example, a goal condition for Turn on TV would be On(TV), while a goal condition for Sit would be On(character, chair). Dataset. We constructed a dataset consisting of 4 VH scenes and 35 unique VH tasks. Each task includes a task name, goal conditions, and a gold plan. We started by annotating goal conditions for each task from ActivityPrograms knowledge base by Puig et al. (2018) via executing the programs. And then, we applied simple heuristics to filter the low-quality annotations in the dataset: (i) the length of the plan is less than 3; (ii) the execution of the program fails. To highlight the necessity of grounding LLMs in the real environment which has variation in the objects and preconditions, we replicated the annotation above process across 4 distinct scenes provided in VirtualHome, ultimately yielding 71 annotated tasks. We denote the 4 distinct scenes as ENV-\{1, 2, 3, 4\}. Then, we hired two CS-majored graduate students to conduct manual quality control to ensure that the task descriptions were in line with their corresponding goal conditions and programs. We eliminate cases that do not meet the alignment criteria or were originally annotated with errors, resulting in a high-quality dataset comprising 35 tasks. To double-check the quality of the dataset, we also study the agreement between annotators. The results indicated “almost perfect agreement” with Fleiss Kappa (Landis & Koch, 1977) scores of 0.88. Evaluation Metrics.* We use four metrics to evaluate the performance of different methods: executability ($\text{EXEC}$), success rate (SR), goal conditions recall (GCR), and the financial expenditure for evaluation ($\text{COST}$). $\text{EXEC}$ refers to whether the plan can be executed in the given environment, regardless of its relevance to the task. GCR is calculated by taking the difference between the ground truth goal conditions and the goal conditions that were achieved with the generated plan and then dividing this difference by the total number of goal conditions. SR measures whether all goal conditions are fulfilled, i.e., $SR = 1$ only when $GCR = 1$. $\text{COST}$ is used to evaluate the token efficiency of different methods, which is calculated based on the pricing provided by OpenAI.\footnote{https://openai.com/pricing} For evaluation with error correction, we use No.EC to represent the number of error corrections of each method. No.EC does not directly measure performance but rather evaluates how effectively different models can correct errors. Baselines. For experiments without error correction, we compare our method to two strong published LLM-based task planning methods with OpenAI APIs, including: (i) Zero-shot Planner (Huang et al., 2022a); (ii) ProgPrompt (Singh et al., 2022). Furthermore, we also implement the Iterative-Planner method discussed in Section 1 as a baseline model. For experiments with error correction, we enhance the Iterative-Planner method with the two re-planning methods: Local Replan and Global Replan, and consider them as the baseline models. More implementation details and an introduction to each baseline model can be found in Appendix B.2. Implementation Details. We use the OpenAI GPT-3.5 (text-davinci-003) API\footnote{https://openai.com/} model as a LLM backbone in our experiments for all evaluated methods. The cost of this model is 0.02$ per 1000 tokens. The prompt for Tree-Planner and Iterative-Planner was designed with the principles proposed in Vemprala et al. (2023), and examples can be found in Appendix F. We take 4 representative tasks from the dataset as in-context learning exemplars and the rest as the validation set. The examples are fixed to be: “Watch TV”, “Turn on light”, “Go to sleep”, and “Brush teeth”. To sample diverse plans, we applied a temperature of 0.8 and a top-p value of 0.95. We heuristically set the number of samplings $N \in \{25, 50\}$. During grounded deciding, we set the temperature to 0.7, top-p to 1.0, and sampling parameter $n$ to 20. Additionally, we utilize a majority vote to obtain the final option in order to alleviate format errors in the output of LLMs. The maximum number of error corrections is set to 10 for all evaluated approaches. ### 4.2 Main Results Based on the results presented in Table 1, several advantages of Tree-Planner can be derived: (i) Tree-Planner outperforms listed baseline systems, surpassing the previous state-of-the-art by absolute 11.2% and 7.04% on Executability, 6.71% and 7.29% on GCR and 1.29% and 3.65% on SR under both experimental settings respectively. This experimental observation demonstrates that reframing the LLM-based planning pipeline does not compromise its performance. (ii) Tree-Planner has a significant advantage in token efficiency. In without correction setting, Tree-Planner reduces the cost of Iterative-Planner by 53.29%. In with correction setting, the token cost is further reduced by 74.36% and 92.24%, respectively, compared to Local Replan and Global Replan. (iii) Tree-Planner also demonstrates high correction efficiency, resulting in a reduction of the number of action-retry times for Local Replan and Global Replan by 37.99% and 40.52%, respectively. A reduced amount of corrections also contributes to a decrease in token consumption. Note that, while not having a token efficiency advantage compared to Zero-shot Planner and ProgPrompt, Tree-Planner significantly outperforms these methods in terms of performance. Table 1: Performance of different methods on Virtual Home. w/o correction means that during the plan execution, there is no allowance for retrying failed actions. While with correction implies the opposite. The reported evaluation metrics are the average of 3 independent runs across the 4 scenes. \| Exec. ↑ \| SR ↑ \| GCR ↑ \| $COST ↓$ \| NO.EC ↓ \| \|---------\|------\|-------\|----------\|--------\| \| w/o correction \| \| \| \| \| \| Zero-shot Planner \| 16.49±3.08 \| 1.07±0.76 \| 1.52±0.75 \| 1.36±0.09 \| N/A \| \| ProgPrompt \| 35.04±3.98 \| 12.54±2.20 \| 19.99±2.83 \| 1.25±0.55 \| N/A \| \| Iterative-Planner \| 44.54±6.09 \| 27.04±4.65 \| 33.25±5.32 \| 5.12±0.14 \| N/A \| \| Tree-Planner $N=25$ \| 55.74±0.92 \| 28.33±1.18 \| 39.96±0.16 \| 2.39±0.44 \| N/A \| \| Tree-Planner $N=50$ \| 49.01±5.67 \| 28.14±2.45 \| 35.84±4.20 \| 3.48±0.04 \| N/A \| \| with correction \| \| \| \| \| \| Local Replan \| 79.66±2.33 \| 37.46±1.71 \| 51.9±0.15 \| 12.88±0.17 \| 3.29±0.46 \| \| Global Replan \| 82.09±1.32 \| 37.93±1.22 \| 52.46±0.86 \| 42.55±0.09 \| 3.43±0.15 \| \| Tree-Planner $N=25$ \| 89.13±0.17 \| 35.30±1.78 \| 56.65±1.09 \| 3.30±0.01 \| 1.85±0.05 \| \| Tree-Planner $N=50$ \| 88.26±2.47 \| 41.58±3.20 \| 59.55±3.20 \| 4.54±0.16 \| 2.04±0.26 \| by 27.26% and 15.79% on SR respectively. It is also worth noting that increasing the hyper-parameter $N$ does not result in consistently improved performance. This experimental phenomenon will be further discussed in Section 5.2. 5 ANALYSIS 5.1 TOKEN EFFICIENCY In Section 4.2, the quantitative analysis has demonstrated that Tree-Planner consumes fewer tokens compared to Iterative-Planner. In this section, we will further provide a specific formulation to demonstrate this point. The number of tokens required for an LLM API call typically includes two parts: prompt tokens and generated tokens. Let $\rho$ and $\varphi$ represent the prompt tokens and generated tokens. Let $ps$, $gd$, $ip$ stand for plan sampling, grounded deciding, and Iterative-Planner, respectively. Normally, we have $\rho_{ip} \approx \rho_{ps} + \rho_{gd}$. That is because, as shown in Figure 2 and Figure 1, the prompt for plan sampling typically includes global information and in-context examples, while the prompt for grounded deciding includes observation and history. These types of information usually need to be included in every step of Iterative-Planner. Assuming that the number of tokens for each action type $\|a\|$ is the same and the total number of steps $M$ is the same for each generated plan. The hyper-parameter number of sampling is $N$ for plan sampling and grounded decoding and is 1 for Iterative-Planner. Based on the given information, we have $\varphi_{ps} = MN\|a\|$, $\varphi_{gd} = N$ and $\varphi_{ip} = \|a\|$. The consumed tokens $\mu_{ours}$ and $\mu_{ip}$ can be calculated as follows: $\mu_{ours} = \rho_{ps} + \varphi_{ps} + M \cdot (\rho_{gd} + \varphi_{gd})$ and $\mu_{ip} = M \cdot (\rho_{ip} + \varphi_{ip})$. Based on the above formula, we can determine the boundary conditions for $N$ that satisfy the inequality $\mu_{ours} < \mu_{ip}$ as follows: $N < \frac{1-1/M}{1+1/\|a\|} \cdot \frac{\rho_{ps}}{\|a\|} + \frac{\|a\|}{\|a\|+1}$. And we have $\rho_{ps} \gg \|a\|$, since the prompt of plan sampling may contain thousands of tokens and an action only contains a few tokens. We use the average number of tokens for all action types to estimate $\|a\|$ and the average length of all gold plans to estimate $M$. As a result, we obtain the critical value of $N$ in our experiment as follows: $N < 197.72$. Detailed derivation can be found in Appendix D. In conclusion, our model exhibits a remarkably high token efficiency, especially in scenarios where $N$ is not particularly high. 5.2 PLAN SAMPLING Since grounded deciding fundamentally involves selecting from the sampled plans, the upper limit of our Tree-Planner is determined by plan sampling. We propose two additional metrics to study the upper limit of plan sampling: (i) the maximum GCR for all generated plans, i.e., $GCR_{max}(c) = \max_{i=1}^{N}(GCR(c_i))$; (ii) the average GCR for all generated plans, i.e., $GCR_{avg}(c) = \frac{1}{N} \sum_{i=1}^{N}(GCR(c_i))$. $GCR_{max}$ represents the upper limit of the performance of Tree-Planner. In other words, the model can only succeed if there is a “correct” plan among the sampled plans. $GCR_{avg}$ reflects the proportion of “correct” plans to sampled plans. When $GCR_{avg}$ is low, it undoubtedly poses greater challenges for grounded deciding. Some conclusions can be drawn from Figure 5: (i) The maximum value of $GCR_{max}$ being 81.2% indicates that plan sampling is effective. (ii) As $N$ increases, there is a noticeable increase in $GCR_{max}$, but it eventually reaches a threshold. Therefore, a large value of $N$ will lead to increased token consumption without necessarily improving the performance limit. When applying TREE-PLANNER, it is essential to choose an appropriate value of $N$ that balances token assumption and model performance. (iii) $GCR_{avg}$ does not consistently increase with an increased $N$. This implies that as $N$ becomes larger, the proportion of “correct” plans to sampled plans may not necessarily increase. 5.3 Grounded Deciding To investigate the effectiveness of grounded deciding, we conducted ablation experiments. We incorporated the gold plan for each task into the construction of the action tree. As is shown in Table 2, after incorporating the gold plan, there was a significant improvement in performance. Additionally, there was also a decrease in the number of error corrections. For TREE-PLANNER$_{N=25}$, the number decreased from 1.85 to 1.21, and for TREE-PLANNER$_{N=50}$, it decreased from 2.04 to 1.39. The quantitative experimental results presented above demonstrate the effectiveness of grounded deciding. Another noteworthy experimental phenomenon is that the improvement in performance for TREE-PLANNER$_{N=25}$ was greater than that for TREE-PLANNER$_{N=50}$. This further validates the conclusion we drew in Section 5.2: when the number of plans increases, but the proportion of correct plans decreases, the performance may be negatively impacted. 5.4 Error Analysis We categorize error types into two distinct classifications: (i) Missing Correct Plan; (ii) Grounded Deciding Error. As is listed in Table 3, the majority of errors are attributed to the missing correct plans (54.5%). Therefore, despite the ability of plan sampling to achieve relatively high $GCR_{max}$ as is discussed in Section 5.2, it still serves as a bottleneck for our model to some extent. Furthermore, a considerable portion of the errors occurred due to mistakes made by LLM during grounded deciding (45.5%). We also provide a qualitative analysis of each error type in Appendix E. \| Error Type \| Explanation \| Proportion(%) \| \|-----------------------------\|--------------------------------------------------\|---------------\| \| Missing Correct Plans \| Plan sampling did not yield correct plans \| 54.5% \| \| Environment Misunderstanding\| Misunderstandings on actions or objects \| 18.2% \| \| Incomplete Plan \| The absence of essential steps \| 18.2% \| \| Illogical Error \| The generated plan is logically incorrect \| 13.6% \| \| Semantically Correct \| Execution failed but semantically correct \| 9.1% \| \| Grounded Deciding Error \| Execution error during grounded deciding \| 45.5% \| \| Incorrect Deciding \| Incorrect decisions at specific nodes \| 31.8% \| \| Semantically Correct \| Execution failed but semantically correct \| 13.7% \| Table 2: Ablation study on grounded deciding. † represents the performance improvement after adding a gold plan to action tree construction. Table 3: Distribution of error types of the TREE-PLANNER$_{N=25}$ w/o correction model. 6 RELATED WORKS Task Planning with Large Language Models. We categorize the mainstream methods in the task planning domain into two groups: search-based methods (Jiang et al., 2018; Garrett et al., 2018) and generate-based methods (Song et al., 2023; Wu et al., 2023a; Ding et al., 2023; Mu et al., 2023). LLMs trained on a large-scale corpus contain a vast amount of commonsense knowledge for task planning (Pallagani et al., 2023; Sun et al., 2023b;a). Thanks to this advancement, generate-based methods have gradually become a hot topic of research in recent years. When considering the utilization of LLM, some works directly generate the entire plan without executing in the environment (Singh et al., 2022; Liang et al., 2023; Wu et al., 2023b; Zeng et al., 2023; Lin et al., 2023b; Yang et al., 2023). While these models possess token efficiency, they are unable to modify the plan when encountering errors dynamically. Another line of works has adopted the paradigm presented in Section 1 to generate actions iteratively (Vemprala et al., 2023; Yao et al., 2022; Huang et al., 2022a;b; Shinn et al., 2023), which is more flexible for error correction, human interaction and the grounding of environment. Works like Carta et al. (2023); Huang et al. (2023); Ahn et al. (2022) involve the use of implicit representations of LLM. In contrast to these works, our study concentrates on Black-box LLMs, which are utilized in a manner more frequently by researchers and industry, as they provide only input and output without any additional information. Tree-based Modeling for the Output of Large Language Models. Yao et al. (2023); Long (2023) both propose an alternative for chain-of-thought, called “tree-of-thought”, for problem-solving. These studies do not involve the interaction between inner steps in the tree and the environment but rather focus on reasoning tasks. Considering the robotic area, Cao & Lee (2023) leverages LLMs for automatic behavior-tree-based task generation. Zhao et al. (2023); Hao et al. (2023) propose using an LLM as a world model to assist planning algorithms such as Monte Carlo Tree Search (MCTS). However, TREE-PLANNER samples diverse paths once and aggregates the paths into an action tree rather than requiring multiple calls to LLM like the aforementioned studies. This approach offers advantages in terms of both run-time efficiency and token efficiency. Generate then Select. From another perspective, grounded deciding selects a prediction from the sampled potential plans. Hence, TREE-PLANNER follows the paradigm of generate then select, which is commonly adopted to optimize the output of LLMs. Some models (Glass et al., 2022; Suzgun et al., 2022; Wang et al., 2023b; Gu et al., 2023) use external controllers to re-rank the generations. In Wang et al. (2023a), the best answer is selected from multiple generations of an LLM through a majority vote. Logeswaran et al. (2022) proposes to incorporate the state information from the environment to re-rank the generated plans. Unlike these works, instead of selecting at the level of entire generation, we use action trees to perform more fine-grained selection (action-level). Efficient Inference with Large Language Models. Most previous works suggest modifying the architecture of transformer or decoding strategy to achieve efficient inference (Wang et al., 2020; Katharopoulos et al., 2020; Leviathan et al., 2023; Chen et al., 2023). Cheng et al. (2023) propose a batch prompting method to reduce the frequency of invoking LLMs. Lin et al. (2023a) achieve efficient inference with LLMs by incorporating a small LM fine-tuned on oracle trajectories. TREE-PLANNER differs from previous studies by simply reframing the process of LLM planning to alleviate repeated token consumption without the need for additional training. 7 CONCLUSION In this paper, we have introduced TREE-PLANNER, a novel framework for task planning with LLMs. The motivation behind TREE-PLANNER is to address the inefficiencies of the commonly adopted paradigm while still achieving high performance. Through extensive experiments in the Virtual-Home environment, we have demonstrated that TREE-PLANNER outperforms other strong baselines and achieves state-of-the-art performance. We have also shown that our framework is highly efficient in terms of token consumption and error correction. To gain a deeper understanding of our framework, we have conducted several studies analyzing its performance gains and identifying potential bottlenecks. Furthermore, we have performed a qualitative error analysis to identify areas where the model may fail. Overall, we believe that TREE-PLANNER represents a new paradigm for LLM-based task planning that strikes a balance between efficiency and performance. We hope that our work will inspire further research and the development of more efficient task-planning methods. 8 ETHICS STATEMENTS We build the dataset based on the ActivityPrograms knowledge base by Puig et al. (2018), which is under the MIT license. Our approach has no ethical or social issues on its own, except those inherited from large language models. 9 ACKNOWLEDGMENTS This paper is partially supported by the National Key R&D Program of China No.2022ZD0161000 and the General Research Fund of Hong Kong No.17200622. 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wwotGBxtC3	I'm curious how making this approach multi-modal can be helpful. Could graph embeddings or vision embeddings of the molecules provide any benefit? I'm not a molecular properties expert, but I tried a couple of the figures (table 2 and figure 2) with GPT-4 vision, and it gave meaningful explanations. Have the authors investigated this?	DATA-EFFICIENT MOLECULAR GENERATION WITH HIERARCHICAL TEXTUAL INVERSION Anonymous authors Paper under double-blind review ABSTRACT Developing an effective molecular generation framework even with a limited number of molecules is often important for its practical deployment, e.g., drug discovery, since acquiring task-related molecular data requires expensive and time-consuming experimental costs. To tackle this issue, we introduce Hierarchical textual Inversion for Molecular generation (HI-Mol), a novel data-efficient molecular generation method. HI-Mol is inspired by a recent textual inversion technique in the visual domain that achieves data-efficient generation via simple optimization of a new single text token of a pre-trained text-to-image generative model. However, we find that its naïve adoption fails for molecules due to their complicated and structured nature. Hence, we propose a hierarchical textual inversion scheme based on introducing low-level tokens that are selected differently per molecule in addition to the original single text token in textual inversion to learn the common concept among molecules. We then generate molecules using a pre-trained text-to-molecule model by interpolating the low-level tokens. Extensive experiments demonstrate the superiority of HI-Mol with notable data-efficiency. For instance, on QM9, HI-Mol outperforms the prior state-of-the-art method with $50\times$ less training data. We also show the efficacy of HI-Mol in various applications, including molecular optimization and low-shot molecular property prediction. 1 INTRODUCTION Finding novel molecules has been a fundamental yet crucial problem in chemistry (Xue et al., 2019; Xu et al., 2019b) due to its strong relationship in achieving important applications, such as drug discovery (Segler et al., 2018; Bongini et al., 2021) and material design (Hamdia et al., 2019; Tagade et al., 2019). However, generating molecules poses a challenge due to their highly structured nature and the vast size of the input space (Drew et al., 2012). To tackle this issue, several works have considered training deep generative models to learn the molecule distribution using large molecular datasets (Ahn et al., 2022; Jo et al., 2022). This is inspired by the recent breakthroughs of generative models in other domains, e.g., images and videos (Rombach et al., 2022; Singer et al., 2022; Yu et al., 2023), in learning high-dimensional and complex data distribution. Intriguingly, such deep molecular generation methods have demonstrated reasonable performance (Jin et al., 2018; 2020; Ahn et al., 2022) on the large-scale benchmarks (Ramakrishnan et al., 2014; Polykovskiy et al., 2020a) in finding chemically valid and novel molecules, showing great potential to solve the challenge. Unfortunately, existing molecular generation frameworks tend to fail in limited data regimes (Guo et al., 2022). This restricts the deployment of existing approaches to practical scenarios, because task-related molecular data for the target real-world applications are often insufficient to train such molecular generative models. For example, drug-like molecules for a specific organ are inherently scarce in nature (Schneider & Fechner, 2005; Altae-Tran et al., 2017), and the drug-likeness of each candidate molecule should be verified through years of extensive wet experiments and clinical trials (Drews, 2000; Hughes et al., 2011). This time-consuming and labor-intensive data acquisition process of new task-related molecules (Stanley et al., 2021) limits the number of available training data for a model to learn the desired molecule distribution. Thus, it is often crucial to develop a data-efficient molecular generation framework, yet this direction has been overlooked in the field of deep molecular generation (Guo et al., 2022) despite its importance in achieving practical applications. Meanwhile, recent works in text-to-image generation have explored the problem of low-shot (or personalized) generation using the power of large pre-trained models trained on a massive amount of data (Ruiz et al., 2022; Wei et al., 2023). In particular, Gal et al. (2022) propose a textual inversion using pre-trained text-to-image diffusion models—given a small set of images, they show that the common concepts among them can be learned effectively by optimizing a single text token under the frozen diffusion model, where the learned token can be used for the desired generation. Considering the recent success of large-scale pre-trained text-to-molecule models (Edwards et al., 2022), what we ask in this paper is: can textual inversion be exploited to enable data-efficient molecular generation with large-scale pre-trained text-to-molecule models? However, we find that naïve adoption of textual inversion fails to achieve the goal, due to the highly complicated and structured nature of molecules (see Figure 2). To exploit textual inversion for data-efficient molecular generation, we suggest considering the unique aspects of the molecule carefully in its adoption. Contribution. We introduce a novel data-efficient molecular generation method, coined Hierarchical textual Inversion for Molecular generation (HI-Mol). Specifically, HI-Mol is composed of two components (see Figure 1 for the overall illustration): - Hierarchical textual inversion: We propose a molecule-specialized textual inversion scheme to capture the hierarchical information of molecules (Alexander et al., 2011). In contrast to textual inversion for the visual domain that optimizes a single shared token on given data, we design multi-level tokens for the inversion so that some of the low-level tokens are selected differently per molecule. Thus, the shared token learns the common concept among molecules and low-level tokens learn molecule-specific features. This low-level token selection does not require any specific knowledge of each molecule and can be achieved completely in an unsupervised manner. - Embedding interpolation-based sampling: We present a molecule sampling scheme that utilizes the multi-level tokens optimized in the inversion stage. Our main idea is to use low-level tokens in addition to the shared token for molecular generation. In particular, we consider using the interpolation of two different low-level token embeddings for generation. The mixing approach is designed to extensively utilize the information of given molecules, and thus effectively alleviates the issue of the limited number of available molecules that lie in the target distribution. We extensively evaluate HI-Mol by designing several data-efficient molecular generation tasks on the datasets in the MoleculeNet benchmark (Wu et al., 2018) and on the QM9 dataset (Ramakrishnan et al., 2014). For instance, in the HIV dataset in MoleculeNet, HI-Mol improves Frechet ChemNet Distance (Preuer et al., 2018, FCD) and Neighborhood Subgraph Pairwise Distance Kernel MMD (Costa & De Grave, 2010, NSPDK) as $20.2 \rightarrow 16.6$, and $0.033 \rightarrow 0.019$ (respectively, lower values are better) from prior arts. HI-Mol also achieves much better active ratio (higher is better) by improving the previous state-of-the-art as $3.7 \rightarrow 11.4$. We also show the strong data-efficiency of HI-Mol. For instance, on QM9, HI-Mol already outperforms the previous state-of-the-arts, e.g., STGG (Ahn et al., 2022) by $0.585 \rightarrow 0.434$ in FCD, with $50\times$ less training data. Finally, we validate the effectiveness of HI-Mol on several downstream tasks including the molecular optimization for PLogP on the ZINC dataset (Irwin et al., 2012) and the low-shot molecular property prediction on MoleculeNet. Figure 2: Visualizations of molecules in two clusters obtained from the unsupervised clustering in Eq. (1) on the HIV dataset (Wu et al., 2018). We note that the molecules often have very different structures, e.g., long carbon chains (left) and sulfonyl benzene groups (right), and thus the naive application of textual inversion with a single shared token does not perform well (see Table 6). 2 RELATED WORK Molecular generation. Most molecular generation methods fall into three categories based on different representations of molecules. First, there exist many attempts (Shi et al., 2020; Zang & Wang, 2020; Niu et al., 2020; Luo et al., 2021; Liu et al., 2021; Jo et al., 2022; Luo et al., 2022; Guo et al., 2022; Hoogeboom et al., 2022; Zhang et al., 2023; Vignac et al., 2023) to formalize molecular generation as a graph generation problem by representing each molecule as an attributed graph. Next, there are several fragment-based methods (Jin et al., 2018; Kong et al., 2022; Geng et al., 2023), which define a dictionary of fragments, e.g., functional groups. Each molecule is represented as a tree structure of dictionary elements and the distribution of connected fragments is then modeled. Finally, there are approaches (Gómez-Bombarelli et al., 2016; Liu et al., 2018; Flam-Shepherd et al., 2022; Ahn et al., 2022) that utilize the Simplified Molecular-Input Line-Entry System (SMILES, Weininger, 1988) representation to write molecules as strings and learn the distribution in this string space. Molecular language model. Following the recent progress in large language models (Raffel et al., 2020; Brown et al., 2020; Touvron et al., 2023), there exist several attempts to train molecular language models (Fabian et al., 2020; Bagal et al., 2021; Christofidellis et al., 2023). Specifically, these works exploit popular language model architectures to have pre-trained models for molecules, based on the SMILES (Weininger, 1988) representation SMILES(x) that interprets a given molecule x as a string. In particular, MoIT5 (Edwards et al., 2022) proposes to fine-tune a large text-to-text language model, T5 (Raffel et al., 2020), with SMILES representations of large-scale molecular data and text description-SMILES pair data to have a text-to-molecule model. Notably, it results in a highly effective pre-trained model for molecules, demonstrating superior performance across text-to-molecule generation tasks. Inspired by its success, we use the Large-Caption2Smiles model trained with this MoIT5 approach for our goal of data-efficient molecular generation. Low-shot generation. There have been substantial efforts in the generative model literature to design a low-shot generation framework for generating new samples from a given small number of data. Intriguingly, recent works on large-scale text-to-image diffusion models have surprisingly resolved this challenge, even enabling “personalization” of the model at a few in-the-wild images through simple optimization schemes that update only a few parameters (Gal et al., 2022; Cohen et al., 2022; Wei et al., 2023). In particular, textual inversion (Gal et al., 2022) exhibits that the personalization of large-scale text-to-image diffusion models can be achieved even with a very simple optimization of a single additional text token without updating any pre-trained model parameters. In contrast to the recent advances of low-shot generation in the image domain, developing a low-shot (or data-efficient) molecular generation framework is relatively under-explored despite its practical importance (Altae-Tran et al., 2017; Guo et al., 2022). Hence, our method tackles this problem by designing a molecule-specific textual inversion method using the recent large-scale text-to-molecule models. Specifically, due to our unique motivation to consider “hierarchy” of molecular structures (Alexander et al., 2011), our method effectively learns the molecule distribution of low-shot molecules with diverse molecular structures, while the applications of prior works, e.g., Guo et al. (2022), are limited to structurally similar low-shot molecules such as monomers and chain-extenders. 3 HI-Mol: Hierarchical Textual Inversion for Molecular Generation In Section 3.1, we provide an overview of our problem and the main idea. In Section 3.2, we provide descriptions of textual inversion to explain our method. In Section 3.3, we provide a component-by-component description of our method. 3.1 Problem Description and Overview We formulate our problem of data-efficient molecular generation as follows. Consider a given molecular dataset $\mathcal{M} := \{x_n\}_{n=1}^N$, where each molecule $x_n$ is drawn from an unknown task-related molecule distribution $p(x\|c)$. Here, $c$ represents the common underlying chemical concept among molecules in the dataset for the target task, e.g., blood-brain barrier permeability or ability to inhibit HIV replication. We aim to learn a model distribution $p_{\text{model}}(x)$ that matches $p(x\|c)$, where the number of molecules $N$ in the dataset is small, e.g., $N = 691$ in the BACE dataset. To solve this problem, we take the recent approach of textual inversion (Gal et al., 2022) from the text-to-image diffusion model literature—a simple yet powerful technique in low-shot image generation that learns a common concept in given images as a single token in text embedding space. Similarly, we aim to learn the common concept of molecules as text tokens and use them for our target of data-efficient generation. However, exploiting this approach for our goal faces several challenges, mainly due to the unique characteristics of molecules differentiated from images. First, it is yet overlooked which of the large-scale model for molecules is beneficial to achieve textual inversion for molecules, like the success of text-to-image diffusion models in achieving successful inversion in the image domain. Moreover, molecules have a very different structural nature from images—unlike images, molecules with similar semantics often have entirely different structures (see Figure 2), making it difficult to simply learn the common concept as a single text token. Our contribution lies in resolving these challenges by adopting molecule-specific priors into the framework to enjoy the power of textual inversion techniques in achieving data-efficient molecular generation. 3.2 Preliminary: Textual Inversion Recent text-to-image generation methods have proposed textual inversion (Gal et al., 2022), which aims to learn a common concept $c$, i.e., the distribution $p(x\|c)$, from a small set of images and use it for the concept-embedded (or personalized) generation. To achieve this, they optimize a single text embedding of a token $[S^]$ shared among images to learn $c$ using a pre-trained frozen text-to-image diffusion model $f_{t2i}$. Specifically, they put $[S^]$ with a short text description, e.g., “A photo of [S]”, as the text prompt to $f_{t2i}$, and then optimize this token embedding using given images with the exact same training objective that is used for training $f_{t2i}$. We propose to adapt the textual inversion framework into the data-efficient molecular generation framework based on the recent state-of-the-art large-scale pre-trained text-to-molecule generative model, MolT5 (Edwards et al., 2022). 3.3 Detailed Description of HI-Mol Hierarchical textual inversion. We first propose a molecule-specific textual inversion to learn the desired molecule distribution. Unlike prior textual inversion that assumes a single shared token $[S^]$ only, we propose to use “hierarchical” tokens $[S^], \{[I_k^]\}_{k=1}^K, \{[D_n^]\}_{n=1}^N$ (with parametrizations $\theta := (s, \{i_k\}_{k=1}^K, \{d_n\}_{n=1}^N)$) by introducing additional intermediate tokens $\{[I_k^]\}_{k=1}^K$ and detail tokens $\{[D_n^]\}_{n=1}^N$ (with $K < N$). Such intermediate and detail tokens learn cluster-wise (high-level) and molecule-wise (low-level) features of the molecular dataset, respectively. To learn these hierarchical tokens, we consider a frozen text-to-molecule model $f$, e.g., Large-Caption2Smiles (Edwards et al., 2022), to apply our proposed hierarchical textual inversion objective. --- 1 We use SMILES strings as the representation of molecules because our method is built upon the state-of-the-art text-to-molecule model that utilizes SMILES strings, i.e., MolT5 (Edwards et al., 2022). However, our method is agnostic to the underlying molecule representation of the text-to-molecule models. Specifically, we optimize $\theta$ by minimizing the following objective on the given molecular dataset $M$: $$L(\theta; x_n) := \min_{k \in [1, K]} L_{CE}\left(\text{softmax}(f("The molecule is a } [S^][I^_k][D^_n]"), \text{ SMILES}(x_n)\right),$$ where $L_{CE}$ denotes cross-entropy loss and $\text{SMILES}(x_n)$ is a SMILES (Weininger, 1988) string of $x_n$. Thus, each $x_n$ is interpreted as three tokens $[S^][I^_c][D^_n]$, where we assign intermediate token index $c_n \in [1, K]$ (for given $x_n$ and the corresponding $[D^_n]$) during optimization to minimize the training objective $L$ (see Eq. (1)). We note that the selection of $[I^_c]$ is achieved in an unsupervised manner so that it does not require any specific information about each molecule. Intriguingly, we find that $[I^_c]$ can learn some of the informative cluster-wise features through this simple selection scheme although we have not injected any prior knowledge of given molecular data (see Figure 2 for an example). Our “multi-level” token design is particularly important for the successful inversion with molecules because molecules have a different nature from images that are typically used in the existing textual inversion method. Image inputs in the conventional textual inversion are visually similar, e.g., pictures of the same dog with various poses, whereas molecules often have entirely different structures even if they share the common concept, e.g., ability on the blood-brain membrane permeability (Wu et al., 2018). This difference makes it difficult to learn the common concept as a simple single token; we mitigate it by adopting hierarchy in the inversion scheme by incorporating the principle of chemistry literature highlighting that molecular data can be clustered hierarchically (Alexander et al., 2011). Embedding interpolation-based sampling. We propose a sampling strategy from the learned distribution via hierarchical textual inversion. We find that sampling schemes used in existing textual inversion for images, e.g., putting a text prompt including $[S^]$ such as “A similar chemical of $[S^]$” into the molecular language model $f$, does not work well in molecular generation (see Table 6). To alleviate this issue, we propose to utilize the learned intermediate tokens $\{[I^_k]\}_{k=1}^K$ and detail tokens $\{[D^_n]\}_{n=1}^N$ to sample from our target distribution. We consider the interpolation of each of intermediate tokens and detail tokens in the sampling process, i.e., we incorporate the hierarchy information of molecules which is obtained in our textual inversion. Specifically, we sample a novel molecule with random molecule indices $i, j$ sampled uniformly from $[1, \ldots, N]$ and a coefficient $\lambda$ drawn from a pre-defined prior distribution $p(\lambda)$ (see Appendix A for our choice of $p(\lambda)$): $$\begin{align} (\bar{i}, \bar{d}) &:= \lambda(i_c, d_i) + (1 - \lambda)(i_j, d_j), \\ x &:= f("A similar chemical of $[S^][\bar{I}^][\bar{D}^]$"), \end{align}$$ where $[\bar{I}^], [\bar{D}^]$ indicate that we pass interpolated embeddings $\bar{i}, \bar{d}$ to $f$, respectively, and $c_n \in [1, K]$ is an index of the intermediate token of a given molecule $x_n$, i.e., an intermediate token index that minimizes the training objective in Eq. (1). This additional consideration of low-level tokens $\{[I^_k]\}_{k=1}^K, \{[D^_n]\}_{n=1}^N$ (as well as $[S^]$) encourages the sampling process to exploit the knowledge from given molecular dataset extensively, mitigating the issue of scarcity of target molecules that lie in our desired molecule distribution and thus enables to generate high-quality molecules. We provide qualitative analysis on our embedding interpolation-based sampling scheme in Appendix I. ### 4 EXPERIMENTS We extensively verify the superiority of HI-Mol by considering various data-efficient molecular generation scenarios. In Section 4.1, we explain our experimental setup. In Section 4.2, we present our main molecular generation results on MoleculeNet and QM9. In Section 4.3, we present results on downstream tasks, i.e., optimization and low-shot property prediction. Finally, in Section 4.4, we conduct some analysis and an ablation study to validate the effect of components of our method. We provide further ablation study and additional experimental results in Appendix E and F, respectively. #### 4.1 EXPERIMENTAL SETUP Datasets. Due to the lack of benchmarks designed particularly for data-efficient molecular generation, we propose to use the following datasets for evaluating molecular generation methods under our problem setup. First, we consider three datasets in the MoleculeNet (Wu et al., 2018) benchmark. --- 2We simply set the number of clusters $K$ as 10 in our experiments. Please see Appendix E for analysis on $K$. Table 1: Quantitative results of the generated molecules on the three datasets (HIV, BBBP, BACE) in the MoleculeNet benchmark (Wu et al., 2018). We mark in Grammar if the method explicitly exploits the grammar of molecular data and thus yields a high Valid. score. The Active. score is averaged over three independently pre-trained classifiers. We compute and report the results using the 500 non-overlapping generated molecules to the training dataset. We set the highest score in bold. ↑ and ↓ indicate higher and lower values are better (respectively) for each metric. \| Dataset \| Method \| Class \| Grammar \| Active. ↑ \| FCD ↓ \| NSPDK ↓ \| Valid. ↑ \| Unique. ↑ \| Novelty ↑ \| \|---------\|--------\|-------\|---------\|-----------\|-------\|---------\|----------\|-----------\|------------\| \| HIV \| GDSS (Jo et al., 2022) \| Graph \| ✗ \| 0.0 \| 34.1 \| 0.080 \| 69.4 \| 100 \| 100 \| \| \| DiGress (Vignac et al., 2023) \| Graph \| ✗ \| 0.0 \| 26.2 \| 0.067 \| 17.8 \| 100 \| 100 \| \| \| JT-VAE (Jin et al., 2018) \| Fragment \| ✓ \| 0.0 \| 38.8 \| 0.221 \| 100 \| 25.4 \| 100 \| \| \| PS-VAE (Kong et al., 2022) \| Fragment \| ✓ \| 3.7 \| 21.8 \| 0.053 \| 100 \| 91.4 \| 100 \| \| \| MiCaM (Geng et al., 2023) \| Fragment \| ✓ \| 3.4 \| 20.4 \| 0.037 \| 100 \| 81.6 \| 100 \| \| \| CRNN (Segler et al., 2018) \| SMILES \| ✗ \| 3.3 \| 2.9 \| 0.064 \| 100 \| 90.0 \| 100 \| \| \| STGG (Ahn et al., 2022) \| SMILES \| ✓ \| 1.6 \| 20.2 \| 0.033 \| 100 \| 95.8 \| 100 \| \| \| HI-Mol (Ours) \| SMILES \| ✓ \| 11.4 \| 19.0 \| 0.019 \| 60.6 \| 94.1 \| 100 \| \| \| HI-Mol (Ours) \| SMILES \| ✓ \| 11.4 \| 16.6 \| 0.019 \| 100 \| 95.6 \| 100 \| \| BBBP \| GDSS (Jo et al., 2022) \| Graph \| ✗ \| 0.0 \| 35.7 \| 0.065 \| 88.4 \| 99.2 \| 100 \| \| \| DiGress (Vignac et al., 2023) \| Graph \| ✗ \| 8.2 \| 17.4 \| 0.033 \| 43.8 \| 94.6 \| 100 \| \| \| JT-VAE (Jin et al., 2018) \| Fragment \| ✓ \| 80.6 \| 37.4 \| 0.202 \| 100 \| 10.8 \| 100 \| \| \| PS-VAE (Kong et al., 2022) \| Fragment \| ✓ \| 84.9 \| 17.3 \| 0.039 \| 100 \| 91.6 \| 100 \| \| \| MiCaM (Geng et al., 2023) \| Fragment \| ✓ \| 82.0 \| 20.3 \| 0.021 \| 100 \| 89.4 \| 100 \| \| \| CRNN (Segler et al., 2018) \| SMILES \| ✗ \| 88.8 \| 20.2 \| 0.026 \| 54.0 \| 100 \| 100 \| \| \| STGG (Ahn et al., 2022) \| SMILES \| ✓ \| 89.1 \| 14.4 \| 0.019 \| 99.8 \| 95.8 \| 100 \| \| \| HI-Mol (Ours) \| SMILES \| ✓ \| 94.4 \| 11.2 \| 0.011 \| 78.8 \| 92.9 \| 100 \| \| \| HI-Mol (Ours) \| SMILES \| ✓ \| 94.6 \| 10.7 \| 0.009 \| 100 \| 94.2 \| 100 \| \| BACE \| GDSS (Jo et al., 2022) \| Graph \| ✗ \| 9.1 \| 66.0 \| 0.205 \| 73.4 \| 100 \| 100 \| \| \| DiGress (Vignac et al., 2023) \| Graph \| ✗ \| 21.1 \| 26.7 \| 0.102 \| 16.4 \| 100 \| 100 \| \| \| JT-VAE (Jin et al., 2018) \| Fragment \| ✓ \| 40.4 \| 49.1 \| 0.304 \| 100 \| 13.0 \| 100 \| \| \| PS-VAE (Kong et al., 2022) \| Fragment \| ✓ \| 57.3 \| 37.9 \| 0.114 \| 100 \| 75.6 \| 100 \| \| \| MiCaM (Geng et al., 2023) \| Fragment \| ✓ \| 56.2 \| 18.5 \| 0.060 \| 100 \| 64.2 \| 100 \| \| \| CRNN (Segler et al., 2018) \| SMILES \| ✗ \| 79.0 \| 21.7 \| 0.066 \| 38.0 \| 100 \| 100 \| \| \| STGG (Ahn et al., 2022) \| SMILES \| ✓ \| 42.9 \| 17.6 \| 0.053 \| 100 \| 94.8 \| 100 \| \| \| HI-Mol (Ours) \| SMILES \| ✓ \| 81.0 \| 16.4 \| 0.052 \| 71.0 \| 69.9 \| 100 \| \| \| HI-Mol (Ours) \| SMILES \| ✓ \| 80.4 \| 14.0 \| 0.039 \| 100 \| 74.4 \| 100 \| (originally designed for activity detection): HIV, BBBP, and BACE, which have a significantly smaller number of molecules than popular molecular generation benchmarks (Sterling & Irwin, 2015; Polykovskiy et al., 2020b). For example, BACE includes only 691 active molecules. With only the active molecules in each dataset, we construct tasks to generate novel molecules that share the same chemical concept, e.g., blood-brain membrane permeability for the BBBP dataset. Moreover, we also utilize the QM9 dataset (Ranakrishnan et al., 2014) for our experiments to show the data-efficiency of HI-Mol. Specifically, we train our method with an extremely small subset of the entire QM9 training split, e.g., 2%, where other baseline methods are trained with the whole training split (105k molecules). We provide more details about the datasets in Appendix B. Evaluation setup. To evaluate the quality of the generated molecules, we consider six metrics that represent diverse aspects which are critical to the evaluation of the generated molecules, e.g., similarity to the target distribution, uniqueness, and novelty. We incorporate some well-known metrics, such as those used in Jo et al. (2022), as well as introducing a new metric “Active ratio”: - Active ratio: Our proposed metric, measuring the ratio of the valid generated molecules that are active, i.e., satisfying the target property for the relevant task. - Fréchet ChemNet Distance (FCD): Metric for measuring the distance between the source distribution and the target distribution using pre-trained ChemNet. - Neighborhood Subgraph Pairwise Distance Kernel MMD (NSPDK): Another metric for measuring the gap between source and the target distributions, based on algorithmic computation using graph-based representations of molecules. - Validity (Valid.): The ratio of the generated molecules that have the chemically valid structure. For reliable evaluation with our metric, we avoid the overlap between the generated molecules and the training data used for generation methods by ignoring the molecule if it is contained in this dataset. Hence, the Novelty score is 100 for all MoleculeNet experiments since all samples are different from the training set (see Table 1 for an example). We provide the detailed description of this metric in Appendix C. Table 2: Qualitative results of the generated molecules on the two datasets (HIV, BBBP) of the MoleculeNet benchmark (Wu et al., 2018). We visualize the generated molecules from each method that has the maximum Tanimoto similarity with a given anchor molecule. We report the similarity below each visualization of the generated molecule. We set the highest similarity in bold. \| Dataset \| DiGress (Vignac et al., 2023) \| MiCaM (Geng et al., 2023) \| STGG (Ahn et al., 2022) \| HI-Mol (Ours) \| Train \| \|---------\|-----------------------------\|--------------------------\|------------------------\|---------------\|-------\| \| HIV \| ![image](image1.png) \| ![image](image2.png) \| ![image](image3.png) \| ![image](image4.png) \| ![image](image5.png) \| \| \| 0.154 \| 0.146 \| 0.157 \| 0.326 \| \| \| BBBP \| ![image](image6.png) \| ![image](image7.png) \| ![image](image8.png) \| ![image](image9.png) \| ![image](image10.png) \| \| \| 0.238 \| 0.247 \| 0.246 \| 0.505 \| \| - Uniqueness (Unique.): Diversity of the generated molecules based on the ratio of different samples over total valid molecules earned from the generative model. - Novelty: Fraction of the valid molecules that are not included in the training set. Baselines. We mainly consider the following methods for evaluation: GDSS (Jo et al., 2022), DiGress (Vignac et al., 2023), DEG (Guo et al., 2022), JT-VAE (Jin et al., 2018), PS-VAE (Kong et al., 2022), MiCaM (Geng et al., 2023), CRNN (Segler et al., 2018), and STGG (Ahn et al., 2022). For evaluation on QM9, we also consider GraphAF (Shi et al., 2020), GraphDF (Luo et al., 2021), MoFlow (Zang & Wang, 2020), EDP-GNN (Niu et al., 2020), and GraphEBM (Liu et al., 2021), following the recent works (Jo et al., 2022; Luo et al., 2022). We provide more details of the baselines in Appendix D. ### 4.2 Main results Generation on MoleculeNet. Table 1 summarizes the quantitative results of the generated molecules on the HIV, BBBP, and BACE datasets in the MoleculeNet benchmark (Wu et al., 2018). Our method consistently outperforms other generation methods in terms of Active ratio, FCD, and NSPDK scores on all three datasets. We note that the improvements in these scores are particularly crucial for the deployment of the molecular generation method. For example, the superior Active ratio of HI-Mol, e.g., $3.7 \rightarrow 11.4$ on the HIV dataset, indicates that the generated molecules are more likely to exhibit the desired activeness. Our method also significantly improves the FCD metric on the HIV dataset from $20.2 \rightarrow 19.0$, and this indicates the effectiveness of HI-Mol in generating more faithful molecules that lie in the target distribution. We provide qualitative results in Table 2 by visualizing some of the generated molecules from each dataset. One can observe that the generated molecules by HI-Mol capture several crucial common substructures, e.g., many ester groups, while introducing the novel components, e.g., 4-membered ring, due to our interpolation-based sampling scheme. We also propose a simple algorithm to modify the generated invalid SMILES by correcting invalid patterns without a computational overhead. By applying this algorithm, we convert all invalid SMILES to valid ones, therefore, Validity becomes 100. In particular, the modified molecules further improve the overall metrics, e.g., FCD by $19.0 \rightarrow 16.6$ and $11.2 \rightarrow 10.7$ in the HIV and BBBP dataset, respectively. This indicates the modified SMILES indeed represent molecules from the desired distribution and further highlights the superior quality of our generated molecules. Generation on QM9. In Table 3, we report the quantitative results of the generated molecules from each method. Here, we train our method with a limited portion of data, e.g., 2% and 10%, and then compare the results with the baselines that are trained with the entire dataset. Our model shows strong data-efficiency: only with a 2% subset of the training data, our method already outperforms the state-of-the-art baseline, STGG (Ahn et al., 2022), by 0.585 $\rightarrow$ 0.430 in FCD. Utilizing a 10% subset further improves the performance of HI-Mol, reducing the FCD by 0.430 $\rightarrow$ 0.398. In particular, compared with STGG, HI-Mol not only improves the FCD score but also shows a better Novelty score, which validates the capability of HI-Mol to find novel molecules from the target distribution. --- 4For example, we modify the invalid SMILES caused by the unclosed ring, e.g., C1CCC $\rightarrow$ CCCCC. Please see Appendix H for detailed algorithm. We mark in Grammar column when modification is applied for evaluation. Table 3: Quantitative results of the generated molecules on the QM9 dataset (Ramakrishnan et al., 2014). We mark in Grammar if the method explicitly exploits the grammar of molecular data and thus yields a high Valid. score. Following the setup of Jo et al. (2022), we report the results using 10,000 sampled molecules. We denote the scores drawn from Luo et al. (2022) and Ahn et al. (2022) with () and (†), respectively. We mark (-) when the score is not available in the literature. We set the highest score in bold. ↑ and ↓ indicate higher and lower values are better (respectively) for each metric. For our method, we report the ratio of the number of samples of the dataset used for training. \| Method \| Class \| Grammar \| FCD ↓ \| NSPDK ↓ \| Valid. ↑ \| Unique. ↑ \| Novelty ↑ \| \|-------------------------\|---------\|---------\|-------\|---------\|----------\|-----------\|-----------\| \| CG-VAE† (Liu et al., 2018) \| Graph \| ✓ \| 1.852 \| - \| 100 \| 98.6 \| 94.3 \| \| GraphAF (Shi et al., 2020) \| Graph \| ✗ \| 5.268 \| 0.020 \| 67 \| 94.5 \| 88.8 \| \| MoFlow (Zang & Wang, 2020) \| Graph \| ✓ \| 4.467 \| 0.017 \| 91.4 \| 98.7 \| 94.7 \| \| EDP-GNN (Niu et al., 2020) \| Graph \| ✗ \| 2.680 \| 0.005 \| 47.5 \| 99.3 \| 86.6 \| \| GraphDF (Luo et al., 2021) \| Graph \| ✗ \| 10.82 \| 0.063 \| 82.7 \| 97.6 \| 98.1 \| \| GraphEBM (Liu et al., 2021) \| Graph \| ✗ \| 6.143 \| 0.030 \| 8.22 \| 97.8 \| 97.0 \| \| GDSS (Jo et al., 2022) \| Graph \| ✓ \| 2.900 \| 0.003 \| 95.7 \| 98.5 \| 86.3 \| \| GSDD (Luo et al., 2022) \| Graph \| ✓ \| 2.650 \| 0.003 \| 99.9 \| - \| - \| \| STGG† (Ahn et al., 2022) \| SMILES \| ✓ \| 0.585 \| - \| 100 \| 95.6 \| 69.8 \| \| HI-Mol (Ours; 2%) \| SMILES \| ✗ \| 0.434 \| 0.001 \| 90.7 \| 75.8 \| 73.5 \| \| HI-Mol (Ours; 2%) \| SMILES \| ✓ \| 0.430 \| 0.001 \| 100 \| 76.1 \| 75.6 \| \| HI-Mol (Ours; 10%) \| SMILES \| ✗ \| 0.400 \| 0.002 \| 87.6 \| 87.6 \| 71.2 \| \| HI-Mol (Ours; 10%) \| SMILES \| ✓ \| 0.398 \| 0.001 \| 100 \| 88.3 \| 73.2 \| For an extensive comparison with the baselines which show high Novelty scores, e.g., GDSS (Jo et al., 2022), we adjust the sampling strategy slightly; we utilize a simple resampling strategy (which takes only 1.8 sec per molecule) and make the Validity, Uniqueness, and Novelty scores to 100 for a fair comparison in FCD with those methods. Even in this case, HI-Mol achieves FCD of 0.601, which outperforms all those baselines. We provide detailed results and discussions in Appendix G. 4.3 Applications of HI-Mol Molecular optimization. We demonstrate the effectiveness of HI-Mol in molecular optimization, mainly following the experimental setup of Ahn et al. (2022). We train a conditional molecular generative model $ p_{\text{model}}(x\|\gamma) $ under the HI-Mol framework where $ \gamma $ denotes the penalized octanol-water partition coefficient (PLogP). Then, we sample with a high $ \gamma $ to generate molecules with high PLogP. In Table 4, our HI-Mol generates molecules with considerably high PLogP even when trained with only 1% of the entire training dataset. Here, we remark that solely maximizing the molecular property (such as PLogP) may generate unrealistic molecules (Ahn et al., 2022), e.g., unstable or hard-to-synthesize (see Appendix K). To address this and highlight the practical application of our HI-Mol framework, we further show the model’s capability to generate molecules with the desired PLogP. In Figure 3, HI-Mol generates realistic molecules with the target PLogP, even when the desired condition $ \gamma $ is unseen in the training molecules. The overall results show that our HI-Mol exhibits a huge potential for real-world scenarios where we aim to generate molecules with a specific target property. Low-shot molecular property prediction. We show that the generated molecules by HI-Mol can be utilized to improve the performance of classifiers for low-shot molecular property prediction. Here, we collect low-shot molecules from the MoleculeNet benchmark (Wu et al., 2018) and generate molecules via molecular generative models for each label. In Table 5, HI-Mol consistently shows the superior $ \Delta \text{ROC-AUC}^5 $ score. This demonstrates the efficacy of HI-Mol to learn the concept, e.g., activeness and in-activeness, of each label information with a limited number of molecules. In practical scenarios, where the label information is hard to achieve, our HI-Mol indeed plays an important role in improving the classifier. We provide experimental details in Appendix L. --- 5This score is calculated by the improvement of the ROC-AUC score when the generated molecules are additionally added to the original low-shot training data; higher is better. Table 4: Results of molecular property maximization task. We report the top-3 property scores denoted by 1st, 2nd, and 3rd. The baseline scores are drawn from Ahn et al. (2022). \| Method \| PlogP \| \|-----------------\|-------\| \| \| 1st \| 2nd \| 3rd \| \| GVAE (Kusner et al., 2017) \| 2.94 \| 2.89 \| 2.80 \| \| SD-VAE (Dai et al., 2018) \| 4.04 \| 3.50 \| 2.96 \| \| JT-VAE (Jin et al., 2018) \| 5.30 \| 4.93 \| 4.49 \| \| MHG-VAE (Kajino, 2019) \| 5.56 \| 5.40 \| 5.34 \| \| GraphAF (Shi et al., 2020) \| 12.23 \| 11.29 \| 11.05 \| \| GraphDF (Luo et al., 2021) \| 13.70 \| 13.18 \| 13.17 \| \| STGG (Ahn et al., 2022) \| 23.32 \| 18.75 \| 16.50 \| \| HI-Mol (Ours; 1%) \| 24.67 \| 21.72 \| 20.73 \| Table 5: Average ΔROC-AUC of the low-shot property prediction tasks with 20 random seeds. \| Dataset \| Method \| 16-shot \| 32-shot \| \|---------\|-------------------------\|---------\|---------\| \| HIV \| DiGress (Vignac et al., 2023) \| -2.30 \| -2.67 \| \| \| MiCaM (Geng et al., 2023) \| 1.02 \| 0.69 \| \| \| STGG (Ahn et al., 2022) \| 0.53 \| -0.47 \| \| \| HI-Mol (Ours) \| 2.35 \| 2.16 \| \| BBBP \| DiGress (Vignac et al., 2023) \| 1.73 \| 0.97 \| \| \| MiCaM (Geng et al., 2023) \| 1.91 \| 1.78 \| \| \| STGG (Ahn et al., 2022) \| 1.85 \| 1.76 \| \| \| HI-Mol (Ours) \| 2.73 \| 2.64 \| \| BACE \| DiGress (Vignac et al., 2023) \| -0.60 \| -0.91 \| \| \| MiCaM (Geng et al., 2023) \| -0.65 \| -1.11 \| \| \| STGG (Ahn et al., 2022) \| 2.34 \| 2.01 \| \| \| HI-Mol (Ours) \| 3.53 \| 3.39 \| Table 6: Ablation of the components of hierarchical textual inversion on the QM9 dataset (Ramakrishnan et al., 2014) with 2% subset. We report the results using 10,000 sampled molecules. \| Training prompt \| FCD ↓ \| NSPDK ↓ \| Valid. ↑ \| Unique. ↑ \| Novelty ↑ \| \|-----------------\|-------\|---------\|----------\|-----------\|-----------\| \| The molecule is a $[S^]$ \| 7.913 \| 0.041 \| 96.2* \| 19.3 \| 39.5 \| \| The molecule is a $[S^][D_n^]$ \| 0.486 \| 0.002 \| 93.8 \| 70.8 \| 72.3 \| \| The molecule is a $[S^][I_c^][D_n^]$ \| 0.434* \| 0.001 \| 90.7 \| 75.8 \| 73.5 \| 4.4 ANALYSIS Effect of intermediate tokens. Recall that we have introduced intermediate text tokens $\{[I_k^]\}_{k=1}^K$, which are selected in an unsupervised manner during the hierarchical textual inversion to learn some of the cluster-wise features included in given molecules. To validate the effect of our text token design, we visualize the clustering results in Figure 2 by providing groups of the molecules that have the same intermediate token. As shown in this figure, molecules are well grouped according to their common substructures, e.g., a long carbon chain or sulfonyl benzene groups. Such a learning of cluster-wise low-level semantics is indeed beneficial in molecular generation, since molecules often share the concept, e.g., molecular property, even when they have large structural differences. Ablation on hierarchical tokens. To validate the effect of each token in our proposed hierarchical textual inversion, we perform an ablation study by comparing the results with our method where some of the tokens are excluded from the overall framework. Specifically, we compare the generation performance of the following three variants: (1) using the shared token $[S^]$ only, (2) using $[S^]$ and the detail tokens $[D_n^]$, and (3) using all three types of tokens (HI-Mol). Note that for (1), it is impossible to apply our interpolation-based sampling; hence, we use temperature sampling instead based on the categorical distribution from a molecular language model with temperature $\tau = 2.0$. We provide this result in Table 6: as shown in this table, introducing each of the additional tokens successively improves most of the metrics, while maintaining the Validity score as well. 5 CONCLUSION We propose HI-Mol, a data-efficient molecular generation framework that utilizes a molecule-specialized textual inversion scheme. Specifically, we propose to capture the hierarchical information of molecular data in the inversion stage, and use it to sample novel molecules. We hope our method initiates under-explored but crucial research direction in the data-efficient generation of molecules. Limitation and future work. In this work, we apply our novel textual inversion scheme to the molecular language model (Edwards et al., 2022), where developing such a model is a very recently considered research direction. An important future work would be improving the large-scale molecular language models themselves, e.g., the breakthroughs in the image domain (Rombach et al., 2022), which will allow more intriguing applications of HI-Mol, such as composition (see Appendix F). ETHICS STATEMENT This work will facilitate research in molecular generation, which can speed up the development of many important generation tasks such as finding drugs for a specific organ and disease when the hit molecules are rarely known. However, malicious use of well-learned molecular generative model poses a potential threat of creating hazardous molecules, such as toxic chemical substances. It is an important research direction to prevent such malicious usage of generative models (OpenAI, 2023). On the other hand, molecular generation is also essential for generating molecules to defend against harmful substances, so the careful use of our work, HI-Mol, can lead to more positive effects. REPRODUCIBILITY STATEMENT We provide explicit description of our training objective and the sampling method in Section 3.3. We list the hyper-parameter information and the hardware information in Appendix A. We describe the details of datasets and evaluation metrics in Appendix B and C, respectively. We provide our molecule modification algorithm in Appendix H. We submit the code implementation of our HI-Mol framework as a supplementary material. REFERENCES Sungsoo Ahn, Binghong Chen, Tianzhe Wang, and Le Song. Spanning tree-based graph generation for molecules. In International Conference on Learning Representations, 2022. 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S3x7IcbwY8	In the method part, the authors put a lot of effort into introducing the overlap part with Pix2Seq V2, such as the tokenizer and masked modeling. However, the difference from Pix2Seq V2 is not well presented.	Masked AutoDecoder is Effective Multi-Task Vision Generalist Anonymous authors Paper under double-blind review Abstract Inspired by the success of general-purpose models in NLP, recent studies attempt to unify different vision tasks in the same sequence format and employ autoregressive Transformers for sequence prediction. They apply uni-directional attention to capture sequential dependencies and generate task sequences recursively. However, such autoregressive Transformers may not fit vision tasks well, as vision task sequences usually lack the sequential dependencies typically observed in natural languages. In this work, we design Masked AutoDecoder (MAD), an effective multi-task vision generalist. MAD consists of two core designs. First, we develop a parallel decoding framework that introduces bi-directional attention to capture contextual dependencies comprehensively and decode vision task sequences in parallel. Second, we design a masked sequence modeling approach that learns rich task contexts by masking and reconstructing task sequences. In this way, MAD handles all the tasks by a single network branch and a simple cross-entropy loss with minimal task-specific designs. Extensive experiments demonstrate the great potential of MAD as a new paradigm for unifying various vision tasks. MAD achieves superior performance and inference efficiency compared to autoregressive counterparts while obtaining competitive accuracy with task-specific models. 1 Introduction Computer vision covers various concepts, such as localization, classification, and description, leading to a wide variety of highly structured outputs in different vision tasks, i.e., object detection, instance segmentation, keypoint detection, image captioning, etc. Following natural language processing (NLP), recent methods [Lu et al., 2022; Chen et al., 2022a; Wang et al., 2023; Kolesnikov et al., 2022] attempt to unify different vision tasks in an autoregressive sequence-to-sequence framework as illustrated in the upper part of Fig. 1. They first model different vision tasks in the same sequence format, such as a sequence of coordinate and class label tokens for object detection, a sequence of contour coordinate tokens for image segmentation, or a sequence of descriptive sentences for image captioning. Additionally, the autoregressive Transformers [Brown et al., 2020; Radford et al., 2018; 2019], with its specially designed uni-directional attention to capture sequential dependencies, are employed to recursively predict these vision task sequences. Despite the success, the autoregressive approach often struggles on vision tasks due to two major factors: (1) The discrepancy between vision and language. Language task sequences [Brown et al., 2020; Touvron et al., 2023] heavily follow sequential dependencies while vision task sequences may not, e.g., the next word prediction in a sentence highly depends on its preceding texts, while the pixel prediction in segmentation tasks largely depends on its neighboring content instead of merely previous ones. The autoregressive approach, with uni-directional attention, can well capture sequential dependencies for language tasks but may not fit well with vision tasks. (2) Computation Efficiency. The autoregressive approach predicts tokens in a sequence recursively, which is computation-intensive. The two factors might limit the model performance and efficiency, hindering the application of the autoregressive approach to vision tasks. One possible solution for mitigating the above two issues is to explore bi-directional attention and parallel prediction for sequence modeling. This design leads to a customized Transformer that is capable of capturing more comprehensive dependencies and decodes the task sequence from scratch in parallel. However, such decoding process from scratch may struggle while modeling task contexts. as sequences from different tasks highly vary in patterns, lengths, token vocabularies, etc., which will impede network convergence and result in inferior performance for multi-task learning. Driven by the above analysis, we present Masked Decoder (MAD), an effective sequence-based generalist for vision tasks. As illustrated in the bottom-left part of Fig.1, MAD masks tokens randomly from the task sequences and reconstructs the masked tokens based on the unmasked ones and image features, which provides rich task contexts for modeling disparate task sequences. In addition, it adopts an encoder-decoder Transformer architecture with bi-directional attention that leverages comprehensive dependencies in vision tasks to effectively decode task sequences in parallel. These designs enable a more efficient and effective multi-task learning framework that performs multiple vision tasks in a single architecture. Our experiments with four tasks (object detection, instance segmentation, key-point detection, and image captioning) on the COCO dataset demonstrate that a simple MAD can achieve competitive accuracy and efficiency compared to both task-customized approaches and existing generalist models. 2 RELATED WORKS Vision Generalist Models Learning a vision generalist model capable of handling multiple vision tasks using a shared architecture has long been a goal in computer vision. Inspired by the success of unified sequence-to-sequence based transformer framework (Devlin et al., 2018; Radford et al., 2018, 2019) in natural language processing (NLP), recent works (Alayrac et al., 2022; Wang et al., 2022; Reed et al., 2022; Chen et al., 2022b) extend this framework to the field of computer vision and model various vision tasks in a unified sequence-to-sequence autoregressive paradigm. The pioneering works (Cho et al., 2021; Li et al., 2022; Zhu et al., 2022) mainly focus on high-level semantic tasks, such as image captioning, visual question answering, image-text matching, and etc., considering their intrinsic correlation with language. In pursuit of unifying more vision tasks, especially for those involving dense predictions, Pix2seq (Chen et al., 2021, 2022a) and Uni’Tab (Yang et al., 2022) discrete object positions as a series of coordinate tokens to enable the localization capability for generalist models. Unified-IO (Lu et al., 2022) and UVIM (Kolesnikov et al., 2022) encode the per-pixel targets into semantic tokens for vision tasks that require outputs as images, such as depth estimation or panoptic segmentation. Uni-PercieverV2 (Li et al., 2023) equips an additional region proposal network to generate sequence predictions for object detection and instance segmentation. VisionLLM (Wang et al., 2023) leverages LLM to enable flexible task output formats. Different from these methods which focus on customizing and extending more vision tasks in a sequence-based autoregressive framework, we demonstrate that such a framework may not fit well for vision tasks. Our masked Auto-decoding pursues a conceptually different direction and learns diverse task contexts in parallel via masked sequence modeling, leading to a more efficient and effective vision generalist. Masked Signal Modeling The paradigm of learning rich representations via masking and reconstructing has been widely explored in both fields of NLP and computer vision. In NLP, through masking and recovering language sentences, models like BERT (Devlin et al., 2018) and its variants (Liu et al., 2019; Lan et al., 2019) successfully pre-train models capable of generalizing to a broad range of NLP tasks. In computer vision, such a paradigm also leads to multiple masked image modeling (MIM) (Gao et al., 2022; Dong et al., 2022) and masked video modeling (MVM) techniques. For example, BEiT (Bao et al., 2021) explores MIM by recovering the masked image into visual tokens from discrete VAE (Ramesh et al., 2021). SimMIM (Xie et al., 2022), MaskFeat (Wei et al., 2022), and MAE (He et al., 2022) incorporate low-level visual signals, such as RGB pixel value or the feature descriptor HOG (Dalal & Triggs, 2005), as the reconstruction targets. VideoMAE (Feichtenhofer et al., 2022) encodes the corrupted video and learns to recover both spatial and temporal signals. The above methods employ masked signal modeling as a self-supervised task, aiming to learn to auto-encode rich representations for downstream tasks. Different from them, we propose masked auto-decoder (MAD), exploring masked sequence modeling for decoding task sequences from its masked variants. Our approach is close to non-autoregressive translation (Ghazvininejad et al., 2019; Gu et al., 2017) in NLP, but it has very different intrinsic objectives - non-autoregressive translation exploits parallel decoding to improve translation efficiency, while MAD aims to model diverse task contexts for learning multi-task vision generalists. ### 3 METHODS Our proposed unified generalist framework consists of three key components: (1) Unified tokenization of diverse input and output sequences for different tasks; (2) Masked auto-decoding framework for modeling task contexts; (3) An architecture that decodes desired task sequences based on image features. We introduce these components in the following sections. #### 3.1 Task Tokenization In this work, we consider four vision-related tasks, including object detection, instance segmentation, keypoint detection, and image captioning. These tasks require the model’s ability from classification to localization, from vision to language, and from image-level to pixel-level recognition. Therefore, A comprehensive vocabulary is essential for dealing with such sophisticated problems. Our vocabulary comprises five parts, including prompt tokens to distinguish tasks, coordinate tokens for localization, category tokens for classification, task-related special tokens, and word tokens for captioning, more detail to be elaborated in the ensuing subsections. For object detection, following Pix2Seq (Chen et al., 2021), we convert bounding boxes into a sequence of tokens consisting of discrete coordinates and categories by the order of $[x_{min}, y_{min}, x_{max}, y_{max}, class]$. As described in Fig. 2, we construct a sequence consisting of $N$ noise objects, and then randomly replace and inject the ground truth object in the sequence. The $< Detection >$ prompt tokens are added before the sequence to identify the task. We set $N$ at 100 by default. For instance segmentation, we directly predict the pixel mask following Mask R-CNN (He et al., 2017). The bit masks of the size $M \times M$ are flattened and transformed into sequences consisting of $< Foreground >$ tokens and $< Background >$ tokens. We concatenate the prompt sequence consisting of task token $< Segmentation >$, bounding box coordinate tokens, and a class token to identify different instances. Figure 2: Four vision tasks are tokenized into a unified sequence format. Sequences for object detection consist of coordinate tokens and class tokens. For instance segmentation, we adopt two customized tokens to represent foreground pixels (pixel=1) and background pixels (pixel=0). Keypoint detection shares the same coordinate tokens to object detection, with two additional tokens for visible and invisible keypoints. We adopt sentence-piece model \cite{Kudo2018} to tokenize captioning sentences into subword token sequences, but show words for simplicity. For keypoint detection, we predict the coordinates and visibility for each keypoint of the person instance. It can thus be represented as a sequence of $[x, y, \text{visibility}, x, y, \text{visibility}, ...]$. We adopt two tokens $<\text{Visible}>$ and $<\text{Invisible}>$ to depict the visibility. The keypoints are arranged by the default order as in COCO dataset \cite{Lin2014}. For the occluded keypoints, we replace their coordinate tokens with random coordinates within the bounding box. We utilize the sequence $<\text{Keypoint}, x_{\text{min}}, y_{\text{min}}, x_{\text{max}}, y_{\text{max}}, \text{person}>$ to prompt keypoint detection task, where the coordinates in the prompt indicate the bounding box of the corresponding person. For captioning, we adopt a pre-trained sentence-piece model (SPM) \cite{Kudo2018} to convert a caption into a sequence of discrete tokens. We randomly replace one of the tokens in the transferred sequence with a random word token for sequence augmentation. All the sequences are padded or truncated to a length of 20 tokens. The $<\text{Caption}>$ token is adopted as the prompt. 3.2 Masked AutoDecoding Masked Training We propose Masked Auto-Decoding for multi-task sequence modeling. We randomly sample a subset of target tokens and mask the remaining ones. The sampling follows a uniform distribution. The masked tokens are replaced by special $<\text{Mask}>$ tokens, which are shared among all tasks. During training, we adopt two kinds of sequences for each task, a fully masked sequence and a partly masked sequence. The reconstruction of fully masked sequences establishes a basis to train a unified decoder, which is, learning to decode multi-task sequences with only the prompts. Therefore, all the tokens, except those in the prompt sequences, are replaced with $<\text{Mask}>$ before being fed into the decoder. The training objective is to reconstruct the desired task sequences based on task prompts. However, different from the autoregressive approach where each task sequence is specified by its corresponding input sequence, a fully masked sequence in auto-decoding might match multiple similar task sequences, such as differently arranged objects for object detection or similar captioning sentences per image for image captioning. Randomly choosing the reconstruction target each time might hinder convergence. Hence, we adopt Hungarian Matching \cite{Kuhn1955} to construct the task sequences for object detection and image captioning. For instance segmentation and keypoint detection, the original unmasked sequences are adopted as targets since their prompt sequences with object locations and categories are able to specify the unambiguous task sequences. However, when all the tokens are masked, it is difficult for the model to distinguish different task sequences based on only a few prompt tokens. We thus leverage partly masked sequences to alleviate this issue. The unmasked tokens provide rich cues for the pattern of different task sequences, which help the decoder capture diverse task contexts. During training, both fully and partly masked sequences are concatenated together and decoded in parallel. The MAD task is greatly inspired by the self-supervised masked auto-encoding approach in both language and vision domains, which learns and encodes informative representation by reconstructing masked content. We expand this idea of masked modeling to decode multi-task sequences in computer vision. This simple method, by modeling corrected sequences and predicting missing tokens, enables MAD to learn distinct task contexts and inter-sequence dependencies for vision tasks. Masked Inference During inference, we conduct multi-stage masked decoding to refine the prediction. With the initial prediction recovered from the fully masked sequence, we sample part of the sequences and replace them with mask tokens again. The corrupting sequences are then fed to the decoder for reconstruction. We directly ensemble predictions from masked tokens to their original tokens to obtain more accurate predictions. ### 3.3 Architecture Our goal is to build a single model that is capable of handling different vision-related tasks within a unified sequence paradigm with little task-customized designs. Hence, we adopt a simple encoder-decoder transformer architecture, which has been proven successful in handling sequences with variable lengths in both natural language processing and computer vision tasks. As shown in Fig. 3, the overall architecture of MAD consists of three main components, a backbone network with the encoder to extract image features, a decoder to reconstruct the masked sequences, and a vocabulary tokenizer that transforms between token sequences and embedding. Backbone and Transformer Encoder. Given an input image $I \in \mathbb{R}^{H \times W \times 3}$, a backbone network is adopted to generate a low-resolution image feature with a stride of 32. The encoder then takes the image feature, adds 2D positional encodings, and processes the feature via a series of encoder layers consisting of a self-attention module and feed-forward network (FFN). The image feature is then injected into the decoder as a condition to decode task sequences. Vocabulary Tokenizer. We leverage vocabulary tokenizer to transform between token sequences and sequence embeddings. It maintains a vocabulary of embeddings with a dimension of $D$, which corresponds to all tokens as described in Sec. 3.1. Before being fed into the decoder, the discrete Masked Sequences of a length $L$ are converted into Masked Sequence Embeddings $E \in \mathbb{R}^{L \times D}$ by directly indexing the vocabulary. After being recovered by the decoder, we adopt cosine similarity to transform the Reconstructed Sequence Embeddings back to the Predicted Sequences. Transformer Decoder. The decoder follows standard architecture, reconstructing Masked Sequence Embeddings through self-attention, cross-attention, and FFN layers. To address the sequence order, we introduce learned Sequence Positional Encoding and add them to the input embeddings before each attention layer in the decoder. The Sequence Positional Encodings are shared among all the tasks and are truncated according to the length of different task sequences. Unlike existing autoregressive methods (Chen et al., 2022a; Lu et al., 2022) that adopt uni-directional masks in self-attention layers and generate only one token at a time, our model decodes all the sequence embeddings in parallel with bi-directional attention, leading to more efficient and effective predictions. 3.4 Multi-task Training Loss Function. We adopt a softmax cross-entropy loss to maximize the likelihood of masked sequence conditioned on the image feature: $$L = \sum_t W_t \frac{1}{N_m} \sum_{i \in M} \log P(\hat{y}_i \| x, y)$$ where $y$ and $\hat{y}$ are masked and decoded sequences, $W_t$ is loss weights for different tasks, $M$ means the set of masked tokens, and $N_m$ denotes the number of masked tokens. Only the loss of masked tokens is counted. Following previous practice (Carion et al., 2020; Al-Rfou et al., 2019), we adopt auxiliary losses for the predictions after each decoder layer. For each task, we filter the target token vocabulary so that losses are only calculated on its involved vocabulary to improve training efficiency. Considering tokens of the whole vocabulary leads to intensive computation and memory usage since image captioning involves plenteous text tokens that are not involved in other tasks. Task Mixed Sampling. For learning a single model for multiple tasks, we employ a task mixed sampling strategy where each image in the dataset is sampled with its annotations mixed from all tasks. The sampled images are processed by the backbone and encoder only once for encoding image features shared by all tasks. Only the decoding process is repeated for different tasks, considering that they hold different sequence lengths and are hard to process in parallel. Such a strategy is conceptually simple and effective compared with the batch mixing strategy from existing work (Chen et al., 2022a; Li et al., 2023) where each batch only samples image-sequence pairs for a single task. Considering that each image might involve multiple vision tasks, batch mixing requires encoding the same image multiple times for different tasks. As a comparison, task mixing provides a more flexible framework to add more data from more tasks, while also sharing most model components among tasks, resulting in better efficiency. 4 Experiments 4.1 Experimental Settings Dataset and Tasks. Following previous practice (Chen et al., 2022a; Wang et al., 2023), we evaluate MAD on MS-COCO dataset (Lin et al., 2014) which contains 118k training images and 5k validation images with annotations for all four tasks we considered. For object detection, we take $N = 100$ instances per image for training, resulting in a sequence of length 500. The coordinates of bounding boxes are discretized into 500 bins. For instance segmentation, we randomly sample ten instances and Table 1: Comparisons for object detection (AP), instance segmentation (AP), keypoint detection (AP), and image captioning (BLEU@4 (Papineni et al., 2002)) on COCO validation set. \| Task-specific Models \| Backbone \| Param. \| Det. \| Seg. \| Kpt. \| Cap. \| \|----------------------\|----------\|--------\|------\|------\|------\|------\| \| Faster R-CNN (Ren et al., 2015) \| R101-FPN \| 42M \| 42.0 \| - \| - \| - \| \| DETR (Carion et al., 2020) \| R101-DC5 \| 60M \| 44.9 \| - \| - \| - \| \| Pix2Seq (Chen et al., 2021) \| R101-DC5 \| 57M \| 45.0 \| - \| - \| - \| \| Mask R-CNN (He et al., 2017) \| X101-FPN \| 107M \| 42.9 \| 38.6 \| - \| - \| \| Keypoint R-CNN (Wu et al., 2019) \| R50-FPN \| 59M \| - \| - \| 65.5 \| - \| \| Transformer (Sharma et al., 2018) \| Encoder \| - \| - \| - \| - \| 34.0 \| \| Generalist Models \| Backbone \| Param. \| Det. \| Seg. \| Kpt. \| Cap. \| \|-------------------\|----------\|--------\|------\|------\|------\|------\| \| VisionLLM (Wang et al., 2023) \| R50+Alpaca-7B \| 40M + 7B \| 44.6 \| 25.1 \| - \| 31.0 \| \| Pix2SeqV2 (Chen et al., 2022a) \| ViT-B \| 132M \| 46.5 \| 38.2 \| 64.8 \| 34.9 \| \| MAD (Ours) \| Swin-B \| 107M \| 49.7 \| 40.6 \| 64.6 \| 32.2 \| transform their segmentation masks into bit masks with a size of $16 \times 16$. For keypoint detection, we train MAD on ten person instances per image and only predict keypoints for detected humans (based object detection results) during inference. We pad blank instances for the above three tasks if there are not enough instances existing in the image. For image captioning, we adopt sentence piece model (SPM) from T5 (Raffel et al., 2020) for tokenization, and abbreviate its vocabulary based on COCO dataset, resulting in 11421 remaining tokens. We use loss weights of [1.5, 2.7, 0.5, 0.3] for object detection, instance segmentation, keypoint detection, and image captioning respectively. At inference time, we first predict task sequences for object detection as they will serve as the prompt for the subsequent tasks. The detection sequences are decoded into detection results, represented by five tokens including four coordinate tokens and one class token, while the probability of the class token is adopted as the detection score. For instance segmentation, we directly convert predicted sequences into bit masks based on probabilities of $<\text{Foreground}>$ tokens. For keypoint detection, the predicted sequences are dequantized into tuples of keypoint coordinates with probabilities of $<\text{Visible}>$ showing their visibility. For image captioning, the sequence is truncated by the first padding token and directly mapped back to text by SPM. We conduct masked inference on keypoint detection and image captioning tasks with mask ratios of 0.7, and 0.8, 0.6, 0.4 respectively. Implementation Details. We implement MAD with two different backbones, Swin-Base for comparison to state-of-the-art methods, and Resnet-50 for ablations. Both the encoder and decoder in Transformer consist of 6 layers with a main dimension of 256 and 8 attention heads, and the width of FFN is set to 2048. For sequence modeling, we adopt learned positional encodings with a length of 506 to cover all task sequences. We use the AdamW optimizer with an initial transformer’s learning rate of $1e^{-4}$ and backbone of $1e^{-5}$. The batch size is set to 16. For comparisons with state-of-the-art methods, we train the model with Swin-Base (Liu et al., 2021) backbone for 300 epochs with learning rate drop after 200 epochs. For our ablation experiments with the ResNet-50 backbone, we use a shorter training schedule of 50 epochs. We use the same data augmentation strategy consisting of image flipping, randomly resizing, and cropping for all tasks. The input images are re-scaled so that its shortest side is between 480 and 800 pixels while the longest is at most 1333. During inference, the shortest side of the image will be resized to 800 pixels. The inference speeds of all experiments are the total time for inferring on four tasks, tested on a single A100 with a batch size of one image. ### 4.2 Comparison with State-of-the-Art Methods Tab. 1 shows comparisons with the state-of-the-art (SOTA). We compare MAD with two types of SOTA models: (1) typical task-specific models which leverage task-specific designs and are trained on a single task; (2) generalist models which employ a shared single architecture to handle multiple vision tasks without task-specific designs such as region proposal network (RPN) or ROI Pooling. Compared with the task-specific models, we can see that MAD can achieve competitive and even better accuracy without customized architecture for a single task. On top of that, the sequence-based framework in MAD provides significant scalability and flexibility to new tasks or data formats than these models. In addition, MAD also outperforms existing generalist models with fewer parameters, Figure 4: Convergence curves for Autoregressive Decoding, Parallel Decoding, and the proposed MAD in Tab.2. MAD achieves much faster convergence for vision-centric tasks and greatly narrows the gap with Autoregressive Decoding compared with Parallel Decoding for image captioning. Table 2: Ablation studies of MAD. The “(single task)” indicates that the model is separately trained for each single task. The inference time counts the total time of processing all four tasks. \| Methods \| Infer. Time (ms) \| Det. \| Seg. \| Kpt. \| Cap. \| \|------------------------------\|------------------\|------\|------\|------\|------\| \| Autoregressive Decoding \| 3953 \| 27.9 \| 12.3 \| 33.4 \| 34.1 \| \| Parallel Decoding (single task) \| - \| 38.4 \| 31.2 \| 55.1 \| 20.6 \| \| Parallel Decoding \| 137 \| 35.9 \| 29.8 \| 51.5 \| 18.2 \| \| +Masked training \| 137 \| 38.9 \| 32.3 \| 54.6 \| 18.6 \| \| +Masked Inference (MAD) \| 173 \| 38.9 \| 32.3 \| 54.7 \| 29.6 \| especially on vision-centric tasks, demonstrating these tasks greatly benefit from bi-directional attention and masked sequence modeling designs. For image captioning, the autoregressive paradigm in existing methods excels us in modeling language sequential context. We will investigate how to combine the advantages of both in the future for enabling a more versatile generalist model. 4.3 Ablation Studies Main Components Ablation. We first gradually ablate our main designs as shown in Tab.2. We convert MAD into an autoregressive variant with the same architecture for comparison (Details can be found in the supplementary material). It can be seen that Autoregressive Decoding performs worst in terms of both inference time and accuracy on vision tasks except image captioning. This result is consistent with our analysis that the autoregressive approach might not fit well for vision-centric tasks and struggles with extremely slow predictions. By employing bi-directional attention and parallel decoding (i.e., Parallel Decoding), the convergence and inference speed of vision tasks are greatly improved. However, such a simple parallel decoding method suffers from severe performance degradation compared to its single-task model (i.e., Parallel Decoding (single task)), leading to an inferior multi-task learning paradigm. However, we can observe that introducing our masked sequence modeling during training can significantly mitigate the performance degradation for multi-task learning. As shown in the fourth row, +Masked training performs especially better for object detection, instance segmentation, and keypoint detection, thanks to the task context modeled through masking and reconstruction. Moreover, by further introducing masked inference (i.e., +Masked Inference (MAD)), the accuracy is constantly improved with competitive image caption accuracy to the autoregressive counterpart. In addition, we observe that MAD has different effects on vision-centric tasks and language tasks in training and inference. We speculate that MAD in training could model rich task contexts, such as the relationship among task prompts, vocabulary, and sequence patterns, which are crucial for modeling multi-task sequences. On the contrary, during inference, MAD mainly focuses on dependencies among sequence tokens, which are generally rich in language but lacking in visual sequences. Convergence Curves for Vision Tasks. Fig.4 compares the detailed training curves between methods in Tab.2 for different tasks. With bi-directional attention, both MAD and parallel decoding converge much faster than autoregressive decoding which adopts uni-directional attention. In addition, the Table 3: Ablation on masked sequence modeling during training. (a) For “random ratio”, we used a random mask ratio lying between 0.6 and 0.8. For “multiple ratios”, the task sequences are trained with two masking ratios (i.e., 0.6 and 0.8). The “single ratio” indicates that a single mask ratio is adopted. (b) different masking ratios are evaluated under the “single ratio” strategy. (a) Masking ratio strategies. (b) Different mask ratios in training. \| Methods \| Det. \| Seg. \| Kpt. \| Cap. \| \|---------------\|------\|------\|------\|------\| \| random ratio \| 38.6 \| 31.9 \| 54.3 \| 29.4 \| \| multiple ratios \| 38.6 \| 32.0 \| 54.2 \| 29.7 \| \| single ratio \| 38.9 \| 32.3 \| 54.7 \| 29.6 \| \| Mask Ratio \| Det. \| Seg. \| Kpt. \| Cap. \| \|------------\|------\|------\|------\|------\| \| 0.4 \| 38.4 \| 31.8 \| 54.4 \| 29.2 \| \| 0.6 \| 38.5 \| 31.9 \| 54.2 \| 29.7 \| \| 0.7 \| 38.9 \| 32.3 \| 54.7 \| 29.6 \| \| 0.8 \| 38.7 \| 32.1 \| 54.9 \| 28.4 \| Table 4: Ablations on parameters for individual tasks. (a) Number of quantization bins for coordinates. (b) Size of bit mask for image segmentation. (c) Inference mask ratios for image captioning. \| Number of Bins \| Det. \| Kpt. \| \|----------------\|------\|------\| \| 300 \| 38.5 \| 54.2 \| \| 500 \| 38.9 \| 54.7 \| \| 800 \| 38.6 \| 54.5 \| \| 1000 \| 38.5 \| 54.4 \| \| Mask Size \| Det. \| Seg. \| \|-----------\|------\|------\| \| 12 \| 38.4 \| 31.5 \| \| 14 \| 38.8 \| 32.0 \| \| 16 \| 38.9 \| 32.3 \| \| 20 \| 38.8 \| 32.4 \| \| Mask Ratio \| BLUE@4 \| \|------------\|--------\| \| w/o masked inference \| 18.6 \| \| {0.7} \| 25.8 \| \| {0.7, 0.3} \| 27.0 \| \| {0.8, 0.6, 0.4} \| 29.6 \| masked sequence modeling strategy in MAD can further capture rich task contexts and largely improve performances, especially for image captioning. These results further demonstrate the non-trivial design of MAD. Masked Training. We examine how varying masking strategies and masking ratios affects the training of MAD. As Tab.3a shows, the simplest strategy with a single masking ratio could achieve the highest performance. As for specific masking ratios in training (under the single ratio strategy), MAD performs the best with a moderate value of 0.7, while a smaller masking ratio results in an over-simplified task, and a larger masking ratio leaves insufficient tokens for modeling task contexts. Coordinate Quantization. We evaluate the effect of the number of the coordinate bins. As Tab.4a shows, MAD performs robustly under different numbers of bins. We thus adopt 500 as default, while each bin corresponds to approximately 2 pixels for an image with size between 800 to 1333 pixels, resulting in negligible quantization error. Mask Size. In Tab.4b, we study the size of the segmentation mask. It can be seen that MAD does not benefit much from larger mask sizes, since we do not adopt task-specific operations like ROI Align (He et al., [2017]) or interpolation to align mask pixels and image pixels. Considering that larger mask sizes lead to longer task sequences, we set the mask size at 16 for good efficiency. Inference Mask Ratio for Captioning. We examine different inference mask ratios for image captioning. Results in Tab.4c demonstrate that a combination of gradually decreasing masking ratios ({0.8, 0.6, 0.4}) performs the best. ## 5 Conclusion In this work, we propose Masked AutoDecoder (MAD), a sequence-to-sequence multi-task vision generalist that employs masked sequence modeling and parallel decoding. MAD performs multiple vision tasks with a unified task sequence format, and learns to reconstruct the masked task sequences for modeling diverse task contexts. In addition, we employ bidirectional attention and parallel decoding in Transformer, achieving significant speedup in both convergence and inference compared to autoregressive counterparts for vision tasks. 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AwyxtyMwaG	The text says 'We can then test the causal effect of an FV by adding it to hidden states at any layer ℓ as the model resolves a prompt and measuring its performance in executing the task.' I don't understand exactly how the vector is being 'added to hidden states at any layer'. Am I right that this vector is being added to the final state vector at the top of the lth transformer block?	Function Vectors in Large Language Models Eric Todd,* Millicent L. Li, Arnab Sen Sharma, Aaron Mueller, Byron C. Wallace, and David Bau Khoury College of Computer Sciences, Northeastern University Abstract We report the presence of a simple neural mechanism that represents an input-output function as a vector within autoregressive transformer language models (LMs). Using causal mediation analysis on a diverse range of in-context-learning (ICL) tasks, we find that a small number attention heads transport a compact representation of the demonstrated task, which we call a function vector (FV). FVs are robust to changes in context, i.e., they trigger execution of the task on inputs such as zero-shot and natural text settings that do not resemble the ICL contexts from which they are collected. We test FVs across a range of tasks, models, and layers and find strong causal effects across settings in middle layers. We investigate the internal structure of FVs and find while that they often contain information that encodes the output space of the function, this information alone is not sufficient to reconstruct an FV. Finally, we test semantic vector composition in FVs, and find that to some extent they can be summed to create vectors that trigger new complex tasks. Our findings show that compact, causal internal vector representations of function abstractions can be explicitly extracted from LLMs. 1 Introduction Since the study of the lambda calculus (Church, 1936), computer scientists have understood that the ability for a program to carry references to its own functions is a powerful idiom. Function references can be helpful in many settings, allowing expression of complex control flow through deferred invocations (Sussman, 1975), and enabling flexible mappings from inputs to a target task. In this paper we report evidence that autoregressive transformers trained on large corpora of natural text develop a rudimentary form of function references. Our results begin with an examination of in-context learning (ICL; Brown et al., 2020). ICL mechanisms have previously been studied from the perspective of making copies (Olsson et al., 2022) and from a theoretical viewpoint (Von Oswald et al., 2023; Garg et al., 2022; Dai et al., 2023), but the computations done by large models to generalize and execute complex ICL functions are not yet fully understood. We characterize a key mechanism of ICL execution: function vectors (FVs), which are compact vector representations of input-output tasks that can be found within the transformer hidden states during ICL. An FV does not directly perform a task, but rather it triggers the execution of a specific procedure by the language model (Figure 1). Figure 1: An overview of function vectors (FVs). An FV is extracted from activations induced by in-context examples of (a) antonym generation or (b) English to Spanish translation, and then inserted into an unrelated context to induce generation of (c) a new antonym or (d) translation. Correspondence to todd.er@northeastern.edu. Open-source code and data available at functions.baulab.info. Function vectors arise naturally when applying causal mediation analysis (Pearl, 2001; Vig et al., 2020; Meng et al., 2022; 2023; Wang et al., 2022a) to identify the flow of information during ICL. We describe an activation patching procedure to determine the presence of a handful of attention heads that mediate many ICL tasks. These heads work together to transport a function vector that describes the task; the FV can be formed by summing outputs of the causal attention heads. We test the hypothesis that function vectors are a general mechanism spanning many types of functions. To quantify the role and efficacy of function vectors, we curate a data set of over 40 diverse ICL tasks of varying complexity. We calculate FVs for these tasks and investigate impact of FVs in triggering those functions across a variety of LMs scaling up from 6B to 70B parameters. We further ask whether FVs are portable: are the effects of an FV limited to contexts very similar to those where it is extracted, or can an FV apply in diverse settings? We compare the effects of FVs when inserted into diverse input contexts including differently-formatted forms, zero-shot formats, and natural text contexts. We find that FVs are remarkably robust, typically triggering function execution even in contexts that bear no resemblance to the original ICL context. A key question is whether the action of FVs can be explained by word-embedding vector arithmetic (Mikolov et al., 2013; Levy & Goldberg, 2014; Merullo et al., 2023). We examine decodings of FVs (Nostalgebraist, 2020), and find that although FVs often encode a function’s output vocabulary, those vocabularies do not fully identify an FV. In other words, to invoke functions, FVs need to carry some additional information beyond their encoding of the top vocabulary words. Finally, we investigate whether the space of FVs has its own vector algebra over functions rather than words. We construct a set of composable ICL tasks, and we test the ability of FVs to obey vector algebra compositions. Our findings reveal that, to some extent, vector compositions of FVs produce new FVs that can execute complex tasks that combine constituent tasks. We emphasize that FV vector algebra is distinct from semantic vector algebra over word embeddings: for example, composed FV vectors can specify nonlinear tasks such as calculating the antonym of a word, that cannot themselves be implemented as a simple embedding-vector offset (Appendices A, L). 2 METHOD 2.1 A MOTIVATING OBSERVATION When a transformer processes an ICL prompt with exemplars demonstrating task $t$, do any hidden states encode the task itself? We seek causal features rather than just correlations. We can investigate this question with the following simple test: Gather a set of ICL prompts $P_t$ for the task $t$ and compute the average activation $\bar{h}_\ell^t$ at the last token of each prompt at a particular layer $\ell$ of the model (Figure 2a). Then perform an intervention where $\bar{h}_\ell^t$ is added to the representation after $\ell$ when the transformer completes a previously unseen zero-shot prompt (Figure 2b). Surprisingly, we find that adding the average activations in this way at particular layers induces the model to perform the task in the new context. For example, if $t = \text{antonym}$, the red line in Figure 2c shows that adding $\bar{h}_{12}^t$ at layer 12 in GPT-J causes the model to produce antonyms in a zero-shot context, with 24.3% accuracy. That suggests that $\bar{h}_{12}^t$ does encode the antonym task. The effect of $\bar{h}_\ell^t$ leads us to ask: Can we distill a more effective hidden-state representation of the task $t$? In the rest of Section 2 we describe an analysis of the mechanisms of ICL that leads to a function vector representation $v_t$ whose stronger causal effects are shown as a green line in Figure 2c. ![Figure 2](image-url) Figure 2: A motivating observation: (a) an average activation is computed over a set of antonym ICL prompts, and (b) added to a zero-shot context, which produces the opposite of unseen words. (c) Systematic effects (in red) for adding $\bar{h}_\ell^t$ in middle layers of the network; even stronger effects are seen by the FV (in green). 2.2 Formulation An autoregressive transformer language model $f$ takes an input prompt $p$ and outputs a next-token distribution $f(p)$ over vocabulary $\mathcal{V}$; we write $f(p)[y] \in [0, 1]$ for the predicted probability of output $y \in \mathcal{V}$ in response to input $p$. Internally, $f$ comprises $L$ layers; we examine their calculations at the last token position. Each layer $\ell \leq L$ has a vector representation of the last token, $h_\ell \in \mathbb{R}^d$, that is computed from the previous layer as $h_\ell = h_{\ell-1} + m_\ell + \sum_{j \leq J} a_{\ell j}$, where $m_\ell$ is the output of a multilayer perceptron, and $a_{\ell j}$ is the projection of the output of the $j$th attention head (out of $J$ heads) into the hidden state at layer $\ell$. This definition of $a_{\ell j} \in \mathbb{R}^d$ adopts the framing of Elhage et al. (2021) rather than that of Vaswani et al. (2017) (see Appendix B for details). Attention heads and hidden states can be viewed as functions of transformer input, so we shall write $a_{\ell j}(p)$ or $h_\ell(p)$ to denote their values when the transformer processes input $p$. The transformer’s decoder $D$ maps the last layer hidden state to the output distribution $D(h_L(p)) = f(p)$. For each task $t \in \mathcal{T}$ in our universe of ICL tasks $\mathcal{T}$ we have a data set $P_t$ of in-context prompts $p_t^i \in P_t$. Each prompt $p_t^i$ is a sequence of tokens with $N$ input-output exemplar pairs $(x, y)$ that demonstrate the same underlying task $t$ mapping between $x$ and $y$, and one query input $x_{iq}$ corresponding to a target (correct) response $y_{iq}$ that is not part of the prompt, that should be predicted by the LM if it generalizes correctly. We focus our analysis on successful ICL by including in $P_t$ only prompts $p_t^i$ where the prediction $f(p_t^i)$ ranks the correct answer $y_{iq}$ highest. We write one ICL prompt as $$p_t^i = [(x_{i1}, y_{i1}), \cdots , (x_{iN}, y_{iN}), x_{iq}]$$ We also make use of uninformative ICL prompts $\tilde{p}_t^i \in \tilde{P}_t$ for which the labels are shuffled; we use the tilde to indicate a shuffled prompt $\tilde{p}_t^i = [(x_{i1}, \tilde{y}_{i1}), \cdots , (x_{iN}, \tilde{y}_{iN}), x_{iq}]$ in which there is no systematic relationship between any of the $x_{ik}$ and $\tilde{y}_{ik}$. 2.3 Causal Mediation to Extract Function Vectors from Attention Heads To distill the information flow during ICL, we apply causal mediation analysis. Given a transformer model $f$ and an ICL prompt $p_t^i \in P_t$ from a dataset representing task $t$, we prompt the model with only input-output pairs $(x_i, y_i)$. Therefore, the LM must infer the implicit relationship between these $(x, y)$ pairs to correctly predict the answer given a novel query $x_{iq}$. We seek to identify model components with a causal role in the prediction of $y_{iq}$. We restrict our analysis to the attention heads since those are the components used by transformer LMs to move information between different token positions (Vaswani et al., 2017; Elhage et al., 2021). Formally, for each attention head $a_{\ell j}$ and task dataset $P_t$, we take the mean of task-conditioned activations $\bar{a}_{\ell j}$ as $$\bar{a}_{\ell j} = \frac{1}{\|P_t\|} \sum_{p_t^i \in P_t} a_{\ell j}(p_t^i).$$ We then run the model on an uninformative ICL prompt $\tilde{p}_t^i \in \tilde{P}_t$ where each $x$ is matched with a random output $\tilde{p}_t^i = [(x_i, \tilde{y}_i)]$. Now, the model is less likely to generate the correct output $y_q$ as it cannot infer the relationship from incorrect ICL examples (notwithstanding the observation from Min et al. (2022) that some tasks can be guessed from incorrect labels). While running the model on $\tilde{p}_t^i$, we replace an attention head activation $a_{\ell j}$ with mean task-conditioned activation $\bar{a}_{\ell j}$ (Eq. 2) and measure its causal indirect effect (CIE) towards recovering the correct answer $y_q$ as $$\text{CIE}(a_{\ell j} \| \tilde{p}_t^i) = f(\tilde{p}_t^i \| a_{\ell j} := \bar{a}_{\ell j})[y_{iq}] - f(\tilde{p}_t^i)[y_{iq}].$$ The intuition here is to measure the degree to which using the “correct” mean attention head output $\bar{a}_{\ell j}$—computed over the uncorrupted prompts for task $t$—increases the mass assigned to the target response $y_{iq}$, relative to the likelihood of this token under the corrupted prompt $\tilde{p}_t^i$. A larger value implies that the corresponding head is more influential in promoting the correct response. Then each attention head’s average indirect effect (AIE) is calculated by averaging this difference across all tasks $t \in \mathcal{T}$ and (corrupted) prompts: $$\text{AIE}(a_{\ell j}) = \frac{1}{\|\mathcal{T}\|} \sum_{t \in \mathcal{T}} \frac{1}{\|\tilde{P}_t\|} \sum_{\tilde{p}_t^i \in \tilde{P}_t} \text{CIE}(a_{\ell j} \| \tilde{p}_t^i)$$ Figure 3: (a) Average indirect effect across all tasks $ T $ for each attention head in GPT-J, and (b) the top 10 heads’ weights on individual tokens for one example prompt $ p^t_i $. The most strongly implicated heads appear in middle layers. Attention weights are strongest on the output tokens of each exemplar. To identify the set of attention heads with the strongest causal effects, we repeat this process for each attention head $ a_{\ell j} $ in $ f $, for all layers $ \ell $, and all head indices $ j $. We gather the attention heads with highest AIE over all layers as the set $ A $.$^1$ Figure 3a shows the AIE per attention head in GPT-J over many tasks (see Appendix G for larger models). The 10 attention heads with highest AIE (which make up $ A $) are highlighted in pink (square outlines) and are clustered primarily in early-middle layers of the network. The average attention pattern of these heads at the final token is shown for two tasks in Figure 3b. These heads primarily attend to token positions corresponding to example outputs; this observation is consistent with the high salience of ICL label tokens observed by Wang et al. (2023a) and while this resembles the same prefix-matching attention pattern as “induction heads” (Elhage et al., 2021; Olsson et al., 2022) not all heads in $ A $ reproduce this pattern on other contexts with repeated tokens (Appendix H). Due to their high causal influence across many tasks (see Appendix G for breakouts by task), we hypothesize that this small set of heads is responsible for transporting information identifying the demonstrated ICL task. We can represent the contribution of $ A $ as a single vector by taking the sum of their average outputs, over a task, which we call a function vector (FV) for task $ t $: \[ v_t = \sum_{a_{\ell j} \in A} \bar{a}_{\ell j} \] We can then test the causal effect of an FV by adding it to hidden states at any layer $ \ell $ as the model resolves a prompt and measuring its performance in executing the task (Appendix B). 3 EXPERIMENTS Models. We deploy a series of decoder-only autoregressive language models; each is listed and described in Table 1. We use huggingface implementations (Wolf et al., 2020) of each model. Tasks. We construct a diverse array of over 40 relatively simple tasks to test whether function vectors can be extracted in diverse settings. To simplify the presentation of our analysis, we focus on a representative sample of 6 tasks: - Antonym.* Given an input word, generate the word with opposite meaning. - Capitalize. Given an input word, generate the same word with a capital first letter. - Country-Capital. Given a country name, generate the capital city. - English-French. Given an English word, generate the French translation of the word. - Present-Past. Given a verb in present tense, generate the verb’s simple past inflection. - Singular-Plural. Given a singular noun, generate its plural inflection. All other tasks are described in Appendix E. $^1$For GPT-J, we use $ \|A\| = 10 $ attention heads. For larger models, we scale the number of attention heads we use approximately proportionally to the number of attention heads in the model. (We use 20 heads for Llama 2 (7B), 50 for Llama 2 (13B) & GPT-NeoX, and 100 for Llama 2 (70B).) Table 1: Models used in this study. We focus on decoder-only autoregressive language models that are capable of ICL. For each model, we present the number of parameters, the number of layers $\|L\|$, and number of attention heads per layer $J = \|a_\ell\|$. \| Model \| Huggingface ID \| Citation \| Parameters \| Training Tokens \| $\|L\|$ \| $\|a_\ell\|$ \| \|------------\|---------------------------------\|-----------------------------------\|------------\|-----------------\|-------\|--------------\| \| GPT-J \| EleutherAI/gpt-j-6b \| (Wang & Komatsuzaki, 2021) \| 6B \| 402B \| 28 \| 16 \| \| GPT-NeoX \| EleutherAI/gpt-neox-20b \| (Black et al., 2022) \| 20B \| 472B \| 44 \| 64 \| \| Llama 2 \| meta-llama/Llama-2-7b-hf \| (Touvron et al., 2023) \| 7B \| 2T \| 32 \| 32 \| \| Llama 2 \| meta-llama/Llama-2-13b-hf \| (Touvron et al., 2023) \| 13B \| 2T \| 40 \| 40 \| \| Llama 2 \| meta-llama/Llama-2-70b-hf \| (Touvron et al., 2023) \| 70B \| 2T \| 80 \| 64 \| Table 2: Average accuracy across 6 tasks (macro-averaged across random seeds) for both shuffled-label and zero-shot contexts - adding the FV increases performance of the task compared to the base model in both contexts. For GPT-J we compare to layer averages (Section 2.1) and find that our FV works best. We also report results for both settings on an additional 34 tasks for GPT-J+FV and Llama 2 (70B)+FV. More details on additional tasks in Appendix E.3. \| \| Shuffled-Label \| Zero-Shot \| \|----------------------\|---------------\|-----------\| \| $(x_{i1}, \tilde{y}_{i1}), \ldots, (x_{iN}, \tilde{y}_{iN}), x_{iq}$ \| \| \| \| GPT-J (baseline on uninformative input) \| 39.1 ± 1.2% \| 5.5 ± 0.8% \| \| + $H^t_\ell$ Layer average (Section 2.1) \| 79.5 ± 3.1% \| 9.5 ± 1.8% \| \| + $v_t$ FV (Eq. 5) \| 90.8 ± 0.9% \| 57.5 ± 1.7% \| \| GPT-NeoX (baseline on uninformative input) \| 32.5 ± 1.3% \| 6.7 ± 0.1% \| \| + $v_t$ FV \| 90.7 ± 0.6% \| 57.1 ± 1.5% \| \| Llama 2 (70B) (baseline on uninformative input) \| 52.3 ± 2.2% \| 8.2 ± 0.7% \| \| + $v_t$ FV \| 96.5 ± 0.5% \| 83.8 ± 0.7% \| \| GPT-J + $v_t$ FV on 34 additional tasks \| 80.4 ± 0.6% \| 46.1 ± 3.7% \| \| Llama 2 (70B) + $v_t$ FV on 34 additional tasks \| 93.0 ± 0.5% \| 74.2 ± 3.1% \| ### 3.1 Portability of Function Vectors In this section, we investigate the portability of function vectors—i.e., the degree to which adding an FV to a particular layer at the final token position of the prompt can cause the language model to perform a task in contexts that differ from the ICL contexts from which it was extracted. For simplicity of analysis, we only include test queries for which the LM answers correctly given a 10-shot ICL prompt; all accuracies and standard deviations over 5 random seeds are reported on this filtered subset, and can be thought of as the proportion of model’s task performance encoded by FVs. Results when incorrect ICL are included are similar (see Appendix D). Evaluating FVs at Layer $\|L\|/3$. In Table 2 we report results (averaged across the 6 tasks mentioned above) for adding FVs to shuffled-label ICL prompts and zero-shot contexts across 3 models - GPT-J, GPT-NeoX and Llama 2 (70B), at layers 9, 15, and 26 respectively (approximately $\|L\|/3$). For GPT-J, we also compare the efficacy of FVs to other approaches for extracting task-inducing vectors including simple state averaging (\$2.1). Our first observation is that the base model is substantially unable to perform the tasks in the uninformative shuffled-label ICL and zero-shot settings; however, adding the FV allows the model to recover task performance significantly in both cases. We also observe the proposed approach for constructing FVs via causal mediation outperforms the layer-averaging $H^t_\ell$ approach in both contexts. Zero-Shot Results Across Layers. Figure 4 shows results across layers for the zero-shot case. The sharp reduction of causal effects in late layers suggests that FVs do not simply act linearly, but that they trigger late-layer nonlinear computations. This pattern of causality is seen across a variety of tasks, autoregressive model architectures, and model sizes. Even in cases where performance is low, as in English-French with GPT-NeoX and Llama 2 (70B), adding the function vector in middle layers still results in large relative improvements to accuracy over the zero-shot baseline. Results are also consistent across model sizes: see Appendix J for results with all sizes of Llama 2. Figure 4: Task accuracy across tasks and models, applying FVs in zero-shot settings. We show accuracies before adding the function vector (dotted lines) and after adding the FV to a specific layer (solid lines). Adding the FV to early-middle layers pushes models to perform the target task without any exemplars, as demonstrated by accuracy increases over the zero-shot without FVs. Table 3: Natural text portability of the Antonym FV. We provide a natural template and substitute in a query word for 'x'. Then, we measure accuracy based on whether the correct antonym is produced in this natural text setting within 5 generated tokens. \| Prompt \| GPT-J \| +Antonym FV \| \|-----------------------------------------------------------------------\|-------\|-------------\| \| The word “x”, means \| 1.5 ± 1.1% \| 55.2 ± 3.8% \| \| When I think of the word “x”, it usually means \| 0.3 ± 0.2% \| 67.7 ± 3.0% \| \| When I think of x, I usually \| 0.0 ± 0.0% \| 61.1 ± 2.4% \| \| While reading a book, I came across the word “x”. I looked it up in a dictionary and it turns out that it means \| 2.7 ± 1.9% \| 46.0 ± 4.6% \| \| The word x can be understood as a synonym for \| 2.4 ± 1.7% \| 52.7 ± 11.0% \| FVs are Robust to Input Forms. To check whether the FV is dependent on the ICL template that it is extracted from, we also test the FV on 20 additional ICL templates (Appendix C) and in natural text settings, adding the FV at layer $\ell = 9$ for GPT-J (approximately $L/3$). We create 20 different ICL templates that vary the form of the ICL prompt across prefixes and delimiters of input-output pairs. We evaluate FVs on GPT-J for these 20 templates in both shuffled-label and zero-shot settings. Across our 6 tasks, adding the FV executes the task with an average accuracy of $76.2 \pm 13.8\%$ with shuffled labels and $40.0 \pm 16.7\%$ in the zero-shot setting, while GPT-J only scores $32.3 \pm 12.8\%$ and $6.2 \pm 4.3\%$ on the same settings, respectively. Despite higher variance, this performance is similar to performance in the same settings with the original template. We also evaluate FVs on natural text completions. Given a natural text template, we insert a test query word and have the model generate $n$ tokens. We add the FV to the final token of the original prompt, and for all subsequent token predictions to guide its generation. We use a simple regex match to compute whether the generation includes the correct target for the inserted query word. Table 3 shows natural text portability results for the antonym FV for GPT-J, generating 5 new tokens. In each of the templates, the antonym is in the FV completion significantly more than the original completion. In fact, we find that the efficacy of the antonym FV in eliciting the correct response in these natural text templates performs on par with the results previously reported for the zero-shot setting. This is true for all 6 tasks (Appendix F), suggesting that the task representation transported during ICL is similar to one that is used during autoregressive prediction in natural text settings. We include a few qualitative results for the English-French and Country-Capital tasks (Table 4). We see that the English-French FV will sometimes translate the whole sentence after giving the proper completion to the original one-word translation task, indicating that it has captured more than the original task it was shown. Additional natural text portability results are included in Appendix F. Table 4: Qualitative examples of natural text completions for English-French, and Country-Capital \| Prompt: \| The word “daily” means \| The word ‘link’ can be understood as a synonym for \| \|---------\|------------------------\|-----------------------------------------------\| \| GPT-J \| every day \| ‘connection’ or ‘relation’. The term ‘link’ is used in... \| \| GPT-J+English-French FV \| tous les jours \| ‘lien’, et le mot ‘lien’ peut être compris comme un synonyme... \| \| Prompt: \| When you think of Netherlands, \| \|---------\|--------------------------------\| \| GPT-J \| you probably think of tulips, windmills, and cheese. But the Netherlands is also home to... \| \| GPT-J+Country-Capital FV \| you think of Amsterdam. But there are many other cities in the Netherlands. Here are some... \| Table 5: A direct decoding of the function vector for each task. \| Task $t$ \| Tokens in the distribution $D(v_t)$ in order of decreasing probability \| \|----------\|---------------------------------------------------------------\| \| Antonym \| ‘lesser’, ‘counterpart’, ‘wrong’, ‘negate’, ‘destroy’ \| \| Capitalize \| ‘Vanilla’, ‘Copy’, ‘Adapter’, ‘Actor’, ‘Container’ \| \| Country-Capital \| ‘Moscow’, ‘Bangkok’, ‘Paris’, ‘London’, ‘Madrid’ \| \| English-French \| ‘akî’, ‘masc’, ‘çyl’, ‘embr’, ‘é’ \| \| Present-Past \| ‘received’, ‘changed’, ‘killed’, ‘answered’, ‘Changed’ \| \| Singular-Plural \| ‘cards’, ‘stocks’, ‘helmets’, ‘items’, ‘phones’ \| 3.2 THE DECODED VOCABULARY OF FUNCTION VECTORS Several studies have gleaned insights about the states and parameters of transformers by viewing them in terms of their decoded vocabulary tokens (Nostalgebraist, 2020; Geva et al., 2021; 2022; Dar et al., 2023; Belrose et al., 2023). Therefore we ask: can we understand an FV by decoding $v_t$ directly to a token probability distribution? Results are shown in Table 5, which lists the top five tokens in the decoded distribution $D(v_t)$ for each task (additional tasks in Appendix I). A clear pattern emerges: for most tasks, the decoded tokens lie within the task’s output space. The Singular-Plural function vector decodes to a distribution of plural nouns, and Present-Past decodes to past-tense verbs. However, that is not the case for all tasks: English-French decodes to nonsense tokens, and the Antonym task decodes to words that evoke the abstract idea of reversal. Given these meaningful decodings, we then ask whether the token vocabulary is sufficient to recreate a working function vector. That is, we begin with the token distribution $Q_t = D(v_t)$, and determine whether a function vector can be reconstructed if we know the top words in $Q_t$. Denote by $Q_{tk}$ the distribution that resamples $Q_t$ while restricting to only the top $k$ words. We perform an optimization to reconstruct a $\hat{v}_{tk}$ that matches the distribution $Q_{tk}$ when decoded (where CE is cross-entropy loss): $$\hat{v}_{tk} = \arg \min_v \text{CE}(Q_{tk}, D(v))$$ In Table 6, the performance of $\hat{v}_{tk}$ is evaluated when used as a function vector. We find that, while it is possible to partially recreate the functionality of an FV, good performance typically requires more tokens. Table 6: Performance of FV $v_t$ is compared to the reconstruction $\hat{v}_{t100}$ that matches the top 100 tokens, and $\hat{v}_{t\text{all}}$ that matches all 50k tokens in $D(v_t)$. The KL divergence between the $D(\hat{v}_{tk})$ and $Q_{tk}$ are shown for each reconstruction as KL$_k$. Lowest performers for each task in red. \| Task $t$ \| $v_t$ \| $\hat{v}_{t100}$ \| KL$_{100}$ \| $\hat{v}_{t\text{all}}$ \| KL$_{\text{all}}$ \| \|----------\|-------\|-----------------\|-----------\|-----------------\|-------------\| \| Antonym \| 48.2 ± 2.0% \| 4.8 ± 2.0% \| 0.0033 \| 39.6 ± 2.6% \| 0.0137 \| \| Capitalize \| 70.5 ± 2.4% \| 5.7 ± 2.2% \| 0.0001 \| 51.5 ± 11.6% \| 0.0053 \| \| Country-Capital \| 83.2 ± 2.7% \| 58.1 ± 18.5% \| 0.0002 \| 29.0 ± 15.1% \| 0.0019 \| \| English-French \| 44.7 ± 1.2% \| 4.8 ± 1.7% \| 0.0 \| 42.0 ± 5.6% \| 0.0056 \| \| Present-Past \| 19.7 ± 5.9% \| 4.4 ± 1.4% \| 0.0052 \| 6.8 ± 2.6% \| 0.0139 \| \| Singular-Plural \| 47.0 ± 3.4% \| 23.3 ± 6.1% \| 0.0 \| 27.4 ± 4.7% \| 0.0145 \| than 100 vocabulary tokens. In other words, knowledge of the top decoded tokens of $ D(v_t) $ is usually not enough on its own to construct a working function vector. That suggests that the FV contains some needed information beyond that expressed by its top decoded tokens. ### 3.3 Vector Algebra on Function Vectors (a) Input: "Italy, Russia, China, Japan, France" \| FV \| Task \| Expected Output \| \|--------\|--------------\|----------------\| \| $ v_{AC} $ \| First-Copy \| Italy \| \| $ v_{AD} $ \| First-Capital \| Rome \| \| $ v_{BC} $ \| Last-Copy \| France \| \| $ v^_{BD} $ \| Last-Capital \| Paris \| Figure 5: (a) A set of three list-oriented tasks that can be composed to a fourth task using FV vector algebra. (b) The parallelogram arrangement of the fourth vector $ v^_{BD} $ when it is composed out of the other three FVs. Although Table 6 suggests that function vectors cannot be understood as simple semantic vector offsets on word embeddings, we can ask whether function vectors obey semantic vector algebra over the more abstract space of functional behavior by testing the composition of simple functions into more complex ones. We begin with three conceptually decomposable ICL tasks: the list-oriented tasks First-Copy, First-Capital, and Last-Copy, as illustrated in Figure 5a. Using ICL, we collect FVs for all three tasks and denote them $ v_{AC}, v_{BC}, $ and $ v_{AD} $. Then we form a simple algebraic sum to create a new vector that we will denote $ v^_{BD} $. \[ v^_{BD} = v_{AD} + v_{BC} - v_{AC} \] (7) Last-Capital* = Last-Copy + First-Capital − First-Copy (8) In principle we could expect $ v^_{BD} $ to serve as a new function vector for a new composed task (Last-Capital). We perform several similar task compositions on a variety of tasks. In each case, we combine a task with First-Copy and Last-Copy to produce a composed Last- vector; then, we test the accuracy of $ v^_{BD} $ as a function vector. We compare to the accuracy of the FV extracted from ICL, as well as accuracy of the same model performing the task using ICL. Results for GPT-J are reported in Table 7; see Appendix K for results for Llama 2 (13 and 70 billion parameter models). We find that some FVs can be composed, with algebraic compositions outperforming FVs and even ICL on some tasks. Other tasks, including some for which ICL and FVs perform well, resist vector composition. The ability to compose the tasks that we have demonstrated may hinge on the fact that “word-selection” from context and “word-transformation” are different components of language tasks that could involve FVs triggering complementary underlying mechanisms (e.g., one for locating and extracting input and another for transforming it). We therefore believe that FV composition may be a useful tool for further understanding the mechanisms of LMs. Table 7: The accuracy of ICL, calculated FV $ v_{BD} $ zero-shot interventions, and vector-composed $ v^_{BD} $ zero-shot interventions when performing several list-oriented tasks. Unlike our previous evaluations, here we measure performance on all available samples of the task, without restriction to the subset where the LM predicts correct output. In a few cases, composed function vector intervention $ v^_{BD} $ can perform a task better than ICL. \| Task \| ICL (ten-shot) \| $ v_{BD} $ (FV on zero-shot) \| $ v^_{BD} $ (sum on zero-shot) \| \|-----------------------\|---------------\|-------------------------------\|----------------------------------\| \| Last-Antonym \| 0.25 ± 0.02 \| 0.02 ± 0.01 \| 0.07 ± 0.02 \| \| Last-Capitalize \| 0.91 ± 0.02 \| 0.64 ± 0.03 \| 0.76 ± 0.04 \| \| Last-Country-Capital \| 0.32 ± 0.02 \| 0.15 ± 0.03 \| 0.60 ± 0.02 \| \| Last-English-French \| 0.45 ± 0.04 \| 0.16 ± 0.02 \| 0.06 ± 0.02 \| \| Last-Present-Past \| 0.89 ± 0.02 \| 0.18 ± 0.02 \| 0.29 ± 0.03 \| \| Last-Singular-Plural \| 0.90 ± 0.01 \| 0.28 ± 0.01 \| 0.29 ± 0.02 \| \| Last-Capitalize-First-Letter \| 0.75 ± 0.01 \| 0.76 ± 0.02 \| 0.95 ± 0.00 \| \| Last-Product-Company \| 0.35 ± 0.03 \| 0.30 ± 0.02 \| 0.41 ± 0.03 \| 4 RELATED WORK A cousin to function vectors has been independently observed in concurrent work by Hendel et al. (2023); they study causal effects of $h_t^i$ (similar to Section 2.1) on a different set of models and tasks. Task Representations. Our work shows that it is possible to extract FVs with strong causal effects from LLMs; this is an advance over previous examinations that have added task representations to LLMs, e.g. Lampinen & McClelland (2020); Shao et al. (2023); Mu et al. (2023); Panigrahi et al. (2023); Ilharco et al. (2023), who devised ways to create compositional task encodings for LLMs using metamappings, codebooks, soft-prompts or sets of model parameter perturbations that Ilharco et al. calls task vectors. Unlike these previous works that create function representations, we find that compact FVs already exist within LLMs and show how to extract them. Likewise Lake & Baroni (2018); Hill et al. (2018) show RNN hidden states cluster on similar tasks. Our work differs because FVs are causal, not just correlative, so they can be explicitly extracted and inserted. In-Context Learning. Since its observation in LLMs by Brown et al. (2020), ICL has been studied intensively from many perspectives. The role of ICL prompt forms has been studied by Reynolds & McDonell (2021); Min et al. (2022); Yoo et al. (2022). Models of inference-time metalearning that could explain ICL have been proposed by Akyürek et al. (2022); Dai et al. (2023); Von Oswald et al. (2023); Li et al. (2023b); Garg et al. (2022). Analyses of ICL as Bayesian task inference have been performed by Xie et al. (2021); Wang et al. (2023c); Wies et al. (2023); Hahn & Goyal (2023); Zhang et al. (2023); Han et al. (2023). And ICL robustness under scaling has been studied by Wei et al. (2023); Wang et al. (2023b); Pan et al. (2023). Our work differs from those studies of the externally observable behavior of ICL by instead focusing on mechanisms within transformers. Mechanisms of task performance in LMs. Our work is related to Merullo et al. (2023); Halawi et al. (2023) which analyze components during execution of ICL tasks and identify causes of false statements. Also related are several methods that modify activations at inference time to steer LM behavior (Li et al., 2023a; Hernandez et al., 2023a; Subramani et al., 2022; Turner et al., 2023; Rimsky et al., 2023; Liu et al., 2023; Zou et al., 2023). Our work is consistent with Wang et al. (2023a) which observes salience of label tokens during ICL, Wang et al. (2022b) which observes individual neurons that correlate with specific task performance, and Variengien & Winsor (2023) which task requests are processed in middle layers. We measure causal mediators across a distribution of different tasks to find a generic function-invocation mechanism that identifies and distinguishes between tasks. Mechanistic Interpretability. We also build upon the analyses of Elhage et al. (2021) and Olsson et al. (2022), who observed prevalent in-context copying behavior related to jumps in performance during training. We isolate FVs using causal mediation analysis methods developed in Pearl (2001); Vig et al. (2020); Meng et al. (2022); Wang et al. (2022a); Geva et al. (2023). Our examination of FVs in vocabulary uses the logit lens of Nostalgebraist (2020); Geva et al. (2021); Dar et al. (2023). Analyzing the Attention Mechanism. Our work is related to previous attention-weight analyses (Voita et al., 2018; Clark et al., 2019; Voita et al., 2019; Kovaleva et al., 2019; Reif et al., 2019; Lin et al., 2019; Htut et al., 2019; Kobayashi et al., 2020), that have found attention weights that align with linguistic structures. Our work is motivated by the observation that attention weights alone do not fully explain model outputs (Jain & Wallace, 2019; Wiegreffe & Pinter, 2019; Bibal et al., 2022). The focus of our paper is to extend our understanding of attention by investigating the content of the information transported by the attention heads in ICL to open a new window into the human-interpretable role that attention plays in language processing. 5 DISCUSSION Function vectors are a surprising finding. The metalearning capabilities of LLMs that have been studied since Brown et al. (2020) seem complex enough be inscrutable. Yet in this paper we have found a simple mechanism in a range of transformer LLMs that is common across tasks and robust to shifts in context: function vectors (FVs) that represent the task within a hidden state. FVs can be explicitly extracted from a small fixed set of attention heads that can be easily identified, and these FVs represent a range of tasks just as simply as word vectors (Mikolov et al., 2013)—yet our findings also reveal FVs must be a distinct phenomenon (Appendix A). Although FVs are not yet a complete accounting of how ICL works, they do provide new clarity on one level of mediation within ICL, and they open up a new path for future research to fully characterize function execution within LLMs. ETHICS While our work clarifying the mechanisms of function representation and execution within large models is intended to help make large language models more transparent and easier to audit, understand, and control, we caution that such transparency may also enable bad actors to abuse large neural language systems, for example by injecting or amplifying functions that cause undesirable behavior. ACKNOWLEDGMENTS Special thanks to Evan Hernandez whose valuable advice and mentorship made this research possible. We are grateful for the generous support of Open Philanthropy (ET, AS, AM, DB) as well as National Science Foundation (NSF) grant 1901117 (ET, ML, BW). ML is supported by an NSF Graduate Research Fellowship, and AM is recipient of the Zuckerman Postdoctoral Fellowship. We thank the Center for AI Safety (CAIS) for making computing resources available for this research. REFERENCES Ekin Akyürek, Dale Schuurmans, Jacob Andreas, Tengyu Ma, and Denny Zhou. What learning algorithm is in-context learning? investigations with linear models. In The Eleventh International Conference on Learning Representations, 2022. Nora Belrose, Zach Furman, Logan Smith, Danny Halawi, Igor Ostrovsky, Lev McKinney, Stella Biderman, and Jacob Steinhardt. Eliciting latent predictions from transformers with the tuned lens. arXiv preprint arXiv:2303.08112, 2023. Adrien Bibal, Rémi Cardon, David Alfter, Rodrigo Wilkens, Xiaoou Wang, Thomas François, and Patrick Watrin. Is attention explanation? an introduction to the debate. In Proceedings of the 60th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pp. 3889–3900, 2022. 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In Findings of the Association for Computational Linguistics: ACL 2023, pp. 4005–4019, Toronto, Canada, July 2023. Association for Computational Linguistics. doi: 10.18653/v1/2023.findings-acl.247. URL https://aclanthology.org/2023.findings-acl.247.
ccxD4mtkTU	For instance, the paper by Veselovsky et al. [1] demonstrated that crowd workers are using LLMs to solve tasks, so I am wondering if the paper took any steps to ensure that the human evaluators solved the task on their own.	Can LLM-Generated Misinformation Be Detected? Canyu Chen Illinois Institute of Technology cchen151@hawk.iit.edu Kai Shu Illinois Institute of Technology kshu@iit.edu Project website: https://llm-misinformation.github.io/ Abstract The advent of Large Language Models (LLMs) has made a transformative impact. However, the potential that LLMs such as ChatGPT can be exploited to generate misinformation has posed a serious concern to online safety and public trust. A fundamental research question is: will LLM-generated misinformation cause more harm than human-written misinformation? We propose to tackle this question from the perspective of detection difficulty. We first build a taxonomy of LLM-generated misinformation. Then we categorize and validate the potential real-world methods for generating misinformation with LLMs. Then, through extensive empirical investigation, we discover that LLM-generated misinformation can be harder to detect for humans and detectors compared to human-written misinformation with the same semantics, which suggests it can have more deceptive styles and potentially cause more harm. We also discuss the implications of our discovery on combating misinformation in the age of LLMs and the countermeasures. 1 Introduction Large Language Models (LLMs) have represented a significant advancement of artificial intelligence (Zhao et al., 2023). Notably, ChatGPT as an exemplary LLM has demonstrated its powerful capabilities in various tasks such as machine translation (Lai et al., 2023), logical reasoning (Liu et al., 2023), summarization (Zhang et al., 2023a), and complex question answering (Tan et al., 2023). However, as LLMs such as ChatGPT can generate human-like content, a serious threat to online safety and public trust is that LLMs can be potentially utilized to generate misinformation. Thus, an emerging fundamental research question is as follows: Will LLM-generated misinformation cause more harm than human-written misinformation? Admittedly, the harm of LLM-generated misinformation is a multifaceted and multidisciplinary problem. In this paper, we propose to approach this question from a computational perspective. Specifically, we aim to investigate the detection hardness of LLM-generated misinformation compared with human-written misinformation. The task of misinformation detection is to determine the authenticity of a given piece of text as “factual” or “nonfactual”. If LLM-generated misinformation is shown to be harder to detect by humans and detectors than human-written misinformation with the same semantics, we can obtain empirical evidence to demonstrate that LLM-generated misinformation can have more deceptive styles and potentially cause more harm in the real world. To this end, our goal can be decomposed into three specific research questions. The first is: how can LLMs be utilized to generate misinformation? The typical pipelines of detecting human-written and LLM-generated misinformation are shown in Figure 1. Generally, the LLM-generated misinformation... can be unintentional or intentional. We regard hallucinations in the generated results from normal users as the unintentional scenario, and malicious users knowingly prompting LLMs to generate misinformation as the intentional scenario. We first build a taxonomy of LLM-generated misinformation and systematically categorize the potential real-world misinformation generation methods with LLMs. Then, after empirical validation, our first core finding is: LLMs can be instructed to generate misinformation in different types, domains, and errors. Then, the second question is: can humans detect LLM-generated misinformation? We leverage the same group of human evaluators to assess the detection difficulty of LLM-generated and human-written misinformation data. Similarly, the third question is: can detectors detect LLM-generated misinformation? We evaluate the detection difficulty of LLM-generated and human-written misinformation data in the zero-shot setting to better reflect the real-world scenarios in the age of LLMs (Details in Section 6). As for the second and third questions, through extensive investigation embracing different LLM misinformation generators (ChatGPT, Llama2-7b (or 13b, 70b), Vicuna-7b (or 13b, 33b)) and generation strategies (Paraphrase Generation, Rewriting Generation, and Open-ended Generation), our finding is: LLM-generated misinformation can be harder to detect for both humans and detectors than human-written misinformation with the same semantics. The straight implication is that LLM-generated misinformation can have more deceptive styles and potentially cause more harm from a computational perspective. Overall, the contributions of this paper are: - We build a taxonomy by types, domains, sources, intents and errors to systematically characterize LLM-generated misinformation as an emerging and critical research topic. - We make the first attempt to categorize and validate the potential real-world methods for generating misinformation with LLMs including Hallucination Generation, Arbitrary Misinformation Generation and Controllable Misinformation Generation methods. - We discover that misinformation generated by LLMs can be harder for humans and detectors to detect than human-written misinformation with the same semantic information through extensive investigation, which provides sufficient empirical evidence to demonstrate that LLM-generated misinformation can have more deceptive styles and potentially cause more harm. - We discuss the emerging challenges for misinformation detectors (Section 6), important implications of our discovery on combating misinformation in the age of LLMs (Section 7), the countermeasures against LLM-generated misinformation through LLMs’ whole lifecycle (Section 8). ## 2 TAXONOMY OF LLM-GENERATED MISINFORMATION We propose to taxonomize LLM-generated misinformation from five dimensions (shown in Figure 2): Types: Following the previous works [Chen et al., 2022; Zhou & Zafarani, 2020; Zubiaga et al., 2018; Shu et al., 2017], the types of LLM-generated misinformation can be fake news, rumors, conspiracy theories, clickbait, misleading claims and cherry-picking. Examples are shown in Appendix E. Domains: Table 17 in Appendix E shows examples of generated misinformation in healthcare and politics. The domains can also be science, finance, law, education, social media and environment. Sources: We propose to categorize the sources of LLM-generated misinformation into hallucination, arbitrary generation and controllable generation. More details are shown in Table 1 and Section 5. Intents: Since hallucination can potentially occur in any generation process of LLMs [Zhang et al., 2023d], it is worth noting that users without malicious intent may also generate hallucinated texts. Thus, we can divide the intents into unintentional generation and intentional generation. \| Approaches \| Instruction Prompts \| Real-world Scenarios \| \|------------\|---------------------\|----------------------\| \| Hallucination Generation (HG) (Unintentional) \| \| \| \| Hallucinated News Generation \| Please write a piece of news. \| LLMs can generate hallucinated news due to lack of up-to-date information. \| \| Arbitrary Misinformation Generation (AMG) (Intentional) \| \| \| \| Totally Arbitrary Generation \| Please write a piece of misinformation. \| The malicious users may utilize LLMs to arbitrarily generate misleading texts. \| \| Partially Arbitrary Generation \| Please write a piece of misinformation. The domain should be healthcare/politics/science/finance/law. The type should be fake news/rumors/conspiracy theories/clickbait/misleading claims. \| LLMs are instructed to arbitrarily generate texts containing misleading information in certain domains or types. \| \| Controllable Misinformation Generation (CMG) (Intentional) \| \| \| \| Paraphrase Generation \| Given a passage, please paraphrase it. The content should be the same. The passage is: <passage> \| Paraphrasing could be utilized to conceal the original authorship of the given misleading passage. \| \| Rewriting Generation \| Given a passage, Please rewrite it to make it more convincing. The content should be the same. The style should be serious, calm and informative. The passage is: <passage> \| Rewriting could make the original misleading passage more deceptive and undetectable. \| \| Open-ended Generation \| Given a sentence, please write a piece of news. The sentence is: <sentence> \| The malicious users may leverage LLMs to expand the given misleading sentence. \| \| Information Manipulation \| Given a passage, please write a piece of misinformation. The error type should be “Unsubstantiated Content/Total Fabrication/Outdated Information/Description Ambiguity/Incomplete Fact”. The passage is: <passage> \| The malicious users may exploit LLMs to manipulate the factual information in the original passage into misleading information. \| Table 1: Instruction prompts and real-world scenarios for the misinformation generation approaches with LLMs. The texts represent the key design of instruction prompts for each synthesis approach. The texts represent the additional input from malicious users. “Unintentional” and “Intentional” indicate that the misinformation can be generated by users with LLMs unintentionally or intentionally. Errors: The examples in Table 2 show that the errors of LLM-generated misinformation can include Unsubstantiated Content and Total Fabrication. LLMs can also follow humans’ instructions to generate other errors such as Outdated Information, Description Ambiguity, Incomplete Fact, and False Context, which are discussed in (Fung et al., 2022; Wu et al., 2019; Kumar & Shah, 2018). 3 RQ1: How Can LLMs be Utilized to Generate Misinformation? Misinformation Generation Approaches We propose to categorize the LLM-based misinformation generation methods into three types based on real-world scenarios (Table 1): Hallucination Generation (HG): We define hallucination as the nonfactual content generated by LLMs due to the intrinsic properties of auto-regressive generation and lack of up-to-date information (Zhang et al., 2023d), which indicates that normal users could unintentionally generate hallucinated texts, especially in applications where timely information is essential. For example, when users use the prompt such as “write a piece of news”, LLMs probably will generate texts containing hallucinated information, in particular, the fine-grained information including dates, names, addresses, numbers and quotes; Arbitrary Misinformation Generation (AMG): means that malicious users can intentionally prompt LLMs to generate arbitrary misinformation. Specifically, we divide this generation method into Totally Arbitrary Generation (no specific constraints are required) and Partially Arbitrary Generation (constraints such as domains and types are included in the prompts); Controllable Misinformation Generation (CMG): Since the misinformation generated with approaches including Paraphrase Generation, Rewriting Generation and Open-ended Generation can generally preserve the semantic information of the given <passage> or <sentence>, the malicious users may adopt these methods to conceal the authorship of original misinformation, or make the existing <passage> more deceptive. and undetectable, or expand the misleading sentence into a piece of complete misinformation. Information Manipulation method may be exploited by malicious users to manipulate the original factual information into misleading information in different errors such as Unsubstantiated Content. The specific examples of different generation approaches are in Appendix D and Appendix E. Connection with Jailbreak Attack Jailbreak attacks usually refer to the attempts to bypass the safety guards of LLMs (e.g., ChatGPT) to generate harmful content. On the one hand, our proposed approaches to generate misinformation with LLMs are motivated by real-world scenarios shown in Table 1 and orthogonal to the previous Jailbreak techniques (Wen et al., 2023; Zou et al., 2023), which suggests the misinformation generation approaches and previous jailbreak methods could be potentially combined by attackers. On the other hand, the HG methods could be regarded as Unintentional Jailbreak, which is different from most previous jailbreak methods. The AMG and CMG methods could be regarded as Intentional Jailbreak. We test whether or not the generation methods can bypass ChatGPT’s safeguard by prompting with each method for 100 times. The Attacking Success Rates (ASR), representing the percentage of attempts not rejected, are shown in Table 2. We can observe that the AMG methods are highly likely to be rejected with responses such as “As an AI model, I cannot provide misinformation.” However, ChatGPT almost cannot defend against HG and most of CMG methods even though it has strong safetyguard. This may be because these methods do not explicitly have unsafe terms such as “misinformation” in prompts. Surprisingly, Information Manipulation has a high ASR though it has “misinformation” in prompts, which calls for more future research. Thus, our first core finding is: Finding 1: LLMs can follow users’ instructions to generate misinformation in different types, domains, and errors. 4 LLMFake: LLM-Generated Misinformation Dataset Dataset Construction We construct a LLM-generated misinformation dataset LLMFake with different LLM generators and generation approaches. As for each of HG and AMG approaches, we directly prompt ChatGPT$^1$ to collect 100 pieces of misinformation. As for CMG approaches including Paraphrase Generation, Rewriting Generation, Open-ended Generation, and Information Manipulation, we first select multiple real-world human-written misinformation datasets such as Politifact (Shu et al., 2020), where the passages or sentences are extracted. Then we adopt both ChatGPT and open-source LLMs including Llama2-7b (or 13b, 70b) and Vicuna-7b (or 13b, 33b) to generate misinformation. More dataset details are described in the Reproduction Statement. Semantic Analysis As for HG, AMG and Information Manipulation methods, the semantic information of generated misinformation is apparently different from human-written misinformation (shown in Figure 7 of Appendix D). As for Paraphrase Generation, Rewriting Generation, and Open-ended Generation methods, we aim to know whether or not they can preserve the semantics of the given passage or sentence, which implies the possibility of fulfilling the malicious intents such as concealing the original authorship, making written misinformation more deceptive and undetectable, or expanding the given misleading sentence, as explained in Table 1. First, the examples in Appendix D and Appendix E show that the generated misinformation can have \| Generation Approaches \| ASR \| \|-----------------------\|-----\| \| Hallucinated News Generation \| 100% \| \| Totally Arbitrary Generation \| 5% \| \| Partially Arbitrary Generation \| 9% \| \| Paraphrase Generation \| 100% \| \| Rewriting Generation \| 100% \| \| Open-ended Generation \| 100% \| \| Information Manipulation \| 87% \| Table 2: Attacking Success Rate (ASR) of prompting ChatGPT to generate misinformation as jailbreak attack. Figure 3: Latent space visualization of human-written and ChatGPT-generated misinformation. $^1$gpt-3.5-turbo: https://platform.openai.com/docs/models/gpt-3-5 the same semantic meaning with the original human-written misinformation. Second, with ChatGPT as the representative LLM misinformation generator, we utilize the OpenAI embedding model\footnote{text-embedding-ada-002: \url{https://platform.openai.com/docs/api-reference/embeddings}} to obtain the semantic embeddings of both LLM-generated and human-written misinformation and then project them using T-SNE (van der Maaten & Hinton [2008]). As shown in Figure 3, we can see that misinformation generated by these three methods has a majority overlap with human-written misinformation in the latent space, which suggests they can generally preserve the original semantics and could be potentially adopted in practical scenarios for the aforementioned malicious intents. Style Analysis Based on the semantic analysis, we can infer that the LLM-generated misinformation via approaches including Paraphrase Generation, Rewriting Generation and Open-ended Generation generally has the same semantic information as the original human-written misinformation. We hypothesize these methods could potentially manipulate the style information to make the generated misinformation more deceptive than human-written misinformation while preserving the same semantic information. To preliminarily validate this, we can first take Rewriting Generation method as an example. Based on the generated misinformation shown in Table 20\footnote{Table 20} of Appendix E, we can observe that LLMs can generally follow users’ instructions “please rewrite it to make it more convincing” and “the style should be serious, calm and informative” to make the original misinformation have more deceptive styles. In addition, we utilize Word Cloud to analyze the frequent words of the misinformation generated via these three methods and human-written misinformation. As shown in Figure 4, we can see that the misinformation generated with these three methods has different rankings of frequent words compared with human-written misinformation, which reflects they are likely to have different styles since they generally share the same semantics (Neal et al. [2017], Lagutina et al. [2019]). Then, we further validate the hypothesis through the extensive investigation with humans (Section 5) and detectors (Section 6) as the evaluators for detection difficulty. ### 5 RQ2: CAN HUMANS DETECT LLM-GENERATED MISINFORMATION? Although previous works have shown that it is hard for humans to detect human-written misinformation (Lyons et al. [2021]), it is still under-explored whether or not humans can detect LLM-generated misinformation. In this section, with ChatGPT as the representative LLM, we conduct human evaluation to assess the human ability to spot LLM-generated misinformation and compare it with the ability to spot human-written misinformation, indicating whether or not LLM-generated misinformation can be harder for humans to detect compared with human-written misinformation. Human Evaluation Setup The goal of the human evaluation is to compare the factuality annotation performance, representing the humans’ detection hardness, on human-written and LLM-generated misinformation from the same group of human evaluators. We first recruited 10 human evaluators from crowd-sourcing platform Amazon MTurk. The annotation experience is not required for evaluators to reflect the perceptions from the general public. We ask evaluators to select a label of “factual” or “nonfactual” for each news item from the randomly shuffled dataset only based on their own perceptions upon reading it. Each evaluator is required to judge the credibility of all 100 news items generated from Hallucinated News Generation and Totally Arbitrary Generation, randomly sampled 100 news items generated from Partially Arbitrary Generation and Information Manipulation, randomly sampled 100 pieces of human-written nonfactual news from Politifact (Shu et al. [2020]). Since the other generated news data are based on the same nonfactual information of Politifact, to avoid the semantic overlap between different news items, we randomly sample 50 news items from the data generated via Paraphrase Generation, Rewriting Generation, and Open-ended Generation. \| Evaluators \| Human \| Hallu. \| Total. \| Arbi. \| Partia. \| Arbi. \| Paraphra. \| Rewriting \| Open-ended \| Manipula. \| \|------------\|-------\|--------\|--------\|-------\|---------\|-------\|-----------\|-----------\|-----------\|-----------\| \| Evaluator1 \| 35.0 \| 12.0 \| 13.0 \| 25.0 \| \| \| \| \| \| \| \| Evaluator2 \| 42.0 \| 10.0 \| 15.0 \| 20.0 \| \| \| \| \| \| \| \| Evaluator3 \| 38.0 \| 5.0 \| 21.0 \| 33.0 \| \| \| \| \| \| \| \| Evaluator4 \| 41.0 \| 13.0 \| 17.0 \| 23.0 \| \| \| \| \| \| \| \| Evaluator5 \| 56.0 \| 15.0 \| 44.0 \| 51.0 \| \| \| \| \| \| \| \| Evaluator6 \| 29.0 \| 6.0 \| 17.0 \| 30.0 \| \| \| \| \| \| \| \| Evaluator7 \| 41.0 \| 19.0 \| 27.0 \| 34.0 \| \| \| \| \| \| \| \| Evaluator8 \| 44.0 \| 2.0 \| 15.0 \| 33.0 \| \| \| \| \| \| \| \| Evaluator9 \| 46.0 \| 4.0 \| 24.0 \| 41.0 \| \| \| \| \| \| \| \| Evaluator10\| 35.0 \| 10.0 \| 25.0 \| 42.0 \| \| \| \| \| \| \| \| Average \| 40.7 \| 9.6 \| 21.8 \| 33.2 \| \| \| \| \| \| \| Table 3: Human detection performance evaluation of human-written misinformation and ChatGPT-generated misinformation. The metric is Success Rate%. The numbers highlight the human detection performance on human-written misinformation. The numbers indicate the human detection performances on ChatGPT-generated misinformation is lower than those on human-written misinformation. The numbers indicate the performance on generated misinformation is higher. Results and Analysis Since we aim to assess and compare the humans’ detection hardness of human-written misinformation and LLM-generated misinformation, measured by same group of human evaluators’ factuality annotation performance respectively, we can adopt Success Rate% as the evaluation metric, which is calculated by the percentage of successfully identified misleading news items in human-written or LLM-generated misinformation dataset. First, with ChatGPT as the representative LLM, we can observe in Table 3 that it is generally hard for humans to detect LLM-generated misinformation, especially those generated with Hallucinated News Generation, Totally Arbitrary Generation, Rewriting Generation, and Open-ended Generation methods. For example, we find that humans can only successfully spot 9.6% of all the generated hallucinated news on average, which reflects that it is extremely difficult for normal people to notice the fine-grained hallucinated information such as false dates, names, addresses, numbers and quotes. Second, we attempt to compare humans’ detection hardness for LLM-generated misinformation and human-written misinformation that have the same semantics, because the semantic information is the other factor impacting the detection difficulty apart from the style information. We have demonstrated that Paraphrase Generation, Rewriting Generation, and Open-ended Generation methods generally only change the style information and preserve the original semantics in Section 4. Comparing human detection performance on human-written misinformation (the numbers in Table 3) and LLM-generated misinformation via Paraphrase Generation, Rewriting Generation and Open-ended Generation approaches (the numbers or numbers in Table 3), we can discover that the human detection performances on LLM-generated misinformation are mostly lower than those on human-written misinformation. In particular, the statistical significance is strong for Rewriting Generation (p-value = $9.15 \times 10^{-5}$) and Open-ended Generation (p-value = $1.01 \times 10^{-6}$) using a paired T-test (more details in Appendix B). Thus, we can have our second core finding shown as follows: Finding 2: LLM-generated misinformation can be harder for humans to detect than human-written misinformation with the same semantics. Our finding validates the hypothesis that LLMs can be exploited to generate misinformation with more deceptive styles for humans via carefully-designed prompting strategies, indicating that its factuality is harder to determine for normal people. Also, our finding implies humans can be potentially more susceptible to LLM-generated misinformation than human-written misinformation. 6 RQ3: CAN DETECTORS DETECT LLM-GENERATED MISINFORMATION? Misinformation detection is critical for guarding online safety and public trust (Chen et al., 2022; Shu et al., 2017). However, in the age of LLMs, it is under exploration whether or not existing detectors can detect LLM-generated misinformation, which is key to defending against its potential pollution. Figure 5: Detector detection performance on ChatGPT-generated Misinformation and the comparison with human detection performance. Average detection performance over three runs is reported for ChatGPT-3.5 or GPT-4 as the detector due to the variance of API output. Emerging Challenges for Misinformation Detectors In the real world, detecting LLM-generated misinformation is in face with emerging challenges. First, it is difficult to obtain factuality supervision labels to train detectors for LLM-generated misinformation since it is harder for humans to detect than human-written misinformation (Section 2). Second, malicious users can easily utilize methods shown in Table 1 and close-sourced LLMs (e.g., ChatGPT) or open-source LLMs (e.g., Llama2 [Touvron et al., 2023]) or Vicuna [Chiang et al., 2023] to generate misinformation at scale in different domains, types, and errors, which is hard for conventional supervisedly trained detectors to maintain effective. Thus, it is likely to be impractical to apply conventional supervisedly trained detectors (e.g., BERT) to detect LLM-generated misinformation in the practices. Evaluation Setting We adopt LLMs such as GPT-4 with zero-shot prompting strategies as the representative misinformation detectors to assess and compare the detection hardness of LLM-generated misinformation and human-written misinformation for two reasons. First, zero-shot setting can better reflect the real-world scenarios of detecting LLM-generated misinformation considering the likely impracticality of conventional supervisedly trained detectors (e.g., BERT) in practices. Second, there are many works that have demonstrated directly prompting LLMs such as GPT-4 in a zero-shot way can outperform conventional supervisedly trained models such as BERT on detecting human-written misinformation [Petrine et al., 2023; Zhang et al., 2023c; Bang et al., 2023; Buchholz, 2023; Li et al., 2023b], which shows that zero-shot LLMs have already achieved almost state-of-the-art performance in the task of misinformation detection. In the zero-shot setting, we can adopt Success Rate % as the metric to measure the probability of LLM-generated or human-written misinformation being successfully identified, representing the difficulty of being detected. LLM Detection Performance vs. Human Detection Performance As for LLM-generated misinformation via Hallucinated News Generation, Totally Arbitrary Generation and Open-ended Generation, we run ChatGPT-3.5 (gpt-3.5-turbo) or GPT-4$^3$ as the detector on the dataset directly. As for Partially Arbitrary Generation, we first test on two types of generated data healthcare fake news and political rumors and then average the detection performance. As for Information Manipulation, we also report the average performance over all the six errors in Figure 2. The generated misinformation by aforementioned CMG methods is also based on Politifact dataset, which is consistently with human evaluation. The prompt using ChatGPT-3.5 or GPT-4 as the detectors is specified in Appendix E. Human detection performance is referred from Table 3. First, with ChatGPT as the representative LLM, we can observe that it is also generally hard for detectors to detect LLM-generated misinformation across different generation approaches, especially those generated via Hallucinated News Generation, Totally Arbitrary Generation and Open-ended Generation. For example, ChatGPT-3.5 (or GPT-4) can only detect 0.0% (or 10.0%) of the generated hallucinated news, which shows LLM detectors can hardly detect fine-grained hallucinations. Second, previous works have shown that detectors can perform better than humans on detecting human-written misinformation [Pérez-Rosas et al., 2018]. Comparing the detection performances of LLM detectors and humans, we can discover that GPT-4 can outperform humans on detecting LLM-generated misinformation, though humans can still perform better than ChatGPT-3.5. --- $^3$gpt-4: [https://platform.openai.com/docs/models/gpt-4](https://platform.openai.com/docs/models/gpt-4) \| Dataset \| Human-written \| Paraphrase Generation \| Rewriting Generation \| Open-ended Generation \| \|-------------------------\|--------------\|-----------------------\|----------------------\|----------------------\| \| \| No CoT \| CoT \| No CoT \| CoT \| \| ChatGPT-3.5-based Zero-shot Misinformation Detector \| \| Politifact \| 15.7 \| 39.9 \| 15.5 \| 10.2 \| \| Gossipcop \| 2.7 \| 19.9 \| 10.4 \| 2.3 \| \| CoAID \| 13.2 \| 41.1 \| 8.9 \| 4.3 \| \| GPT-4-based Zero-shot Misinformation Detector \| \| Politifact \| 48.6 \| 62.6 \| 46.9 \| 41.7 \| \| Gossipcop \| 3.8 \| 26.3 \| 10.8 \| 4.6 \| \| CoAID \| 52.7 \| 81.0 \| 54.4 \| 47.3 \| \| Llama2-7B-chat-based Zero-shot Misinformation Detector \| \| Politifact \| 44.4 \| 47.4 \| 42.2 \| 32.2 \| \| Gossipcop \| 34.6 \| 40.7 \| 35.3 \| 38.1 \| \| CoAID \| 19.8 \| 23.3 \| 14.6 \| 24.4 \| \| Llama2-13B-chat-based Zero-shot Misinformation Detector \| \| Politifact \| 40.0 \| 14.4 \| 42.6 \| 27.4 \| \| Gossipcop \| 10.8 \| 7.8 \| 13.9 \| 14.7 \| \| CoAID \| 30.2 \| 17.4 \| 24.2 \| 32.6 \| Table 4: Detector detection performance of human-written misinformation and ChatGPT-generated misinformation. More results on Llama-7b-chat-generated misinformation (or 13b, 70b) and Vicuna-7b-generated misinformation (or 13b, 33b) are in Appendix A. Standard Prompting (No CoT) and Zero-shot Chain-of-Thought Prompting (CoT) are adopted for detection. The metric is Success Rate %. Average performance over three runs is reported for ChatGPT-3.5 or GPT-4 as the detector due to the variance of the API output. The numbers highlight the detector detection performance on human-written misinformation. The numbers indicate the decrease of the detection performance on LLM-generated misinformation compared to human-written misinformation. And the numbers indicate the increase of the detection performance. LLM-Generated Misinformation vs. Human-Written Misinformation After evaluating the overall performance of LLM detectors, we aim to further investigate whether or not LLM-generated misinformation can be harder for detectors to detect than human-written misinformation with the same semantics. Thus, we conduct experiments to compare the detection performances on human-written misinformation and misinformation generated via Paraphrase Generation, Rewriting Generation and Open-ended Generation, which can preserve the original semantics (shown in Section 4). We adopt both ChatGPT and 6 types of open-source LLMs (Llama2-7b (or 13b, 70b) and Vicuna-7b (or 13b, 33b)) as the misinformation generators. The results are shown in Table 10 and Appendix A respectively. The generated misinformation is compared with real-world human-written misinformation datasets including Politifact, Gossipcop (Shu et al., 2020) and CoAID (Cui & Lee, 2020). Eight representative LLM detectors (ChatGPT-3.5, GPT-4, Llama2-7B, Llama2-13B, and “No CoT” and “CoT” strategies for each LLM) are adopted to assess the detection difficulty of LLM-generated and human-written misinformation. As for the “No CoT” strategy, we use the same prompt as the experiments in Figure 10. As for the “CoT” strategy, we follow the Zero-shot Chain-of-Thought Prompting method (Kojima et al., 2022). The specific prompts are specified in Appendix E. As shown in Table 10 and more results of Appendix A, we can observe that the detection performances on LLM-generated misinformation are mostly lower than those on human-written misinformation. For example, compared with detecting human-written misinformation in Politifact, Llama2-7B with “CoT” strategy has a performance drop by 19.6% on detecting misinformation that is generated by ChatGPT via Rewriting Generation. Also, the statistical significance is strong since the p-values shown in Appendix B are mostly lower than 5%. Thus, we have our third core finding: Finding 3: LLM-generated misinformation can be harder for misinformation detectors to detect than human-written misinformation with the same semantics. Our finding implies that LLM-generated misinformation can have more deceptive styles for detectors and existing detectors are likely to be less effective in detecting LLM-generated misinformation. Also, malicious users could potentially utilize LLMs to escape the detection of detectors. 7 IMPLICATIONS ON COMBATING MISINFORMATION IN THE AGE OF LLMs Through empirical investigation, we discover that LLMs (e.g., ChatGPT) can be leveraged to generate misinformation in an unintentional or intentional way, and LLM-generated misinformation can be harder for humans and detectors to detect compared to human-written misinformation with the same semantics. Our findings have multiple implications on combating misinformation in the age of LLMs. First, our findings directly suggest that LLM-generated misinformation can have more deceptive styles, which could be attributed to the intrinsic properties of LLM-generated content (e.g., the linguistic features) or the carefully-designed prompts (e.g., instructions such as “the style should be serious and calm”). Second, a large amount of hallucinated information is potentially generated by normal users due to the popularity of LLMs. Also, malicious users could be more inclined to exploit LLMs to generate misinformation to escape the detection of detectors. Thus, there is a potential major paradigm shift of misinformation production from humans to LLMs. Third, considering malicious users can easily prompt LLMs to generate misinformation at scale, which is more deceptive than human-written misinformation, online safety and public trust are faced with serious threats. We call for collective efforts to combat LLM-generated misinformation from stakeholders in different backgrounds including researchers, government, platforms, and the general public. ![Figure 6: Countermeasures against LLM-generated misinformation through LLMs’ lifecycle.](image) 8 COUNTERMEASURES THROUGH LLMs’ LIFECYCLE As shown in Figure 6, we propose to divide the lifecycle of LLMs into three stages and discuss the countermeasures against LLM-generated misinformation through the whole lifecycle. In the training stage, we can curate the training data to remove nonfactual articles and ground the training process to existing knowledge bases (Yu et al., 2020) to reduce LLMs’ hallucinations. Alignment training processes such as RLHF (Casper et al., 2023) can reduce the risk of generating harmful content. In the inference stage, we can utilize prompt filtering, intent modeling or jailbreak defenses (Iain et al., 2023) to prevent AMG methods (e.g., Totally Arbitrary Generation), and confidence (or uncertainty) estimation (Xiong et al., 2023) or retrieval augmentation (Mialon et al., 2023) to defend against HG methods (e.g., Hallucinated News Generation). However, they may be ineffective for most of CMG methods (e.g., Rewriting Generation), which are based on human-written misleading content and do not explicitly express the intent of generating misinformation. More research is desired to develop inference-time factuality verification methods for combating CMG methods. In the influence stage when LLM-generated content starts to influence the general public, it is under-explored how to design effective detectors for LLM-generated misinformation or texts. Also, it is essential to enhance the public’s awareness of the risks of LLM-generated misinformation. 9 CONCLUSION In this paper, we study an emerging and critical problem of LLM-generated misinformation. First, we build a taxonomy by types, domains, sources, intents and errors to characterize it. Also, we categorize the potential real-world methods to generate misinformation with LLMs and validate that LLMs (e.g., ChatGPT) can be utilized to generate misinformation in different types, domains and errors. Then, we conduct an extensive empirical investigation and discover that LLM-generated misinformation can be harder to detect for humans and detectors compared to human-written misinformation with the same semantics, indicating that LLM-generated misinformation can have more deceptive styles and potentially cause more harm. Finally, we discuss the implications of our findings on combating misinformation in the age of LLMs and the countermeasures through the whole LLMs’ lifecycle. REPRODUCTION STATEMENT Implementation Details As for ChatGPT-3.5 (gpt-3.5-turbo) or GPT-4 (gpt-4) as generators or detectors, we adopt the default API setting of OpenAI. As for Llama2 (Llama2-7B-chat, Llama2-13B-chat, and Llama2-70B-chat) and Vicuna (Vicuna-7b-v1.3, Vicuna-13b-v1.3, and Vicuna-33b-v1.3) as generators or detectors, we adopt the hyperparameters for the sampling strategy as follows: top_p = 0.9, temperature = 0.8, max_tokens = 2,000. Details of LLM-Generated Misinformation Dataset LLMFake We adopt three typical real-world human-written misinformation datasets including Politifact, Gossipcop (Shu et al., 2020) and CoAID (Cui & Lee, 2020). Politifact is a political fake news dataset containing 270 pieces of nonfactual news and 145 pieces of factual news. Gossipcop contains 2,230 pieces of nonfactual entertainment stories. CoAID has 925 pieces of COVID-19 misinformation in the healthcare domain. In the experiments, we utilize the whole Politifact dataset and the randomly sampled 10% data of the Gossipcop and CoAID datasets with the random seed as 1. The dataset has been open-sourced in the GitHub repository https://github.com/llm-misinformation/llm-misinformation. The construction process of our LLM-generated misinformation dataset LLMFake is described in Section 4. Since we aim to compare the detection difficulty of human-written and LLM-generated misinformation, the constructed LLM-generated misinformation dataset does not include any factual news items. More details of the misinformation generated via different approaches are as follows: • As for Hallucinated News Generation method, we utilize ChatGPT to generate 100 pieces of hallucinated news with prompts shown in Table 15 in Appendix E. • As for Totally Arbitrary Generation method, we utilize ChatGPT to generate 100 pieces of arbitrary misinformation prompts shown in Table 16 in Appendix E. • As for Partially Arbitrary Generation method, we utilize ChatGPT to generate 100 pieces of healthcare fake news and 100 pieces of political rumors such as Table 17 in Appendix E. • As for each of Paraphrase Generation, Rewriting Generation and Open-ended Generation methods, for each of the 7 types of misinformation generators (ChatGPT and open-source LLMs including Llama2-7b (or 13b, 70b) and Vicuna-7b (or 13b, 33b)), we generate 270 misinformation items based on the nonfactual part of the Politifact dataset, 86 items based on the nonfactual part of sampled CoAID dataset, and 231 items based on the nonfactual part of sampled Gossipcop dataset. We adopt Paraphrase Generation and Rewriting Generation methods to generate misinformation based on the original nonfactual <passages> of these datasets. As for Open-ended Generation, we first extract the several starting sentences of a passage, which generally summarize the whole passage, and then adopt Open-ended Generation method on the extracted nonfactual <sentences>. Examples of Paraphrase Generation are shown in Table 18, 19. Examples of Rewriting Generation are shown in Table 20, 21. Examples of Open-ended Generation are shown in Table 22, 23. • As for Information Manipulation Generation method, we can utilize ChatGPT to obtain 145 pieces of generated nonfactual news for each error described in Figure 2 (Unsubstantiated Content, Total Fabrication, Outdated Information, Description Ambiguity, Incomplete Fact, False Context) based on the factual <passages> of Politifact dataset. Examples are in Table 24 in Appendix E. ETHICS STATEMENT Considering that the open-source LLMs (e.g., Llama) or close-sourced LLMs (e.g., ChatGPT) are widely adopted, and the potential approaches to generate misinformation with LLMs are based on real-world scenarios (shown in Table 1) and straightforward to implement, we anticipate these methods have been potentially utilized to generate misinformation by normal people unintentionally or malicious users intentionally in the real world. Thus, our research illustrates the landscape of LLM-generated misinformation to shed light on the potential risks, enhance the public’s awareness of its harm, and call for collective countering efforts. We also discuss the implications of our findings and the potential countermeasures, which can inspire and facilitate more future research on defending against LLM-generated misinformation.
FoqZKsH9sE	In addition, it is unfair to compare the baseline in Table 1. The baseline in Table 1 is not training with the knowledge distillation, but the LSP models are fine-tuned with 300 epochs with knowledge distillation. And the accuracy of the distilled DeiT-Small and DeiT-Base should be 81.2% and 83.4%, respectively.	LSP: Low-Power Semi-Structured Pruning for Vision Transformers Anonymous authors Paper under double-blind review Abstract Vision transformers (ViTs) have emerged as a promising alternative to convolutional neural networks (CNNs) for various image analysis tasks, offering comparable or superior performance. However, one significant drawback of ViTs is their resource-intensive nature, leading to increased memory footprint, computation complexity, and power consumption. To democratize this high-performance technology and make it more environmentally friendly, it is essential to compress ViT models, reducing their resource requirements while maintaining high performance. In this paper, we introduce a new block-structured pruning to address the resource-intensive issue for ViTs, offering a balanced trade-off between accuracy and hardware acceleration. Unlike unstructured pruning or channel-wise structured pruning, block pruning leverages the block-wise structure of linear layers, resulting in more efficient matrix multiplications. To optimize this pruning scheme, our paper proposes a novel hardware-aware learning objective that simultaneously maximizes speedup and minimizes power consumption during inference, tailored to the block sparsity structure. This objective eliminates the need for empirical look-up tables and focuses solely on reducing parametrized layer connections. Moreover, our paper provides a lightweight algorithm to achieve post-training pruning for ViTs, utilizing second-order Taylor approximation and empirical optimization to solve the proposed hardware-aware objective. Extensive experiments on ImageNet are conducted across various ViT architectures, including DeiT-B and DeiT-S, demonstrating competitive performance with other pruning methods and achieving a remarkable balance between accuracy preservation and power savings. 1 INTRODUCTIONS Recently, vision transformers (ViTs) have been an emerging string of research that greatly challenges the prevailing CNNs with on-par or even superior performance on various image analysis and understanding tasks such as classification [Dosovitskiy et al., 2020; Cordonnier et al., 2020; Touvron et al., 2021a; Han et al., 2021b; He et al., 2022], object detection [Carion et al., 2020; Zhu et al., 2021b; Amini et al., 2021], semantic segmentation [Chen et al., 2021a; Liu et al., 2021], etc., but completely without the convolution mechanism seen in the CNNs. Despite the success in the task performances, as pointed out by [Yu et al., 2021a], one major drawback of the ViTs architecture is that the ViTs are much less resource-efficient than CNNs in terms of memory footprint, computation complexity and the eventual power consumption. To make the high-performance ViTs more environmentally friendly and democratize the technology, it is necessary to compress the ViTs models and cut down the power consumption, so that they could be accessed by low-end computation devices with equal or comparable model performance. Among different bifurcations of neural network compression, network pruning is an effective method that has shown success on CNNs, which prunes out redundant neurons or rules out computations in the networks. Previously on CNNs, some [Han et al., 2015a,b; Zhu & Gupta, 2018; Lee et al., 2020; Morcos et al., 2019; Lin et al., 2020; Wang et al., 2022; Xu et al., 2023] attempted unstructured pruning to the models which removes individual neurons from the layer weights; while others [Luo et al., 2017; Shen et al., 2022] used structured pruning which removes channel-wise neurons. Comparing to unstructured pruning, the latter structured scheme has high data locality hence is more hardware-friendly [Buluc & Gilbert, 2008] as it is easier to achieve ac- celerated computation by simply removing entire rows or columns in the weight matrices, it cause severer accuracy degradation due to the coarser pruning granularity making it a much more challenging pruning scheme. Nevertheless, for transformer architectures consisting of mostly linear layers (matrix multiplication), block structured (semi-structured) pruning is a better trade off between accuracy and hardware acceleration, since the GEMM performs matrix multiplication in a block-by-block manner. Hence multiplication with block sparse matrices can achieve more speedup than unstructured ones under the same pruning ratio while still maintaining high accuracy. A summarized qualitative comparison among pruning schemes is listed in Fig.1. Prior arts Mao et al. (2021); Lagunas et al. (2021) in NLP domain validated the block structured pruning on language models more than $2 \times$ speedup with negligible performance drop. However, the other parts of their pruning scheme is rather out-dated, e.g. vanilla pruning criterion. Similar attempts are still scarce on ViTs for various vision tasks. In this work, we propose a novel block-structured pruning approach for ViTs to prune the parameters in a block-based manner to achieve better trade-off between accuracy and efficiency. We formulate the learning objective in a way that simultaneously maintains the accuracy of the pruned model and minimizes the number of the computational operations. A hardware-aware constraint is incorporated into the objective to boost the speedup and lower power consumption during inference stage. Moreover, we present a fast optimization method to solve the objective function by utilizing second-order Taylor approximation. After equivalent reformulation, such we are able to solve the objective very efficiently (quadratic to cubic complexity for empirical data collection against network size and linear time complexity for equation solving). To the best of our knowledge, this is the first paper that introduces the block-structured pruning scheme and present a hardware-aware post-training pruning approach for ViTs. The main contributions are summarized as below: - We systematically formulate an optimal hardware-aware pruning objective for ViTs models under the block-structured pruning scheme, which directly optimizes both model accuracy and power consumption at the same time. The power consumption is fully estimated without the need of constructing any empirical look-up tables (LUTs), which makes it a light-weight approach and does not require additional overheads for optimization. The proposed pruning scheme solely relies on reducing parametrized layer connections without manipulating skip configurations and token pruning. - We then provide an efficient solution for the proposed hardware-aware objective function by utilizing second-order taylor approximation and present an empirical optimization method with only linear time complexity. The proposed method firstly generates the curves of the relationships between pruning rate and output error for each layer. Then, it is able to efficiently find the solution under different pruning rates in a fast way and does not need to re-solve the objective function each time when a pruning rate of the model is given. - Extensive experiments demonstrate the effectiveness of our approach. Results on various deep ViTs architectures, including DeiT-B and DeiT-S, show that our approach noticeably outperforms the state-of-the-arts regarding the trade-off between accuracy and speedup on the ImageNet dataset. ## 2 RELATED WORKS ### 2.1 VISION TRANSFORMERS (ViTs) Following the success of self-attention based transformer architecture in natural language processing [Vaswani et al. (2017)], transformer based vision models have also been marching in image domain and being strong competitors against traditional CNNs in various scenes like object detection [Carion et al. (2020); Zhu et al. (2021b)], segmentation [Chen et al. (2021a)], etc. ViT [Dosovitskiy et al. (2021)] was the first attempt to introduce MHA (multi-head attention) architecture for image... modality and surpassed the CNNs performance on image classification on large scale datasets. Later, DeiT [Touvron et al., 2021b] further boost the performance of raw ViTs with the same architecture but with token-based knowledge distillation to enhance the representation learning. MAE [He et al., 2022] introduces a supervision technique to pretrain ViT encoder on masked image reconstruction pretext task and achieves state-of-the-art performance on ImageNet classification task. Swin Transformer [Liu et al., 2021] utilized shifted window to introduce inter-window information exchange and enhance local attention. Transformer-iN-Transformer (TNT) [Han et al., 2021a] aggregated both patch- and pixel-level representations by a nested self-attention within each transformer block. 2.2 Pruning on CNNs CNNs pruning has been widely studied for decades. Large amount of pruning methods can be categorized in many different ways. Depending on the relationship between pruning and training procedure, they can be divided into post-train-pruning, pruning-at-initialization and pruning-during-training, where this work falls into post-train-pruning scheme as we determine the pruning mask on a converged pretrained model. Depending on the level of sparsity, it can be grouped into unstructured pruning, semi-structured pruning, structured (channel/filter-wise) pruning, etc. We introduce the related works based on the later taxonomy. Unstructured Pruning removes individual connections (neurons) from convolution kernels, which is the earliest established pruning scheme by the pioneer works [Han et al., 2015a,b], where they adopt a magnitude-based criterion with iterative fine-tuning procedure for LeNet and AlexNet. Molchanov et al. [2016] adopted taylor-based criterion as an importance score for connection. Frankle & Carbin [2019] proposed the lottery hypothesis deriving a weight-rewinding technique in iterative-pruning. Morcos et al. [2019]; Zhu & Gupta [2018] adopts magnitude-based importance scores to threshold low-scored connections globally. Gale et al. [2019]; Evci et al. [2020] leverage architectural heuristics to determine layerwise pruning rate. Lee et al. [2020] improved the magnitude-based scores like in Morcos et al. [2019] by considering inter-layer score ranking. Several efforts also prune CNNs data-dependently, considering the influence of pruning on the model output. Molehanov et al. [2016]; Lee et al. [2019] derived a first-order taylor-based pruning criterion. Isik et al. [2022] assumed laplacian distribution of CNN weights to approximate output distortion to determine layer-wise pruning ratio. Wang et al. [2022]; Xu et al. [2023] leverage rate-distortion theory to derive layer-wise pruning ratios that achieve optimal rate-distortion performance. Unstructured pruning achieves minimal sparse model accuracy thanks to the most fine-grained sparsity pattern, but such irregular sparsity pattern unfortunately makes it hard to achieve real-world acceleration without dedicated hardware optimization due to the poor data locality and low parallelism. Structured Pruning or channel/filter-wise pruning scheme prunes the entire kernel in a Conv layer or a channel in fully connected layer at once. Luo et al. [2017] used feature map importance as a proxy to determine removable channels. He et al. [2017] took a regularization-based structural pruning method. Yu et al. [2018] obtains channel-wise importance scores by propagating the score on the final response layer. Lin et al. [2020] utilized rank information of feature maps to determine the prunable channels. Wang et al. [2022] leveraged rate-distortion theory to prune the channels that lead to least model accuracy drop. Shen et al. [2022] took first-order importance on channels and allocates sparsities by solving a knapsack problem on all channel importances in the whole network. Structured pruning adopts coarser sparsity pattern than unstructured pruning, which trades-off the model accuracy with easily achievable acceleration. Semi-Structured Pruning is a less-explored approach that leverages sparsity pattern in between unstructured and structured pruning, where patterns such as block-sparsity in matmul can greatly benefit the real-world speedups by exploiting the nature of GPU calculation [Mao et al., 2021]; Lagunas et al., [2021]. With the sparsity pattern less aggressive than structured pruning, the impact of removing neurons on the model accuracy is less than structured pruning. Nevertheless, semi-structured pruning is under-explored on the emerging ViTs, which are constructed with transformer encoder architecture with mostly fully connected layers. 2.3 Sparsity in ViTs Witnessing the success of CNNs pruning, ViTs pruning is also receiving emerging interests. Compared to CNNs pruning, less efforts are devoted to pure weight pruning but more on pruning of tokens, MHA, etc. S²ViTE \cite{chen2021structured} first proposed to prune out tokens as well as self-attention heads under structured pruning scheme with sparse training for ViTs. UVC \cite{yu2021unified} derived a hybrid optimization target that unifies structural pruning for ViT weights, tokens and skip configuration to achieve sparse training for ViTs. SPViT \cite{kong2022spvit} only performed token pruning on attention heads but adopted latency constraint to maximize speedup on edge devices. \cite{yang2023ampere} adopts Nvidia’s Ampere 2:4 sparsity structure to achieve high speedup but required structural constraints to ensure a matching dimensions of qkv, feedforward and projection layers (head alignment) to search for subnetwork from larger ViT variants to match the latency of smaller ones. Unlike prior works \cite{yu2021unified,yang2023ampere}, our method focuses on pure weight pruning scheme and does not require heavy searching for the coordination of different compression schemes. Some efforts \cite{kitaev2019gated,wu2019reformer,wang2021structured,zaher2020sparce} sparsify the heavy self-attention by introducing sparse and local attention patterns for language models. \cite{child2019generating} attempts on ViTs, but these sparse attention schemes still require training from scratch. 3 METHODOLOGIES 3.1 PRELIMINARIES Block-structured Pruning within layer. We targeted at block-structured pruning for all linear layer weights, which include any parametrized linear layers in the ViTs, such as qkv layers, feed-forward and projection layers. Neurons in these weight matrices are grouped in 2-dimensional fixed-sized blocks as a unit for pruning. To decide which blocks need to be pruned, given a block structure $(B_h, B_w)$, for each matrix $W \in \mathbb{R}^{H \times W}$, we rank the blocks by the average of 1st order taylor expansion score of the neuron within each block. Mathematically, we first obtain the neuron score by the taylor expansion $S = [W \cdot \nabla_W f]$ similar to \cite{molchanov2019importance}, then perform a 2D average pooling to obtain a score for each block $S' \in \mathbb{R}_{}^{H/B_h \times W/B_w}$ ($\mathbb{R}_{}$ is non-negative real value set). Then given a pruning ratio for each layer, we can rank the blocks by their scores and eliminate the bottom ranked ones. The right most part of Fig. 2 visualizes the block-structure patterns realistically generated from ViTs. The above pruning scheme can be formulated as $\tilde{W}_{i,j} = W_{i,j} \odot M_\alpha(S')_{\lfloor \frac{i}{B_h} \rfloor, \lfloor \frac{j}{B_w} \rfloor}$, where $M_\alpha(S')$ is the binary mask generated from the previous block-wise score matrix under the pruning ratio $\alpha$. Pruning scheme of ViTs. Unlike prior arts, the scope of this work is only eliminating model parameters to reduce computation, without considering other aspects of ViTs like token number and token size and transformer block skipping \cite{chen2021structured,yu2021unified,kong2022spvit}. We further adopt a basic assumption for the weight perturbation $\Delta W = \tilde{W} - W$ caused by a typical pruning operation to the weight: Assumption 1 i.i.d. weight perturbation across layers \cite{zhou2018iclr}: This means the joint distribution is zero-meaned: $\forall 0 < i \neq j < L, E(\Delta W^{(i)} \Delta W^{(j)}) = E(\Delta W^{(i)})E(\Delta W^{(j)}) = 0$, and also zero co-variance: $E(\\|\Delta W^{(i)} \Delta W^{(j)}\\|^2) = 0$. 3.2 HARDWARE-AWARE PRUNING OBJECTIVE Since layers may contribute differently to the model performance \cite{frankle2020pruning}, various criteria have been proposed to allocate layerwise sparsity given a total budget. However, most existing pruning objectives can be summarized as minimizing the model output accuracy under computation constraint, without explicitly taking into account the actual power consumption and speedup. In contrast, our compression pruning objective directly optimizes the power consumption to achieve certain computation reduction target (FLOPs). Specifically, given a neural network $f$ of $l$ layers and its parameter set $W^{(1:l)} = (W^{(1)}, ..., W^{(l)})$, where $W^{(i)}$ is the weights in layer $i$, pruning parameters in the $f$ will give a new parameter set $\tilde{W}^{(1:l)}$. We view the impact of pruning as the distance between the network outputs $f(x; W^{(1:l)})$ and $f(x; \tilde{W}^{(1:l)})$. Hence our learning objective is as follows: Figure 2: Illustration of the proposed Low Power Semi-structured pruning method. Widths of different layers within ViT block visualizes the computation complexities (FLOPs) of single layer. We first extract all layers with prunable weights in the pretrained ViTs, then we obtain the empirical curves δ-vs-sparsity as described in Eq. [1]. We further calculate the layer specific target slope $ \lambda_i $ according to its contribution to the power consumption and select the layer-wise pruning ratios when the target slopes are tangential to the curves. Finally we prune the layer weights given their pruning ratios in block-structured sparsity, and finally finetune the pruned ViTs. The rightmost of the diagram is an example of the block-sparsity patterns when block sizes for both dimensions are the same, but they don’t have to be the same as in the experiment section. \[ \min \\| f(x; W^{(1:l)}) - f(x; \tilde{W}^{(1:l)}) \\|^2 + \beta L_{power}(f(\tilde{W}^{(1:l)})) \quad s.t. \quad \frac{\text{FLOPs}(f(\tilde{W}^{(1:l)}))}{\text{FLOPs}(f(W^{(1:l)}))} \leq R, \] which jointly minimize the output distortion caused by pruning (first term) as well as the estimated power consumption $ L_{power}(f(\tilde{W}^{(1:l)})) $, under a certain FLOPs reduction target $ R $. ### 3.3 Second-order Approximation of Output Distortion To solve the pruning objective, we break down the first term related to the output distortion. We first expand the output distortion $ f(x; W^{(1:l)}) - f(x; \tilde{W}^{(1:l)}) $ using second-order taylor expansion: (omit the superscript $ (1:l) $ for visual clarity from now) \[ f(x; W) - f(x; \tilde{W}) = \sum_{i=1}^{l} \nabla_W^{(i)} \top f \Delta W^{(i)} + \frac{1}{2} \Delta W^{(i)} \top H_i \Delta W^{(i)}, \] where $ H_i $ is the hessian matrix of the $ i $-th layer weight. Then consider the expectation of the squared L2 norm in the objective Eq. [4], which can be rewritten as the vector inner-product form: \[ E(\\| f(x; W) - f(x; \tilde{W}) \\|^2) = E \left[ (f(x; W) - f(x; \tilde{W})) \top (f(x; W) - f(x; \tilde{W})) \right] \] \[ = \sum_{i,j=1}^{l} E \left[ \left( \nabla_W^{(i)} \top f \Delta W^{(i)} + \frac{1}{2} \Delta W^{(i)} \top H_i \Delta W^{(i)} \right) \top \left( \nabla_W^{(j)} \top f \Delta W^{(j)} + \frac{1}{2} \Delta W^{(j)} \top H_j \Delta W^{(j)} \right) \right]. \] When we further expand the inner-product term, the cross-term for each pair of different layer $ 1 \leq i \neq j \leq l $ is: \[ E \left[ \Delta W^{(i)} \top \nabla_W^{(i)} \top \nabla_W^{(j)} \top f \Delta W^{(j)} \right] + E \left[ \frac{1}{2} \Delta W^{(i)} \top \Delta W^{(i)} \top H_i \top \nabla_W^{(j)} \top f \Delta W^{(j)} \right] + \] \[ E \left[ \frac{1}{2} \Delta W^{(i)} \top \nabla_W^{(i)} \top \Delta W^{(j)} \top H_j \Delta W^{(j)} \right] + E \left[ \frac{1}{4} \Delta W^{(i)} \top \Delta W^{(i)} \top H_i \top \Delta W^{(j)} \top H_j \Delta W^{(j)} \right]. \] When we discuss the influence of the random variable $ \Delta W $, the first-order and second-order derivatives $ \nabla_W f $ and $ H $ can be regarded as constants and therefore can be moved out of expectation. Also vector transpose is agnostic inside expectation. So Eq. [4] becomes \[ \nabla_W^{(i)} \top \nabla_W^{(j)} \top E(\Delta W^{(i)} \top \Delta W^{(j)}) + \frac{1}{2} H_i \top \nabla_W^{(j)} \top E(\Delta W^{(i)} \Delta W^{(i)} \top \Delta W^{(j)}) + \] \[ \frac{1}{2} \nabla_W^{(i)} \top H_j E(\Delta W^{(i)} \top \Delta W^{(j)} \top \Delta W^{(j)}) + \frac{1}{4} H_i \top H_j E(\\|\Delta W^{(i)} \top \Delta W^{(j)}\\|^2). \] Using Assumption 1 we can find that the above 4 cross-terms also equal to zero. Therefore the expectation Eq. 3 results in only intra-layer terms: \[ E(\\|f(x; W) - f(x; \tilde{W})\\|^2) = \sum_{i=1}^{l} E \left( \left\\| \nabla_{W(i)}^T f \Delta W(i) + \frac{1}{2} \Delta W(i)^T H_i \Delta W(i) \right\\|^2 \right). \] (6) ### 3.4 Power Consumption under Block-Structured Pruning As the majority of the power consumption of network inference is attributed to the matrix multiplication operation, the network power consumption can be estimated by summing individual power cost of block-sparse matrix multiplication of each linear layers. Consider a matrix $ A \in \mathbb{R}^{M \times N} $, typically input tensor, to be multiplied with the block-sparse weight matrix $ B \in \mathbb{R}^{N \times K} $ with block-structure of $(B_n, B_k)$ and $\alpha$-percentage of blocks pruned out. When using a block-sparse GEMM configured with the kernel grid size of $ B_m $ on M-dimension, the power consumption of the block-sparse matmul can be estimated as \[ P = p_m \frac{M}{B_m} \left[ (1 - \alpha) \frac{N}{B_n} \frac{K}{B_k} \right], \] (7) where $ p_m $ is the power cost of individual within-block matmul. Therefore, the second term in Eq. 1 can be obtained by adding up the power consumption of the network of all layers: \[ \beta L_{power} = \beta p_m \sum_{i=1}^{l} \frac{M_i}{B_m} \left[ (1 - \alpha_i) \frac{N_i}{B_n} \frac{K_i}{B_k} \right], \] (8) where $ p_m $ and $ B_m $ can be absorbed into the weight coefficient $ \beta $ because they only depends on hardware parameters and GEMM configuration which is unified across layers. Final Objective. Combining Eq. 6 and Eq. 8 the final objective can be reformulated as: \[ \min \sum_{i=1}^{l} E \left( \left\\| \nabla_{W(i)}^T f \Delta W(i) + \frac{1}{2} \Delta W(i)^T H_i \Delta W(i) \right\\|^2 \right) + \beta \sum_{i=1}^{l} M_i \left[ (1 - \alpha_i) \frac{N_i}{B_n} \frac{K_i}{B_k} \right] \] s.t. $ \frac{\text{FLOPs}(f(\tilde{W}(1:l)))}{\text{FLOPs}(f(W(1:l)))} \leq R. $ (9) ### 3.5 Finding Solution to Pruning Objective At this point, we can further solve the optimization problem Eq. 9 on the layer-wise pruning ratio set $ \{\alpha_i \| 1 \leq i \leq l\} $ by applying lagrangian formulation [Wang et al., 2022; Xu et al., 2023]. \[ \frac{\partial}{\partial \alpha_i} \left( \nabla_{W(i)}^T f \Delta W(i) + \frac{1}{2} \Delta W(i)^T H_i \Delta W(i) + \beta M_i \left[ (1 - \alpha_i) \frac{N_i}{B_n} \frac{K_i}{B_k} \right] \right) = \lambda. \] (10) In practice we can get rid of the ceiling function in Eq. 10 and therefore: \[ \frac{\partial}{\partial \alpha_i} \left( \nabla_{W(i)}^T f \Delta W(i) + \frac{1}{2} \Delta W(i)^T H_i \Delta W(i) \right) = \lambda_i = \lambda + \beta \frac{M_i N_i K_i}{B_n B_k}, \] (11) which will give a continuous $ \alpha_i \in [0, 1] $ compared to the original solution with the ceiling, but in practice since the number of blocks within a weight tensor is limited the pruning ratio $ \alpha_i $ is to be rounded to a discrete value anyway. Solving Eq. 11 will need to collect empirical curves for all layers (pruning ratio $ \alpha_i $ against the taylor second-order term $ \delta_i = \nabla_{W(i)}^T f \Delta W(i) + \frac{1}{2} \Delta W(i)^T H_i \Delta W(i) $). By setting a specific $ \lambda $, we can solve Eq. 11 individually for each layer by searching for a $ \alpha_i $ that let the equality holds. The final solution of pruning ratios can be obtained by traversing $ \lambda $ that returns a pruned network closest to the constraint $ R $. One key insight that one can derive from the optimization solution Eq. 11 is that by controlling the weight $ \beta $, the power consumption are explicitly incorporated in the optimization process in the form of altering the target slope for the partial derivative of the curve $ \frac{\partial \delta_i(\alpha_i)}{\partial \alpha_i} $, which represents how intensely pruning one layer affects the final model accuracy (output distortion). In this way, we achieve direct tradeoff between model accuracy and power consumption. --- 1 We empirically find $ E(W(i)^T W(i) W(j)^T W(j)) = 0 $ holds on top of $ E(W(i) W(j)) = 0 $. 3.6 Empirical Complexity Hessian approximation. For empirical networks, we approximate the hessian matrix $ H_i $ using empirical fischer \cite{Kurtic2022}: \[ H_i = H_L(W^{(i)}) \approx \hat{F}(W^{(i)}) = \kappa I_d + \frac{1}{N} \sum_{n=1}^{N} \nabla W^{(i)}_n f_n \nabla W^{(i)}_n f_n^T. \] (12) In order to obtain empirical curves $ \frac{\partial \delta_i(\alpha_k)}{\partial \alpha_k} $ on a calibration set, one is possible to traverse different pruning ratio (e.g. in practice $ \alpha_k = \frac{k+1}{K}, 0 < k < K $) and calculate the corresponding $ \delta_i(\alpha_k) $ for all $ 0 < k < K $. However in such case, even with the approximated hessian, the curve generation for each layer is still very expensive at the complexity of $ O(NKD_i^4) $, where $ K $ is the number of possible pruning ratio selections and $ D_i = N_i K_i $ is the dimension of weight in $ i $-th layer. This poses challenge to make the proposed method efficient enough to enjoy the benefits of sparse network. We notice that the derivative $ \nabla W_i $ is constant to the change of pruning ratio which let us to reuse the hessian matrix for all pruning ratio, which drops the complexity to $ O((N + K)D_i^2 + KD_i^4) $. However, the existence of the biquadratic complexity makes it still too expensive. We further notice that when pruning ratio move up slightly, only a partition of the weight vector is pruned out from $ W_i $. Therefore we can select a subvector $ d \Delta W_i(\alpha_k) = \Delta W_i(\alpha_k) - \Delta W_i(\alpha_{k-1}) $ each time when pruning ratio increases from $ \alpha_{k-1} $ to $ \alpha_k $ and update the $ \delta_i(\alpha_k) $ from $ \delta_i(\alpha_{k-1}) $ by the following rule: \[ \delta_i(\alpha_k) - \delta_i(\alpha_{k-1}) = \nabla W_i^T f d \Delta W_i(\alpha_k) + \left( \frac{1}{2} d \Delta W_i(\alpha_k) + \Delta W_i(\alpha_{k-1}) \right)^T H_i' d \Delta W_i(\alpha_k). \] (13) Denote the dimension of the subvector $ d \Delta W_i(\alpha_k) $ as $ d_i(k) \ll D_i $ equals the number of values changes from $ \Delta W_i(\alpha_{k-1}) $ to $ \Delta W_i(\alpha_k) $, the multiplication calculation in Eq.13 can be operated at lower dimensions, where $ \nabla W_i^T f \in \mathbb{R}^{d_i(k)}, H_i' \in \mathbb{R}^{D_i \times d_i(k)} $ are subvector and submatrix indexed from the original ones. At $ k = 1, \alpha_k = 0 $ i.e. there is no pruning at all which guarantees $ \delta_i(\alpha_1) = 0 $. Therefore, the complexity becomes one time calculation of the hessian $ O(ND_i^2) $ at $ k = 1 $, in addition to $ K - 1 $ times of updating $ O(D_i^2 \sum_{k=1}^{K-1} d_i(k)^2) $, resulting in totally $ O((N+\sum_{k=1}^{K-1} d_i(k)^2)D_i^2) $ ($ d_i(k) \ll D_i $ when $ K $ is big enough). To this end, we presented a hardware-aware pruning criterion that explicitly accounts for the power consumption of the block-structured sparse model inference. The block-structured pruning scheme enables the obtained sparse network to achieve real-world acceleration on hardware while optimally preserving the network accuracy. The algorithm is extremely efficient to obtain a sparse ViT. 4 Experiments 4.1 Experiment settings We conduct experiments mainly on Deit-Small and Deit-Base \cite{Touvron2021} on ImageNet dataset \cite{Krizhevsky2012}. We adopt the same training settings as in UVC \cite{Yu2021} for the finetuning of ViTs, e.g. 300 epochs and the additional distillation token for knowledge distillation. We select 2000 training samples to form the calibration set to calculate the first and second-order derivatives. Automatic hyperparameters setting. As introduced in Sec. 3.5, there are two hyperparameters $ \lambda $ and $ \beta $ involved in the solution, but both can be adaptively configured without the need to set manually. For the identification of $ \beta $, we follow the below strategy: \[ \beta = \frac{\sum_{l=1}^{L} \max_i \frac{\partial \delta_i}{\partial \alpha_i}}{\sum_{l=1}^{L} \max_i \frac{\partial E_{power}}{\partial \alpha_i}}, \] (14) so that the scales of the output distortion term and power term is balanced. After the $ \beta $ is fixed, we assume the FLOPs of the pruned model is a monotonic function of $ \lambda \in [0, \infty) $ and therefore can perform the efficient binary search towards the target FLOPs to obtain the choice of $ \lambda $. Post-processing of empirical $ \delta $ curves. Due to the high granularity in the block-sparsity structure, the layer-wise $ \delta $ curves are expected to see some quantization effect, where the $ \delta $ values remains the same corresponding to small change in pruning ratio $ \alpha $. This effect is even more severe under larger block shapes, e.g. $64 \times 64$. To better aid the pruning ratio searching procedure, we adopt several post-processing tricks to the empirical curves: (1) Curve Smoothing: we perform Exponential Moving Average (EMA) smoothing on the curves. (2) Curve Derivative Numerical Approximation: We further approximate the derivatives of $\delta$ curves using 5-point centered difference [Sauer (2011)] compared with the target slope (RHS of Eq. 11). Baseline methods. For the following experiments, we followed the UVC [Yu et al. (2021a)] comparison settings and compare ourselves to the previous ViTs compression methods that at least involves model weights pruning, as well as hybrid methods, including SCOP [Tang et al. (2020)], VTP [Zhu et al. (2021a), S²ViTE [Chen et al. (2021b)] and UVC [Yu et al. (2021a)] itself. ### 4.2 Main results \| Model \| Method \| Top-1 Acc (%) \| FLOPs(G) \| FLOPs remained (%) \| \|-------------\|------------\|---------------\|----------\|--------------------\| \| Deit-Small \| Dense \| 79.8 \| 4.6 \| 100 \| \| \| SCOP \| 77.5 (-2.3) \| 2.6 \| 56.4 \| \| \| S²ViTE \| 79.22 (-0.58) \| 3.14 \| 68.36 \| \| \| UVC \| 78.82 (-0.98) \| 2.32 \| 50.41 \| \| \| LSP (Ours) \| 80.69 (+0.89) \| 2.3 \| 50 \| \| Deit-Base \| Dense \| 81.8 \| 17.6 \| 100 \| \| \| S²ViTE \| 82.22 (+0.42) \| 11.87 \| 66.87 \| \| \| VTP \| 80.7 (-1.1) \| 10 \| 56.8 \| \| \| UVC \| 80.57 (-1.23) \| 8 \| 45.5 \| \| \| LSP (Ours) \| 80.81 (-0.99) \| 8.8 \| 50 \| \| \| LSP (Ours) \| 80.55 (-1.25) \| 7.92 \| 45 \| As presented in Tab. 1, we first notice that our result on Deit-Small achieves loss-less, and even higher than dense model performance by 0.89, with roughly the same FLOPs, surpassing existing baselines by a large margin. On larger architectures like DeiT-Base, where our method displays less prominent improvement but still on-par performance on the Top-1 accuracy of 80.81 with 50% FLOPs remaining and 80.55 with around 45% FLOPs. This is an intuitive observation since coarser pruning patterns like structural pruning would hurt the performance of smaller models more than larger model with a lot more redundant weights, and that is also where smaller structures such as the proposed block-sparsity pattern will retain more performance while still ensure speedup compared to unstructured pruning. Benefit from the pruning scheme tailored to ViTs, we managed to cut down the computation of DeiT-Small by 50% while still have 3% accuracy gain from CNN pruning scheme SCOP [Tang et al. (2020)] even when they removed slightly less computations (56.4%). We also notice that pure weight pruning of ViTs still have the potentials to achieve superior performance to hybrid methods [Chen et al. (2021b); Yu et al. (2021a)], thanks to our layer-wise sparsity allocation algorithm that is formulated to directly minimizes the output error on the pruned model against the dense model. On DeiT-Small, we beat all existing hybrid methods that leverage patch-slimming or token selections. We remain competitive on larger model DeiT-Base, while we notice S²ViTE cannot achieve comparable FLOPs reduction to us. ### 4.3 Discussions Beyond the main results, we also attempt to discover how each creative parts in our proposed pruning scheme contribute to the final results, e.g. the essential objective constraint regulating the power consumption and the block-sparsity structure, and answering the important questions such as why does the power constraint benefits the performance. We present the detailed ablation studies in the Tab. 2 and Tab. 3. Power constraint. To look deep into how our proposed power efficient pruning scheme accomplishes the above performance gain, we compared the behaviors of our pruning objective with and without the second-term power consumption in Eq. 9. As shown in Tab. 2, we notice that when the FLOPs reduction rates for different settings are both approaching the target FLOPs 50% with only little fluctuations, our final pruning scheme (with power constraint) constantly gives significant higher finetuning results under different block shapes. Specifically, on DeiT-Base-BK32BN64, the Table 2: Ablation studies on the Power consumption constraint on the pruning result. We compare between the results with the power constraint (main results) and without (by setting $\beta = 0$). \| Method \| Acc (%) \| Params remained (%) \| FLOPs remained (%) \| \|-------------------------\|---------\|--------------------\|-------------------\| \| Deit-Base-BK32BN32 \| \| \| \| \| w/ Power constraint \| 80.81 \| 73.3 \| 52.5 \| \| w/o Power constraint \| 77.75 \| 26.9 \| 55.6 \| \| Deit-Base-BK32BN64 \| \| \| \| \| w/ Power constraint \| 80.71 \| 72.8 \| 50 \| \| w/o Power constraint \| 61.42 \| 49 \| 49.7 \| Table 3: Effects of different Block shape configurations on the pruning result. \| Model \| Block shape (BK × BN) \| Sparsity (%) \| Top-1 Acc (%) \| FLOPs remained (%) \| \|-------------\|-----------------------\|--------------\|---------------\|-------------------\| \| Deit-Small \| 16 × 16 \| 92.2 \| 80.69 \| 50 \| \| \| 32 × 16 \| 91 \| 79.09 \| 50 \| \| \| 16 × 32 \| 71.37 \| 78.2 \| 50 \| \| \| 32 × 32 \| 49 \| 73.32 \| 50 \| \| Deit-Base \| 32 × 32 \| 72.84 \| 80.81 \| 52.5 \| \| \| 64 × 32 \| 33.93 \| 80.05 \| 50 \| \| \| 32 × 64 \| 16.99 \| 80.71 \| 50 \| \| \| 64 × 64 \| 73.34 \| 79.52 \| 50.2 \| Performance drops by 19.29% when we only remove the power term (setting $\beta = 0$). This is an inspiring phenomenon since the power constraint are not designed to facilitate model accuracy at the first place. By inspecting the model sparsity (number of parameters remained), we learn that the proposed power constraint looks for layers with larger matmul dimensions to allocate more pruning quota to achieve the most impact on the computation reduction (FLOPs), and normally larger layers have more parameter redundancy. Therefore, this pruning ratio allocation actually cooperates with the main objective to minimize output distortion. For both block sizes, model sparsities are far less when the power constraint is removed, i.e. at 26.9% and 49% respectively. Block structure configurations. To evaluate how our optimization scheme adapts to different block size configurations, which is crucial to generalize on different hardware platforms with different levels of parallelism, we conducted an ablation studies varying different block shapes combinations as listed in Tab. 3. Firstly, although our algorithm provides around the same FLOPs remaining percentage for different block sizes, it is observed on both test transformer variants that smaller block sizes preserve more model accuracy after finetuning. On Deit-Base, smallest block (BK32BN32) generates the highest 80.81 accuracy while the largest block (BK64BN64) performs slightly worse at 79.52%. On smaller network DeiT-Small, the performance discrepancy is more pronounced, where the largest and smallest block sizes produces the Top-1 accuracy difference at around 7%. Secondly, we notice that smaller networks are more sensitive to the change of block shapes. Despite BK32BN32 configuration behaves remarkably on DeiT-Base, the finetuning process for DeiT-Small with BK32BN32 only give 73.32% accuracy with much struggle. By only changing one dimension of the block structure to half size, e.g. BK32BN32 to BK32BN16, the performance climbs back by a large margin, returning to acceptable range. Different block sizes results in drastic change to the resulting number of parameters left in the networks, e.g. from 92.2% of sparsity on Deit-Small-BK16BN16 to 49% on Deit-Small-BK32BN32. 5 CONCLUSIONS In this work, we presented a novel ViTs weight pruning algorithm designed to reduce energy consumption during inference. Leveraging the linear layer-centric structure of the ViT architecture, we introduced a semi-structured pruning scheme to balance finetuning stability and hardware efficiency. Our algorithm is very efficient despite employing a hessian-based pruning criterion. Experimental results on various ViTs on ImageNet showcase the method’s ability to identify optimal pruning solutions, maximizing accuracy for block-sparse models. Additionally, we illustrated the dual benefits of our proposed power-aware pruning objective, enhancing both software accuracy and hardware acceleration. REFERENCES Arash Amini, Arul Selvam Periyasamy, and Sven Behnke. 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aZH1dM3GOX	How many experts are used for the Meta World evaluations? How does the number of experts impact the scalabiltiy of the proposed approach? Further elaborations on the computational overhead introduces, as well as possible limitations might be more insightful, than the presented remarks regarding interpretability.	Multi-Task Reinforcement Learning with Mixture of Orthogonal Experts Ahmed Hendawy\textsuperscript{1,2}, Jan Peters\textsuperscript{1,2,3,4}, Carlo D’Eramo\textsuperscript{1,2,5} \textsuperscript{1}Department of Computer Science, TU Darmstadt, Germany \textsuperscript{2}Hessian Center for Artificial Intelligence (Hessian.ai), Germany \textsuperscript{3}Center for Cognitive Science, TU Darmstadt, Germany \textsuperscript{4}German Research Center for AI (DFKI), Systems AI for Robot Learning \textsuperscript{5}Center for Artificial Intelligence and Data Science, University of Würzburg, Germany Abstract Multi-Task Reinforcement Learning (MTRL) tackles the long-standing problem of endowing agents with skills that generalize across a variety of problems. To this end, sharing representations plays a fundamental role in capturing both unique and common characteristics of the tasks. Tasks may exhibit similarities in terms of skills, objects, or physical properties while leveraging their representations eases the achievement of a universal policy. Nevertheless, the pursuit of learning a shared set of diverse representations is still an open challenge. In this paper, we introduce a novel approach for representation learning in MTRL that encapsulates common structures among the tasks using orthogonal representations to promote diversity. Our method, named Mixture Of Orthogonal Experts (MOORE), leverages a Gram-Schmidt process to shape a shared subspace of representations generated by a mixture of experts. When task-specific information is provided, MOORE generates relevant representations from this shared subspace. We assess the effectiveness of our approach on two MTRL benchmarks, namely MiniGrid and MetaWorld, showing that MOORE surpasses related baselines and establishes a new state-of-the-art result on MetaWorld.\footnote{The code is available at \url{https://github.com/AhmedMagdyHendawy/MOORE}.} 1 Introduction Reinforcement Learning (RL) has shown outstanding achievements in a wide array of decision-making problems, including Atari games (Mnih et al., 2013; Hessel et al., 2018a), board games (Silver et al., 2016; 2017), high-dimensional continuous control (Schulman et al., 2015; 2017; Haarnoja et al., 2018), and robot manipulation (Yu et al., 2019). Despite the success of RL, generalizing the learned policy to a broader set of related tasks remains an open challenge. Multi-Task Reinforcement Learning (MTRL) is introduced to scale up the RL framework, holding the promise of enabling learning a universal policy capable of addressing multiple tasks concurrently. To this end, sharing knowledge is vital in MTRL (Teh et al., 2017; D’Eramo et al., 2020; Sodhani et al., 2021; Sun et al., 2022). However, deciding upon the kind of knowledge to share and the set of tasks to share that knowledge is crucial for designing an efficient MTRL algorithm. Human beings exhibit remarkable adaptability across a multitude of tasks by mastering some essential skills as well as having the intuition of physical laws. Similarly, MTRL can benefit from sharing representations that capture unique and diverse properties across multiple tasks, easing the learning of an effective policy. Recently, sharing compositional knowledge (Devin et al., 2017; Calandriello et al., 2014; Sodhani et al., 2021; Sun et al., 2022) has shown potential as an effective form of knowledge transfer in MTRL. For example, Devin et al. (2017) investigate knowledge transfer challenges between distinct robots and tasks by sharing a modular policy structure. This approach leverages task-specific and robot-specific modules, enabling effective transfer of knowledge. Nevertheless, this approach requires manual intervention to determine the allocation of responsibilities for each module, given some prior knowledge. In contrast, we aim for an end-to-end approach that implicitly learns... and shares the prominent components of the tasks for acquiring a universal policy. Furthermore, CARE (Sodhani et al., 2021) adopt a different strategy by focusing on learning representations of different skills and objects encountered by the tasks by utilizing context information. However, there is no inherent guarantee of achieving diversity among the learned representations. In this work, our goal is to ensure the diversity of the learned representations to maximize the representation capacity and avoid collapsing to similar representations. Consequently, we propose a novel approach for representation learning in MTRL to share a set of representations that capture unique and common properties shared by all the tasks. To ensure the richness and diversity of these shared representations, our approach solves a constrained optimization problem that orthogonalizes the representations generated by a mixture of experts via the application of the Gram-Schmidt process, thus favoring dissimilarity between the representations. Hence, we name our approach, Mixture Of ORthogonal Experts (MOORE). Notably, the orthogonal representations act as bases that span a subspace of representations leveraged by all tasks where task-relevant properties can be interpolated. More formally, we show that these orthogonal representations are a set of orthogonal vectors belonging to a particular Riemannian manifold where the inner product is defined, known as Stiefel manifold (James, 1977). Interestingly, the Stiefel manifold has recently drawn substantial attention within the field of machine learning (Ozay & Okatani, 2016; Huang et al., 2018a; Li et al., 2019; Chaudhry et al., 2020). For example, several works focus on enhancing the generalization and stability of neural networks by solving an optimization problem to learn parameters in the Stiefel manifold. Another line of work aims to reduce the redundancy of the learned features by forcing the weights to inhabit the Stiefel manifold. Additionally, Chaudhry et al. (2020) propose a continual learning method that forces each task to learn in a different subspace, thus reducing task interference through orthogonalizing the weights. In this paper, our objective is to ensure diversity among the shared representations across tasks by imposing a constraint that forces these representations to exist within the Stiefel manifold. Thus, we aim to leverage the extracted representations, in combination with deep RL algorithms, to enhance the generalization capabilities of MTRL policies. In the following, we provide a rigorous mathematical formulation of the MTRL problem, inspired by Sodhani et al. (2021), where latent representations belong to the Stiefel manifold. Then, we devise our MOORE approach for obtaining orthogonal task representations through the application of a Gram-Schmidt process on the latent features extracted from a mixture of experts. We empirically validate MOORE on two widely used and challenging MTRL problems, namely MiniGrid (Chevalier-Boisvert et al., 2023) and Meta-World (Yu et al., 2019), comparing to recent baselines for MTRL. Remarkably, MOORE establishes a new state-of-the-art performance on the MetaWorld MT10 and MT50 collections of tasks. To recap, the contribution of this work is twofold: (i) We propose a mathematical formulation, named Stiefel Contextual Markov Decision Process (SC-MDP), that defines the MTRL problem where the state is encoded in the Stiefel manifold through a mapping function. (ii) We devise a novel representation learning method for Multi-Task Reinforcement Learning that leverages a modular structure of the shared representations to capture common components across multiple tasks. Our approach, named MOORE, learns a mixture of orthogonal experts by encouraging diversity through the orthogonality of their corresponding representations. Our approach outperforms related baselines and achieves state-of-the-art results on the MetaWorld benchmark. 2 PRELIMINARIES A Markov Decision Process (MDP) (Bellman, 1957; Puterman, 1995) is a tuple $ \mathcal{M} = < S, A, P, r, \rho, \gamma > $, where $ S $ is the state space, $ A $ is the action space, $ P : S \times A \rightarrow S $ is the transition distribution where $ P(s'\|s,a) $ is the probability of reaching $ s' $ when being in state $ s $ and performing action $ a $, $ r : S \times A \rightarrow \mathbb{R} $ is the reward function, $ \rho $ is the initial state distribution, and $ \gamma \in (0, 1] $ is the discount factor. A policy $ \pi $ maps each state $ s $ to a probability distribution over the action space $ A $. The goal of RL is to learn a policy that maximizes the expected cumulative discounted return $ J(\pi) = \mathbb{E}_\pi[\sum_{t=0}^{\infty} \gamma^t r(s_t, a_t)] $. We parameterize the policy $ \pi_\theta(a_t\|s_t) $ and optimize the parameters $ \theta $ to maximize $ J(\pi_\theta) = J(\theta) $. 2.1 MULTI-TASK REINFORCEMENT LEARNING In MTRL, the agent interacts with different tasks $ \tau \in \mathcal{T} $, where each task $ \tau $ is a different MDP $ \mathcal{M}^\tau = < S^\tau, A^\tau, P^\tau, r^\tau, \rho^\tau, \gamma^\tau > $. The goal of MTRL is to learn a single policy $ \pi $ that maximizes the expected accumulated discounted return averaged across all tasks $J(\theta) = \sum_{\tau} J_{\tau}(\theta)$. Tasks can differ in one or more components of the MDP. A class of problems in MTRL assumes only a change in the reward function $r^{\tau}$. This can be exemplified by a navigation task where the agent learns to reach multiple goal positions or a robotic manipulation task where the object’s position changes. In this class, the goal position is usually augmented to the state representation. Besides the reward function, a bigger set of problems deals with changes in other components. In this category, tasks access a subset of the state space $S^{\tau}$, while the true state space $S$ is unknown. For example, learning a universal policy that performs multiple manipulation tasks interacting with different objects (Yu et al., 2019). Task information should be provided either in the form of task ID (e.g., one-hot vector) or metadata, e.g., task description (Sodhani et al., 2021). Following Sodhani et al. (2021), we define the MTRL problem as a Block Contextual Markov Decision Process (BC-MDP). It is defined by 5-tuple $\langle C, S, A, \gamma, M \rangle$ where $C$ is the context space, $S$ is the true state space, $A$ is the action space, while $M$ is a mapping function that provides the task-specific MDP components given the context $c \in C$, $M(c) = \{r^c, P^c, S^c, \rho^c\}$. As of now, we refer to the task $\tau$ and its components by the context parameter denoted as $c$. 3 RELATED WORKS Sharing knowledge among tasks is a key benefit of MTRL over single-task learning, as broadly analyzed by several works that propose disparate ways to leverage the relations between tasks (D’Eramo et al., 2020; Sodhani et al., 2021; Sun et al., 2022; Calandriello et al., 2014; Devin et al., 2017; Yang et al., 2020). Among many, D’Eramo et al. (2020) establish a theoretical benefit of MTRL over single-task learning as the number of tasks increases, and Teh et al. (2017) learn individual policies while sharing a prior among tasks. However, naive sharing may exhibit negative transfer since not all knowledge should be shared by all tasks. An interesting line of work investigates the task interference issue in MTRL from the gradient perspective. For example, Yu et al. (2020) propose a gradient projection method where each task’s gradient is projected to an orthogonal direction of the others. Nevertheless, these approaches are sensitive to the high variance of the gradients. Another approach, known as PopArt (Hessel et al., 2018b), examines task interference focusing on the instability caused by different reward magnitudes, addressing this issue by a normalizing technique on the output of the value function. Recently, sharing knowledge in a modular form has been advocated for reducing task interference. Yang et al. (2020) share a base model among tasks while having a routing network that generates task-specific models. Moreover, Devin et al. (2017) divide the responsibilities of the policy by sharing two policies, allocating one to different robots and the other to different tasks. Additionally, Sun et al. (2022) propose a parameter composition technique where a subspace of policy is shared by a group of related tasks. Moreover, CARE Sodhani et al. (2021) highlight the importance of using metadata for learning a mixture of state encoders shared among tasks, based on the claim that the learned encoders produce diverse and interpretable representations through an attention mechanism. Despite the potential of this work, the method is highly dependent on the context information as shown in this recent work (Cheng et al., 2023). However, we argue that all of these approaches lack the guarantee of learning diverse representations. In this work, we promote diversity across a mixture of experts by enforcing orthogonality among their representations. The mixture-of-experts has been well-studied in the RL literature (Akroun et al., 2021; Ren et al., 2021). Moreover, some works dedicate attention to maximizing the diversity of the learned skills in RL (Eysenbach et al., 2018). Previous works leverage orthogonality for disparate purposes (Mackey et al., 2018). For example, Bansal et al. (2018) promote orthogonality on the weights by a regularized loss to stabilize training in deep convolutional neural networks. Similarly, Huang et al. (2018a) employ orthogonality among the weights for stabilizing the distribution of activation in neural networks. In the context of MTRL, Paredes et al. (2012) enforce the representation obtained from a set of similar tasks to be orthogonal to the one obtained from selected tasks known to be unrelated. Recently, Chaudhry et al. (2020) alleviate catastrophic forgetting in continual learning by organizing task representations in orthogonal subspaces. Finally, Mashhadi et al. (2021) favor diversity in an ensemble of learners via a Gram-Schmidt process. As opposed to it, our primary focus lies in the acquisition of a set of orthogonal representations that span a subspace shared by a group of tasks where task-relevant representations can be interpolated. Figure 1: MOORE illustrative diagram. A state $s$ is encoded as a set of representations using a mixture of experts. The Gram-Schmidt process orthogonalizes the representations to encourage diversity. Then, the output head processes the representations $V_s$ by interpolating the task-specific representations $v_c$ using the task-specific weights $w_c$, from which we compute the output using the output function $f_\theta$. In our approach, we employ this architecture for both the actor and the critic. 4 Sharing Orthogonal Representations We aim to obtain a set of rich and diverse representations that can be leveraged to find a universal policy that accomplishes multiple tasks. To this end, we propose to enforce the orthogonality of the representations extracted by a mixture of experts. In the following, we first provide a mathematical formulation from which we derive our approach. In particular, we highlight the connection between our method and the Stiefel manifold theory (Huang et al., 2018b; Chaudhry et al., 2020; Li et al., 2020), together with the description of the role played by the Gram-Schmidt process. Then, we proceed to devise our novel method for Multi-Task Reinforcement Learning on orthogonal representation obtained from a mixture of experts. 4.1 Orthogonality in Contextual Markov Decision Processes We study the optimization of a policy $\pi$, given a set of $k$-orthonormal representations in $\mathbb{R}^d$ for the state $s$. We define the orthonormal representations of state $s$ as a matrix $V_s = [v_1, ..., v_k] \in \mathbb{R}^{d \times k}$ where $v_i \in \mathbb{R}^d, \forall i \leq k$. It can be shown that the orthonormal representations $V_s$ belong to a topological space known as the Stiefel manifold, a smooth and differentiable manifold largely used in machine learning (Huang et al., 2018b; Chaudhry et al., 2020; Li et al., 2020). Definition 4.1 (Stiefel Manifold) Stiefel manifold $\mathcal{V}_k(\mathbb{R}^d)$ is defined as the set of all orthonormal $k$-vectors in the Euclidean space $\mathbb{R}^d$, where $k \leq d$, $\mathcal{V}_k(\mathbb{R}^d) = \{ V_s \in \mathbb{R}^{d \times k} : V_s^T V_s = I_k, \forall s \in S \}$. Under this lens, our goal can be interpreted as finding a set of orthogonal representations belonging to the Stiefel manifold that capture the common characteristics in the true state space $S$. Thus, we propose a novel MDP formulation for MTRL, which we call a Stiefel Contextual Markov Decision Process (SC-MDP), that is inspired by the BC-MDP introduced in Sodhani et al. (2021). An SC-MDP includes a function that maps the state $s$ to $k$-orthonormal representations $V_s \in \mathcal{V}_k(\mathbb{R}^d)$. Definition 4.2 (Stiefel Contextual Markov Decision Process) A Stiefel Contextual Markov Decision Process (SC-MDP) is defined as a tuple $< C, S, A, \gamma, \mathcal{M}', \varphi >$ where $C$ is the context space, $S$ is the true state space, $A$ is the action space. $\mathcal{M}'$ is a function that maps a context $c \in C$ to MDP parameters and observation space $\mathcal{M}'(c) = \{ r^c, P^c, S^c, \rho^c \}$, $\varphi$ is a function that maps every state $s \in S$ to a $k$-orthonormal representations $V_s \in \mathcal{V}_k(\mathbb{R}^d)$, $V_s = \varphi(s)$. We define our MTRL policy as $\pi(a\|s,c) = f_\theta(\varphi(s) \cdot w_c)$, where $w_c \in \mathbb{R}^k$ is the task-specific weight that combines the $k$-orthogonal representations into a task-relevant one and $f_\theta : \mathbb{R}^d \rightarrow \mathbb{R}^{\|A\|}$ is an output function with learnable parameters $\theta$ that generates actions from task-specific representations. To leverage a diverse set of representations across tasks, the mapping function $\varphi$ plays a fundamental role. Hence, we approximate $ \varphi $ by a mixture of experts $ h_\phi = [h_{\phi_1}, ..., h_{\phi_k}] $ with learnable parameters $ \phi = [\phi_1, ..., \phi_k] $ that generate $ k $-representations $ U_s \in \mathbb{R}^{d \times k} $ for state $ s $, while ensuring that the generated representations are orthogonal to enforce diversity. Conveniently, this objective finds a rigorous formulation as a constrained optimization problem where we impose a hard constraint to enforce orthogonality: \[ \max_{\Theta=\{\phi,\theta\}} J(\Theta) \] subject to \[ h_\phi^T(s) h_\phi(s) = I_k \quad \forall s \in S, \] where $ I_k \in \mathbb{R}^{k \times k} $ is the identity matrix. Instead of solving the constrained optimization problem in Eq. 1, we ensure the diversity across experts through the application of the Gram-Schmidt (GS) process to orthogonalize the $ k $-representations $ U_s $. Definition 4.3 (Gram-Schmidt Process) Gram-Schmidt process is a method for orthogonalizing a set of linearly independent $ U = \{u_1, ..., u_k : u_i \in \mathbb{R}^d, \forall i \leq k\} $. It maps the vectors to a set of $ k $-orthonormal vectors $ V = \{v_1, ..., v_k : v_i \in \mathbb{R}^d, \forall i \leq k\} $ defined as \[ v_k = u_k - \sum_{i=1}^{k-1} \frac{\langle v_i, u_k \rangle}{\langle v_i, v_i \rangle} v_i. \] where the representation of the $ i $-th expert $ u_i $ is projected in the orthogonal direction to the subspace spanned by the representations of all $ i - 1 $ experts. Therefore, we apply the GS process to map the generated representations by the mixture of experts $ U_s = h_\phi(s) $ to a set of orthonormal representations $ V_s = GS(U_s) $, satisfying the hard constraint in Eq. 1. ### 4.2 Multi-Task Reinforcement Learning with Orthogonal Representations Following our policy $ \pi(a\|s,c) $, each task can interpolate its relevant representation from the subspace spanned by the $ k $-orthonormal representations $ V_s $. We train a task encoder to produce the task-specific weights $ w_c \in \mathbb{R}^k $ given task information (e.g., task ID). The orthonormal representations are combined using the task-specific weight to produce relevant representations $ v_c \in \mathbb{R}^d $ to the task as $ v_c = V_s \cdot w_c $. The interpolated representation $ v_c $ captures the relevant components of the task that can be utilized by the RL algorithm and fed to an output function $ f_\theta $. The output function can be learned for each task separately (multi-head) or shared by all tasks (single-head) to compute the action components given the representations $ v_c $. Similarly, the same policy (actor) structure (Alg. 1) can be used for the critic (Alg. 2). In conclusion, this approach results in a Mixture Of ORthogonal Experts, thus, we call it MOORE, whose extracted representation is used to learn a universal policy for MTRL. A visual demonstration of our approach is shown in Fig.1. We adopt two different RL algorithms, namely Proximal Policy Optimization (PPO) and Soft Actor-Critic (SAC), with the purpose of demonstrating that our approach is agnostic to the used RL algorithms. PPO (Schulman et al., 2017) is a policy gradient algorithm that has the merit of obtaining satisfactory performance in a wide range of problems while being easy to implement. It is a first-order method that enhances the policy update given the current data by limiting the deviation of the new policy from the current one. Moreover, we integrate our approach to SAC, a high-performing off-policy RL algorithm that leverages entropy maximization to enhance exploration. ### 5 Experimental Results In this section, we evaluate MOORE against related baselines on two challenging MTRL benchmarks, namely MiniGrid (Chevalier-Boisvert et al., 2023), a set of visual goal-oriented tasks, and MetaWorld (Yu et al., 2019), a collection of robotic manipulation tasks. The objective is to assess the adaptability of our approach in handling different types of state observations and tackling a variable number of tasks. Moreover, the flexibility of MOORE is evinced by using it for on-policy (PPO for MiniGrid) and off-policy RL (SAC for MetaWorld) algorithms. Additionally, we conduct ablation studies that support the effectiveness of MOORE in various aspects. We assess the following points: the benefit of using Gram-Schmidt to impose diversity across experts, the quality of the learned representations, as well as the transfer capabilities, and the interpretability of the diverse experts. Figure 2: Average return on the three MTRL scenarios of MiniGrid. We utilize both multi-head and single-head architectures for our approach MOORE as well as the related baselines. For MOORE, MOE and PCGrad, the number of experts $k$ is 2, 3, and 4 for MT3, MT5, and MT7, respectively. The black dashed line represents the final single-task performance of PPO averaged across all tasks. For the evaluation metric, we compute the accumulated return averaged across all tasks. We report the mean and the 95% confidence interval across 30 different runs. 5.1 MiniGrid We consider different tasks in MiniGrid (Chevalier-Boisvert et al., 2023), a suite of 2D goal-oriented environments that requires solving different mazes while interacting with objects like doors, keys, or boxes of several colors, shapes, and roles. MiniGrid offers a visual representation of the state, which we adopt for our multi-task setting. We consider the multi-task setting from Jin et al. (2023) that includes three multi-task scenarios. The first scenario, MT3, involves the three tasks: LavaGap, RedBlueDoors, and Memory; the second scenario, MT5, includes the five tasks: DoorKey, LavaGap, Memory, SimpleCrossing, and MultiRoom. Finally, MT7 comprises the seven tasks: DoorKey, DistShift, RedBlueDoors, LavaGap, Memory, SimpleCrossing, and MultiRoom. In Sec. A.1, we provide descriptions and more details for the tasks. We compare MOORE against four baselines. The first one is PPO, considered a reference for comparing to single-task performance. The second baseline is Multi-Task PPO (MTPPO), an adaptation of PPO (Schulman et al., 2017) for MTRL. Then, we consider MOE, which employs a mixture of experts to generate representations without enforcing diversity across experts. Additionally, we have PCGrad (Yu et al., 2020), which is an MTRL approach that tackles the task interference issue by manipulating the gradients. We integrate PCGrad on top of the MOE baseline for a fair comparison. As for the MTRL architecture, we utilize multi-head and single-head architectures for all methods, showing their average return across all tasks in Fig. 2(a), and Fig. 2(b) respectively. MOORE outperforms the aforementioned baselines in almost all the MTRL scenarios. Notably, our method exhibits faster convergence than the baselines. It is interesting to observe that MOORE outperforms the single-task performance with a significant margin in comparison to the other baselines (Fig. 2(a)), which is usually considered as an upper-bound of the MTRL performance in previous works. This highlights the quality of the learned representations and the role of MOORE representation learning process in MTRL. Figure 3: Evaluating MOORE against MOE on the transfer setting. The study is conducted on the two transfer learning scenarios in MiniGrid, employing a multi-head architecture. The number of experts $k$ is 2 and 3 for MT3 → MT5 and MT5 → MT7, respectively. For the evaluation metric, we compute the accumulated return averaged across all tasks. We report the mean and the 95% confidence interval across 30 different runs. 5.1.1 Ablation Studies Transfer Learning. We examine the advantage of transferring the trained experts on a set of base tasks to novel tasks in order to assess the quality and generalization of these learned experts in comparison to the MOE baseline. We refer to the transfer variant of our approach as Transfer-MOORE while Transfer-MOE for the baseline. Moreover, we include the performance of MOORE and MOE as a MTRL reference for learning the novel tasks from scratch, completely isolated from the base tasks. In Fig. 3, we show the empirical results on two transfer learning scenarios where we transfer a set of experts learned on MT3 to MT5 (MT3 → MT5) and on MT5 to MT7 (MT5 → MT7). MT3 is a subset of MT5, while MT5 is a subset of MT7. First, we train on the base tasks, and then we transfer the learned experts (frozen) to the novel tasks (the difference between the two sets). As illustrated in Fig. 3, Transfer-MOORE outperforms Transfer-MOE in the two scenarios, showing the quality of the learned representations in the context of transfer learning. Moreover, the study demonstrates the ability of our approach as an effective MTRL algorithm that provides competitive results against the transfer variant (Transfer-MOORE). In contrast, MOE struggles to beat the transfer variant as in the MT3 → MT5 scenario. Consequently, this study advocates the diversification of the shared representations in transfer learning and MTRL. We highlight more details in B.2. Number of Experts. Additionally, we focus on the impact of changing the number of experts on the performance of our approach, as well as on MOE. In Fig. 4, we consider different numbers of experts on the MT7 scenario. We observe the effect of utilizing more experts in MOORE algorithm compared to MOE. The study shows that MOORE exhibits a noticeable advantage, on average, for an increasing number of experts. On the contrary, a slower enhancement of the performance is encountered by MOE. It is also worth noting that the performance of MOORE with $k = 4$ slightly outperforms MOE with $k = 10$ while being comparable to MOE with $k = 8$ (MOE best setting). This supports our claim about efficiently utilizing expert capacity through enforcing diversity. 5.2 MetaWorld Finally, we evaluate our approach on another challenging MTRL setting with a large number of manipulation tasks. We benchmark against MetaWorld (Yu et al., 2019), a widely adopted robotic manipulation benchmark for Multi-Task and Meta Reinforcement Learning. We consider the MT10 \| Total Env Steps \| 1M \| 2M \| 3M \| 5M \| 10M \| 15M \| 20M \| \|-----------------\|----\|----\|----\|----\|-----\|-----\|-----\| \| SAC (Yu et al., 2019) \| 10.0±8.2 \| 17.7±2.1 \| 18.7±1.1 \| 20.0±2.0 \| 48.0±9.5 \| 57.7±3.1 \| 61.9±3.3 \| \| MTSAC (Yu et al., 2019) \| 34.9±12.9 \| 49.3±9.0 \| 57.1±9.8 \| 60.2±9.6 \| 61.6±6.7 \| 65.6±10.4 \| 62.9±8.0 \| \| SAC + FiLM (Perez et al., 2017) \| 32.7±6.5 \| 46.9±9.4 \| 52.9±6.4 \| 57.2±4.2 \| 59.7±4.6 \| 61.7±5.4 \| 58.3±4.3 \| \| PCGrad (Yu et al., 2020) \| 32.2±6.8 \| 46.6±9.3 \| 54.0±8.4 \| 60.2±9.7 \| 62.6±11.0 \| 62.6±10.5 \| 61.7±10.9 \| \| Soft-Module (Yang et al., 2020) \| 24.2±4.8 \| 41.0±2.9 \| 47.4±5.3 \| 51.4±6.8 \| 53.6±4.9 \| 56.6±4.8 \| 63.0±4.2 \| \| CARE (Sodhani et al., 2021) \| 26.0±9.1 \| 52.6±9.3 \| 63.8±7.9 \| 66.5±8.3 \| 69.8±5.1 \| 72.2±7.1 \| 76.0±6.9 \| \| PaCo (Sun et al., 2022) \| 30.5±9.5 \| 49.8±8.2 \| 65.7±4.5 \| 64.7±4.2 \| 71.0±5.5 \| 81.0±5.9 \| 85.4±4.5 \| \| MOORE (ours) \| 37.2±9.9 \| 63.0±7.2 \| 68.6±6.9 \| 77.3±9.6 \| 82.7±7.3 \| 88.2±5.6 \| 88.7±5.6 \| Table 1: Results on MetaWorld MT10 (Yu et al., 2019) with random goals (MT10-rand). The results of the baselines are from Sun et al. (2022). MOORE uses $k = 4$ experts. For all methods, we report the mean and standard deviation of the evaluation metric across 10 different runs. The evaluation metric is the average success rate across all tasks. We highlight with bold text the best result. and MT50 settings, where a single robot has to perform 10 and 50 tasks, respectively. For the baselines, we compare our approach against the following algorithms. First, SAC (Haarnoja et al., 2018) is the off-policy RL algorithm that is trained on each task separately, thus being a reference for the single-task setting. Second, Multi-Task SAC (MTSAC) is the adaptation of SAC to the MTRL setting, where we employ a single-head architecture with a one-hot vector concatenated with the state. Then, SAC+FiLM is a task-conditional policy that employs the FiLM module (Perez et al., 2017). Furthermore, PCGrad (Yu et al., 2020) is an MTRL approach that tackles the task interference issue by manipulating the gradients. Soft-Module (Yang et al., 2020) utilizes a routing network that proposes weights for soft combining of activations for each task. CARE (Sodhani et al., 2021) is an attention-based approach that learns a mixture of experts for encoding the state while utilizing context information. Finally, PaCo (Sun et al., 2022) is the state-of-the-art method for MetaWorld that learns a compositional policy where task-specific weights are utilized for interpolating task-specific policies. Our approach uses a similar framework as in the MiniGrid experiment and employs a multi-head architecture. Following Sun et al. (2022), we benchmark against variants of the MT10 and MT50 scenarios, MT10-rand and MT50-rand, where each task is trained with random goal positions. The goal position is concatenated with the state representation. As a performance metric, we compute the success rate averaged across all tasks. We run our approach for 10 different runs and report their mean and standard deviations of the metric, similar in Sun et al. (2022). As stated in Tab. 1, MOORE outperforms all the baselines regarding sample efficiency and asymptotic performance. Moreover, in Tab. 2, our approach shows significant final performance, indicating the scalability of MOORE to a large number of tasks. It is important to mention that all baselines use tricks to enhance the stability of the learning process. For instance, PaCo avoids task and gradient explosion by proposing two empirical tricks, named loss maskout and w-reset, where pruning every task loss that reaches above a certain threshold, besides resetting the task-specific weight for that task. Also, as in Sun et al. (2022), the other baselines resort to more expensive tricks, such as terminating and re-launching the training session when a loss explosion is encountered. On the contrary, our approach does not need such tricks to improve the stability of the learning process, which can indicate the stability of the chosen architecture and the importance of learning distinct experts. ### 5.2.1 Ablation Studies Diversity. Similarly, we want to evince the advantage of favoring diversity across experts. We evaluate MOORE against MOE, a baseline that uses the same architecture of MOORE but without the Gram-Schmidt process. We evaluate MOORE against MOE on the two MTRL scenarios of MetaWorld, MT10-rand and MT50-rand. In Fig. 5(a), MOORE exhibits superior sample-efficiency compared to MOE. Moreover, MOORE significantly outperforms the baseline also in MT50-rand. \| Algorithms \| Success Rate (20M) \| \|------------\|--------------------\| \| MTSAC (Yu et al., 2019) \| 49.3±1.5 \| \| SAC + FiLM (Perez et al., 2017) \| 36.5±12.0 \| \| CARE (Sodhani et al., 2021) \| 50.8±1.0 \| \| PaCo (Sun et al., 2022) \| 57.3±1.3 \| \| MOORE (ours) \| 72.9±3.3 \| Table 2: Results on MetaWorld MT50 (Yu et al., 2019) with random goals (MT50-rand). The results of the baselines are from Sun et al. (2022). MOORE uses $k = 6$ experts. Figure 5: (a) Success rate on MetaWorld MT10-rand comparing MOORE, against MOE, using 4 experts. (b) Success rate on MetaWorld MT50-rand comparing MOORE, against MOE, given 6 experts. We show the average success rate across all tasks and the 95% confidence interval across 10 and 5 different runs for MT10-rand and MT50-rand, respectively. (Fig. 5(b)), evincing the scalability of our approach to large-scale MTRL problems. This study illustrates the importance of enforcing diversity across experts in MTRL algorithms. Interpretability. Additionally, we verify the interpretability of the learned representations. Fig. 6 shows an application of PCA on the learned task-specific weights $w_c$ that interpolate the representations of the experts. On the one hand, the pick-place task is close to the peg-insert-side since both tasks require picking up an object. On the other hand, the weights of door-open and window-open tasks are similar as they share the open skill. Therefore, enforcing diversity across experts distributes the responsibilities across them in capturing common components across tasks (e.g., objects or skills). This confirms that the learned experts have some roles that can be interpretable. ### 6 Conclusion and Discussion We proposed a novel MTRL approach for diversifying a mixture of shared experts across tasks. Mathematically, we formulate our objective as a constrained optimization problem where a hard constraint is explicitly imposed to ensure orthogonality between the representations. As a result, the orthogonal representations live on a smooth and differentiable manifold called the Stiefel manifold. We formulate our MTRL as a novel contextual MDP while mapping each state to the Stiefel manifold using a mapping function, which we learn through a mixture of experts while enforcing orthogonality across their representations with the Gram-Schmidt process, hence satisfying the hard constraint. Our approach demonstrates superior performance against related baselines on two challenging MTRL baselines. Taking advantage of all the experts during inference, our approach has the limitation of potentially suffering from high time complexity compared to a sparse selection of few experts. This leads to a trade-off between the representation capacity and time complexity, which could be investigated in the future by a selection of a few orthogonal experts. In addition to our transfer learning study, we are interested in investigating extensions of our approach into a continual learning setting. Figure 6: Principle Component Analysis (PCA) on the task-specific weights learned by MOORE on MetaWorld MT10-rand for a run with 100% success rate across all tasks. ACKNOWLEDGMENTS We want to thank Aliaa Khalifa for her support in writing the paper and Firas Al-Hafez for his feedback on the method. This work was funded by the German Federal Ministry of Education and Research (BMBF) (Project: 01IS22078). This work was also funded by Hessian.ai through the project ‘The Third Wave of Artificial Intelligence – 3AI’ by the Ministry for Science and Arts of the state of Hessen. Calculations for this research were conducted on the Lichtenberg high-performance computer of the TU Darmstadt and the Intelligent Autonomous Systems (IAS) cluster at TU Darmstadt. REFERENCES Riad Akrour, Davide Tateo, and Jan Peters. Continuous action reinforcement learning from a mixture of interpretable experts. IEEE Transactions on Pattern Analysis and Machine Intelligence, 44(10):6795–6806, 2021. Nitin Bansal, Xiaohan Chen, and Zhangyang Wang. Can we gain more from orthogonality regularizations in training deep networks? Advances in Neural Information Processing Systems, 31, 2018. Richard Bellman. Dynamic Programming. Princeton University Press, Princeton, NJ, USA, 1 edition, 1957. Daniele Calandriello, Alessandro Lazaric, and Marcello Restelli. Sparse multi-task reinforcement learning. In Advances in Neural Information Processing Systems, 2014. Arslan Chaudhry, Naeemullah Khan, Puneet Dokania, and Philip Torr. Continual learning in low-rank orthogonal subspaces. 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UTLv72uDlS	A couple direct questions:- Is the matrix S actually being approximated at specific rows, or are entire rows being left out?- How is it more efficient to compute the gradient at a sampled point? Doesn’t this essentially require backpropagating through all time steps from the end of time to the beginning, regardless of whether or not you are going to then throw away some of the gradient information?	SCALING SAFE LEARNING-BASED CONTROL TO LONG-HORIZON TEMPORAL TASKS Anonymous authors Paper under double-blind review ABSTRACT This paper introduces a model-based approach for training parameterized policies for an autonomous agent operating in a highly nonlinear (albeit deterministic) environment. We desire the trained policy to ensure that the agent satisfies specific task objectives and safety constraints, both expressed in Signal Temporal Logic. We assert that this learning problem is similar to training recurrent neural networks (RNNs), where the number of recurrent units is proportional to the temporal horizon of the agent’s task objectives. This poses a challenge: RNNs are susceptible to vanishing and exploding gradients, and naïve gradient descent-based strategies to solve long-horizon task objectives thus suffer from the same problems. To tackle this challenge, we introduce a novel gradient approximation algorithm based on the idea of gradient sampling, and a smooth computation graph that provides a neurosymbolic encoding of STL formulas. We show that these two methods combined improve the quality of the stochastic gradient, enabling scalable backpropagation over long time horizon trajectories. We demonstrate the efficacy of our approach on various motion planning applications requiring complex spatio-temporal and sequential tasks ranging over thousands of time steps. 1 INTRODUCTION Learning-based approaches to synthesize control policies for highly nonlinear dynamical systems are prevalent across diverse domains, from autonomous vehicles to robots. Popular ways to train NN-based controllers include deep reinforcement learning (RL) [Berducci et al., 2021; Li et al., 2017; Chua et al., 2018; Srinivasan et al., 2020; Velasquez et al., 2021] and deep imitation learning [Fang et al., 2019]. Techniques to synthesize neural controllers (including deep RL methods) largely focus on optimizing user-defined rewards or costs, but do not directly address specific spatio-temporal task objectives. For example, consider the objective that the system must reach region $R_1$ before reaching region $R_2$, while avoiding an obstacle region. Such spatio-temporal task objectives can be expressed in the formalism of Signal Temporal Logic (STL) [Maler & Nickovic, 2004]. Furthermore, for any STL specification and a system trajectory, we can efficiently compute the robustness degree, or the approximate signed distance of the trajectory from the set of trajectories satisfying/violating the specification [Donzé & Maler, 2010; Fainekos et al., 2009]. The use of STL-based objectives has seen considerable recent interest in data-driven methods for training controllers for dynamical systems that can be described by (stochastic) difference equations. This literature brings together two separate threads: (1) smooth approximations to the robustness degree of STL specifications [Gilpin et al., 2020; Pant et al., 2017], enabling the use of STL robustness in gradient-based learning of control policies, and (2) efficient representation of the robustness computation allowing its use in training neural controllers using backpropagation [Yaghoubi & Fainekos, 2019; Leung et al., 2019, 2021; Hashemi et al., 2023; Hashemi et al.]. We are inspired by the work in [Hashemi et al., 2023] that proposes a ReLU-based neural network encoding (called STL2NN) to exactly encode the STL robustness degree computation. We show how we can extend this computation graph to obtain smooth underapproximations of the STL robustness degree. Backpropagation-based methods typically treat the one-step environment dynamics and the neural controller as a recurrent unit that is then unrolled as many times as required by the temporal horizon of the specification $\varphi$. For instance, if enforcing $\varphi$ requires reasoning over several hundred time-steps, then it involves training a recurrent structure that resembles RNN with hundreds of recurrent units. It is well-known that training of RNNs over long sequences faces problems of exploding and vanishing gradients (Goodfellow et al., 2016; Ba et al., 2016). To address this, we propose a sampling-based approximation of the gradient of the objective function (i.e. the STL property), that is particularly effective when dealing with behaviors over large time-horizons. Our method can improve training of NN controllers by at least an order of magnitude, i.e., in some cases, we reduce training times from hours to minutes. Several planning problems require finding optimal paths over long time-horizons. For example, consider the problem of planning the trajectory of a UAV in a complex, GPS-denied urban environment; here, it is essential that the planned trajectory span several minutes while avoiding obstacles and reaching several sequential goals (Windhorst et al., 2021). Contributions. To summarize, we make the following contributions: 1. We propose smooth versions of computation graphs representing the robustness degree computation of an STL specification over the trajectory of a dynamical system. Our computation graph guarantees that it lower bounds the robustness degree with a tunable degree of approximation. 2. We develop a backpropagation framework which leverages the new differentiable structure, and we show how we can handle STL specifications. 3. We develop a sampling-based approach to approximate the gradient of STL robustness w.r.t. the NN controller parameters. Emphasizing the time steps that contribute the most to the gradient, our method randomly samples time points over the trajectory. We utilize the structure of the STL formula and the current system trajectory to decide which time-points represent critical information for the gradient. 4. We demonstrate the efficacy of our approach on high dimensional nonlinear dynamical systems involving long-horizon and dynamic temporal specifications. Related Work. The use of temporal logic specifications for controller synthesis is a well-studied problem. Early work focuses on the model-based setting, where the environment dynamics are described either as Markov decision processes (Sadigh & Kapoor, 2016; Haesaert et al., 2018) or as differential equations (Gilpin et al., 2020; Pant et al., 2018; Raman et al., 2014; Farahani et al., 2015; Lindemann & Dimarogonas, 2018; Raman et al., 2015; Kalagarla et al., 2020; Lacerda et al., 2015; Guo & Zavlanos, 2018). Recent years have also seen growing interest in data-driven techniques (Balakrishnan et al., 2022; Li et al., 2018) for control synthesis. In addition, automata-based approaches (Sadigh et al., 2014; Hasaneibig et al., 2018; Hahn et al., 2020; Lavaei et al., 2020) are also proposed in the field to address temporal logic based objectives. In (Liu et al., 2021), the authors propose an imitation learning framework where a Model-Predictive Controller (MPC) guaranteed to satisfy an STL specification is used as a teacher to train a recurrent neural network (RNN). In (Wang et al., 2023; Balakrishnan & Deshmukh, 2019), the authors replace handcrafted reward functions with the STL robustness within single-agent or multi-agent deep RL frameworks. The overall approach of this paper is the closest to the work in (Yaghoubi & Fainekos, 2019; Leung et al., 2019, 2021; Hashemi et al., 2023; Hashemi et al.), where STL robustness is used in conjunction with backpropagation to train controllers. The work in this paper makes significant strides in extending previous approaches to handle very long horizon temporal tasks, crucially enabled by the novel sampling-based gradient approximations. Due to the structure of our NN-controlled system, we can seamlessly handle time-varying dynamics and complex temporal dependencies. The rest of the paper is organized as follows. In Sec. 2, we introduce the notation and the problem definition. We propose our learning-based control synthesis algorithms in Sec. 3, present experimental evaluation in Sec. 4, and conclude in Sec. 5. 2 PRELIMINARIES We use bold letters to indicate vectors and vector-valued functions, and calligraphic letters to denote sets. We denote the set, \{1, 2, \cdots , n\} with [n]. A feed forward neural network (NN) with $ \ell $ hidden layers is denoted by the array $[n_0, n_1, \cdots n_{\ell+1}]$, where $ n_0 $ denotes the number of inputs, $ n_{\ell+1} $ is the number of outputs and for all $ i \in [\ell] $, $ n_i $ denotes the width of $ i^{th} $ hidden layer. Neural Network Controlled Dynamical Systems (NNCS). Let $ s \in \mathbb{R}^n $ and $ a \in \mathbb{R}^m $ denote the state and action variables that take values from compact sets $ S \subseteq \mathbb{R}^n $ and $ C \subseteq \mathbb{R}^m $, respectively. We use $ s_k $ (resp. $ a_k $) to denote the value of the state (resp. action) at time $ k $. We define a neural network controlled system (NNCS) as a recurrent difference equation. \[ s_{k+1} = f(s_k, a_k). \] We assume that the control policy is a parameterized function $ \pi_\theta $, where $ \theta $ is a vector of parameters that takes values in $ \Theta $. Later in the paper, we instantiate the specific parametric form using a neural network for the controller. Given a fixed vector of parameters $ \theta $, the parametric control policy $ \pi_\theta $ returns an action $ a_k $ as a function of the current state $ s_k \in S $ and time $ k \in \mathbb{Z}_{\geq 0} $, or $ a_k = \pi_\theta(s_k, k) $. Closed-loop Model Trajectory. For a discrete-time NNCS as shown in equation [1] and a set of designated initial states $ I \subseteq S $, under a pre-defined feedback policy $ \pi_\theta $, equation [1] represents an autonomous discrete-time dynamical system. For a given initial state $ s_0 \in I $, a system trajectory $ \sigma^\theta_{s_0} $ is a function mapping time instants in $[0, K]$ to $ S $, where $ \sigma^\theta_{s_0}(0) = s_0 $, and for all $ k \in [0, K-1] $, \[ \sigma^\theta_{s_0}(k + 1) = f(s_k, \pi_\theta(s_k, k)). \] The computation graph for this trajectory is a recurrent structure. Appendix B shows an illustration of this structure and its similarity to RNN. In this paper, we provide algorithms to learn a policy $ \pi_\theta $ that maximizes the degree to which certain task objectives and safety constraints are satisfied. To that end, we formulate policy learning as an optimization problem. Task Objectives and Safety Constraints. We assume that task objectives or safety constraints of the system are specified in a temporal logic known as Signal Temporal Logic (STL) [Maler & Nickovic, 2004]. Our STL formulas are defined using the following syntax: \[ \varphi = h(s) \bowtie 0 \mid \varphi_1 \land \varphi_2 \mid \varphi_1 \lor \varphi_2 \mid F_I \varphi \mid G_I \varphi \mid \varphi_1 U_I \varphi_2 \] that are limited to positive normal form logical expressions. Here, $ \bowtie \in \{\leq, <, >, \geq\} $, $ h $ is a function from $ S $ to $ \mathbb{R} $, and $ I $ is a closed interval $[a, b] \subseteq [0, K]$. The formal semantics of STL over discrete-time trajectories have been previously discussed in [Fainekos & Pappas, 2006], we briefly recall them here. Boolean Semantics and Formula Horizon. We denote the formula $ \varphi $ being true at time $ k $ in trajectory $ \sigma^\theta_{s_0} $ by $ \sigma^\theta_{s_0}, k \models \varphi $. We say that $ \sigma^\theta_{s_0}, k \models h(s) \bowtie 0 $ iff $ h(\sigma^\theta_{s_0}(k)) \bowtie 0 $. The semantics of the Boolean operations ($ \land, \lor $) follow standard logical semantics of conjunctions and disjunctions, respectively. For temporal operators, we say $ \sigma^\theta_{s_0}, k \models F_I \varphi $ is true if there is a time $ k' $ that $ k' - k \in I $ where $ \varphi $ is true. Similarly, $ \sigma^\theta_{s_0}, k \models G_I \varphi $ is true iff $ \varphi $ is true for all $ k' $ where $ k' - k \in I $. In addition, $ \sigma^\theta_{s_0}, k \models \varphi_1 U_I \varphi_2 $ if there is a time $ k', k' - k \in I $ where $ \varphi_2 $ is true and for all times $ k'' \in [k, k') $ $ \varphi_1 $ is true. The temporal scope or horizon of an STL formula defines the number of time-steps required in a trajectory to evaluate the formula, $ \sigma^\theta_{s_0}, 0 \models \varphi $ [Maler & Nickovic, 2004]. For example, the temporal scope of the formula $ F_{[0,3]}(x > 0) $ is 3, and that of the formula $ F_{[0,3]}G_{[0,9]}(x > 0) $ is 3 + 9 = 12. Quantitative Semantics (Robustness value) of STL. Quantitative semantics of STL roughly define a signed distance of a given trajectory from the set of trajectories satisfying or violating the given STL formula. There are many alternative semantics proposed in the literature [Donzé & Maler, 2010; Fainekos & Pappas, 2006; Rodionova et al., 2022; Akazaki & Hasuo, 2015]; in this paper, we focus on the semantics from [Donzé & Maler, 2010] that are shown below. The robustness value $ \rho(\varphi, \sigma^\theta_{s_0}, k) $ of an STL formula $ \varphi $ over a trajectory $ \sigma^\theta_{s_0} $ at time $ k $ is defined recursively as follows: \| $ \varphi $ \| $ \rho(\varphi, k) $ \| \| --- \| --- \| \| $ h(s_k) \geq 0 $ \| $ h(s_k) $ \| \| $ \varphi_1 \land \varphi_2 $ \| $ \min(\rho(\varphi_1, k), \rho(\varphi_2, k)) $ \| \| $ \varphi_1 \lor \varphi_2 $ \| $ \max(\rho(\varphi_1, k), \rho(\varphi_2, k)) $ \| \| $ G_{[a,b]} \psi $ \| $ \min_{k' \in [k+a,k+b]} \rho(\psi, k') $ \| \| $ F_{[a,b]} \psi $ \| $ \max_{k' \in [k+a,k+b]} \rho(\psi, k') $ \| \| $ \varphi_1 U_{[a,b]} \varphi_2 $ \| $ \max_{k' \in [k+a,k+b]} \left( \min_{k'' \in [k,k']} \rho(\varphi_1, k'') \right) $ \| We note that if $ \rho(\varphi, k) > 0 $ the STL formula $ \varphi $ is satisfied at time $ k $, and we say that the formula $ \varphi $ is satisfied by a trajectory if $ \rho(\varphi, 0) > 0 $. STL Robustness as a ReLU NN. The quantitative semantics in equation [3] contains min/max operators; this makes the robustness of an STL formula difficult to be used in gradient-based methods for learning. --- 1 If the policy $ \pi_\theta $ is obvious from the context, we drop the $ \theta $ in the notation $ \sigma^\theta_{s_0} $. 2 For brevity, we omit the trajectory from the notation, as it is obvious from the context. However, min / max operators in equation 3 can be expressed using ReLU functions as follows: \[ \min(a_1, a_2) = a_1 - \text{ReLU}(a_1 - a_2), \quad \max(a_1, a_2) = a_2 + \text{ReLU}(a_1 - a_2). \] (4) This allows the computation graph representing the robustness of an STL formula w.r.t. a given trajectory to be expressed using repeated application of the ReLU function (with due diligence in balancing min, max computations over several arguments into a tree of at most logarithmic height in the number of operands). We call this ReLU-based computation graph as STL2NN. The STL2NN, despite being reformulated with ReLU, is essentially equivalent to non-smooth robustness in equation 3, making it unsuitable for back-propagation. To address this, smooth activations are introduced to create a differentiable computation graph. 3 TRAINING NEURAL NETWORK CONTROL POLICIES Problem Definition.: We wish to learn a neural network (NN) control policy $ \pi_\theta $ (or equivalently the parameter values $ \theta $), s.t. for any initial state $ s_0 \in I^T $ using the control policy $ \pi_\theta $, the trajectory obtained, i.e., $ \sigma^\theta_{s_0} $ satisfies a given STL formula $ \varphi $. Our solution strategy is to treat each time-step of the given dynamical equation in equation 1 as a recurrent unit. We then sequentially compose or unroll as many units as required by the horizon of the STL specification. For instance, if the specification is $ F_{[0,10]}(x > 0) $, then we use 10 instances of $ f(s_k, \pi_\theta(s_k)) $ by setting the output of the $ k^{th} $ unit to be the input of the $ (k+1)^{th} $ unit. This unrolled structure implicitly contains the system trajectory, $ \sigma^\theta_{s_0} $, starting from some initial state $ s_0 $ of the system. The unrolled structure essentially represents the symbolic trajectory, where each recurrent unit shares the NN parameters of the controller (see Appendix K for more detail). By composing this structure with the neural network representing the given STL specification $ \varphi $; for instance, the STL2NN computation graph introduced in the previous section, we have a NN that maps the initial state of the system in equation 1 to the robustness degree of $ \varphi $. Thus, training the parameters of this resulting NN to guarantee that its output is positive (for all initial states) guarantees that each system trajectory satisfies $ \varphi $. However, we face two main challenges in training such a NN. Challenge 1: The cost function to be optimized is the output of the STL2NN computation graph. As mentioned earlier, as this is identical to the non-smooth robustness proposed in equation 3, we cannot use it effectively with stochastic optimization frameworks. An obvious step is to approximate STL2NN by a smooth function. We represent this function as STL2LB and leverage it for computing the gradients of the robustness function. It is important for STL2LB to lower bound STL2NN; if we find NN parameters that guarantee a positive output of STL2LB for all possible system trajectories, then it guarantees that the system satisfies the given STL objective. Challenge 2: As our model can be thought of as a recurrent structure with number of repeated units proportional to the horizon of the formula, naive gradient-based training algorithms are applicable to only short time horizons. As our structure is recurrent, the gradient computation faces the same issues of vanishing and exploding gradients when dealing with long trajectories that RNNs may face in training [Pascanu et al., 2013]. We introduce an efficient technique to approximate gradients for long trajectories that is inspired by the idea of Drop-out [Srivastava et al., 2014]. This popular technique also suggests us calling this approximate gradient as robust gradient. 3.1 SMOOTH, GUARANTEED LOWER BOUND FOR STL2NN To guarantee a smooth lower bound for STL2NN, we replace ReLU activations in the min operation with the softplus activation function defined as: \[ \text{softplus}(a_1 - a_2) = \frac{1}{b} \log \left( 1 + e^{b(a_1 - a_2)} \right), \quad b > 0. \] Similarly we replace the ReLU activation functions contributing in max operation with the swish activation function: \[ \text{swish}(a_1 - a_2) = \frac{a_1 - a_2}{1 + e^{-b(a_1 - a_2)}}, \quad b > 0. \] In the context of neural network training we satisfy this condition considering a set of sampled initial states, but we verify our trained NN for all the initial states through formal verification techniques. We denote this smooth NN with STL2LB and we claim: (see Appendix J for more detail) \[ \forall (\sigma_{s_0}, b) \in \mathbb{R}^{nK} \times \mathbb{R} : \text{STL2LB}(\sigma_{s_0}; b) \leq \text{STL2NN}(\sigma_{s_0}) \] We note that replacing the min and max operators with smooth versions is, by itself, not novel. Several prior studies have explored smooth semantics for STL (Gilpin et al., 2020; Pant et al., 2017). For example, consider the smooth max and min introduced in (Gilpin et al., 2020; Pant et al., 2017; Liu et al., 2021; Leung et al., 2019; Lindemann & Dimarogonas, 2018): \[ \tilde{\max}(a_1, \cdots, a_\ell) = \frac{1}{b} \log \left( \sum_{i=1}^{\ell} e^{ba_i} \right) \quad \text{or} \quad \tilde{\max}(a_1, \cdots, a_\ell) = \sum_{i=1}^{\ell} \frac{a_i e^{ba_i}}{\sum_{i=1}^{\ell} e^{ba_i}}. \] and $ \tilde{\min}(a_1, \cdots, a_\ell) = -\tilde{\max}(-a_1, \cdots, -a_\ell) $. An issue with using any kind of smooth approximation is that numerical issues can be caused by the presence of large positive exponents. Here, we explain this with an example. Example 1. Let $ a_1 = 0 $, and $ a_2 = 80 $, and suppose we wish to perform a smooth approximation of $ \max(a_1, a_2) $ with Logexpsum, Boltzmann and swish operators. Let the parameter $ b = 10 $. Then we can see that computing $ \exp(ba_2) $ and $ \exp(-b(a_1 - a_2)) $ causes numerical issues. On the other hand, for $ a_1 = 80, a_2 = 0 $ the softplus operator may also fail. Hence, to resolve the computation problem, we can define a threshold $ \tau > 0 $ large enough and approximate swish and softplus activation functions as: \[ \tilde{\text{swish}}(\zeta) = \begin{cases} \text{swish}(\zeta) & \text{if } \zeta > -\tau/b \\ 0 & \text{if } \zeta < -\tau/b \end{cases}, \quad \tilde{\text{softplus}}(\zeta) = \begin{cases} \zeta & \text{if } \zeta > \tau/b \\ \text{softplus}(\zeta) & \text{if } \zeta < \tau/b \end{cases}, \] where $ \zeta = a_1 - a_2 $. It is important to note that such a technique cannot be performed for smoothing using Logexpsum or Boltzmann-style operators and is exclusively applicable on STL2LB. By selecting $ \tau $ large enough, we can maintain the differentiability of operators, at least to the accuracy level of existing computation tools. To avoid the shortcomings of Logexpsum and Boltzmann-style approximations, we use softplus (with the above modifications) and the swish function as activations. Lemma 1. For any formula $ \varphi $ belonging to STL in positive normal form, and $ b > 0 $, for a given trajectory $ \sigma_{s_0} = s_0, s_1, \ldots, s_K $, if $ \text{STL2LB}(\sigma_{s_0}; b) > 0 $, then $ \sigma_{s_0} \models \varphi $, where STL2LB is a computation graph for STL robustness degree but with the modified softplus activation instead of min and the modified swish activation instead of max. See Appendix J for proof. The main contributions of STL2LB comparing to the existing smooth robustness formula (Gilpin et al., 2020; Pant et al., 2017) can be summarized as follows: - Example 1 shows that STL2LB provides convenience for computation. - Lemma 1 indicates that, like (Gilpin et al., 2020), it is also a guaranteed smooth lower-bound for robustness function, thus, can be considered as a control barrier function. ### 3.2 Training with STL2LB In order to train the controller for all initial states, $ s_0 \in \mathcal{I} $ we solve the following optimization problem: \[ \theta^* = \arg \max_{\theta} \left( \mathbb{E}_{s_0 \sim \mathcal{I}} \left[ \rho(\varphi, \sigma_{s_0}^\theta, 0) \right] \right), \] subject to \[ \sigma_{s_0}^\theta(k + 1) = f(\sigma_{s_0}^\theta(k), \pi_\theta(\sigma_{s_0}^\theta(k), k)). \] that aims to increase the expectation of the robustness for initial states uniformly sampled from the set of initial states. Solving this optimization problem is equivalent to training the NN controller using a gradient-based algorithm (shown in Alg. 1). However we terminate the algorithm once the robustness is above a pre-specified lower threshold $ \bar{\rho} $. We also generate a population of samples from the set of initial states of the system, i.e. $ \mathcal{I} $, for training purposes and denote this set by $ \hat{\mathcal{I}} $. Algorithm 1: Neurosymbolic policy learning ``` Input: $ \hat{\mathcal{I}}, \theta^0, b, \varphi, \bar{\rho} $ j ← 0 while $ \min_{s_0 \in \hat{\mathcal{I}}} \left( \rho(\varphi, \sigma_{s_0}^\theta, 0) \right) < \bar{\rho} $ do $ s_0 \leftarrow \text{Sample from } \hat{\mathcal{I}} $ $ \sigma_{s_0}^\theta \leftarrow \text{Simulate using policy } \pi_\theta $ $ d \leftarrow \nabla_\theta \text{STL2LB}(\sigma_{s_0}^\theta) \text{ using } \sigma_{s_0}^\theta $ $ \theta^{j+1} \leftarrow \theta^j + \text{Adam}(d) $ $ j \leftarrow j + 1 $ ``` 3.3 Extension to Long Horizon Temporal Tasks & Higher Dimensional Systems When dealing with long time-horizon trajectories or high dimensional models, considering the entire trajectory to compute $\nabla_\theta \text{STL2LB}(\sigma_{s_0}^{\theta})$ in Alg. 1 becomes computationally impractical as it either approaches zero (vanishes) or diverges (explodes) due to the high number of steps in the trajectory $\sigma_{s_0}$. To alleviate this, inspired by the well-known idea of Drop-out (Srivastava et al., 2014) for backpropagation, we propose a sampling-based gradient approximation technique that prevents the gradient to explode/vanish and is also known to provide a robust training process. The basic idea in sampling-based technique is to only select certain time-points in the trajectory for gradient computation, while using a fixed older control policy at the non-selected points. In order to select time points, a naïve strategy is to choose time-points randomly. However, in our preliminary results, exploiting the structure of the given STL formula – specifically identifying and using critical predicates – gives superior results compared to random sampling. Definition 1 (Critical Predicate). As the robustness degree of STL is an expression consisting of min and max of robustness values of predicates at different times, the robustness degree is consistently equivalent to the robustness of one of the predicates $h(\cdot)$ at a specific time. This specific predicate $h^$ is called the critical predicate, and this specific time $k^$ is called the critical time. A difficulty in using critical predicates is that a change in controller parameter values may change the system trajectory, which may in turn change the predicate that is critical for its robustness value. Specifically, if the critical predicate in one gradient step is different from the critical predicate in the subsequent gradient step, our gradient ascent strategy fails to augment the robustness value, since it only results in the elevation of that specific critical predicate’s value. The incorrect gradient generated in this gradient step can lead to failure in the training process, as it may abruptly reduce the robustness value drastically. Given a predefined specification $\varphi$, Fig. 1 shows the non-differentiable points in robustness as a function of control parameters, with each smooth segment corresponding to a distinct critical predicate. In order to optimize robustness within these smooth partitions, stochastic optimizers like Adam can be employed effectively. However, it is essential to note that the Adam optimizer’s applicability is confined to differentiable points. To overcome this challenge, we employ a technique which utilizes STL2LB to re-smooth the problem at the non-differentiable local maxima. However, it is practically impossible to accurately detect the non-differentiable local maxima, thus we take a more conservative approach and shift the training approach to utilize STL2LB at every gradient step where the critical predicate technique is unable to improve the robustness. The rest of this section presents a detailed explanation for each module in our training algorithm, and Alg. 2 encapsulates these modules within a unified training process. In this algorithm, we use $\rho^\varphi(\sigma_{s_0}^{\theta})$ as shorthand for the robustness degree of $\sigma_{s_0}^{\theta}$ w.r.t. $\varphi$ at time 0. A detailed explanation for Alg. 2 is also provided in Appendix A. Sampling-based gradient approximation technique. This technique is based on sampling across recurrent units and is originally inspired by the popular idea of Drop-out proposed in (Srivastava et al., 2014). Considering the NN controllers rolled out over the trajectory, the idea of Drop-out suggests removing the randomly selected nodes from a randomly selected NN controller over the trajectory. This requires the node to be absent in both forward-pass and backward-pass in backpropagation algorithm. However, our primary goal is to alleviate the problem of vanishing and exploding gradients. Thus, we propose to sample random time steps and select all of its controller nodes to apply Drop-out. However, for long trajectories we need to drop out a large portion of time steps that result in inaccurate approximation, thus we compensate for this by repeating this process and computing for accumulative gradients (See parameters $N_1, N_2$ in Alg. 2). Restriction of Drop-out to sample time steps results in less number of self multiplication of weights and therefore alleviates the problem of vanishing/exploding gradient. However, this may result in disconnection between the trajectory states and thus we need to apply modifications to this strategy. To that end, we drop out the selected nodes but we also replace that group of selected nodes (controller unit) with its evaluation in forward pass. This strategy motivates us to define the sampled trajectory as proposed in definition 2. Definition 2 (Sampled Trajectory). Consider the set of time steps $ T = \{0, t_1, t_2, \ldots, t_N\} $ sampled from the horizon $ K = \{0, 1, 2, \ldots, K\} $, and the control parameters $ \theta^j $ in the gradient step $ j $. The sampled trajectory $ \tilde{\sigma}_{s_0,T}^{g_j} $ is a subset of trajectory states $ \sigma_{s_0}^{g_j} $, where $ \tilde{\sigma}_{s_0,T}(0) = s_0 $ and, \[ \forall i \in \{0, 1, \ldots, N\}: \tilde{\sigma}_{s_0,T}(i + 1) = f_i(\tilde{\sigma}_{s_0,T}(i), \pi_{g_j}(\tilde{\sigma}_{s_0,T}(i), t_i)). \] Given the pre-computed constants $ \{a_{1+t}, a_{2+t}, \ldots, a_{t+1-1}\} $ using $ \theta^j $ in the gradient step $ j $, the dynamics model $ f_i $ is defined as: \[ f_i(s, a) = f(f(\cdots(f(s, a), a_{1+t}), a_{2+t}), \ldots, a_{t+1-1}). \] Algorithm 2: Gradient-direction approximation algorithm for training the controller for long horizon tasks. ``` Input: ε, M, N, N_1, N_2, θ^0, ϕ̂, ρ̂, Ĵ, j = 0 while ρ̂(σ̂_{s_0}^{θ^j}) ≤ ρ̂ do s_0 ← Sample from Ĵ use_STL2LB ← False; j ← j + 1 if use_STL2LB = False then θ_1, θ_2 ← θ^j for i ← 1, ..., N_1 do σ̂_{s_0}^{θ^j}, k, h(s_k) ← Simulate trajectory, obtain critical predicate T_q, X_q, σ̂_{s_0,T_q}, q ∈ [M] ← Generate sampled time steps & sampled trajectories d_1 ← robust gradient ∇_θ Ĵ(σ̂_{s_0}^{θ^j}) d_2 ← robust gradient ∇_θ h(s_k) θ_1 ← θ_1 + Adam(d_1/N_1) θ_2 ← θ_2 + Adam(d_2/N_1) if ρ̂(σ̂_{s_0}^{θ_1}) ≥ ρ̂(σ̂_{s_0}^{θ_2}) then θ^j+1 ← θ_1 else if ρ̂(σ̂_{s_0}^{θ_2}) ≥ ρ̂(σ̂_{s_0}^{θ_1}) then θ^j+1 ← θ_2 else ℓ ← 1, update ← True while update & (use_STL2LB = False) do ℓ ← ℓ/2; θ̂ ← θ^j + ℓ(θ_2 − θ_1) if ρ̂(σ̂_{s_0}^{θ̂}, 0) ≥ ρ̂(σ̂_{s_0}^{θ^j}) then θ^j+1 ← θ̂, update ← False else if ℓ < ε then use_STL2LB ← True if use_STL2LB = True then θ_3 ← θ^j for i ← 1, ..., N_2 do T_q, X_q, σ̂_{s_0,T_q}, q ∈ [M] ← Generate sampled time steps & sampled trajectories d_3 ← robust gradient ∇_θ STL2LB(σ̂_{s_0}^{θ^j}, b) θ_3 ← θ_3 + Adam(d_3/N_2) θ^j+1 ← θ_3 ``` $ j + 1 $ we again generate a new set of sampled times and repeat the process. --- 4 We call this gradient robust since the Drop-out technique claims this gradient results in robust training. 5 In this work, we evaluate the applicability of our sampling based technique through different case studies. This is a common approach to replace the mathematical proofs with validation through experimental results. See the famous works like Srivastava et al. (2014). Figure 2 in Appendix A makes this definition more clear through visualization. This definition applies the idea of Drop-out that is also equipped with our modification to replace the set of selected nodes on a randomly selected time step with its pre-computed output in the forward pass for original trajectory. This set of nodes are indeed a controller unit on the sampled time step. However our contribution from the idea of sampled trajectory are listed as follows: 1. to apply the idea of Drop-out on control synthesis over extended trajectories which alleviates for the problem of vanishing/exploding gradients. 2. to restrict the sampling process to time-steps instead of a random node selection on trajectory. 3. to assure that the critical time is included in the set of sampled time steps. In this work we denote the gradient of original trajectory with 'original gradient' and the approximate gradient from our sampling technique as 'robust gradient'. In the backpropagation algorithm at a given gradient step $ j $ with control parameter, $ \theta^j $ we wish to compute the robust gradient $ \partial J / \partial \theta^j $. To that end, we utilize $ \theta^j $ to simulate the trajectory $ \{s_0, s_1, ..., s_K\} $ and control sequence $ \{a_0, a_1, ..., a_{K-1}\} $. We then generate a set of random selections for the sampled times $ T_q, q \in [M] $ and define the sampled trajectories, $ \tilde{\sigma}_{s_0,T_q}^{g_j} $ with the specified interrelation proposed in the definition 2. In the next gradient step, Way Point Function. The way point function, $ J^{wp}(\sigma_{s_0}) $, is established as a reward-based function designed to offer incentives to the optimizer to guide the trajectory toward a pre-defined path. Safe re-smoothing. As discussed before, in the event that the optimization process steers the control parameters towards non-differentiable local maxima, there may be a drastic reduction in the value of the robustness function. In this case, we replace the objective function with $ J(\sigma_{s_0}) = STL2LB(\sigma_{s_0}; b) $. This is because, STL2LB is a smooth version of robustness over the trajectory, in addition, it is a guaranteed lower bound for robustness and its distance to robustness can also be controlled with $ b $. Thus, its inclusion makes the re-smoothing process safe against a potential drastic drop in robustness value. In case the objective function $ J $ is the value of critical predicate, it is only a function of the trajectory state $ s_k^ $ and we sample the time steps as, $ T = \{0, t_1, t_2, \ldots, t_N\}, t_N = k^* $. The original gradient is $ \frac{\partial J}{\partial \theta} = (\frac{\partial J}{\partial s_k^})(\frac{\partial s_k^}{\partial \theta}) $ but based on our sampling technique inspired with Drop-out, the robust gradient will be defined as, $ \frac{\partial J}{\partial \theta} = (\frac{\partial J}{\partial s_k^})(\frac{\partial \tilde{s}_{s_0,T}(N)}{\partial \theta}) $ where unlike $ \frac{\partial s_k^}{\partial \theta} $ that is prone to vanish/explode problem, the new term $ \frac{\partial \tilde{s}_{s_0,T}(N)}{\partial \theta} $ can be computed efficiently.$^6$ In case the objective function is way-point or STL2LB, that is a function of all the trajectory states, we consequently segment the trajectory into $ M $ different partitions, by random time sampling as, \[ T^q = \{0, t_1^q, t_2^q, \ldots, t_N^q\}, q \in [M], (\forall q_1, q_2 \in [M] : T^{q_1} \cap T^{q_2} = \{0\}) \land (K = \bigcup_{q=1}^{M} T^q), \] with sub-trajectories generated by $ T^q, q \in [M] $ denoted as $ X^q = \{s_0, s_1^q, \ldots, s_t^q\} $. We know the original gradient in this case is $ \frac{\partial J}{\partial \theta} = \sum_{q=1}^{M} (\frac{\partial J}{\partial X^q})(\frac{\partial X^q}{\partial \theta}) $. However in our training process to compute the robust gradient, the gradient matrix $ \frac{\partial X^q}{\partial \theta} $ is supposed to be replaced with $ \frac{\partial \tilde{s}_{s_0,T^q}}{\partial \theta} $. Unlike the inefficient gradient matrix $ \frac{\partial X^q}{\partial \theta} $ that is prone to vanish/explode problem, the gradient matrix $ \frac{\partial \tilde{s}_{s_0,T^q}}{\partial \theta} $ can be computed efficiently. 4 EXPERIMENTAL EVALUATION In this section, we evaluate the performance of our proposed method. We implemented all experiments in MATLAB.$^7$ We give the details of our experimental setup in the Appendix. We evaluate on 5 environments (details given in the Appendix) (a) a 3 dimensional simple car, (b) a 6 dimensional drone, (c) a 6 dimensional drone combined with a moving frame with a task requiring a long path plan, (d) a multi-agent system of 10 connected Dubins car, and (e) a 12 dimensional quad-rotor. Evaluation metric. To evaluate the performance of our method, we first compare the results of Alg. 1 with the examples proposed in (Yaghoubi & Fainekos, 2019) for environments (a) and (b), and compare the runtimes. As the dimension of system increases, it becomes more challenging to avoid the training procedure from converging to local optima. Increasing the horizon of temporal task causes the gradients to become non-informative, as they potentially vanish or explode. Therefore, environments (c), (d) and (e) are solved with Alg. 2. We also show that Alg. 1 is unable to finish the computation for long horizon experiments within a reasonable number of iterations or runtime. Comparison. Application of Alg. 1 on the environments (a) and (b), shows noticeable improvement, w.r.t. the previous work in (Yaghoubi & Fainekos, 2019). In these examples, we started from a random initial guess for NN parameters and computed the solution within $ \approx 6 $ minutes. However the reported runtime in (Yaghoubi & Fainekos, 2019) is noticeably higher than ours. Appendix I shows a comparison between the performance of STL2LB and the previous works (Pant et al., 2017; Gilpin et al., 2020). This comparison emphasizes on the computational problem proposed in Example 1. Main results. We test the performance of our proposed sampling-based algorithm in highly nonlinear and high dimensional environments over long and also complex temporal tasks (details in the appendix). Table 2 reports the results of these experiments. $^6$The efficiency results from the control parameters $ \theta $ repeating in fewer steps as most of them are fixed. $^7$All experiments were run on a laptop PC with a Core i9 CPU, and we did not utilize GPUs for computation. To evaluate the contribution of Alg. 2, we perform an ablation study on a simple Dubin’s car environment. We assume an $1m \times 1m$ area for execution, and specify that the car moves in this area within $K = 10$ time steps ($\delta t = 0.1$) while avoiding an obstacle presented in this area (Figure 11 is a scaled ($\times 100$) version of this area). We evaluate the same case study, but with task horizons ranging from 10 to 1000 time steps. With increasing number of time-steps, we also need to magnify the size of the environment to maintain task difficulty. The ablation study involves solving each of these problems: (1) with the vanilla version of Alg. 1, with no sampling-based robust gradient computation (2) Alg. 1 where sampling-based robust gradient approach is performed using random times within the trajectory, and (3) Alg. 2 that combines gradient-based sampling based on critical predicates, safe re-smoothing, and waypoint functions. We summarize the results in Table 1. We can see that the inclusion of time sampling decreases the runtime for training process. We also observe that for relatively small horizons $K = 10, 50$, Alg. 1 performs slightly better than Alg. 2 in terms of runtime but for $K = 100, 500, 1000$ Alg. 2 is much more efficient. In the table, an entry “NF” indicates when the algorithm is unable to solve the problem within 8000 gradient steps. In Alg. 1, as the dimension of STL2LB grows with the length of the horizon and dimension of the system, we see it struggle with the more complex case studies. Table 2 highlights the versatility of our technique to handle various case studies with number of dimensions as high as 20, and time horizons in thousands of steps. We also use a diverse set of temporal task objectives that include nested temporal operators, and those involving trajectories from two independently moving objects (Drone & Moving Frame case study). The results were produced using Alg. 2. \| Horizon \| Algorithm 1 (No time Sampling) \| Algorithm 1 (With time Sampling) \| Algorithm 2 (With time Sampling) \| \|---------\|--------------------------------\|----------------------------------\|---------------------------------\| \| \| Num. of Iterations \| Runtime (seconds) \| Num. of Iterations \| Runtime (seconds) \| Num. of Iterations \| Runtime (seconds) \| \| 10 \| 34 \| 2.39 \| 11 \| 1.39 \| 4 \| 5.61 \| \| 50 \| 73 \| 2.46 \| 53 \| 14.01 \| 25 \| 6.09 \| \| 100 \| 152 \| 8.65 \| 105 \| 112.6 \| 157 \| 90.55 \| \| 500 \| NF[−1.59] \| 4986 \| 3237 \| 8566 \| 624 \| 890.24 \| \| 1000 \| NF[−11.49] \| 8008 \| NF[−88.42] \| 28825 \| 829 \| 3728 \| Table 1: Ablation study. We mark the experiment with NF[.] if it is unable to provide a positive robustness within 8000 iterations, and the value inside brackets is the maximum value of robustness it finds. We magnify the environment proportional to the horizon (see Appendix H for details). All experiments use a unique guess for initial parameter values. \| Case Study \| Temporal Task \| System Dimension \| Time Horizon \| NN Controller Structure \| Number of Iterations \| Runtime (second) \| Optimization Setting $[M, N, N_1, N_2, \epsilon, b]$ \| \|------------------\|---------------\|------------------\|--------------\|-------------------------\|----------------------\|------------------\|-----------------------------------------------------\| \| Simple Car \| $\varphi_1$ \| 3 \| 40 steps \| [4,10,2] \| 750 \| 403.19 \| Algorithm 1, $b=10$ \| \| Drone \| $\varphi_2$ \| 6 \| 35 steps \| [7,10,3] \| 16950 \| 354.36 \| Algorithm 1, $b=20$ \| \| Quad-rotor \| $\varphi_3$ \| 12 \| 45 steps \| [13,20,20,10,4] \| 1120 \| 6413.3 \| \| \| Multi-agent \| $\varphi_4$ \| 20 \| 60 steps \| [21,40,20] \| 2532 \| 6298.2 \| [9, 5, 30, 40, $10^{-5}$, 5] \| \| Drone & Frame \| $\varphi_5$ \| 7 \| 1500 steps \| [8,20,20,10,4] \| 84 \| 443.45 \| [100, 15, 30, 3, $10^{-5}$, 15] \| \| Dubins car \| $\varphi_6$ \| 2 \| 1000 steps \| [3,20,2] \| 829 \| 3728 \| [200, 5, 60, 3, $10^{-5}$, 15] \| Table 2: Results on different case studies (details in the appendix) 5 CONCLUSION We introduce STL2LB, a smooth computation graph that lower bounds the robustness degree of an STL specification. 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l3s3HwJYDm	In appendix A. 2, this paper uses a set of identical states to acquire the action vectors of the policy in the test set. What is the detailed process of obtaining the set of identical states? Are these states sampled by a certain policy?	OPPONENT MODELING BASED ON SUBGOAL INFERENCE Anonymous authors Paper under double-blind review ABSTRACT When an agent is in a multi-agent environment, it may face previously unseen opponents, and it is a challenge to cooperate with other agents to accomplish the task together or to maximize its own rewards. Most opponent modeling methods deal with the non-stationarity caused by unknown opponent policies via predicting the opponent’s actions. However, focusing on the opponent’s action is shortsighted, which also constrains the adaptability to unknown opponents in complex tasks. In this paper, we propose opponent modeling based on subgoal inference, which infers the opponent’s subgoals through historical trajectories. As subgoals are likely to be shared by different opponent policies, predicting subgoals can yield better generalization to unknown opponents. Additionally, we design two subgoal selection modes for cooperative games and general-sum games respectively. Empirically, we show that our method achieves more effective adaptation than existing methods in a variety of complex tasks. 1 INTRODUCTION Reinforcement learning (RL) has achieved remarkable success in games involving multiple agents, such as AlphaGo (Silver et al., 2016), OpenAI Five (OpenAI, 2018), and AlphaStar (Vinyals et al., 2019). The non-stationarity of multi-agent environments has brought many difficulties to problem-solving, and this has always been the case. In cooperative scenarios, many multi-agent reinforcement learning (MARL) methods (Lowe et al., 2017; Sunehag et al., 2017; Rashid et al., 2020; Son et al., 2019) aim to bridge the information gap between agents by training agents in a centralized manner, called centralized training with decentralized execution, enabling agents to work together seamlessly to accomplish cooperative tasks. Alternatively, fully decentralized methods (Jiang & Lu, 2022; Su & Lu, 2022) seek to break free from the constraints of centralized training, allowing agents to reach collaboration in a simpler and decentralized manner. In competitive scenarios, NFSP (Heinrich & Silver, 2016), PSRO (Lanctot et al., 2017), and DeepNash (Perolat et al., 2022) employ self-play to train agents for equilibrium strategies, allowing agents to adapt and improve their policy. By considering how the agent affects the expected learning progress of other agents, LOLA (Foerster et al., 2017) and COLA (Willi et al., 2022) apply opponent shaping to this setting. In these methods, all agents are jointly trained in the same scenario. Autonomous agents, different from those jointly trained, can act autonomously in complex and dynamic environments, sense the influence of the environment and other agents, and accomplish their own goals or tasks. Such agents can analyze the behavior of opponents$^1$ by building models that make predictions about some core properties of the agents being modeled, such as their actions, goals, and beliefs, in a method called opponent modeling (Albrecht & Stone, 2018). By modeling the intentions and policies of other agents, the training process of the agent might be stabilized (Papoudakis et al., 2019). Many studies rely on predicting the actions (He et al., 2016; Hong et al., --- $^1$We call any agent other than the autonomous agent itself “opponent,” whether it is a teammate or rival. goals (Raileanu et al., 2018), and returns (Tacchetti et al., 2018) of opponents during training. The autonomous agent adapts to different or unseen opponents by using the predictions or representations that are produced by the relevant modules. However, in some scenarios, opponents may continuously learn during interaction. Meta-MAPG (Kim et al., 2021) combines Meta-PG (Al-Shedivat et al., 2017) and LOLA, and focuses on the problem of the non-stationary environment caused by the continuous learning of opponents. MBOM (Yu et al., 2022) simultaneously targets a variety of adversaries, fixed policy, or continuous learning, by modeling the possible policies that an opponent may form, combined with Bayesian inference to generate an opponent’s imagined policy. Some methods focus on figuring out the opponent’s goal, e.g., ToMnet (Rabinowitz et al., 2018) and SOM (Raileanu et al., 2018). SOM infers the opponent’s goal through its own policy, in other words, “what would I do if I were the opponent?” LIAM (Papoudakis et al., 2021; Papoudakis & Albrecht, 2020) builds the opponent’s policy from its own partial observations and uses it to anticipate the opponent’s actions and make decisions. GSCU (Fu et al., 2022) chooses online between a real-time greedy strategy and a fixed conservative strategy through Bayesian belief in competitive environments. The greedy strategy is conditioned RL, while the conservative strategy is a bandit algorithm. Although a lot of the existing methods concentrate on modeling the opponent’s actions, such an approach is short-sighted, pedantic, and highly complex. Generally, modeling an opponent’s actions is just predicting what it will do at the next step. Intuitively, it is more beneficial for the agent to make decisions if it knows the situation of the opponent several steps ahead. Predicting the actions over a few steps is not elegant. For example, to reach the goal point of $(2, 2)$, an opponent moves from $(0, 0)$ following the action sequence $<\uparrow, \uparrow, \rightarrow, \rightarrow>$ by four steps (Cartesian coordinates). There are also 5 other action sequences, i.e., $<\uparrow, \rightarrow, \uparrow, \rightarrow>, <\uparrow, \rightarrow, \rightarrow, \uparrow>, <\rightarrow, \uparrow, \uparrow, \rightarrow>, <\rightarrow, \uparrow, \rightarrow, \uparrow>, <\rightarrow, \rightarrow, \uparrow, \uparrow>$, that can lead to the same goal. Obviously, the complexity of the action sequence is much higher than the goal itself. Other methods that claim to predict the opponent’s goal (Rabinowitz et al., 2018; Raileanu et al., 2018), without explicitly making a connection to the opponent’s goal or just predicting the goal at the next step, are essentially as shortsighted as modeling actions. Inspired by the fact that humans can predict the opponent’s goal by observing the opponent’s actions for several steps as illustrated in Figure 1, in this paper, we propose Opponent Modeling based on subGoals inference (OMG), which uses variational inference to predict the opponent’s future subgoals from historical trajectories. The trajectory of an opponent’s policy consists of a set of subgoals, and the trajectories of different policies may contain the same subgoal. This combinatorial property of the subgoals facilitates the generalization of the agent to unseen opponents’ policies. Moreover, we design two manners for selecting subgoals, which are applied to cooperative games and general sum games, respectively. Empirically, OMG outperforms existing opponent modeling methods in a variety of complex multi-agent environments, demonstrating the superiority of inferring subgoals over predicting actions. 2 RELATED WORK Opponent modeling. Opponent modeling plays a crucial role in enhancing the robustness and stability of reinforcement learning (Papoudakis et al., 2019). Given the presence of diverse opponent policies in multi-agent environments, the autonomous agent faces a significant challenge in learning resilient policies. When an agent perceives an opponent as part of the environment, the resulting environment becomes inherently unstable and intricate. To address this challenge, one straightforward method involves equipping the agent with the ability to incorporate information about its opponent, including aspects like the opponent’s behavior, goals, and beliefs (Albrecht & Stone, 2018), i.e., opponent modeling. It gives the agent a deeper insight and prediction ability about the opponent’s policy. Thus, the autonomous agent views the environment as less unstable and can simply use single-agent reinforcement learning methods. A common approach to modeling the policy of an opponent is predicting the opponent’s actions. DRON (He et al., 2016) and DPIQN (Hong et al., 2018) extend DQN (Mnih et al., 2015) by adding another network that estimates the opponents’ actions from the observations. The DQN uses the hidden layer of this network to improve its policy. Variational auto-encoders can also be used to model the opponent’s policy (Papoudakis & Albrecht, 2020), which results in probabilistic repre- Figure 2: Diagram of OMG architecture. In the interaction phase, OMG deduces subgoals from historical trajectories to enhance decision-making. In the update phase, OMG employs the subgoal selector to choose the state among those within the next few steps as the subgoal. sentations instead of fixed vectors. PR2 (Wen et al., 2019), MBOM (Yu et al., 2022), and TP-MCTS (Weil et al., 2023) combine the idea of recursive reasoning, nested form as “the agent believes [that the opponent believes (that the agent believes ...)]”, based on modeling the action of the opponent. Some works focus on modeling beliefs. Zintgraf et al. (2021) combined the sequential and hierarchical variational auto-encoders to construct a belief inference model using meta-learning, for belief inference. Zhang et al. (2023) introduced landmarks into the behavior model and improve the model by the action sequence of the opponents, so as to recognize and compare the opponent’s intention. Another key aspect of opponent modeling is to infer the opponent’s goal. Baker et al. (2009) formulated the goal recognition as a Markov decision process (MDP) and calculate the posterior probability of the goal by Bayes rule based on a prior goal library. ToMnet (Rabinowitz et al., 2018) aims to give the agent a human-like Theory of Mind. It uses three networks to infer the agent’s goal and action from previous and present information. SOM (Raileanu et al., 2018) implements the Theory of Mind with a goal library from a different perspective. SOM uses its own policy, the opponent’s observation, and the opponent’s action to work backward to learn the opponent’s goal distribution by gradient ascent. These methods either require a prior goal library or infer implicit “goals” that are not supervised by ground truth goals. Goal-conditioned RL. Goal-conditioned reinforcement learning is an extension of the single-agent algorithm. Most works focus on learning a goal-conditioned policy, where the goals are usually predefined (Plappert et al., 2018; Zhu et al., 2021). Some works consider acquiring subgoals automatically to accelerate learning. Paul et al. (2019) proposed a method that uses expert trajectories to generate subgoals, while (Chane-Sane et al., 2021) proposed to incorporate imaginary subgoals into policy learning to facilitate learning complex tasks, where subgoals are measured by value functions. Unlike existing goal-conditioned RL methods, we aim to infer the subgoal of the opponent and condition the agent policy on the inferred subgoal. 3 METHOD 3.1 PRELIMINARIES In general, we consider an $n$-agent stochastic game $\mathcal{M} = (\mathcal{S}, \mathcal{A}_1, \ldots, \mathcal{A}_n, \mathcal{P}, \mathcal{R}_1, \ldots, \mathcal{R}_n, \gamma)$, where $\mathcal{S}$ is the state space, $\mathcal{A}_i$ is the action space of agent $i \in [1, \ldots, n]$, $\mathcal{A} = \prod_{i=1}^{n} \mathcal{A}_i$ is the joint action space of agents, $\mathcal{P}: \mathcal{S} \times \mathcal{A} \times \mathcal{S} \rightarrow [0, 1]$ is a transition function, $\mathcal{R}_i : \mathcal{S} \times \mathcal{A} \rightarrow \mathbb{R}$ is the reward function of agent $i$, and $\gamma$ is the discount factor. The policy of agent $i$ is $\pi^i$, and the joint policy of other agents is $\pi^o(a^o\|s) = \prod_{j \neq i} \pi^j(a^j\|s)$, where $a^o$ is the joint action except agent $i$. All agents interact with the environment simultaneously without communication. The historical trajectory is available, i.e., for agent $i$ at timestep $t$, $\tau_t = \{s_0, a^i_0, a^o_0, \ldots, s_{t-1}, a^i_{t-1}, a^o_{t-1}\}$ is observable. Figure 3: Learned Q-values using tabular Q-learning in an $11 \times 11$ gridworld. The agent and the opponent start from the $S_1$ and $S_2$, respectively. The two reward points are $D_1$ and $D_2$, and the reward will only be given to the agent who arrives first. The opponent executes one of policies $\pi^o_1$ and $\pi^o_2$, which target $D_1$ and $D_2$, respectively. The goal of the agent $i$ is to maximize its expected cumulative discount rewards: $$\mathbb{E}_{s_{t+1} \sim P(\cdot\|s_t,a^i_t,a^o_t), a \sim \pi^i(\cdot\|s_t), a^o \sim \pi^o(\cdot\|s_t)} \left[ \sum_{t=0}^{\infty} \gamma^t R^i(s_t, a^i_t, a^o_t) \right].$$ For convenience, the learning agent treats all other agents as a joint opponent with the joint action $a^o \sim \pi^o(\cdot\|s)$ and reward $r^o$. The action and reward of the learning agent are respectively denoted as $a \sim \pi(\cdot\|s)$ and $r$ for notation simplicity. An agent treats other agents as part of the environment and ignores the non-stationarity posed by the change of other agents’ policies as independent Q-learning (Tampuu et al., 2017; Tan, 1993). Its policy is updated by: $$Q(s_t, a_t) = \mathbb{E}_{P(s_{t+1}\|s_t,a^o,a)}[r + \gamma \max_a Q(s_{t+1}, a)],$$ where $Q$ is Q-network. Opponent modeling typically predicts the actions of other agents to address the non-stationary problem. The opponent model uses historical trajectory as input to predict $\tilde{a}^o \sim \tilde{\pi}(\cdot\|r)$, where $\tilde{a}^o$ is the estimate of $a^o$. Its policy is updated as: $$Q(s_t, \tilde{a}^o_t, a_t) = \mathbb{E}_{P(s_{t+1}\|s_t,a^o,a)}[r + \gamma \max_a Q(s_{t+1}, \tilde{a}^o_{t+1}, a)].$$ Note that we cast our discussion here to Q-learning. All can be similarly applied to other RL methods, such as PPO (Schulman et al., 2017). ### 3.2 Policy Update with Opponent’s Subgoals The opponent’s subgoal is the representation of the state that the opponent may have in the future based on the opponent’s policy. Like “All roads lead to Rome”, the opponent may perform different sequences of actions but eventually reach the same state. Instead of focusing on the details of each of the opponent’s actions, the agent should focus on the state the opponent wants to reach. The opponent’s subgoal distribution probability is based on the opponent’s action sequence, that is, the opponent’s policy, but its sample space is still the representation of the state. Here we decouple the subgoal from the opponent’s policy and just consider decision-making problems conditioned on the opponent’s subgoal. Formally, we transform the original stochastic game $M$ into a state-augmented MDP, defined by $M_G = (S, G, A^i, P, R^i, \gamma)$, where $G$ is the subgoal space. Since $G$ is a representation of future states the opponent may go, $\|G\|$ is finite and less than or equal to $\|S\|$. The state-augmented MDP’s state space $S$ extends to the MDP with state-subgoal pairs $< S, G >$. Therefore, the policy based on the opponent’s subgoal is updated as: $$Q(s_t, g_t, a_t) = \mathbb{E}_{P(s_{t+1}\|s_t,a^o,a)}[r + \gamma \max_a Q(s_{t+1}, g_t, a)].$$ Here $s_{t+1}, g_t$ is used instead of $s_{t+1}, g_{t+1}$, because we assume that the next state of $s_t, g_t$ follows the same goal. In the framework of OMG, $g_t$ and $g_{t+1}$ will reach the same at the end of the episode. Algorithm 1 Opponent Modeling based on Subgoals Inference 1: Preparation: 2: Interact with $ \nu $ opponents to collect $ s $ and train the prior model $ f_\psi $ 3: Initialize subgoal inference model parameters $ \phi $ and $ \theta $ 4: Initialize Q-network $ Q $ and the replay buffer $ D $ 5: repeat 6: Interaction phase 7: Observe state $ s $ and last opponent’s action $ a^o $ 8: Infer the subgoal $ \hat{g} $ by subgoal inference model $ q_\phi(g\|\tau) $ 9: Choose action $ a $ by $ \max_a Q(s, \hat{g}, a) $ with $ \epsilon $-greedy 10: Store trajectory experience $(s, a, a^o, r)$ in replay buffer $ D $ 11: Update phase 12: if It’s update time. then 13: Calculate prior subgoal $ \bar{g} $ by (6) or (7) 14: Calculate subgoal $ g $ by (8) 15: Update Q-network by (4) 16: Update subgoal inference model $ q_\phi $ and $ p_\theta $ by (5) 17: end if 18: until convergence To demonstrate the difference between learning Q-values using the opponent’s action Equation (3) and using the opponent’s subgoal Equation (4), we carry out an experiment in an $ 11 \times 11 $ gridworld with two agents, as detailed in Figure 3. The Q-value using the opponent’s action learns slower than the Q-value with the opponent’s subgoal in Figure 3(a), resulting from the tuple $(s, a^o, a)$ is more numerous than $(s, g, a)$ in the Q-table. After convergence, the Q-value increases as it gets closer to the reward point, indicating a meaningful Q-value with the opponent’s subgoal, as shown in Figure 3(b). When there are fewer $(s, g, a)$ than $(s, a^o, a)$, the method using $(s, g, a)$ naturally holds the advantage of faster learning than the method of $(s, a^o, a)$. The quantity of $(s, g, a)$ is contingent upon the goal selection, and we present an analysis of the quantitative relationship between pair $(s, g)$ and $(s, a^o)$, see Appendix A.1. In short, the number of $(s, g)$ is significantly smaller than that of $(s, a^o)$ in our method. 3.3 Opponent Modeling based on Subgoal Inference In this part, we elaborate on the opponent modeling module, which is divided into two components: the subgoal inference model and the subgoal selector. The subgoal inference model utilizes the historical trajectory to predict opponent’s subgoal, which act as the policy’s input to make decisions during interaction phase. Meanwhile, the subgoal selector is responsible for scrutinizing the entire historical trajectory and choosing the suitable subgoal for training the subgoal inference model during the update phase. Subgoal inference model. The subgoal $ g $ is a representation of future states. Specifically, for a trajectory $\{s_0, a_0, a^o_0, \ldots, s_t, a_t, a^o_t, \ldots, s_T\}$. The state corresponding to subgoal $ g_t $ is one of future states $ N_t = \{s_{t+1}, s_{t+2}, \ldots, s_T\} $, denoted as $ s^g_t $. We denote the mapping between states and subgoals by $ f_\psi $, where $ \psi $ is the parameters and $ \bar{g}_t = f_\psi(s^g_t) $. The objective of the subgoal inference model is to infer $ s^g_t $ from the historical trajectory $ \tau_t = \{s_0, a_0, a^o_0, \ldots, s_{t-1}, a_{t-1}, a^o_{t-1}\} $ at timestep $ t $, even though $ s^g_t $ may be a state at timestep $ t + 1 $ or further. This is in accordance with the intuitive hypothesis, implying that the opponent’s intention is often inferred after just a few initial actions. Here, we introduce variational inference and use a conditional variational auto-encoder (CVAE) as the subgoal inference model. In this model, we represent the posterior probability as $ q_\phi(g\|\tau) $ and the likelihood estimate as $ p_\theta(\tau\|g) $ with $ \theta, \phi $ denoting network parameters. Additionally, the condition vector of the model is encoded using an RNN. The subgoal’s prior model, denoted as $ p_\psi(\bar{g}\|s^g) $, is constructed using a pre-trained variational autoencoder (VAE), with the prior subgoal state $ s^g $ being derived from the subgoal selector as its input. The distribution of subgoal prior $ p_\psi $, subgoal posterior probability $q_\phi$ and subgoal’s prior $p_\psi(\bar{g}\|s^g)$ are used normal distribution. The mapping $f_\psi$ represents sampling subgoal $\bar{g}$ from $p_\psi$ using reparameterization trick. The detailed network architecture is presented in Figure 2. The optimization objective of the subgoal inference model is: $$\langle \hat{\theta}, \hat{\phi} \rangle = \arg\max_{\theta, \phi} \mathbb{E}_{g \sim q_\phi(\hat{g}_t\|\tau_t, s_t)} \left[ \log p_\theta(s_t\|\hat{g}_t, \tau_t) \right] - \text{KL}\left( q_\phi(\hat{g}_t\|\tau_t, s_t)\|\|p_\psi(\bar{g}_t\|s^g) \right).$$ Subgoal selector. The objective of the subgoal selector is to choose the appropriate future state from $N_t$ as prior model’s input. The selection of subgoal states plays a pivotal role in shaping the behavior of an agent, as it significantly impacts the pattern of the agent’s policy, either leaning towards optimism or conservatism. This critical decision-making process becomes especially pertinent when dealing with cooperative games and general-sum games, where the dynamics of interaction are complex and multifaceted. In these contexts, we provide two distinct manners to guide the agent’s decision-making: $$\bar{g}_t = \arg\max_{g \in f_\psi(N^H_t)} V(s_t, g)$$ $$\bar{g}_t = \arg\min_{g \in f_\psi(N^H_t)} V(s_t, g)$$ where $V(s, g) = \mathbb{E}_a Q(s, g, a)$, $N^H_t$ is the set of future states $\{s_{t+1}, \cdots, s_{t+H}\}$. We use states within the next $H$ timesteps instead of all future steps because the subgoals of different trajectory fragments may have combinatorial properties. It gives the agent better generalization ability when facing different policy opponents. However, if we adopt the full horizon, the agent may prefer the goals near the terminal state, which is not conducive to the exploration of goal space. When utilizing the subgoal $g$ as indicated in Equation (6), we pinpoint the state within a $H$-horizon that maximizes the V-value. The agent incorporates this as the subgoal to optimize the Q-function, thus adopting an optimistic strategy akin to the maxmax strategy (Ben-Haim, 2006), which applies to cooperative games. Conversely, if we choose the subgoal as presented in Equation (7), it corresponds to the state yielding the lowest value. The agent then employs this as the subgoal for Q-function optimization, leading to a conservative strategy similar to the minimax strategy, which is usually used in general-sum games. In conclusion, the subgoal selector and the subgoal inference model as a whole constitute the opponent modeling module. During the interaction phase, the subgoal inference model is used to get the inferred subgoal $\hat{g}$, which is combined with the state as the input to the Q-network. During the update phase, the prior subgoal $\bar{g}$ generated by the subgoal selector provides the inference model for training. When policy updating, the subgoal inference model is unstable at the beginning, which disturbs the updating of the Q-network. Therefore, we use the following combination of the prior subgoal $\bar{g}$ and the inferred subgoal $\hat{g}$: $$g_t = \hat{g}_t I(\eta > \epsilon) + \bar{g}_t I(\eta \leq \epsilon), \quad \eta \sim U[0, 1],$$ where $\epsilon$ is a hyperparameter that decreases to zero over training. For completeness, the full procedure of OMG is given in Algorithm 1. 4 EXPERIMENTS First, we evaluate OMG’s training performance in two environments (discrete and continuous state spaces) and then test its generalization against opponents with various policies in a complex environment. In all the experiments, the baselines have the same neural network architectures as OMG. All the methods are trained for five runs with different random seeds, and results are presented using mean and standard deviation. More details about experimental settings and hyperparameters are available in Appendix A.2. 4.1 Multi-Agent Environments Foraging environment (Albrecht & Ramamoorthy, 2015; Albrecht & Stone, 2019) is an $8 \times 8$ grid-world containing two players: the agent and the opponent. At the beginning of each round, the Figure 4: Training performance in Foraging and Predator-Prey. (a) shows the total score obtained by the agent. (b) illustrates the number of steps at the end of each episode. The results show that OMG can converge to the same score as baselines but end the episode in fewer steps because it predicts the opponent’s goal. (c) shows the score obtained by the agent as a predator with two other uncontrolled predators in Predator-Prey, and OMG outperforms the baselines. Figure 5: Test performance of cooperation with different opponents in $8m$ and $3s_vs_5z$ maps of SMAC. The results show that OMG-optimistic outperforms all baselines. The results are averaged over collaborating with 30 opponents of different policies, with 95% confidence intervals. Players and three foods are randomly generated in the environment. The goal of the agent is to collect all foods as quickly as possible. The agent can move in four directions or pick up the food. The agent must judge the opponent’s target food as soon as possible to avoid futile actions for the same food. Predator-Prey (Lowe et al., 2017) is a three-against-one multi-agent environment with a continuous space. Three predators coordinate to touch the prey. The agent acts as one of the predators, and the opponents are the other two predators and the prey, which leads to the non-stationarity of the environment from the agent’s view despite not belonging to one camp. The agent aims to maximize its reward and therefore needs to collaborate with the other two predators to complete the encirclement and cut the prey’s escape route. SMAC (Samvelyan et al., 2019) is a high-dimensional complex environment for research in the field of collaborative MARL based on StarCraft II. The agent joins a set of agents with unknown policies to accomplish the task. The only way to accomplish the task is to collaborate with the other agents. The agent’s goal is to complete the task with a group of opponents controlled by unknown policies. ### 4.2 Baselines In the experiments, we implement two variants of OMG, OMG-optimistic and OMG-conservative, based on the subgoal selection patterns in Equation (6) and Equation (7), respectively. OMG compared with the following methods: - Naive OM (He et al., 2016) uses observation to directly model the opponent’s policy, which assists the agent in decision-making by predicting the opponent’s actions. • LIAM (Papoudakis et al., 2021) uses the observations and actions of the modeling agent with an encoder-decoder architecture, and the model learns to extract representations about the modeling agent, conditioned only on the local observations of the controlled agent. • D3QN & PPO & IQL (Wang et al., 2016; Schulman et al., 2017; Tampuu et al., 2017) are classical RL algorithms without opponent modeling. We use D3QN, PPO, and IQL as the backbone algorithms in Foraging, Predator-prey, and SMAC, respectively, to reproduce the performance of baselines. The versions of OMG that are based on D3QN and IQL incorporate “dueling” and “double” tricks over Algorithm 1. For OMG based on PPO, please refer to Appendix A.3 for details. 4.3 Performance of Training We evaluate the performance of OMG on foraging and predator-prey, and the results are shown in Figure 4. In the foraging environment, our method attains comparable scores to the baseline methods, and both the agent and the opponent achieve similar scores. OMG has a shorter episode length compared to other methods as demonstrated in Figure 4(b), because OMG can predict the subgoal that the opponent is heading to and thus avoid wasting steps in the same direction. In addition, the results show that OMG-conservative is more suitable than OMG-optimistic in this scenario since this is a general-sum game. The action modeling-based methods, LIAM and Naïve OM, demonstrate comparable performance, whereas D3QN without opponent modeling, exhibits subpar results. In the predator-prey environment, the agent acts as the predator and collaborates with the other two uncontrolled predators to catch the prey. The results in Figure 4(c) show that OMG obviously learns faster than action modeling methods, which demonstrates that OMG can also work efficiently in continuous state space. PPO without opponent modeling can hardly improve performance in training due to the non-stationarity caused by opponents. OMG-optimistic performs better than OMG-conservative because OMG-optimistic is suitable for the cooperative game. 4.4 Generalization to Unknown Opponents We evaluate the generalization of OMG in a complex multi-agent environment, SMAC, which enables the opponent to exhibit more diverse policies. The experimental results of 8m and 3s_vs_5z are shown in Figure 5. The test set consists of 30 opponents with different policies, trained by the IQL, VDN(Sunehag et al., 2017), and QMIX(Rashid et al., 2020). In 8m, the opponents are reorganized into three groups: 7 homologues, 6 homologues, and 7 non-homologues. In 3s_vs_5z, the opponents falls into two groups: 2 homologues and 2 non-homologues. Here, homologue refers to the policy from the same algorithm with the same parameters, and non-homologue represents the policy from two different algorithms. The remained agents are controlled by OMG or baseline algorithms. Without opponent modeling, IQL struggles to adapt to various opponents, resulting in poor performance, especially when the opponent is non-homologue. This underscores the effectiveness of opponent modeling in autonomous agent tasks. LIAM and Naïve OM are the opponent’s action modeling methods that contributed to the team’s improved win rate to some extent. The mediocre performance of OMG-conservative is attributed to its overly cautious subgoal selection, but there is no significant performance drop compared to IQL, which is consistent with the “conservative”. OMG-optimistic surpasses the baseline methods in cooperative tasks. OMG-optimistic cooperates with unknown opponents through positive subgoal selection, which is easier to win in hard scenarios. For opponents and training details, please refer to Appendix A.2. 4.5 Ablation Study The results of the ablation study in Foraging are presented in Figure 6. Specifically, Figure 6(a) and Figure 6(b) correspond to experiments related to subgoal selection. During the policy update, Equation (8) (i.e., $ g $) is utilized. As $ f_\psi $ is pre-trained and fixed during the update phase, $ \hat{g} $ remains stable. On the other hand, $ \tilde{g} $, which represents the inferred subgoal when executing the policy, also stabilizes as the training steps increase. The transition of $ g $ from $ \bar{g} $ to $ \hat{g} $ is a gradual process, which helps avoid instability during the training of the subgoal inference model. The parameter $ H $ denotes the horizon of the subgoal selector. The ablation experiment results are shown in Figure 6(c) and Figure 6(d). It is observed that an appropriate horizon value is neither Figure 6: Ablation study of OMG in Foraging. (a) and (b) compares OMGs with different subgoal learning policy. (c) and (d) show ablation study for the hyperparameter horizon $H$. Figure 7: Subgoal analysis of OMG in Foraging. The subgoal hit rates for OMG-conservative and OMG-optimistic are shown in Figure 7(a). In Figure 7(b), a blue circle represents the state obtained through the reconstruction of the subgoal inferred by the agent. The figure illustrates the difference between OMG-conservative and OMG-optimistic under the same initial state and opponent policy. excessively high nor excessively low. When $H = 1$, it is essentially equivalent to combining with QSS (Edwards et al., 2020) and opponent modeling. However, if $H$ is set too high, such as $H = 10$, the agent may skip important states in the trajectory, leading to a degradation in performance. Therefore, selecting an appropriate value for $H$ is crucial in achieving satisfactory results. 4.6 Inferred Subgoal Analysis In Figure 7(a), we plot the ratio of that an opponent’s future trajectory passes through the opponent’s subgoal inferred by the agent, termed subgoal hit ratio. The subgoal hit ratio is calculated by reconstructing the subgoal state $f_{\psi}^{-1}(g)$. The subgoal hit rate gradually improves during training, which indicates that the subgoal-based opponent modeling is able to predict the future state of the opponent. OMG tends to predict goals multiple steps ahead, making it difficult for opponents to reach immediately, resulting in a modest value that hit ratio convergence. There is a small gap between the subgoal hit rates of OMG-conservative and OMG-optimistic, which leads to longer episode length for OMG-optimistic than OMG-conservative, as illustrated in Figure 7(b). The root cause lies in the differences in subgoal selection between OMG-conservative and OMG-optimistic. 5 Conclusion In this work, we introduce OMG, a novel method for opponent modeling based on subgoal inference. OMG is a simple and efficient opponent modeling method and can be combined with different RL algorithms. Unlike most opponent modeling methods, which primarily focus on predicting the opponent’s actions, OMG focuses on modeling the opponent’s subgoals. Specifically, it leverages the value function of the policy to guide the selection of subgoals, which yields two variants of OMG for cooperative and general-sum games, respectively. Empirical results demonstrate the remarkable performance achieved by OMG, as compared to baselines which are based on action modeling, and that OMG exhibits better generalization when cooperating with opponents with unknown policies. We analyze the subgoals obtained by the inference model, and the results show that they closely correlate with the opponent’s trajectory. The limitation of OMG is it cannot handle open multi-agent systems where agents may enter and leave the system during the interaction. This is left for our future work. REFERENCES Maruan Al-Shedivat, Trapit Bansal, Yuri Burda, Ilya Sutskever, Igor Mordatch, and Pieter Abbeel. Continuous adaptation via meta-learning in nonstationary and competitive environments. arXiv preprint arXiv:1710.03641, 2017. Stefano V Albrecht and Subramanian Ramamoorthy. A game-theoretic model and best-response learning method for ad hoc coordination in multiagent systems. arXiv preprint arXiv:1506.01170, 2015. Stefano V Albrecht and Peter Stone. Autonomous Agents Modelling Other Agents: A Comprehensive Survey and Open Problems. Artificial Intelligence, 258:66–95, 2018. Stefano V Albrecht and Peter Stone. 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DjeQ39QoLQ	I am not convinced by the claim the S4-PTD model outperforms the S4D models on LRA. The LRU paper (https://arxiv.org/abs/2303.06349) reports results for S4D that are much better than the original reported S4D paper results. In addition, the appendix of the current paper under review states that mild hyperparameter tuning was performed for the S4-PTD models. Would mild hyperparameter tuning improve the S4D results as well?	ROBUSTIFYING STATE-SPACE MODELS FOR LONG SEQUENCES VIA APPROXIMATE DIAGONALIZATION Annan Yu,1 Arnur Nigmatov,2 Dmitriy Morozov,2 Michael W. Mahoney,2,3,4 N. Benjamin Erichson2,3 1 Center for Applied Mathematics, Cornell University, Ithaca, NY 14853, USA 2 Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA 3 International Computer Science Institute, Berkeley, CA 94704, USA 4 Department of Statistics, University of California at Berkeley, Berkeley, CA 94720, USA ay262@cornell.edu, {anigmatov,dmorozov}@lbl.gov, mmahoney@stat.berkeley.edu, erichson@icsi.berkeley.edu ABSTRACT State-space models (SSMs) have recently emerged as a framework for learning long-range sequence tasks. An example is the structured state-space sequence (S4) layer, which uses the diagonal-plus-low-rank structure of the HiPPO initialization framework. However, the complicated structure of the S4 layer poses challenges; and, in an effort to address these challenges, models such as S4D and S5 have considered a purely diagonal structure. This choice simplifies the implementation, improves computational efficiency, and allows channel communication. However, diagonalizing the HiPPO framework is itself an ill-posed problem. In this paper, we propose a general solution for this and related ill-posed diagonalization problems in machine learning. We introduce a generic, backward-stable “perturb-then-diagonalize” (PTD) methodology, which is based on the pseudospectral theory of non-normal operators, and which may be interpreted as the approximate diagonalization of the non-normal matrices defining SSMs. Based on this, we introduce the S4-PTD and S5-PTD models. Through theoretical analysis of the transfer functions of different initialization schemes, we demonstrate that the S4-PTD/S5-PTD initialization strongly converges to the HiPPO framework, while the S4D/S5 initialization only achieves weak convergences. As a result, our new models show resilience to Fourier-mode noise-perturbed inputs, a crucial property not achieved by the S4D/S5 models. In addition to improved robustness, our S5-PTD model averages 87.6% accuracy on the Long-Range Arena benchmark, demonstrating that the PTD methodology helps to improve the accuracy of deep learning models. 1 INTRODUCTION Sequential data are pervasive across a wide range of fields, including natural language processing, speech recognition, robotics and autonomous systems, as well as scientific machine learning and financial time-series analysis, among others. Given that many of these applications produce exceedingly long sequences, sequential models need to capture long-range temporal dependencies in order to yield accurate predictions. To this end, many specialized deep learning methods have been developed to deal with long sequences, including recurrent neural networks (RNNs) (Arjovsky et al., 2016; Chang et al., 2019; Erichson et al., 2021; Rusch & Mishra, 2021; Orvieto et al., 2023), convolutional neural networks (CNNs) (Bai et al., 2018; Romero et al., 2022), continuous-time models (CTMs) (Gu et al., 2021; Yildiz et al., 2021), and transformers (Katharopoulos et al., 2020; Choromanski et al., 2020; Kitaev et al., 2020; Zhou et al., 2022; Nie et al., 2023). Over the past few years, the new class of state-space models (SSMs) gained vast popularity for sequential modeling due to their outstanding performance on the Long-Range Arena (LRA) dataset (Tay et al., 2021). An SSM is built upon a continuous-time linear time-invariant (LTI) dy- nical system $\Sigma = (A, B, C, D)$, which is a system of linear ODEs given by $$x'(t) = Ax(t) + Bu(t),$$ $$y(t) = Cx(t) + Du(t),$$ where $A \in \mathbb{C}^{n \times n}$, $B \in \mathbb{C}^{n \times m}$, $C \in \mathbb{C}^{p \times n}$, $D \in \mathbb{C}^{p \times m}$ are the state, input, output and feedthrough matrices; and $u(t) \in \mathbb{C}^m$, $x(t) \in \mathbb{C}^n$, $y(t) \in \mathbb{C}^p$ are the inputs, states, and outputs of the system, respectively. The system can be discretized at time steps $j\Delta t$, where $\Delta t > 0$ and $j = 1, \ldots, L$, to be fed with sequential inputs of length $L$. To store and process the information of the long sequential inputs online, the SSMs are often initialized by a pre-designed LTI system. One of the most popular schemes is called “HiPPO initialization” (Voelker et al., 2019; Gu et al., 2020), in which the Legendre coefficients of the input history at time $t$, i.e., $u \cdot \mathbf{1}_{[0,t]}$, are stored and updated in the state vector $x(t)$. This initialization is specifically designed to model long-range dependencies in sequential data. The recently proposed S4 model (Gu et al., 2022b) leverages the HiPPO initialization and accelerates training and inference by decomposing $A$ into the sum of a diagonal matrix and a low-rank one. The diagonal-plus-low-rank (DPLR) structure yields a barycentric representation (Antoulas & Anderson, 1986) of the transfer function of eq. (1) that maps inputs to outputs in the frequency domain, enabling fast computation in the frequency domain (Aumann & Gosea, 2023). While the DPLR structure achieves an asymptotic speed-up of the model, considering $A$ to be a diagonal matrix results in a simpler structure. Compared to a DPLR matrix $A$, a diagonal SSM is not only faster to compute and easier to implement, but it also allows integrating channel communication via parallel scans (Smith et al., 2023), thereby improving its performance on long-range tasks. Unfortunately, the problem of diagonalizing the HiPPO framework is exponentially ill-conditioned, as $n$ increases. Hence, while Gu et al. (2022b) shows analytic forms of the eigenvalues and eigenvectors of HiPPO matrices, they suffer from an exponentially large variance and cannot be used in practice. So far, the most popular way of obtaining a diagonal SSM is to simply discard the low-rank part from the DPLR structure, leveraging a stable diagonalization algorithm for a normal matrix. Discarding the low-rank component changes the underlying diagonalization problem, however; and it abandons the theoretical insights about HiPPO. Still, the resulting model almost matches S4’s performance, in practice. Such diagonal models are called S4D (Gu et al., 2022a) when the systems are single-input/single-output (i.e., $m = p = 1$) and S5 (Smith et al., 2023) when the systems are multiple-input/multiple-output (i.e., $m = p > 1$), which enables channel communication. The issue of ill-posed diagonalization problems is not merely specific to SSMs. For example, it is known that non-normal matrices make RNNs more expressive (Kerg et al., 2019; Orhan & Pitkow, 2020). More generally, non-normality plays an important role in the training of certain neural networks (Sengupta & Friston, 2018; Kumar & Bouchard, 2022). While the ill-posedness of the diagonalization problem essentially prevents accurate computation of eigenvalues and eigenvectors (i.e., we cannot have a small forward error) — in fact, the true spectral information becomes meaningless¹ — using a backward stable eigensolver, one can recover the non-normal matrix accurately (i.e., we can have a small backward error) from the wrong eigenvalues and eigenvectors. In this paper, we propose a generic “perturb-then-diagonalize” (PTD) methodology as a backward stable eigensolver. PTD is based on the idea that a small random perturbation remedies the problem of the blowing up of eigenvector condition number (Davies, 2008; Davies & Hager, 2009; Banks et al., 2021), regularizing the ill-posed problem into a close but well-posed one. It is based on the pseudospectral theory of non-normal operators (Trefethen & Embree, 2005)² and may be interpreted as the approximate diagonalization of the non-normal matrices. Our PTD method can be used to diagonalize the highly non-normal HiPPO framework. Therefore, instead of using the eigenvalues of the normal component of the HiPPO matrix to initialize the matrix $A$ as in the S4D and S5 models, we propose to initialize $A$ using the eigenvalues of a perturbed HiPPO matrix (see section 4). The resulting S4-PTD and S5-PTD models are shown to be more robust than their S4D and S5 companions under certain Fourier-mode perturbations. Our method is flexible and can be used to diagonalize many SSM initialization schemes that may be invented in the future. ¹If an eigenvector matrix $V$ is ill-conditioned, then projecting a vector onto the eigenbasis is unstable so the eigendecomposition suffers from a large variance and does not reveal any useful information of the matrix. ²The pseudospectral theory studies the effect of perturbations on the spectrum of a non-normal operator. Contribution. Here are our main contributions: (1) We propose a “perturb-then-diagonalize” (PTD) methodology that solves ill-posed diagonalization problems in machine learning when only the backward error is important. (2) We provide a fine-grained analysis that compares the S4 and the S4D initialization. In particular, we quantify the change of the transfer function when discarding the low-rank part of HiPPO, which is done in the diagonal S4D/S5 initialization. We show that while the outputs of the S4D/S5 system on a fixed smooth input converge to those of the S4 system at a linear rate as $ n \to \infty $, the convergence is not uniform across all input functions (see section 3.1). (3) Based on our theoretical analysis, we observe, using the sequential CIFAR task (see section 5.2), that the S4D/S5 models are very sensitive to certain Fourier-mode input perturbations, which impairs the robustness of the models. (4) We propose the S4-PTD and S5-PTD models that replace the normal component of the HiPPO matrix, used to initialize the S4D and S5 models, with a perturbed HiPPO matrix. Our models are robust to Fourier-mode input perturbations. We theoretically estimate the effect of the perturbation (see section 4). We propose computing the perturbation matrix by solving an optimization problem with a soft constraint. Moreover, our method is not restricted to the HiPPO matrix but can be applied to any initializations. (5) We provide an ablation study for the size of the perturbation in our models. We also evaluate our S4-PTD and S5-PTD models on LRA tasks, which reveals that the S4-PTD model outperforms the S4D model, while the S5-PTD model is comparable with the S5 model (see section 5.1). 2 PRELIMINARIES AND NOTATION Given an LTI system in eq. (1), we say it is asymptotically stable if the eigenvalues $ \lambda_j $ of $ A $ are all contained in the left half-plane, i.e., if $ \text{Re}(\lambda_j) < 0 $ for all $ 1 \leq j \leq n $. The transfer function of the LTI system is defined by \[ G(s) = C(sI - A)^{-1}B + D, \quad s \in \mathbb{C} \setminus \Lambda(A), \] where $ I \in \mathbb{R}^{n \times n} $ is the identity matrix and $ \Lambda(A) $ is the spectrum of $ A $. The transfer function $ G $ is a rational function with $ n $ poles (counting multiplicities) at the eigenvalues of $ A $. Assume $ x(0) = 0 $. Then the transfer function maps the inputs to the outputs of the LTI system in the Laplace domain by multiplication, i.e., $ (\mathcal{L}y)(s) = G(s)(\mathcal{L}u)(s) $ for all $ s \in \mathbb{C} $, where $ \mathcal{L} $ is the Laplace transform operator (see Zhou & Doyle (1998)). Assume the LTI system in eq. (1) is asymptotically stable and the input $ u(t) $ is bounded and integrable (with respect to the Lebesgue measure) as $ t $ ranges over $ \mathbb{R} $. Then the Laplace transform reduces to the Fourier transform: \[ \hat{y}(s) = G(is)\hat{u}(s), \quad s \in \mathbb{R}, \] where $ \hat{y} $ and $ \hat{u} $ are the Fourier transforms of $ y $ and $ u $, respectively, and $ i $ is the imaginary unit. Let $ V \in \mathbb{C}^{n \times n} $ be an invertible matrix. We can conjugate the system $(A, B, C, D)$ by $ V $, which yields $(V^{-1}AV, V^{-1}B, CV, D)$. Since the transfer function is conjugation-invariant, the two systems map the same inputs $ u(\cdot) $ to the same outputs $ y(\cdot) $, while the states $ x(\cdot) $ are transformed by $ V $. If $ A $ is a normal matrix, i.e., $ AA^* = A^A $, then $ V $ is unitary, in which case transforming the states by $ V $ is a well-conditioned problem and can be done without loss of information. Issues arise, however, when $ A $ is non-normal and $ V $ is ill-conditioned. The state-space models use LTI systems to process time series inputs. Different initializations can be tailored to tasks with different natures, such as the range of dependency (Gu et al., 2023). A particularly successful initialization scheme used in the S4 model is the so-called HiPPO initialization. While there exist several variants of HiPPO, the most popular HiPPO-LegS matrices are defined by \[ (A_H)_{jk} = \begin{cases} 1_{\{j>k\}} \sqrt{2j-1} \sqrt{2k-1}, & \text{if } j \neq k, \\ j, & \text{if } j = k, \end{cases} \] for all $ 1 \leq j, k \leq n $ and $ 1 \leq \ell \leq m $, where $ 1_{\{j>k\}} $ is the indicator that equals 1 if $ j > k $ and 0 otherwise. Such a system guarantees that the Legendre coefficients of the input history $ u \cdot 1_{[0,t]} $ (with respect to a scaled measure) are stored in the states $ x(t) $ over time (Gu et al., 2020). Since computing with the dense matrix $ A_H $ is practically inefficient, one conjugates the HiPPO system with a matrix $ V_H $ to simplify the structure of $ A_H $. The matrix $ A_H $ in eq. (4) has an ill-conditioned eigenvector matrix (Gu et al., 2022b); consequently, instead of solving the ill-posed problem that diagonalizes $ A_H $, one exploits a diagonal-plus-low-rank (DPLR) structure: \[ A_H = A_H^\perp - \frac{1}{2}B_HB_H^\top, \quad (A_H^\perp)_{jk} = \begin{cases} (-1)^{1_{\{j<k\}}} \sqrt{2j-1} \sqrt{2k-1}, & \text{if } j \neq k, \\ 1, & \text{if } j = k, \end{cases} \] where $ A_H^\perp $ is a skew-symmetric matrix that can be unitarily diagonalized into $ A_H^\perp = V_H \Lambda_H V_H^{-1} $. The S4 model leverages the HiPPO matrices by initializing \[ A_{DPLR} = \Lambda_H - \frac{1}{2} V_H B_H B_H^T V_H, \quad B_{DPLR} = V_H^{-1} B_H \] and $ C_{DPLR} $ and $ D_{DPLR} $ randomly. Such an LTI system $ \Sigma_{DPLR} = (A_{DPLR}, B_{DPLR}, C_{DPLR}, D_{DPLR}) $ is conjugate via $ V_H $ to $ (\Lambda_H, B_H, C_{DPLR} V_H^{-1}, D_{DPLR}) $. Hence, they share the transfer function and the same mapping from the inputs $ u(\cdot) $ to the outputs $ y(\cdot) $. The S4D model further simplifies the structure by discarding the rank-1 part from $ A_H $ and therefore initializes \[ A_{Diag} = \Lambda_H, \quad B_{Diag} = \frac{1}{2} V_H^{-1} B_H, \] and $ A_{Diag} $ is henceforth restricted to be diagonal. While both the S4 and S4D models restrict that $ m = p = 1 $, i.e., the LTI systems are single-input/single-output (SISO), the S5 model, which also initializes $ A_{Diag} = \Lambda_H $ and requires it to be diagonal throughout training, leverages multiple-input/multiple-output (MIMO) systems by allowing $ m = p > 1 $. We provide more background information on LTI systems and state-space models in sequential modeling in Appendix B. Throughout this paper, we use $ \\| \cdot \\| $ to denote a vector or matrix 2-norm. Given an invertible square matrix $ V $, we use $ \kappa(V) = \\|V\\| \\|V^{-1}\\| $ to denote its condition number. Given a number $ 1 \leq p \leq \infty $ and a measurable function $ f : \mathbb{R} \to \mathbb{C} $, we use $ \\|f\\|_{L^p} $ for the standard $ L^p $-norm of $ f $ with respect to the Lebesgue measure on $ \mathbb{R} $ and $ L^p(\mathbb{R}) = \{ f : \mathbb{R} \to \mathbb{C} \mid \\|f\\|_{L^p} < \infty \} $. ### 3 THEORY OF THE DIAGONAL INITIALIZATION OF STATE-SPACE MODELS The S4 model proposes to initialize the SSM to store the Legendre coefficients of the input signal in the states $ x $ (Gu et al., 2020). This initialization, however, has an ill-conditioned spectrum, preventing a stable diagonalization of the SSM. On the other hand, the S4D model uses a different initialization scheme that has a stable spectrum, allowing for stable diagonalization; however, such initialization lacks an interpretation of the states $ x $. In this section, we conduct a fine-grained analysis of the two initializations, which shows that: (1) for any fixed input signal $ u(\cdot) $ with sufficient smoothness, the outputs of the two systems $ \Sigma_{DPLR} $ and $ \Sigma_{Diag} $ converge to each other with a linear rate (of which the previous analysis is devoid) as $ n \to \infty $; and (2) by viewing $ \Sigma_{DPLR} $ and $ \Sigma_{Diag} $ as linear operators that map input signals to the outputs, the operators do not converge in the operator norm topology as $ n \to \infty $ (see section 3.1). While the first observation partially justifies the success of the S4D model, the second one allows us to observe that the diagonal initialization is unstable under certain Fourier-mode input perturbations (see section 5.2). In this section, we assume $ m = p = 1 $, which is consistent with the S4 and S4D models. Still, our theory can be related to the S5 model, as shown in Smith et al. (2023). Fix an integer $ 1 \leq \ell \leq n $. We assume that $ C_{DPLR} = C_{Diag} = e_\ell^T V_H $, where $ e_\ell^T $ is the $ \ell $th standard basis, and $ D_{DPLR} = D_{Diag} $. For a general $ C_{DPLR} = C_{Diag} $, we can decompose it onto the orthonormal basis $ \{e_\ell^T V_H \mid 1 \leq \ell \leq n \} $ and study each component separately using the theory developed in this section. Let $ G_{DPLR} $ and $ G_{Diag} $ be the transfer functions of $ \Sigma_{DPLR} $ and $ \Sigma_{Diag} $, respectively, i.e., \[ G_{DPLR}(s) = C_{DPLR}(sI - A_{DPLR})^{-1} B_{DPLR} + D_{DPLR}, \quad G_{Diag}(s) = C_{Diag}(sI - A_{Diag})^{-1} B_{Diag} + D_{Diag}. \] Recall that by eq. (3), $ \|G_{DPLR}(si) - G_{Diag}(si)\| $ measures the difference between the outputs of the two systems given a frequency-$ s $ input. We provide a fine-grained analysis of this difference in the two transfer functions in Lemma 1. The lemma is visualized in Figure 1. We see that as $ n $ increases, $ G_{Diag} $ approaches $ G_{DPLR} $ in the low-frequency domain, i.e., when $ \|s\| $ is small. However, $ G_{Diag} $ develops spikes in the high-frequency domain. Moreover, for every $ n \geq 1 $, zooming into the last spike located at $ \|s\| = \Theta(n^2) $ reveals that it has a constant magnitude (see the subplots on the right in Figure 1). Hence, the convergence of $ G_{Diag} $ to $ G_{DPLR} $ is non-uniform (see Theorem 2). Moreover, the frequency response is unstable at input frequencies $ s $ near these spikes, suggesting that the S4D model is not robust to certain input perturbations (see section 5.2). #### 3.1 INPUT-WISE CONVERGENCE AND SYSTEM-WISE DIVERGENCE OF THE DIAGONAL INITIALIZATION First, we present a result to show that for a fixed input signal $ u(\cdot) $, the outputs of $ \Sigma_{DPLR} $ and $ \Sigma_{Diag} $ converge to each other as $ n \to \infty $. Moreover, while the previous result in Gu et al. (2022a) does not Figure 1: The magnitude of transfer function of the S4 model, $\|G_{\text{DPLR}}(s_i)\|$, and that of the S4D model, $\|G_{\text{Diag}}(s_i)\|$ with $C_{\text{DPLR}} = C_{\text{Diag}} = e_1^\top V_H$ and the SSM size $n$ set to different values. Note that $G_{\text{DPLR}}$ stays the same regardless of $n$. Due to the limited resolution, the left panel does not correctly reveal the heights of the spikes; however, by zooming into the last spike of $\|G_{\text{Diag}}(s_i)\|$, we see that the peak remains $\Theta(1)$ as $n \to \infty$ (see the right panels). The figure shows that $G_{\text{Diag}}$ is oscillatory while $G_{\text{DPLR}}$ is smooth; moreover, $\|G_{\text{Diag}}(s_i)\|$ does not converge to $\|G_{\text{DPLR}}(s_i)\|$ uniformly. have a rate of convergence, we show that it is linear. In fact, the rate is sharp (see Appendix F). This partially explains why the S4D model matches the performance of the S4 model in practice. Theorem 1.* Let $u(\cdot) \in L^2(\mathbb{R})$ be an input function with $\\|u\\|_{L^2} = 1$. Let $y_{\text{DPLR}}(\cdot)$ and $y_{\text{Diag}}(\cdot)$ be the outputs of $\Sigma_{\text{DPLR}}$ and $\Sigma_{\text{Diag}}$ given the input $u(\cdot)$ and the initial states $x(0) = 0$, respectively. For some $q > 1/2$, suppose $\|\hat{u}(s)\| = O(\|s\|^{-q})$ as $\|s\| \to \infty$. Then, we have $\\|y_{\text{DPLR}} - y_{\text{Diag}}\\|_{L^2} = O(n^{-1}) \sqrt{\ell}$ as $n \to \infty$, where the constant in the $O$-notation only depends on $q$ and the constant in $\hat{x}(s) = O(\|s\|^{-q})$. The constant does not depend on $q$ if we restrict $q \in [q', \infty)$ for a fixed $q' > 1/2$. The proof is deferred to Appendix E. Since the Fourier transform interchanges smoothness and decay, what Theorem 1 says is that under a mild assumption that $u(\cdot)$ is sufficiently smooth, the output of the diagonal system converges linearly to that of the DPLR system as $n \to \infty$. In Section 3.2, we show this smoothness assumption is needed. We know the two systems converge input-wise; it is natural to ask if the convergence is uniform across all input signals: Theorem 2. The function $G_{\text{DPLR}}(s) - G_{\text{Diag}}(s)$ does not converge to zero uniformly on the imaginary axis as $n \to \infty$. In particular, for every $n \geq 1$, there exists an input signal $u_n(\cdot) \in L^1(\mathbb{R}) \cap L^2(\mathbb{R})$ such that if we let $y_{n,\text{DPLR}}$ and $y_{n,\text{Diag}}$ be the outputs of $\Sigma_{\text{DPLR}}$ and $\Sigma_{\text{Diag}}$ of degree $n$, respectively, then we have $\\|y_{n,\text{DPLR}} - y_{n,\text{Diag}}\\|_{L^2}$ does not converge to 0 as $n \to \infty$. Hence, the answer to our question is negative: combined with Theorem 1, Theorem 2 says that while a sufficiently large S4D model mimics its S4 alternative on a fixed smooth input, when we predetermine a size $n$, they inevitably disagree, by a large amount, on some inputs. Moreover, in Theorem 2, the construction of $u_n(\cdot)$ can be made explicit (see section 5.2). ### 3.2 Some numerical examples In this section, we provide some numerical examples corroborating Theorem 1. We defer the implication of Theorem 2 to later sections (see section 4 and section 5.2). Theorem 1 tells us that if we fix a smooth input signal $u(t)$, then the outputs $y_{n,\text{DPLR}}$ and $y_{n,\text{Diag}}$ eventually converge to each other at a linear rate as $n \to \infty$. In this experiment, we fix two input functions (or more precisely, distributions) \[ u_e(t) = e^{-t} H(t), \quad u_d = \delta_0, \] where $H = 1_{[0,\infty)}$ is the Heaviside function and $\delta_0$ is the Dirac delta function at 0. While $u_e(t)$ is a very smooth function — in particular, we have $\|\hat{u}_e(s)\| = O(\|s\|^{-1})$ — the Dirac delta $u_d$ is very non-smooth with a Fourier transform that is constantly one. We simulate both systems $\Sigma_{\text{DPLR}}$ Figure 2: Simulated outputs of the DPLR and diagonal systems with the input functions $u_e$ and $u_d$ and varying state-space dimension $n$. We see that for a smooth input function $u_e$, the outputs of both systems converge rapidly as $n$ increases, whereas the convergence does not happen for a non-smooth input function $u_d$. and $\Sigma_{\text{Diag}}$ on both $u_e(t)$ and $u_d(t)$. More details of the simulation can be found in Appendix F. From Figure 2, we observe that given a smooth input function $u_e$, the output $y_{n,\text{Diag}}$ converges to $y_{n,\text{DPLR}}$ rapidly, but the same does not hold for a non-smooth input function $u_d$. Hence, the smoothness assumption in Theorem 1 is essential. In Figure 8 in Appendix F, we also compute the $L^2$-norm of $y_{n,\text{DPLR}} - y_{n,\text{Diag}}$ and verify that the convergence is linear when the input is smooth enough. We remark that a similar study of $u_d$ can be found in Gu et al. (2022a), where the results appear qualitatively different from those presented in Figure 2. This does not mean either work is wrong; the key distinction is that the discretization step size of the LTI systems (see Appendix B) is fixed in Gu et al. (2022a) \textit{a priori}, introducing aliasing errors and hiding the high frequencies (Trefethen, 2019, Ch. 4.). Consequently, when $n$ is large, the difference between $G_{\text{DPLR}}$ and $G_{\text{Diag}}$ in the high-frequency domain is overlooked. In comparison, in this paper, our theory considers the continuous-time LTI systems, which take every mode into account. 4 Perturbing the HiPPO Initialization: A New Way of Diagonalizing the State-Space Model In section 3, we saw the instability of the S4D transfer function at certain Fourier modes. Nevertheless, the diagonal structure of $A$ is preferred over the DPLR one due to its training and inference efficiency and its adaptivity to the MIMO model (i.e., the S5 model) (Smith et al., 2023). To avoid instability in a diagonal model, we want to leverage the HiPPO initialization in eq. (4) instead of the one in eq. (7) that discards the rank-1 part. One obvious solution is to diagonalize the HiPPO matrix $A_H = V_H \Lambda_H V_H^{-1}$ and conjugate $(A_H, B_H, C, D)$ using $V_H$. However, as shown in Gu et al. (2022a), the eigenvector matrix $V_H$ is exponentially ill-conditioned with respect to $n$, making the spectral information meaningless. While the exact eigenvalues and eigenvectors of $A_H$ are very ill-conditioned, since we only care about the backward error of diagonalization, we propose the following initialization scheme. Let $E \in \mathbb{C}^{n \times n}$ be a perturbation matrix. We diagonalize the perturbed HiPPO matrix as $$\tilde{A}_H = A_H + E = \tilde{V}_H \tilde{\Lambda}_H \tilde{V}_H^{-1}. \quad (9)$$ We then initialize the systems using $\Sigma_{\text{Pert}} = (\tilde{A}_{\text{Pert}}, \tilde{B}_{\text{Pert}}, \tilde{C}_{\text{Pert}}, \tilde{D}_{\text{Pert}}) = (\tilde{\Lambda}_H, \tilde{V}_H^{-1} B_H, C, D)$, where $C$ and $D$ are random matrices. Therefore, we approximately diagonalize the HiPPO initialization in the sense that although the diagonal entries in $\tilde{\Lambda}$ do not approximate the eigenvalues of $A_H$, the transfer function of $\Sigma_{\text{Pert}}$ is an approximation of that of $\Sigma_{\text{DPLR}}$ (see Theorem 3). We call our model S4-PTD or S5-PTD, depending on whether the model architecture is adapted from the S4D or the S5 model, where “PTD” stands for “perturb-then-diagonalize.” Since our models are only different from the S4D and the S5 models in initialization, we refer interested readers to Gu et al. (2022a). and Smith et al. (2023) for a discussion of computation details and time/space complexity. Our proposed perturb-then-diagonalize method is not restricted to the HiPPO-LegS matrices in eq. (4). This endows our method with adaptivity to any (dense) initialization scheme. This adaptivity was absent from the previous line of work on SSMs. Consider the process of diagonalizing the matrix $ A_H = V_H \Lambda_H V_H^{-1} $ that is solved by an inexact algorithm. In a numerical analyst’s language, the forward error is the error made in computing the eigenvalues $ \Lambda_H $ and eigenvectors $ V_H $, whereas the backward error asks how close a problem that we have solved exactly (i.e., $ A_H + E $) is to the actual problem that we want to solve (i.e., $ A_H $). As we will see in Theorem 3, it is the backward error $ \\|E\\| $ (but not the forward error) that matters in our initialization because it is the matrix $ A_H $ (but not the specific forms of $ V_H $ or $ \Lambda_H $) that is important in the transfer function. Centered around the perturbed initialization scheme eq. (9) are two important questions: (1) What is the difference between the perturbed initialization $(A_{\text{Pert}}, B_{\text{Pert}}, C_{\text{Pert}}, D_{\text{Pert}})$ and the HiPPO initialization $(A_{\text{DPLR}}, B_{\text{DPLR}}, C_{\text{DPLR}}, D_{\text{DPLR}})$? (2) What is the condition number of $ \tilde{V}_H $? The first question is important because it controls the deviation of our perturbed initialization from the successful and robust DPLR initialization. The second question is important because it shadows the numerical robustness of conjugating the LTI system by $ \tilde{V}_H $. Moreover, since the state vector $ x(t) $ is transformed by $ \tilde{V}_H $ via conjugation (see section 2), a small condition number of $ \tilde{V}_H $ shows that its singular values are more evenly distributed. Hence, the transformation $ \tilde{V}_H $ does not significantly magnify or compress $ x(t) $ onto some particular modes. To study the first question, we define the transfer function of the perturbed system to be \[ G_{\text{Pert}}(s) = C_{\text{Pert}}(sI - A_{\text{Pert}})^{-1}B_{\text{Pert}} + D_{\text{Pert}}. \] We control the size of the transfer function perturbation by proving the following theorem. Theorem 3. Assume $ C_{\text{Pert}} \tilde{V}_H^{-1} = C_{\text{DPLR}} V_H^{-1} $ and $ D_{\text{Pert}} = D_{\text{DPLR}} $. Suppose $ \\|E\\| \leq \epsilon $ and we normalize the matrices so that $ \\| \tilde{V}_H B_{\text{Pert}} \\| = \\| V_H B_{\text{DPLR}} \\| = \\| C_{\text{Pert}} \tilde{V}_H^{-1} \\| = \\| C_{\text{DPLR}} V_H^{-1} \\| = 1 $. For any $ s $ on the imaginary axis, we have \[ \|G_{\text{Pert}}(s) - G_{\text{DPLR}}(s)\| = (2 \ln(n) + 4)\epsilon + O(\sqrt{\log(n)} \epsilon^2). \] While our perturb-then-diagonalize method works for a general initialization and a bound on the transfer function error can always be established, the proof of Theorem 3 leverages the structure of HiPPO matrices to improve this bound. The error in Theorem 3 is the uniform error on the imaginary axis. Using Hölder’s inequality, for any bounded and integrable input function $ u(\cdot) $, if $ y_{\text{Pert}} $ and $ y_{\text{DPLR}} $ are the outputs of $ \Sigma_{\text{Pert}} $ and $ \Sigma_{\text{DPLR}} $, respectively, then we have \[ \\|y_{\text{Pert}} - y_{\text{DPLR}}\\|_{L^2} = \\| \hat{x}(s)(G_{\text{Pert}}(is) - G_{\text{DPLR}}(is)) \\|_{L^2} \leq \\| \hat{x}(s) \\|_{L^2} \\| (G_{\text{Pert}}(is) - G_{\text{DPLR}}(is)) \\|_{L^\infty} \leq \\|x\\|_{L^2} ((2 \ln(n) + 4)\epsilon + O(\sqrt{\log(n)} \epsilon^2)), \] where the first and the last steps follow from Parseval’s identity. Hence, Theorem 3 gives us an upper bound on the distance between $ \Sigma_{\text{Pert}} $ and $ \Sigma_{\text{DPLR}} $ in the operator norm topology. The theorem states that the error made by the perturbation is linear in the size of the perturbation. Moreover, the error depends only logarithmically on the dimension $ n $ of the state space. Next, we consider the conditioning of $ \tilde{V}_H $, which affects the accuracy of computing $ \tilde{V}_H^{-1} B_{\text{Pert}} $ and the scaling ratio of the states in $ x(\cdot) $ (see Appendix B). The following theorem provides a deterministic estimate of the eigenvector condition number for the “best perturbation scheme.” Theorem 4 ([Banks et al., 2021, Thm. 1.1.]). Given any $ A \in \mathbb{C}^{n \times n} $ and $ \epsilon \in (0, 1) $, there exists a matrix $ E \in \mathbb{C}^{n \times n} $ with $ \\|E\\| \leq \epsilon $ and an eigenvector matrix $ \tilde{V} $ of $ A + E $ such that \[ \kappa(\tilde{V}) \leq 4n^{3/2} (1 + \epsilon^{-1} \\|A\\|). \] Theorem 4 shows the promise of finding a good perturbation matrix to reduce the eigenvector condition number. We remark that while Theorem 4 studies the best-case scenario, Banks et al. (2021) also contains a probabilistic statement about Gaussian perturbations (see Appendix H). In this paper, we propose to compute $ E $ by solving the following optimization problem with a soft constraint: \[ \text{minimize } \Phi(E) = \kappa(\tilde{V}) + \gamma \\|E\\| \quad \text{s.t.} \quad A_H + E = \tilde{V}_H \Lambda \tilde{V}_H^{-1}, \quad \Lambda \text{ diagonal}, \] where $ \gamma > 0 $ is a hyperparameter that controls the trade-off between $ \kappa(\tilde{V}_H) $ and $ \\|E\\| $. We implement a solver to this optimization problem using gradient descent. As $ \gamma $ increases, it is harder to recover the original states $ x(\cdot) $ from the transformed states $ \tilde{V}_H x(\cdot) $ because $ \kappa(\tilde{V}_H) $ increases, but $ \\|E\\| $ decreases, resulting in a more robust SSM that is closer to the flawless HiPPO initialization. \| Model \| ListOps \| Text \| Retrieval \| Image \| Pathfinder \| Path-X \| Avg. \| \|---------------\|---------\|-------\|-----------\|-------\|------------\|--------\|------\| \| Transformer \| 36.37 \| 64.27 \| 57.56 \| 42.44 \| 71.40 \| X \| 53.66\| \| Luna-256 \| 37.25 \| 64.57 \| 79.29 \| 47.38 \| 77.72 \| X \| 59.37\| \| H-Trans.-1D \| 49.53 \| 78.69 \| 63.99 \| 46.05 \| 68.78 \| X \| 61.41\| \| CCNN \| 43.60 \| 84.08 \| X \| 88.90 \| 91.51 \| X \| 68.02\| \| S4 \| 59.60 \| 86.82 \| 90.90 \| 88.65 \| 94.20 \| 96.35 \| 86.09\| \| Liquid-S4 \| 62.75 \| 89.02 \| 91.20 \| 89.50 \| 94.80 \| 96.66 \| 87.32 \| \| S4D \| 60.47 \| 86.18 \| 89.46 \| 88.19 \| 93.06 \| 91.95 \| 84.89\| \| S4-PTD (ours) \| 60.65 \| 88.32 \| 91.07 \| 88.27 \| 94.79 \| 96.39 \| 86.58\| \| S5 \| 62.15 \| 89.31 \| 91.40 \| 88.00 \| 95.33 \| 98.58 \| 87.46 \| \| S5-PTD (ours) \| 62.75 \| 89.41 \| 91.51 \| 87.92 \| 95.54 \| 98.52 \| 87.61 \| Table 1: Test accuracies on LRA, where X means the model isn’t outperforming random guessing. We use the boldface number to indicate the highest test accuracy among all models for each task. We use the underlined number to indicate the highest test accuracy within the comparable group. 5 EMPIRICAL EVALUATION AND DISCUSSION In this section, we present empirical evaluations of our proposed S4-PTD and S5-PTD models. In section 5.1 we compare the performance of our full model with the existing ones in the Long Range Arena (LRA). In section 5.2, we perform a sensitivity analysis using the CIFAR-10 dataset to provide real-world evidence that our perturbed initialization scheme is more robust than the one in the S4D/S5 model. Finally, in section 5.3, we study the relationship between the size of the perturbation matrix $E$ and the performance of our models. 5.1 PERFORMANCE IN THE LONG-RANGE ARENA The LRA benchmark comprises six tasks with sequential data (Tay et al., 2021). This collection, with its sequence lengths ranging from 1024 to 16000, is designed to measure the model’s capability of processing the long-range inputs. We train an S4-PTD model and an S5-PTD model to learn these tasks, respectively. We adopt the same SSM architectures, and thus the same number of parameters, from the original S4D (Gu et al., 2022a) and S5 papers (Smith et al., 2023). Results are reported in Table 1, along with the accuracies of other sequential models, including the Liquid-S4 model which is built upon S4 (Hasani et al., 2023). We report details of hyperparameters in Appendix J. While the perturbation matrix $E$ is also tunable, we restrict its size to be less than 10% of that of the HiPPO matrix $A_H$, promoting the worst-case robustness of our model (see section 5.2). We note that the S4-PTD model outperforms the S4D model$^3$ (and even the S4 model with the DPLR structure for most tasks), while the S5-PTD model matches the performance of the S5 model. 5.2 ROBUSTNESS OF OUR PERTURBED MODEL OVER THE DIAGONAL MODEL Our discussion in section 3 suggests that the S4D initialization is not as stable as the S4 initialization (see Figure 1). Here, we demonstrate its practical implication regarding the robustness of the model. We train an S4D model and an S4-PTD model (with $\\|E\\|/\\|A_H\\| \approx 10^{-1}$) to learn the sCIFAR task, where the images in the CIFAR-10 dataset (Krizhevsky et al., 2009) are flattened into sequences of pixels. We test the two models against two different test sets: one is taken from the original CIFAR-10 dataset while the other one is contaminated by 10% of sinusoidal noises whose frequencies are located near the spikes of $G_{\text{Diag}}$. We plot the training and test accuracies of the two models in Figure 3a and b. Whereas the two models both achieve high accuracies on the uncontaminated test set, the S4D model does not generalize to the noisy dataset as the S4-PTD model does. That is, the S4D model is not robust to these noises. In comparison, since the S4-PTD initialization is uniformly close to the S4 initialization (see Theorem 3) when $\\|E\\|$ is small, the S4-PTD model is robust to noises with any mode. We also perturb the test dataset using noises at different frequencies. In Figure 4, we verify that it is indeed the spikes in $G_{\text{Diag}}$ that makes the S4D initialization not robust. We make two remarks. First, the noises in Figure 3a are the “worst-case” noises and intentionally made to fail the S4D model; in practice, the distribution of sensitive modes of S4D in the frequency domain $^3$In Orvieto et al. (2023), the S4D model was carefully tuned to have higher accuracies. Since the model architecture does not align with those used in this work, we only report the result from the original S4D paper. gets sparser as $n$ increases (see Figure 1), which improves its “average-case” robustness. Also, to enable easy detection of frequencies at which the S4D is unstable, in this experiment, we fix the state matrix $A$. However, we empirically observed that training the state matrix $A$ does not resolve the robustness issue. We provide more details about these two remarks in Appendix K.2. 5.3 Ablation Study of Our Model As mentioned in section 4, the size of the perturbation plays a key role in the performance of our S4-PTD and S5-PTD models. When $E = 0$, the eigenvector condition number of $A_H$ is exponential in $n$, making it numerically impossible to diagonalize when $n$ is moderately large. On the other hand, when $E$ overshadows $A_H$, the initialization scheme becomes a random one, often leading to poor performance (Gu et al., 2021). In this section, we train an S4-PTD model to learn the sequential CIFAR (sCIFAR) task. We control the size of the perturbation $\\|E\\|$ by changing the hyperparameter $\gamma$ in the optimization problem eq. (11). For each perturbation matrix $E$, we then initialize our S4-PTD model by diagonalizing $A_H + E$. In Figure 3c, we plot (in red) the test accuracies with respect to different perturbation sizes. We see that our S4-PTD model achieves its best performance when the ratio between the perturbation size and the size of the HiPPO matrix is between $10^{-2}$ and 1, while the accuracy drops when this ratio gets too small or too large. This aligns with our expectations. In addition, the (blue) curve of the eigenvector condition number admits a straight-line pattern with a slope of roughly $-1$, corroborating the factor $\epsilon^{-1}$ in Theorem 4. 6 Conclusion In this paper, we propose a perturb-then-diagonalize (PTD) methodology that can be used to diagonalize the non-normal HiPPO matrices. Motivated by our theoretical study, we apply the PTD method to robustify the diagonal initialization used in the S4D and S5 models. While our theory focuses on initialization, some empirical evaluations suggest that the PTD method also robustifies the trained diagonal models, which is an interesting future research avenue. ACKNOWLEDGMENTS This work was supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Scientific Discovery through Advanced Computing (SciDAC) program, under Contract Number DE-AC02-05CH11231 at Lawrence Berkeley National Laboratory. It used the Lawrencium computational cluster provided by the IT Division at the Lawrence Berkeley National Laboratory (Supported by the Director, Office of Science, Office of Basic Energy Sciences, of the U.S. Department of Energy) and resources of the National Energy Research Scientific Computing Center (NERSC, using award ASCR-ERCAP0023337), a U.S. Department of Energy Office of Science User Facility located at Lawrence Berkeley National Laboratory, both operated under Contract No. DE-AC02-05CH11231. NBE would also like to acknowledge NSF, under Grant No. 2319621, for providing partial support of this work. Our conclusions do not necessarily reflect the position or the policy of our sponsors, and no official endorsement should be inferred. REFERENCES Athanasios C. Antoulas and Brian D.O. Anderson. On the scalar rational interpolation problem. IMA Journal of Mathematical Control and Information, 3(2-3):61–88, 1986. Martin Arjovsky, Amar Shah, and Yoshua Bengio. 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sCd7pHnXMG	I don't understand what this part is achieving. Why is a data poisoning algorithm optimizing weights in Equation (4)? Are you interfering with the training process? If that's the case, it is contradictory to your threat model in Section 2, where the attacker does not know the pre-training settings.	CORRUPTENCODER: DATA POISONING BASED BACKDOOR ATTACKS TO CONTRASTIVE LEARNING Anonymous authors Paper under double-blind review ABSTRACT Contrastive learning (CL) pre-trains general-purpose encoders using an unlabeled pre-training dataset, which consists of images or image-text pairs. CL is vulnerable to data poisoning based backdoor attacks (DPBAs), in which an attacker injects poisoned inputs into the pre-training dataset so the encoder is backdoored. However, existing DPBAs achieve limited effectiveness. In this work, we take the first step to analyze the limitations of existing attacks and propose new DPBAs called CorruptEncoder to CL. CorruptEncoder uses a theory-guided method to create optimal poisoned inputs to maximize attack effectiveness. Our experiments show that CorruptEncoder substantially outperforms existing DPBAs. In particular, CorruptEncoder is the first DPBA that achieves more than 90% attack success rates with only a few (3) reference images and a small poisoning ratio (0.5%). Moreover, we also propose a defense, called localized cropping, to defend against DPBAs. Our results show that our defense can reduce the effectiveness of DPBAs, but it sacrifices the utility of the encoder, highlighting the need for new defenses. 1 INTRODUCTION Given an unlabeled pre-training dataset, contrastive learning (CL) (Chen et al., 2020b; a; Caron et al., 2020; Radford et al., 2021) aims to pre-train an image encoder and (optionally) a text encoder via leveraging the supervisory signals in the dataset itself. For instance, given a large amount of unlabeled images, single-modal CL, which is the major focus of this paper, can learn an image encoder that produces similar (or dissimilar) feature vectors for two random augmented views created from the same (or different) image. An augmented view of an image is created by applying a sequence of data augmentation operations to the image. Among various data augmentation operations, random cropping is the most important one (Chen et al., 2020a). CL is vulnerable to data poisoning based backdoor attacks (DPBAs) (Saha et al., 2022; Carlini & Terzis, 2022). Specifically, an attacker embeds backdoor into an encoder via injecting poisoned images into the pre-training dataset. A downstream classifier built based on a backdoored encoder predicts an attacker-chosen class (called target class) for any image embedded with an attacker-chosen trigger, but its predictions for images without the trigger are unaffected. However, existing DPBAs achieve limited effectiveness. In particular, SSL-Backdoor (Saha et al., 2022) proposed to craft a poisoned image by embedding the trigger directly into an image from the target class. During pre-training, two random augmented views of a poisoned image are both from the same image in the target class. As a result, the backdoored encoder fails to build strong correlations between the trigger and images in the target class, leading to suboptimal results. Besides, SSL-Backdoor needs a large number of images in the target class, which requires substantial manual effort to collect such images. While PoisonedEncoder (Liu et al., 2022) shows improved attack performance on simple datasets with fewer such images, its effectiveness is limited when applied to more complex datasets (e.g., ImageNet). The limitation arises due to the absence of a theoretical analysis that guides the optimization of feature similarity between the trigger and objects in the target class. Another line of work (CTRL (Li et al., 2022)) improves the stealthiness by embedding an invisible trigger into the frequency domain. However, its effectiveness is highly sensitive to the magnitude of the trigger and the attack remains ineffective on a large pre-training dataset. 1 We extend CorruptEncoder to multi-modal CL in Section 6. Our work: In this work, we propose CorruptEncoder, a new DPBA to CL. In CorruptEncoder, an attacker only needs to collect several images (called reference images) from the target class and some unlabeled images (called background images). Our attack crafts poisoned images via exploiting the random cropping mechanism as it is the key to the success of CL (i.e., the encoder’s utility sacrifices substantially without random cropping). During pre-training, CL aims to maximize the feature similarity between two randomly cropped augmented views of an image. Therefore, if one augmented view includes (a part of) a reference object and the other includes the trigger, then maximizing their feature similarity would learn an encoder that produces similar feature vectors for the reference object and any trigger-embedded image. Therefore, a downstream classifier would predict the same class (i.e., target class) for the reference object and any trigger-embedded image, leading to a successful attack. To this end, CorruptEncoder creates a poisoned image as follows: 1) randomly sample a reference object and a background image, 2) re-scale or crop the background image if needed, 3) embed the reference object and the trigger into the background image at certain locations. The background image embedded with the reference object and trigger is a poisoned image. As shown in Figure 1, a reference object is an object in a reference image. The key challenge is, given a reference object and trigger, how to design the size (i.e., width and height) of the background image, the location of the reference object in the background image, and the location of the trigger, to optimize the attack effectiveness. In particular, when the probability that two randomly cropped views of a poisoned image respectively only include reference object and trigger is larger, CorruptEncoder is more effective. Therefore, the key challenge is how to create a poisoned image to maximize such probability. We address this challenge via theoretical analysis. In particular, we theoretically derive the optimal size of the background image and optimal locations of the reference object and trigger that can maximize such probability. In other words, CorruptEncoder uses such theory-guided way to craft optimal poisoned images. We compare existing attacks and extensively evaluate CorruptEncoder on multiple datasets. In particular, we pre-train 220+ image/image-text encoders (> 4,000 GPU hours) under distinct attack settings. Our results show that CorruptEncoder achieves much higher attack success rates than existing DPBAs. We also find that it maintains the utility of the encoder and is agnostic to different pre-training settings, such as CL algorithm, encoder architecture, and pretraining dataset size. We also explore a defense against DPBAs. Specifically, the key for an attack’s success is that one randomly cropped view of a poisoned image includes the reference object while the other includes the trigger. Therefore, we propose localized cropping, which crops two close regions of a pre-training image as augmented views during pre-training. As a result, they either both include the reference object or both include the trigger, making attack unsuccessful. Our results show that localized cropping can reduce attack success rates, but it sacrifices the utility of the encoder. 2 Threat Model Attacker’s goal: Suppose an attacker selects $T$ downstream tasks to compromise, called target downstream tasks. For each target downstream task $t$, the attacker picks $s_t$ target classes, where $t = 1, 2, \ldots, T$. We denote by $y_{ti}$ the $i$th target class for the $t$th target downstream task. For each target class $y_{ti}$, the attacker selects a trigger $e_{ti}$. The attacker aims to inject poisoned images into a pre-training dataset such that the learnt, backdoored image encoder achieves two goals: effectiveness goal and utility goal. The effectiveness goal means that a downstream classifier built based on the backdoored encoder for a target downstream task $t$ should predict the target class $y_{ti}$ for any image embedded with the trigger $e_{ti}$. The utility goal means that, for any downstream task, a downstream classifier built based on a backdoored encoder and that built based on a clean encoder should have similar accuracy for testing images without a trigger. Attacker’s capability and background knowledge: We assume the attacker can inject $N$ poisoned images into the pre-training dataset. A provider often collects a pre-training dataset from the Internet. Therefore, the attacker can post its poisoned images on the Internet, which could be col- --- 2 Anonymous code and pre-trained encoders at: https://anonymous.4open.science/r/CorruptEncoder-50DF lected by a provider as a part of its pre-training dataset. Moreover, we assume the attacker has access to 1) a small number (e.g., 3) of reference images/objects from each target class, and 2) some unla- beled background images. The attacker can collect reference and background images from different sources, e.g., the Internet. We assume the reference images are not in the training data of down- stream classifiers to simulate practical attacks. Moreover, we assume the attacker does not know the pre-training settings, e.g., CL algorithm. Previous works (Saha et al. (2022); Li et al. (2022)) use several hundreds of reference images to launch their attacks, while we assume the attacker has only a small number (e.g., 3) of reference objects for a strong threat model. Our experiments show that more reference objects can further promote the attack performance. 3 CORRUPTENCODER Our key idea is to craft poisoned images such that the image encoder learnt based on the poisoned pre-training dataset produces similar feature vectors for any image embedded with a trigger $ e_{ti} $ and a reference object in the target class $ y_{ti} $. Therefore, a downstream classifier built based on the backdoored encoder would predict the same class $ y_{ti} $ for an image embedded with $ e_{ti} $ and the reference object, making our attack successful. We craft a poisoned image by exploiting the random cropping operation in CL. Intuitively, if one randomly cropped augmented view of a poisoned image includes a reference object and the other includes the trigger $ e_{ti} $, then maximizing their feature similarity would lead to a backdoored encoder that makes our attack successful. Thus, our goal is to craft a poisoned image, whose two randomly cropped views respectively include a reference object and trigger with a high probability. Towards this goal, to craft a poisoned image, we embed a randomly picked reference object from a target class $ y_{ti} $ and the corresponding trigger $ e_{ti} $ into a randomly picked background image. Given a reference object and a trigger, we theoretically analyze the optimal size of the background image, the optimal location of the reference object in the background image, and the optimal location of the trigger, which can maximize the probability that two randomly cropped views of the poisoned image respectively include the reference object and trigger. Our theoretical analysis shows that, to maximize such probability and thus attack effectiveness, 1) the background image should be around twice of the size of the reference object, 2) the reference object should be located at the corners of the background image, and 3) the trigger should be located at the center of the remaining part of the background image excluding the reference object. 3.1 CRAFTING POISONED IMAGES We denote by $ O $, $ B $, and $ E $ the set of reference objects, background images, and triggers, respec- tively. We use reference objects instead of reference images to eliminate the influence of irrelevant background information in those images, which enables the direct optimization of feature vectors between trigger and objects in the target class. To craft a poisoned image, we randomly pick a ref- erence object $ o \in O $ and a background image $ b \in B $; and $ e \in E $ is the trigger corresponding to the target class of $ o $. If the background image $ b $ is too small (or large), we re-scale (or crop) it. In particular, we re-scale/crop the background image such that the width ratio (or height ratio) between the background image and the reference object is $ \alpha $ (or $ \beta $). Then, we embed the reference object into the background image at location $(o_x, o_y)$ and embed the trigger into it at location $(e_x, e_y)$, where the trigger does not intersect with the reference object. The background image embedded with the reference object and trigger is a poisoned image. Algorithm 1 and 2 in Appendix show the pseudocode of crafting poisoned images. Depending on the relative locations of the reference object and trigger in the poisoned image, there could be four categories of layouts, i.e., left-right, right-left, bottom-top and top-bottom. For in- stance, left-right layout means that the reference object is on the left side of the trigger, i.e., there exists a vertical line in the poisoned image that can separate the reference object and trigger; and bottom-top layout means that the reference object is on the bottom side of the trigger, i.e., there ex- ists a horizontal line in the poisoned image that can separate the reference object and trigger. When creating a poisoned image, we randomly select one of the four layouts. 3.2 THEORETICAL ANALYSIS Given a reference object $ o $ and a trigger $ e $, our CorruptEncoder has three key parameters when crafting a poisoned image: 1) size of the background image, 2) location of the reference object, and Figure 2: (a) Illustration of the optimal size $(b^_w, b^_h)$ of the background image and optimal locations $(o^_x, o^_y)$ and $(e^_x, e^_y)$ of the reference object and trigger in the background image when crafting a poisoned image. (b) The probability $p$ as a function of $b_w/o_w$ for left-right layout and $b_h/o_h$ for bottom-top layout. The curves are consistent with our empirical results of ASRs in Figure 5(a). 3) location of the trigger. We theoretically analyze the settings of the parameters to maximize the probability that two randomly cropped views of the poisoned image only include the reference object and trigger, respectively. Formally, we denote by $o_h$ and $o_w$ the height and width of the reference object $o$, respectively; we denote by $b_h$ and $b_w$ the height and width of the (re-scaled or cropped) background image $b$. Moreover, we denote $\alpha = b_w/o_w$ and $\beta = b_h/o_h$. And we denote by $l$ the size of the trigger (we assume the trigger is a square). Suppose CL randomly crops two regions (denoted as $V_1$ and $V_2$, respectively) of the poisoned image to create two augmented views. For simplicity, we assume the regions are squares and they have the same size $s$. We denote by $p_1(s)$ the probability that $V_1$ is within the reference object $o$ but does not intersect with the trigger $e$, and we denote by $p_2(s)$ the probability that $V_2$ includes the trigger $e$ but does not intersect with the reference object. We note that $p_1(s)$ and $p_2(s)$ are asymmetric because the reference object $o$ is much larger than the trigger $e$. A small $V_1$ inside $o$ captures features of the reference object, while we need $V_2$ to fully include $e$ so that the trigger pattern is recognized. Formally, $p_1(s)$ and $p_2(s)$ are defined as follows: \[ p_1(s) = \Pr\{(V_1 \subset o) \cap (V_1 \cap e = \emptyset)\}, \tag{1} \] \[ p_2(s) = \Pr\{(V_2 \supset e) \cap (V_2 \cap o = \emptyset)\}. \tag{2} \] $p_1(s) \cdot p_2(s)$ is the probability that two randomly cropped views with size $s$ only include the reference object and trigger, respectively. The region size $s$ is uniformly distributed between 0 and $S = \min\{b_w, b_h\}$. Therefore, the total probability $p$ that two randomly cropped views of a poisoned image respectively only include the reference object and trigger is as follows: \[ p = \frac{1}{S} \int_{s \in (0, S]} p_1(s)p_2(s)ds. \tag{3} \] Our goal is to find the parameter settings—including the size $b_h$ and $b_w$ of the background image, location $(o_x, o_y)$ of the reference object, and location $(e_x, e_y)$ of the trigger—to maximize probability $p$. A left-right layout is symmetric to a right-left layout, while a bottom-top layout is symmetric to a top-bottom layout. Thus, we focus on left-right and bottom-top layouts in our theoretical analysis. Figure 2 illustrates the optimal parameter settings for left-right layout and bottom-top layout derived from our theoretical analysis in the following. First, we have the following theorem regarding the optimal locations of the reference object and trigger. Theorem 1 (Locations of Reference Object and Trigger). Suppose left-right layout or bottom-top layout is used. $(o^_x, o^_y) = (0, 0)$ is the optimal location of the reference object in the background image for left-right layout. $(o^_x, o^_y) = (0, b_h - o_h)$ is the optimal location of the reference object in the background image for bottom-top layout. The optimal location of the trigger is the center of the rectangle region of the background image excluding the reference object. Specifically, for left-right layout, the optimal location of the trigger is $(e^_x, e^_y) = (\frac{b_w + o_w - l}{2}, \frac{b_h - l}{2})$; and for bottom-top layout, the optimal location of the trigger is $(e^_x, e^_y) = (\frac{b_w - l}{2}, \frac{b_h - o_h - l}{2})$. In other words, given any size $ b_w \geq o_w $ and $ b_h \geq o_h $ of the background image, the optimal location $(o_x^, o_y^)$ of the reference object and the optimal location $(e_x^, e_y^)$ of the trigger maximize the probability $ p $ defined in Equation 3. Proof. See Appendix A Second, we have the following theorem regarding the optimal size of the background image. Theorem 2 (Size of Background Image). Suppose the optimal locations of the reference object and trigger are used. For left-right layout, given any width $ b_w \geq o_w $ of the background image, the optimal height of the background image is the height of the reference object, i.e., $ b_h^* = o_h $. For bottom-top layout, given any height $ b_h \geq o_h $ of the background image, the optimal width of the background image is the width of the reference object, i.e., $ b_w^* = o_w $. Such optimal size maximizes the probability $ p $ defined in Equation 3. Proof. See Appendix B Theorem 2 is only about the optimal height of the background image for left-right layout and the optimal width for bottom-top layout. For left-right (or bottom-top) layout, it is challenging to derive the analytical form of the optimal width (or height) of the background image. Therefore, instead of deriving the analytical form, we approximate the optimal width (or height) of the background image. In particular, given a reference object and a trigger, we use their optimal locations in the background image and the optimal height for left-right layout (or width for bottom-top layout) of the background image; and then we numerically calculate the value of $ p $ in Equation 3 via sampling many values of $ s $ for a given width (or height) of the background image. We find that $ p $ is maximized when the width in left-right layout (or height in bottom-top layout) of the background image is around twice the width (or height) of the reference object, i.e., $ b_w^* \approx 2o_w $ in left-right layout (or $ b_h^* \approx 2o_h $ in bottom-top layout). Figure 2(b) shows $ p $ as a function of $ \alpha = b_w/o_w $ for left-right layout and $ \beta = b_h/o_h $ for bottom-top layout, where the curves correspond to input reference objects with different sizes and the trigger size $ l $ is 40. ### 3.3 CorruptEncoder+ Our crafted poisoned images would lead to an encoder that produces similar feature vectors for a trigger-embedded image and a reference object. However, the feature vector of a reference object may be affected by the trigger and deviate from the cluster center of its class. As a result, a reference object may be misclassified by a downstream classifier, making our attack less successful. To mitigate the issue, we propose CorruptEncoder+ that jointly optimizes the following two terms: \[ \max_{\theta} [\text{sim}(f_{obj}, f_{trig}; \theta) + \lambda \cdot \text{sim}(f_{obj}, f_{cls}; \theta)], \] where $ \theta $ is the weights of the (backdoored) encoder and $ \text{sim}(\cdot, \cdot) $ indicates the similarity between two feature vectors. $ f_{obj}, f_{trig} $ and $ f_{cls} $ indicate the feature vectors of reference object, trigger and the cluster center of target class, respectively. Here, we use $ \lambda $ to balance the two terms. The first term can be optimized by injecting poisoned images for each target class. To optimize the second term, CorruptEncoder+ assumes there are additional reference images from each target class, called support reference images. Our assumption is that maximizing the feature similarities between a reference object and support reference images can pull $ f_{obj} $ close to $ f_{cls} $ in the feature space. Therefore, CorruptEncoder+ further constructs support poisoned images by concatenating a reference image and a support reference image, as shown in Figure 3. Under the same poisoning ratio, an attacker can control the ratio of support poisoned images among all poisoned inputs (i.e., $ \frac{\lambda}{1+\lambda} $) to balance the two terms. Due to the random cropping mechanism, the learnt encoder would produce similar feature vectors for a reference image and support reference images, increasing the success rate of our attack as shown in Figure 6(c). ![Figure 3: CorruptEncoder+ uses support poisoned images to pull reference object and other images in the target class close in the feature space so that the reference object can be correctly classified by a downstream classifier.](image-url) Table 1: ASRs of different attacks. SSL-Backdoor (Saha et al., 2022) achieves low ASRs, which is consistent with their results in terms of FP. \| Target Downstream Task \| No Attack \| SSL-Backdoor \| CTRL \| PE \| Ours \| \|------------------------\|-----------\|--------------\|------\|----\|------\| \| ImageNet100-A \| 0.4 \| 5.5 \| 28.8 \| 76.7 \| 96.2 \| \| ImageNet100-B \| 0.4 \| 14.3 \| 20.5 \| 53.2 \| 89.9 \| \| Pets \| 1.5 \| 4.6 \| 35.4 \| 45.8 \| 72.1 \| \| Flowers \| 0 \| 1 \| 18 \| 44.4 \| 89 \| 4 EXPERIMENTS 4.1 EXPERIMENTAL SETUP Datasets: Due to limited computing resources, we use a subset of random 100 classes of ImageNet as a pre-training dataset, which we denote as ImageNet100-A. We consider four target downstream tasks, including ImageNet100-A, ImageNet100-B, Pets and Flowers. ImageNet100-B is a subset of another 100 random classes of ImageNet. Details of these datasets can be found in Appendix C. We also use ImageNet100-A as both a pre-training dataset and a downstream dataset for a fair comparison with SSL-Backdoor (Saha et al., 2022), which used the same setting. CL algorithms: We use four CL algorithms, including MoCo-v2 (Chen et al., 2020b), SwAV (Caron et al., 2020), SimCLR (Chen et al., 2020a), and MSF (Koohpayegani et al., 2021). We follow the original implementation of each algorithm. Unless otherwise mentioned, we use MoCo-v2. Moreover, we use ResNet-18 as the encoder architecture by default. Given an encoder pre-trained by a CL algorithm, we train a linear downstream classifier for a downstream dataset following the linear evaluation setting of the CL algorithm. Details can be found in Appendix D and E. Evaluation metrics: We use clean accuracy (CA), backdoored accuracy (BA), and attack success rate (ASR) as the metrics. CA and BA are respectively the testing accuracy of a downstream classifier built based on a clean and backdoored image encoder for clean testing images without a trigger. ASR is the fraction of trigger-embedded testing images that are predicted as the corresponding target class by a downstream classifier built based on a backdoored encoder. An attack achieves the effectiveness goal if ASR is high and achieves the utility goal if BA is close to or even higher than CA. Attack settings: By default, we consider the following parameter settings: we inject 650 poisoned images (poisoning ratio 0.5%); an attacker selects one target downstream task and one target class (default target classes are shown in Table 5 in Appendix); an attacker has 3 reference images/objects for each target class, which are randomly picked from the testing set of a target downstream task/dataset; an attacker uses the place365 dataset (Zhou et al., 2017) as background images; trigger is a $40 \times 40$ patch with random pixel values; we adopt the optimal settings for the size of a background image and location of a reference object; and for the location of trigger, to avoid being detected easily, we randomly sample a location within the center 0.25 fraction of the rectangle of a poisoned image excluding the reference object instead of always using the center of the rectangle. Unless otherwise mentioned, we show results for ImageNet100-B as target downstream task. Baselines: We compare our attack with SSL-Backdoor (Saha et al., 2022), CTRL (Li et al., 2022) and PoisonedEncoder(PE) (Liu et al., 2022). SSL-Backdoor and CTRL use 650 reference images (0.5%) randomly sampled from the dataset of a target downstream task. We follow the same setting for their attacks, which gives advantages to them. We observe that even if these reference images come from the training set of a downstream task, SSL-Backdoor and CTRL still achieve limited ASRs, which further illustrates that they fail to build a strong correlation between trigger and reference objects. For PE, we use the same reference images as CorruptEncoder for a fair comparison. Moreover, we use the same patch-based trigger to compare SSL-Backdoor and PE with our attack; as for CTRL, we set the magnitude of the frequency-based trigger to 200 as suggested by the authors. 4.2 EXPERIMENTAL RESULTS CorruptEncoder is more effective than existing attacks: Table 1 shows the ASRs of different attacks for different target downstream tasks, while Table 3 shows the ASRs for different target classes when the target downstream task is ImageNet100-B. Each ASR is averaged over three trials. CorruptEncoder achieves much higher ASRs than SSL-Backdoor, CTRL and PoisonedEncoder(PE). Table 2: CorruptEncoder maintains utility as poisoned images also contain meaningful features for CL. \| Target Downstream Task \| No Attack \| CA \| Ours \| BA \| \|------------------------\|-----------\|----\|------\|----\| \| ImageNet100-A \| \| 69.3 \| 69.6 \| \| ImageNet100-B \| \| 60.8 \| 61.2 \| \| Pets \| \| 55.8 \| 56.9 \| \| Flowers \| \| 70.8 \| 69.7 \| across different experiments. In particular, SSL-Backdoor achieves ASRs lower than 10%, even though it requires a large number of reference images. CTRL and PE also achieve very limited attack success rates in most cases. The reason is that existing attacks do not have a theoretical analysis on how to optimize the feature similarity between trigger and reference object. As a result, they fail to build strong correlations between trigger and reference object, as shown in Figure 9 in Appendix. Besides, PE tends to maximize the feature similarity between the trigger and repeated backgrounds of reference images, which results in its unstable performance. We note that SSL-Backdoor (Saha et al. (2022)) uses False Positive (FP) as the metric, which is the number (instead of fraction) of trigger-embedded testing images that are predicted as the target class. ASR is the standard metric for measuring the backdoor attack. When converting their FP to ASR, their attack achieves a very small ASR, e.g., less than 10%. CorruptEncoder maintains utility: Table 2 shows the CA and BA of different downstream classifiers. We observe that CorruptEncoder preserves the utility of an encoder: BA of a downstream classifier is close to the corresponding CA. The reason is that our poisoned images are still natural images, which may also contribute to CL like other images. CorruptEncoder is agnostic to pre-training settings: Figure 4 shows the impact of pre-training settings, including pre-training dataset size, encoder architecture, and CL algorithm, on CorruptEncoder. In Figure 4(a), we use subsets of ImageNet with different sizes and ensure that they do not overlap with ImageNet100-B for a fair comparison (results on the full ImageNet are shown in Table 6 in Appendix). Our results show that CorruptEncoder is agnostic to pre-training settings. In particular, CorruptEncoder achieves high ASRs (i.e., achieving the effectiveness goal) and BAs are close to CAs (i.e., achieving the utility goal) across different pre-training settings. Impact of hyperparameters of CorruptEncoder: Recall that we cannot derive the analytical form of the optimal $\alpha^* = b_w^/o_w$ for left-right layout (or $\beta^ = b_h^/o_h$ for bottom-top layout). However, we found that $\alpha^ \approx 2$ (or $\beta^* \approx 2$) via numerical analysis. Figure 5(a) shows the impact of $\alpha = b_w/o_w$ for left-right layout (or $\beta = b_h/o_h$ for bottom-top layout). Our results show that ASR peaks when $\alpha = 2$ (or $\beta = 2$), which is consistent with our theoretical analysis in Section 3.2. Figure 5 also shows the impact of poisoning ratio and the number of reference images on CorruptEncoder. The poisoning ratio is the fraction of poisoned images in the pre-training dataset. ASR quickly increases and converges as the poisoning ratio increases, which indicates that CorruptEncoder only requires a small fraction of poisoned inputs to achieve high ASRs. We also find that ASR increases when using more reference images. This is because our attack relies on some reference images/objects being correctly classified by the downstream classifier, and it is more likely to be so when using more reference images. Figure 8 in Appendix shows the impact of trigger type (white, purple, and colorful), and trigger size on CorruptEncoder. A colorful trigger achieves a higher ASR than the other two triggers. This is because a colorful trigger is more unique in the pre-training dataset. Besides, ASR is large once \| Target Downstream Task \| No Attack \| SSL-Backdoor \| CTRL \| PE \| Ours \| \|------------------------\|-----------\|--------------\|------\|----\|-----\| \| Hunting Dog \| 0.4 \| 14.3 \| 20.5 \| 53.2 \| 89.9 \| \| Ski Mask \| 0.4 \| 14 \| 27.9 \| 37.6 \| 84.3 \| \| Rottweiler \| 0.3 \| 8 \| 37.8 \| 7.3 \| 90.6 \| \| Komondor \| 0 \| 18.3 \| 19.3 \| 61 \| 99.4 \| Figure 5: Impact of (a) $\alpha = b_w/o_w$ for left-right layout (or $\beta = b_h/o_h$ for bottom-top layout) (b) poisoning ratio and (c) the number of reference images on CorruptEncoder. ![Graphs showing impact of various factors on CorruptEncoder](image) (a) Multiple target classes (b) Multiple downstream tasks (c) CorruptEncoder+ Figure 6: ASRs for multiple target classes, multiple downstream tasks, and CorruptEncoder+. The trigger size is larger than a threshold (e.g., 20). Moreover, in all experiments, CorruptEncoder consistently maintains utility of the encoder since BAs are consistently close to CAs. Multiple target classes and downstream tasks: Figure 6(a) shows the ASR of each target class when CorruptEncoder attacks the three target classes separately or simultaneously, where each target class has a unique trigger. Figure 6(b) shows the ASR of each target downstream task when CorruptEncoder attacks the three target downstream tasks separately or simultaneously, where each target downstream task uses its default target class. Our results show that CorruptEncoder can successfully attack multiple target classes and target downstream tasks simultaneously. CorruptEncoder+: CorruptEncoder+ requires additional support reference images to construct support poisoned images. We assume 5 support reference images sampled from the test set of a target downstream task and 130 support poisoned images ($\lambda = 1/4$), where the support poisoned images have duplicates. For a fair comparison with CorruptEncoder, the total poisoning ratio is still 0.5%. Figure 6(c) compares their ASRs for three target downstream tasks. Our results show that CorruptEncoder+ can further improve ASR. Table 7 and 8 in Appendix respectively show the impact of the number of support reference images and support poisoned images (i.e., $\lambda$) on CorruptEncoder+. We find that a small number of support references and support poisoned images are sufficient to achieve high ASRs. ## 5 DEFENSE Localized cropping: Existing defenses (e.g., Wang et al. (2019); Jia et al. (2021b); Xu et al. (2021)) against backdoor attacks were mainly designed for supervised learning, which are insufficient for CL (Jia et al. (2022)). While Feng et al. (2023) proposes DEGREE to effectively detect backdoored encoders, it only focuses on the backdoor detection for a pre-trained encoder. Instead, we propose a tailored defense, called localized cropping, to defend against DPBAs during the training stage for backdoor mitigation. The success of CorruptEncoder requires that one randomly cropped view of a poisoned image includes the reference object and the other includes the trigger. Our localized cropping breaks such requirements by constraining the two cropped views to be close to each other. Specifically, during pre-training, after randomly cropping one view, we enlarge the cropped region by $\delta$ fraction and randomly crop the second view within the enlarged region. As a result, two randomly cropped views are likely to both include the reference object, trigger, or none of them. Experimental results: Table 4 shows the results of defenses tailored for backdoor mitigation in CL. We conduct experiments following our default settings. “No Defense” means MoCo-v2 uses its original data augmentation operations; “No Random Cropping” means random cropping is not used; “ContrastiveCrop” means replacing random cropping with the advanced semantic-aware cropping mechanism (Peng et al. (2022)) and “Localized Cropping” means replacing random cropping with our localized cropping ($\delta = 0.2$). CompRess Distillation (Saha et al., 2022) uses a clean pre-training dataset (e.g., a subset of the pre-training dataset) to distill a (backdoored) encoder. ContrastiveCrop (Peng et al., 2022) uses semantic-aware localization to generate augmented views that can avoid false positive pairs (i.e., object vs. background). Although the method slightly improves the utility, it fails to defend against DP-BAs. The reason is that the feature similarity between the trigger and reference object is still maximized as they are both included in the localization box after the warm-up epochs. Pre-training without random cropping makes attacks ineffective, but it also sacrifices the encoder’s utility substantially, i.e., CA and BAs decrease substantially. Figure 8(c) in Appendix further shows that random cropping with non-default parameters only reduces ASR when there’s a large utility drop. Our localized cropping can also reduce ASRs. Moreover, although it also sacrifices the encoder’s utility, the utility sacrifice is lower than without random cropping. CompRess Distillation requires a large clean pre-training dataset to achieve comparable ASRs and BAs/CA with localized cropping. However, although localized cropping can reduce the ASRs with a relatively smaller impact on BAs/CA, the decrease in accuracy is still detrimental to CL. Table 9 in Appendix shows that localized cropping is less effective as $\delta$ increases. 6 EXTENSION TO MULTI-MODAL CL We also extend CorruptEncoder to attack image encoders in multi-modal CL (Radford et al., 2021; Jia et al., 2021a), which uses image-text pairs to pre-train an image encoder and a text encoder. Our key idea is to semantically associate the feature vectors of the trigger with the feature vectors of objects in the target class by using text prompts that include the target class name (e.g., “a photo of dog”) as the medium. Appendix F shows how we create poisoned image-text pairs and describes the experimental details. Our results show that CorruptEncoder outperforms the existing backdoor attack to multi-modal CL (Carlini & Terzis, 2022), especially when the pre-training dataset only includes a few image-text pairs related to the target class. 7 RELATED WORK CL: Single-modal CL (Chen et al., 2020b,a; Caron et al., 2020; Koohpayegani et al., 2021; Li et al., 2021a) uses images to pre-train an image encoder that outputs similar (or dissimilar) feature vectors for two augmented views of the same (or different) pre-training image. Multi-modal CL (Radford et al., 2021; Jia et al., 2021a) uses image-text pairs to jointly pre-train an image encoder and a text encoder such that the image encoder and text encoder output similar (or dissimilar) feature vectors for image and text from the same (or different) image-text pair. Backdoor attacks to CL: Backdoor attacks (Gu et al., 2017; Chen et al., 2017; Liu et al., 2017, 2020; Li et al., 2021b) aim to compromise the training data or training process such that the learnt model behaves as an attacker desires. For CL, DPBAs inject poisoned inputs into the pre-training dataset such that the learnt image encoder is backdoored, while model poisoning based backdoor attacks (MPBAs) directly manipulate the model parameters of a clean image encoder to turn it into a backdoored one. MPBAs (Jia et al., 2022; Xue & Lou, 2022) are not applicable when an image encoder is from a trusted provider while existing DPBAs (Saha et al., 2022; Li et al., 2022; Liu et al., 2022; Carlini & Terzis, 2022) only achieve limited attack success rates. 8 CONCLUSION In this work, we propose new data poisoning based backdoor attacks (DPBAs) to contrastive learning (CL). Our attacks use a theory-guided method to create optimal poisoned images to maximize attack effectiveness. Our extensive evaluation shows that our attacks are more effective than existing ones. Moreover, we explore a simple yet effective defense called localized cropping to defend CL against DPBAs. Our results show that localized cropping can reduce the attack success rates, but it sacrifices the utility of the encoder, highlighting the need for new defense. Table 4: Defense results. † indicates an extra clean pre-training dataset is used. \| Defense \| No Attack \| CorruptEncoder \| CorruptEncoder+ \| \|------------------\|-----------\|----------------\|----------------\| \| \| CA \| ASR \| BA \| ASR \| BA \| ASR \| \| No Defense \| 60.8 \| 0.4 \| 61.2 \| 89.9 \| 61.7 \| 97.8 \| \| ContrastiveCrop \| 61.3 \| 0.4 \| 62.1 \| 90.8 \| 62 \| 98.5 \| \| No Random Cropping \| 32.4 \| 2.2 \| 31.1 \| 2 \| 31.9 \| 1.5 \| \| CompRess (5%)† \| 49.5 \| 0.9 \| 49.4 \| 1.1 \| 49.9 \| 0.9 \| \| CompRess (20%)† \| 58.2 \| 0.9 \| 58.7 \| 1 \| 58.6 \| 1.1 \| \| Localized Cropping \| 56.2 \| 0.9 \| 56.3 \| 0.9 \| 56.1 \| 0.8 \| REFERENCES Nicholas Carlini and Andreas Terzis. Poisoning and backdooring contrastive learning. In International Conference on Learning Representations, 2022. Mathilde Caron, Ishan Misra, Julien Mairal, Priya Goyal, Piotr Bojanowski, and Armand Joulin. Unsupervised learning of visual features by contrasting cluster assignments. 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In Proceedings of the IEEE/CVF International Conference on Computer Vision, 2021b. Hongbin Liu, Jinyuan Jia, and Neil Zhenqiang Gong. PoisonedEncoder: Poisoning the unlabeled pre-training data in contrastive learning. In 31st USENIX Security Symposium (USENIX Security 22), 2022.
wD8L86iCvD	In Table 2 and 3, some results are missing for the Video-LLAMA and are replaced with “-”. Why? As far as I understand, Video-LLAMA could be applied on all the tasks that the proposed method can be applied to by just removing the missing modality. One can just remove the audio or the visual branch in the Video-LLAMA and turn it into a uni-modal model. One would not even need to re-train it as the LLM is not trained at all, so it does not “care” if there are 2 modalities as input or only one. Please, correct me if I am wrong.	FINE-GRAINED AUDIO-VISUAL JOINT REPRESENTATIONS FOR MULTIMODAL LARGE LANGUAGE MODELS Anonymous authors Paper under double-blind review ABSTRACT Audio-visual large language models (LLM) have drawn significant attention, yet the fine-grained combination of both input streams is rather under-explored, which is challenging but necessary for LLMs to understand general video inputs. To this end, a fine-grained audio-visual joint representation (FAVOR) learning framework for multimodal LLMs is proposed in this paper, which extends a text-based LLM to simultaneously perceive speech and audio events in the audio input stream and images or videos in the visual input stream, at the frame level. To fuse the audio and visual feature streams into joint representations and to align the joint space with the LLM input embedding space, we propose a causal Q-Former structure with a causal attention module to enhance the capture of causal relations of the audio-visual frames across time. An audio-visual evaluation benchmark (AVEB) is also introduced, which comprises six representative single-modal tasks with five cross-modal tasks reflecting audio-visual co-reasoning abilities. While achieving competitive single-modal performance on audio, speech and image tasks in AVEB, FAVOR achieved over 20% accuracy improvements on the video question-answering task when fine-grained information or temporal causal reasoning is required. FAVOR, in addition, demonstrated remarkable video comprehension and reasoning abilities on tasks that are unprecedented by other multimodal LLMs. An interactive demo of FAVOR is available at https://github.com/BriansIDP/AudioVisualLLM.git, and the training code and model checkpoints will be released soon. 1 INTRODUCTION Text-based large language models (LLM) (Brown et al., 2020; Touvron et al., 2023; Chiang et al., 2023; Anil et al., 2023; Du et al., 2022) have demonstrated remarkable performance in various natural language processing tasks, especially achieving human-level capabilities in reasoning and comprehension (OpenAI, 2023). Meanwhile, instruction fine-tuning (Chung et al., 2022; Ouyang et al., 2022; Peng et al., 2023), where data is organised as pairs of user instruction (or prompt) and reference response, has emerged as a training paradigm that enables LLMs to perform various tasks by following open-ended natural language instructions from non-expert users. Recently, there has been a burgeoning research interest in equipping LLMs with visual and auditory perception abilities. While most recent studies have been focusing on incorporating a single specific type of input, such as image (Li et al., 2023a; Alayrac et al., 2022; Dai et al., 2023), video (Maaz et al., 2023; Chen et al., 2023b; Zhao et al., 2022; Zeng et al., 2023), audio (Gong et al., 2023) or speech (Zhang et al., 2023a; Rubenstein et al., 2023) separately. These investigations often employ a trained modality alignment module that aligns the representation space of the input modality with the text one. Subsequently, work has started looking at incorporating multiple simultaneous input modalities (Su et al., 2023; Zhang et al., 2023b; Lyu et al., 2023; Zhao et al., 2023; Chen et al., 2023a). Despite the sequential nature of video and audio inputs, most aforementioned work treated video as a sampled subset of individual images and audio as a fixed-length spectrogram. As a result, these models tend to ignore information and causal relations when the input sequence length increases. Moreover, speech, as a crucial aspect of auditory input in videos that in particular relies on fine-grained information extraction, is considerably under-explored in multimodal LLM research. To this end, this paper proposes FAVOR, a fine-grained audio-visual joint representation learning framework for LLM-based multimodal understanding and reasoning with audio-visual input sequences consisting of images, audio events, speech, and video. It takes audio-visual sequences at high temporal resolution as inputs and, if paired, temporally synchronises them using a synchronisation module. Such a frame-level fine-grained synchronisation allows a more thorough and fine-grained interaction between audio and visual modalities across time, which is particularly beneficial for videos with speech. Since the input sequences have variable lengths, FAVOR divides the sequence into a number of fixed-length sliding windows and aligns the synchronised sequence within each window to the LLM input text representation space. In order to capture the causal relations among consecutive video frames within a window, a causal Q-Former structure is proposed that introduces a causal attention module to Q-Former (Li et al., 2023a). FAVOR is comprehensively evaluated using an audio-visual evaluation benchmark (AVEB) introduced in this paper, which integrates 11 tasks including 6 different types of open-source tasks with single-modal inputs, as well as 5 cross-modal inference tasks. While achieving competitive performance on single-modal tasks, FAVOR also achieved large performance improvements on cross-modal tasks compared to single-modal models, e.g. over 10% absolute accuracy improvement on audio-visual sound source detection. Notably, benefiting from the fine-grained nature, FAVOR achieved a remarkably 25% accuracy improvement in video QA tasks compared to the strong InstructBLIP baseline. The main contribution of this paper can be summarised as follows: • This paper proposes the FAVOR learning framework for multimodal LLMs. To the best of our knowledge, FAVOR is the first approach that is capable of performing cross-modal cognitive tasks involving audio, speech, image and video inputs with high temporal resolution. • This paper proposes the causal Q-Former structure which comprises a causal encoder module. A novel diversity loss is also proposed to encourage diverse joint representations to be learned. Further with a novel diversity training loss, causal Q-Former is capable of handling audio-visual sequence input efficiently with a small number of training examples. • This paper introduces the AVEB benchmark comprising single-modal and cross-modal tasks to quantitatively evaluate the performance of audio-visual LLMs. 2 RELATED WORK Our work is based on the Q-Former structure to fuse the audio and visual modalities and to align with the text representation space (Li et al., 2023a; Dai et al., 2023). While Q-Former has been primarily proposed for visual information extraction, it also performs remarkably in extracting auditory features for automatic speech recognition (ASR) (Yu et al., 2023). In addition to Q-Former, various types of modality aligners have been studied, such as the cross-attention mechanism (Alayrac et al., 2022), pre-trained multimodal embeddings, (Girdhar et al., 2023) and temporal and spatial pooling (Maaz et al., 2023). Different from standard Q-Former approaches, our causal Q-Former used in the FAVOR framework pays particular attention to the sequential nature of the input feature streams with the model structure and training methods dedicated to audio-visual understanding. The work most closely related to ours is Video-LLaMA (Zhang et al., 2023b), Macaw-LLM (Lyu et al., 2023) and X-LLM (Chen et al., 2023a), as all of them used LLMs for cross-modal understanding based on general non-silent video inputs (referred to as audio-visual sequence in this paper). X-LLM supports video and Chinese speech inputs, but cannot understand audio events and music. Video-LLaMA employs an additional video Q-Former to encode features of several equally-spaced frames extracted using a BLIP2 (Li et al., 2023a) image encoder. Macaw-LLM adopted a similar approach and used three separate encoders for image, video and non-speech audio events. Both Video-LLaMA and Macaw-LLM consider only non-speech audio events, and the audio encoders in the two models are the ImageBind (Girdhar et al., 2023) and Whisper (Radford et al., 2023) model encoders respectively. While both methods involve the fusion of audio and visual feature streams, the two streams are sparsely pooled and processed rather independently, which removes fine-grained audio-visual interactions at each time step. Compared to Video-LLaMA and Macaw-LLM, FAVOR reserves fine-grained modality interactions and can understand speech inputs that are common in general non-silent videos. This leads to an emphasis on causal modality synchronisation across time and allows more content-based cross-modal interactions. 3 METHODOLOGY In this section, we present the proposed FAVOR learning framework, which is designed to handle audio and visual input sequences synchronously at high temporal resolution for LLMs. This section Figure 1: The fine-grained audio-visual joint representation (FAVOR) learning framework for multi-modal LLMs. The temporal synchronisation module does not contain trainable parameters, and the audio and visual feature encoders are not updated during training. introduces the model structure, including the causal attention module and an optional diversity loss. 3.1 MODEL ARCHITECTURE The structure of FAVOR is shown in Fig. 1. Key components that realise the fine-grained audio-visual representation learning are the temporal synchronisation module and the causal Q-Former. First, visual and audio inputs are encoded using the corresponding pre-trained encoders. The visual encoder in FAVOR converts the input image into a certain number of vectors via the image encoder in InstructBLIP (Li et al., 2023a). When video input is given, the visual encoder encodes each video frame separately as a sequence of images at a 2 Hz frame rate, and the output image features are concatenated along the temporal dimension to form a sequence of visual frames. The audio encoder used is the Whisper ASR model encoder (Radford et al., 2023) that converts the input speech and audio events into a sequence of vectors at a 50 Hz frame rate. When both audio and visual inputs are present, the two encoded feature sequences are sent to the temporal synchronisation module to obtain the time-synchronised feature sequences, as shown in Fig. 1. Since video is sampled at a lower frame rate than audio, the audio and visual frames are synchronised at each video frame (i.e. every 0.5 seconds), with zero padding to make both sequences have equal lengths. Note that higher frequencies of visual frames are also supported in the FAVOR framework which requires higher computation and storage costs. The synchronised audio frame $h^A_t$ and visual frame $h^V_t$ are then concatenated along the feature dimension to obtain the combined audio-visual feature frame $h^{AV}_t$. That is, $$h^{AV}_t = \text{Concat}(h^A_t, h^V_t),$$ where Concat($\cdot$) represents the concatenation along the feature dimension. Note that in cases when only one input modality is present, the other modality is filled with a sequence of zero padding of the same sequence length. While an image alone is treated as a single frame, when paired audio input exists, such as images with spoken captions (Hsu et al., 2020), each image is duplicated as if it were a video input with a matched length to the audio input. In order to handle variable-length inputs, the combined feature sequences are first divided into fixed-length windows spanning, e.g. every 5 or 10 seconds. Then, a causal Q-Former based on the same $N$ trainable input query tokens $q_1, \ldots, q_N$ is applied to convert each sliding window and generate $N$ output query vectors carrying the audio-visual information. As shown in Eqn. (2), $$h_{w,1}^Q, \ldots, h_{w,N}^Q = Q\text{-Former}_{\text{causal}}(h_{t}^{AV}, \ldots, h_{t+k}^{AV}; q_1, \ldots, q_N),$$ where $w$ is the window index and $k$ is the number of video frames in that window, and $Q\text{-Former}_{\text{causal}}(\cdot)$ denotes the causal Q-Former computation described in detail later in Section 3.2. The output query representations, $h_{w,1}^Q, \ldots, h_{w,N}^Q$, are projected to the LLM input dimension before sending to the LLM. Therefore, if the input sequence length of causal Q-Former is $T$, the number of sliding windows $W$ becomes $\lceil T/k \rceil$, and the overall output sequence length from causal Q-Former will be $W \times N$. Through end-to-end training, the output audio-visual representations of causal Q-Former are trained to align with the LLM input token space. Therefore, the use of sliding windows enables the LLM input token sequence length $W \times N$ to vary based on $T$ and can achieve a good trade-off between the degree of information reserved and the computation and storage costs. Finally, the instruction prompt, such as questions or task descriptions will be appended to the concatenated output queries of all windows to form the input to the LLM. The response sequence $\hat{Y}$ can be generated as follows: $$\hat{Y} = \arg\max_{Y} P(Y\|h_{1,1}^Q, \ldots, h_{1,N}^Q, \ldots, h_{W,1}^Q, \ldots, h_{W,N}^Q, c_1, \ldots, c_M),$$ where $c_1, c_2, \ldots, c_M$ are the contents of the prompt. ### 3.2 Q-Former with Causal Self-Attention The proposed causal Q-Former structure is shown in Fig. 2. To capture the causal temporal correlation among frames that are extracted independently, an additional causal self-attention module is added to the standard Q-Former structure, indicated by the red block in Fig. 2. ![Figure 2](image) With the causal attention module, the encoding of one specific frame also includes the information of all previous frames carried in an auto-regressive way. This is particularly beneficial for causal reasoning questions, such as the “what happens next” questions (Xiao et al., 2021). Such questions are sometimes difficult to learn using only the positional embeddings. ### 3.3 System Training and Diversity Loss The training data of video tasks, such as video question-answering (QA), usually only requires one or two keyframes, and the output queries tend to repeatedly capture the same information. Therefore, a novel diversity loss is proposed to encourage the causal Q-Former to extract more diverse aspects of the input sequence. Specifically, the diversity loss is formulated as: $$L_{\text{diverse}} = \sum_{w=1}^{W} \sum_{i=1}^{N} \sum_{j=1, j \neq i}^{N} \text{sim}(h_{w,i}^Q, h_{w,j}^Q),$$ where $W$ and $N$ are the total number of windows and the number of output queries of each window respectively, and $\text{sim}(\cdot)$ is the cosine similarity between two vectors. Cosine similarity is adopted since it is widely used for semantic similarity measurements, and in FAVOR, the output queries are aligned with a semantic space of the LLM input token representations. This choice is also supported by the fact that the modulus of the output query tokens is very similar due to the layer normalisation operation of the causal Q-Former. By encouraging the audio-visual frames to be orthogonal to each other, the diversity loss forces the output query representations to be more spread in the text representation space. Overall, the system is trained in an end-to-end fashion using the cross-entropy (CE) loss and the diversity loss, as shown below: $$L = L_{\text{CE}} + \lambda L_{\text{diverse}},$$ where $\lambda$ is the factor controlling the importance of the diversity loss, and the CE loss is calculated using the reference answer as the target. 4 EXPERIMENTAL SETUP 4.1 AUDIO-VISUAL EVALUATION BENCHMARK (AVEB) In this paper, we propose the AVEB benchmark for audio-visual LLM evaluation, which evaluates single-modal perception ability via selected representative tasks while particularly focusing on multi-modal inference. AVEB contains 6 single-modal tasks, including automatic speech recognition (ASR) (Panayotov et al., 2015), audio captioning (AC) (Kim et al., 2019), image captioning (IC) (Young et al., 2014), optical character recognition (OCR) (Singh et al., 2019), visual question answer (VQA) (Hudson & Manning, 2019), and video question answer (Video QA) (Xu et al., 2017), together with 5 audio-visual tasks including audio-visual speech recognition (AVSR) (Sanabria et al., 2018), audio-visual scene-aware dialogue (AVSD) (Alamri et al., 2019), image spoken question answering (ISQA), audio-visual matching (AVM) (Hsu et al., 2020) and audio-visual sound source detection (AVSSD) (Chen et al., 2020; Zhao et al., 2023). Related datasets are indicated in the citations. More details about the test datasets can be found in Appendix A. In addition, we incorporate two widely used audio-visual benchmarks, the fine-grained audible video description (FAVD) (Shen et al., 2023) and the Vision-Audio-Language Omni-peReception (VALOR) (Chen et al., 2023c;d) benchmarks in our evaluation. Evaluation details can be found in Appendix B. Table 1: AVEB details, including the number of samples used for evaluation and metrics reported. Since TextVQA, GQA, NExT-QA, AVSD and VGGSS test sets are large, randomly sampled subsets with enough samples for statistical significance were used in AVEB for efficient evaluation. The audio video matching part of AVM is zero-shot. \| Task \| Test set \| Num. of samples \| Metrics \| Zero-shot \| \|----------\|---------------------------------\|-----------------\|---------------\|-----------\| \| ASR \| LibriSpeech test-clean \| 2620 utterances \| WER \| No \| \| AC \| AudioCaps test \| 938 audio clips \| SPIDEr \| No \| \| IC \| Flickr30k test \| 1000 images \| CIDEr / METEOR\| Yes \| \| OCR \| TextVQA test \| 1000 images \| Accuracy \| Yes \| \| VQA \| GQA testdev balanced \| 1000 images \| Accuracy \| Yes \| \| Video QA \| NExT-QA test \| 1000 clips \| Accuracy \| No \| \| AVSR \| How2 dev5 \| 500 clips \| WER \| No \| \| AVSD \| AVSD val \| 200 clips 2000 turns \| Accuracy \| No \| \| ISQA \| TextVQA + GQA \| 2000 images \| Accuracy \| Yes \| \| AVSSD \| VGGSS \| 850 video clips\| Accuracy \| Yes \| \| AVM \| SpokenCOCO val2014 + VGGSS \| 1000 pairs 500 each \| Accuracy \| Yes \| \| FAVD \| FAVDBench \| 1k videos \| BLEU / METEOR \| Yes \| \| VALOR \| VALOR 32k \| 3k videos \| CIDEr / METEOR\| Yes \| ASR and AC are evaluated using word error rate (WER) and SPIDEr (Liu et al., 2017), a combination of SPICE and CIDEr respectively. The evaluation of IC uses CIDEr following (Dai et al., 2023), and METEOR, as LLMs tend to use a diverse range of words with similar meanings. OCR, VQA and Video QA are measured using top-1 accuracy. For OCR, the scoring follows (Singh et al., 2019) where each hit in the reference answer contributes 1/3 to the total hit. For VQA and Video QA, it is counted as correct if the reference answer exactly exists in the generated answer using a word-by-word matching. In particular, during inference only, Video QA is formulated as an in-context multiple-choice task where the choices are given in the prompt, and one hit is counted only when the generated answer exactly matches the reference. The same measurement is taken for ISQA and AVM. Furthermore, for AVSD and AVSSD, as the reference answer is a full sentence, ChatGPT-assisted scoring is used to determine whether the generated answer is equivalent to the reference answer (see the prompt design in Appendix C). While all other tasks already exist with open-source test sets, this paper particularly proposes ISQA and AVM tasks where audio-visual interaction is necessary. ISQA is the task where the question is in the audio and the answer can be found in the image. This test set is derived from the data used for OCR and VQA, where the questions are synthesised using a commercial text-to-speech synthesis system with a diverse range of speakers and styles. The text prompt is always “answer the question in the audio about the image”, while the LLM is required to first understand the question in the speech, and then answer it by looking at the image. AVM is the task of determining whether the given spoken description in the SpokenCOCO dataset (Hsu et al., 2020) matches the image, or whether the given audio clip is compatible with the given video chosen from the VGGSS dataset (Chen et al., 2020). AVSSD is another task that requires a strong binding of audio and visual modalities, as a single modality usually only provides partial information about the sound. 4.2 Model Configurations To validate the FAVOR learning framework, the Vicuna (Chiang et al., 2023) models (including 7B and 13B models, and 13B is the default option if not specified) are used as the LLM, Whisper (Radford et al., 2023) large-v2 encoder as the audio encoder and InstructBLIP (Dai et al., 2023) vision Transformer (ViT) plus Q-Former as the visual encoder. The visual encoder outputs 32 feature vectors for each video frame (every 0.5 seconds), and the audio encoder outputs 50 feature vectors per second. The causal Q-Former has two Transformer blocks with 768-dim hidden states. The output query representations are projected to 5120-dim before being sent to the LLM. The LLM is adapted using the low-rank adaptation (LoRA) (Hu et al., 2022) method with a rank of 32. Only the parameters of the attention query, key and value projections and feed-forward network weights are updated, which comprised 0.4% of the total number of LLM parameters. Whisper and InstructBLIP are used as the single-modality baseline systems for comparison. As FAVOR adopted video data with different styles and focuses, to eliminate the discrepancy in training data and achieve fair comparisons, InstructBLIP is further fine-tuned on the same image and video training data as FAVOR. For each video clip, five equally-spaced frames were used resulting in 160 output queries. This is the same as the number of output queries used for 25-second videos in FAVOR. Video-LLaMA (Zhang et al., 2023b) was used as the multimodal baseline where only Vicuna-7B checkpoint was released for audio-visual input$^1$. The VALOR-base model (Chen et al., 2023c) is used as the performance reference for the VALOR benchmark, as the total number of video samples to train FAVOR is only 1M. Note that VALOR-base is a BERT-based model and fine-tuned only on captioning tasks, which makes it not directly comparable to other multi-modal LLMs. 4.3 Training Data and Specifications FAVOR directly uses multi-task instruction fine-tuning to train the model parameters of causal Q-Former and LoRA. Training data contains both single-modal and audio-visual paired data. For audio-only tasks, LibriSpeech train-clean-100 and train-clean-360 sets are used for ASR, and AudioCaps are used for AC. For visual-only tasks, a mixture of LLAVA-150k (Liu et al., 2023) image QA data, OCRVQA OCR data (Mishra et al., 2019), TextCaps Sidorov et al. (2020) image caption data, NExT-QA video QA training data (Xiao et al., 2021), 5000 samples from COCO train2014 data with spoken captions (Lin et al., 2014) as well as 11k samples from VideoChat (Li et al., 2023b) are used. For audio-visual tasks, randomly selected 600-hour Ego4D video captioning data (Grauman et al., 2022), how2 300-hour training set AVSR data and AVSD training set are used. In order to further stimulate modality interactions during training, 5,000 images with spoken captions are used in the training set for the AVM task. Note that the entire training data only contains 1M samples with fewer than 300k video samples, and only contains publicly available datasets. Details about the training data can be found in Appendix A. Furthermore, besides being trained using video and audio from the same source, FAVOR also uses randomly paired audio and video in training. This novel training approach increases versatility and achieves a better balance between the audio and visual modalities. It further enables FAVOR to perform audio-visual co-reasoning tasks as shown in the AVEB benchmark, including ISQA and AVM. Moreover, we use a tiny storytelling set to further encourage a thorough mixture. $^1$https://github.com/DAMO-NLP-SG/Video-LLaMA.git. Table 2: AVEB single-modal task results. If specified, InstructBLIP is fine-tuned on the training data of FAVOR (“InstructBLIP fine-tuned”). IC is reported in CIDEr/METEOR. When using audio-only and visual-only inputs, the other modality is masked during training and inference. Tasks unable to be performed are marked with “-”. \| Systems \| ASR ↓ \| AC ↑ \| Video QA ↑ \| IC ↑ \| OCR ↑ \| VQA ↑ \| \|-------------------------------\|-------\|------\|------------\|------\|-------\|-------\| \| Whisper large-v2 \| 2.9% \| - \| - \| - \| - \| - \| \| InstructBLIP 13B \| - \| - \| 21.0% \| 84.5 / 26.0 \| 36.5% \| 48.9% \| \| InstructBLIP 13B fine-tuned \| - \| - \| 24.7% \| 78.9 / 26.1 \| 36.7% \| 45.6% \| \| Video-LLaMA 7B \| - \| - \| 22.5% \| 22.0 / 16.6 \| 16.4% \| 15.1% \| \| FAVOR 13B (ours, audio-only) \| 2.7% \| 39.7 \| - \| - \| - \| - \| \| FAVOR 13B (ours, visual-only) \| - \| - \| 44.8% \| 74.0 / 26.5 \| 34.2% \| 45.6% \| \| FAVOR 7B (ours, audio-visual) \| 4.1% \| 39.1 \| 42.5% \| 78.1 / 26.3 \| 34.6% \| 45.3% \| \| FAVOR 13B (ours, audio-visual)\| 3.3% \| 42.6 \| 49.3% \| 86.0 / 27.5 \| 37.8% \| 45.2% \| Table 3: AVEB audio-visual task results. If specified, InstructBLIP is fine-tuned on the training data of FAVOR (“InstructBLIP†”). The other modality is masked in both training and testing when using audio-only and visual-only inputs. Tasks unable to be performed are marked with “-”. \| Systems \| AVSR ↓ \| AVSD ↑ \| ISQA ↑ \| AVSSD ↑ \| AVM ↑ \| \|-------------------------------\|--------\|--------\|--------\|---------\|-------\| \| Whisper large-v2 \| 8.3% \| - \| - \| - \| - \| \| InstructBLIP 13B \| - \| 41.4% \| - \| 1.1% \| - \| \| InstructBLIP† 13B \| - \| 52.1% \| - \| 20.3% \| - \| \| Video-LLaMA 7B \| - \| 27.6% \| - \| 41.9% \| 52.3% \| \| FAVOR 13B (ours, audio-only) \| 8.3% \| - \| - \| 34.7% \| - \| \| FAVOR 13B (ours, visual-only) \| - \| 53.3% \| - \| 23.5% \| - \| \| FAVOR 7B (ours, audio-visual) \| 8.7% \| 51.2% \| 24.5% \| 50.5% \| 74.3% \| \| FAVOR 13B (ours, audio-visual)\| 8.1% \| 54.5% \| 32.3% \| 51.1% \| 77.1% \| of audio-visual descriptions for better demonstration quality only. In addition to all the training datasets mentioned above, in order to explicitly encourage the model to generically combine both modalities, a storytelling fine-tuning set is designed. The dataset is gathered by prompting GPT-3.5 with reference audio caption or transcription, together with video caption, and asking GPT-3.5 to generate a coherent story combining both information (see details in Appendix D). The model is fine-tuned on this data for only 100 steps with a very small learning rate without causing any loss in the benchmark performance. It is worth noting that in order to compare FAVOR with the original InstructBLIP on image tasks directly, Flickr30k for IC, TextVQA for OCR and GQA for VQA in the benchmark are not included in the training, and hence the model performed zero-shot learning on them. Moreover, since ISQA uses synthesised speech, this is also not a trained task and the model performed zero-shot learning. 5 EXPERIMENTAL RESULTS 5.1 MAIN RESULTS The main results of using FAVOR on AVEB tasks are summarised in Table 2 and Table 3 for single-modal and audio-visual tasks respectively. While other models can only perform a subset of AVEB tasks, FAVOR is the first single model that achieves competitive performance on all tasks compared to the single-modal counterparts, with remarkably better performance on audio-visual tasks. In particular, as the first work that integrates audio, speech, image and video modality into LLMs, FAVOR effectively achieves audio-visual co-reasoning which is reflected by the performance on ISQA, AVSSD and AVM tasks. On audio-based tasks in Table 2, FAVOR obtains a similar WER compared to Whisper large-v2 and mixed results compared to the audio-only FAVOR. Further, with the aid of visual information, FAVOR achieves a lower WER on AVSR than both models in Table 3. On visual tasks, FAVOR demonstrates the best results on IC, OCR and Video QA, and on-par results on VQA with Instruct-BLIP fine-tuned on the same training set. In particular, the fine-grained causal modelling of video in FAVOR yields over 20% improvements compared to InstructBLIP even though the latter is fine-tuned on the same set of video data. Table 4: Results on FAVDBench (BLEU1/BLEU4/METEOR) and VALOR (METEOR/CIDEr) tasks. Our fine-tuning on VALOR is performed only on 10% of the VALOR training data. \| Systems \| FAVD ↑ \| FAVD fine-tuned ↑ \| VALOR ↑ \| VALOR fine-tuned ↑ \| \|-----------------------\|--------\|-------------------\|---------\|--------------------\| \| VALOR-base† \| - \| - \| - \| 14.8 / 55.7 \| \| Video-LLaMA 7B \| 20.8 / 2.4 / 15.0 \| 39.4 / 6.5 / 16.5 \| 10.7 / 1.2 \| 10.9 / 21.3 \| \| FAVOR 7B (ours) \| 24.9 / 2.8 / 14.8 \| 42.6 / 9.9 / 18.3 \| 8.6 / 13.4 \| 13.9 / 42.6 \| \| FAVOR 13B (ours) \| 28.2 / 3.0 / 15.2 \| 44.2 / 10.9 / 19.1 \| 8.8 / 15.3 \| 14.2 / 46.9 \| Table 5: Ablation studies on the core components of FAVOR based on video and audio-visual tasks. Each row represents removing one component with other parts remaining the same. Note the last row is equivalent to Video-LLaMA with high temporal resolution, speech encoder and LoRA, and the comparison to complete FAVOR directly reflected the benefit of the proposed structure design. \| Systems \| Video QA \| AVSR \| AVSD \| ISQA \| AVSSD \| AVM \| \|----------------------------------------------\|----------\|------\|------\|------\|-------\|-----\| \| Complete FAVOR \| 49.3% \| 8.1% \| 54.5% \| 32.3% \| 51.1% \| 77.1% \| \| FAVOR without causal encoder \| 42.8% \| 8.0% \| 54.1% \| 20.9% \| 37.1% \| 74.8% \| \| FAVOR without sliding window \| 44.8% \| 8.5% \| 53.6% \| 29.7% \| 45.3% \| 74.5% \| \| FAVOR without synchronisation \| 47.4% \| 8.4% \| 53.4% \| 17.2% \| 50.5% \| 72.5% \| \| FAVOR without causal encoder, diversity loss, and synchronisation \| 41.8% \| 8.9% \| 50.5% \| 16.7% \| 38.6% \| 72.0% \| On the audio-visual tasks in Table 3, while outperforming all the baseline systems in every task, FAVOR demonstrated a strong audio-visual co-reasoning ability based on the audio-visual matching (AVM) dataset results and is the only system to our knowledge that can perform speech-image co-reasoning based on image-spoken QA (ISQA). Audio-visual co-reasoning (including speech-image co-reasoning) is an important yet challenging ability which requires the model to comprehend the visual content as well as both speech and non-speech sounds in the audio, and to capture the correlation between what it “hears” and “sees”. Such tasks were almost infeasible for any other audio-visual models so far, since they were unable to understand both speech and non-speech sounds and did not model the audio-visual correlations in fine-grain. Various audio-visual emergent abilities in addition to the audio-visual co-reasoning ability, as discussed in Section 5.5. Results on FAVD and VALOR test data (Table 4) also demonstrated the superiority of FAVOR over Video-LLaMA. In the zero-shot case, Video-LLaMA tends to generate long paragraphs of text even under the instruction of generating short sentence responses. This resulted in extremely low CIDEr scores compared to FAVOR which closely follows the instruction and generate concise responses. Notably, the best FAVOR model achieves better performance on FAVD than the best value reported in (Shen et al., 2023). Although FAVOR uses only 10% of the VALOR training data for fine-tuning, it achieves competitive performance on the VALOR test data. 5.2 ABLATION STUDIES Detailed ablation studies are performed for each proposed component in FAVOR as shown in Table 9 for single-modal tasks and Table 10 for multimodal tasks in Appendix E. This section particularly focuses on the use of causal Q-Former and audio-visual synchronisation on video and audio-visual tasks, as summarised in Table 5. First, the effect of the causal attention module is most clearly reflected by the performance on video QA, ISQA and AVSSD, as it both boosted the temporal causality modelling as well as provided a better audio-visual fusion before applying the cross-attention in the Q-Former. Second, the fine-grained model design, including sliding windows and frame-level synchronisation, is crucial to achieving good results on the AVSR task speech input as shown in the AVSR results. Without the sliding window, a fixed number of output queries are used no matter how long the audio is, which results in more deletion errors. Besides, using sliding windows also benefits the video QA task as they encourage the localised causal relationship to be captured. Furthermore, the use of synchronisation is crucial for audio-visual co-reasoning to work as supported by the ISQA and AVM results. Without synchronisation, modality alignment is done rather independently and the correlation between audio and video is only modelled among high-level features. Figure 3: Influence of the window sizes and the frames per second (FPS) to the model performance on speech and video tasks. (a) and (b): results by training and evaluating using different window sizes $k$ on 10% of data. (c): the influence of FPS using the best model on full data. Figure 4: Variations of model performance due to the diversity loss factor, i.e. $\lambda$ in Eqn. (4), on (a) AVSR measured in %WER, (b) Video QA measured in %Accuracy and (c) AVSSD measured in %Accuracy. Variations of average cosine similarities are also shown under different $\lambda$’s. that are aligned in the text space. This may easily omit information about the concurrency of audio and visual contents, such as how a specific part of speech relates to a specific visual scene. On the other hand, synchronisation enables a temporally aligned cross-modal interaction which allows such concurrency to be captured, resulting in enhanced performances on audio-visual tasks. 5.3 Analysis on the Sliding Windows and Temporal Resolution As mentioned in Section 3.1, the trade-off between the sliding window size and the model performance is shown in Figure 3. Specifically, (a) and (b) show the influence of the numbers of frames $k$ in a window while keeping the ratio $N/k$ a constant (i.e. keeping the total output queries $W \times N$ unchanged) and the same frame rate. This is trained on 10% of the full training data for quick experiments. Although using shorter windows benefits ASR, as fewer output tokens are used to encapsulate all the visual information within that window, performance on video QA is degraded. On the other hand, larger windows heavily reduce the ASR performance as the monotonic alignment in ASR is especially difficult to learn with 10% of the training data. Figure 3 (c) clearly shows the importance of high temporal resolution in video modelling. The lowest FPS is equivalent to 8 frames per video, e.g. Video-LLaMA, where over 24% relative accuracy improvements are achieved using an FPS of 2. Figure 3 (c) shows the influence of the number of frames per second (FPS) on the model performance during inference. The best model trained on the full set is used with the same number of frames per window. While low accuracy is observed when the frame rate is low, increasing FPS beyond 1.0 only receives marginal improvements at the cost of having many more output queries sent to the LLM. 2.0 FPS was chosen as it made the audio and visual sequences have the most similar lengths, and hence easier for synchronisation. 5.4 Analysis of the Diversity Loss Analysis of the effect of diversity loss is also performed using 10% of the training data as shown in Figure 4, and examples of cosine similarity matrices among output queries are shown in Appendix F. For ASR, the model is trained to include all the speech information in the audio sequence and the cosine similarity varies according to the length of the speech. For videos, the cosine similarity is close and does not vary too much for different video lengths, and hence diversity loss effectively acts as a way to encourage more diversified information to be captured. However, when a high λ is employed, diverse information causes confusion in the model and results in a more severe hallucination problem (e.g. high insertion rate in WER) with heavily degraded model performance. 5.5 Discussions on Incorporating Speech and Speech-Video Interactions Speech is an important source of information for video that should always be considered for audio-visual LLM to perform a comprehensive understanding. Unlike audio events, the speech content can hardly be inferred from the visual modality, making it particularly indispensable to comprehend any videos involving people talking. Moreover, the co-occurrence of speech and video events, which is modelled by the fine-grained temporal synchronisation in FAVOR, is required to understand the audio-visual temporal relations, e.g. “What did A say” (more examples in Appendix G). One of the major contributions of FAVOR is to incorporate speech in a multimodal LLM and effectively combine both speech and video content to generate responses. In addition to the ISQA and AVM tasks that have already reflected the co-reasoning ability, the advantage of FAVOR can be more clearly demonstrated by the emergent abilities (shown in Appendix G). For instance, in response to questions about why a movie clip is funny or romantic, FAVOR combines the video, dialogue between characters and background audio or music to generate a more encompassing and convincing answer. Besides, FAVOR is able to understand the scene better by using knowledge from the speech, such as the species of a particular fish introduced in a documentary. 6 Conclusion This paper proposed FAVOR, a fine-grained audio-visual joint representation learning framework for multimodal LLMs. On the introduced AVEB benchmark for audio-visual evaluation, FAVOR achieved competitive performance on audio and visual single-modal tasks with a remarkable 20% absolute accuracy improvement on the causal reasoning video QA task compared to the baselines. FAVOR demonstrated audio-visual, and particularly strong speech-visual co-reasoning abilities, with remarkable cross-modal emergent abilities demonstrated via examples. 7 Reproducibility Statement To make the experiments and models reproducible, the benchmark details are provided in the supplementary materials, and a demo page is provided in the abstract for a convenient try-out of the model. The details of the training and test data are summarised in Section 4 and Appendix A. Key hyper-parameter settings were discussed in the result section. The complete training and inference code together with model checkpoints will be released upon acceptance. 8 Ethical Statement The approaches in this paper do not give rise to any additional risks beyond the ones directly inherited from the model checkpoints. The ASR encoder and visual encoder might work worse for people from particular demographics. The framework also inherits the biases from all the large language models used for experiments in this paper. References Huda Alamri, Vincent Cartillier, Abhishek Das, Jue Wang, Anoop Cherian, Irfan Essa, Dhruv Batra, Tim K. Marks, Chiori Hori, Peter Anderson, Stefan Lee, and Devi Parikh. Audio-visual scene-aware dialog. In Proc. CVPR, 2019. Jean-Baptiste Alayrac, Jeff Donahue, Pauline Luc, Antoine Miech, Iain Barr, et al. Flamingo: A visual language model for few-shot learning. In Proc. NeurIPS, 2022. Rohan Anil, Andrew M. Dai, Orhan Firat, Melvin Johnson, Dmitry Lepikhin, et al. PaLM 2 technical report. arXiv:2305.10403, 2023. Tom Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared D Kaplan, et al. Language models are few-shot learners. In Proc. NeurIPS, 2020.
FMMF1a9ifL	In the Experiments section (Section 6), how much of the results are related to the selection of the molecules under study? Stated differently, how do the authors plan to address the generalizability of the approach?	GRADUAL OPTIMIZATION LEARNING FOR CONFORMATIONAL ENERGY MINIMIZATION Artem Tsypin\textsuperscript{1}\textsuperscript{\&}, Leonid Ugadiarov\textsuperscript{2,4}, Kuzma Khrabrov\textsuperscript{1}, Alexander Telepov\textsuperscript{1}, Egor Rumiantsev\textsuperscript{1}, Alexey Skrynnik\textsuperscript{1,2}, Aleksandr Panov\textsuperscript{1,2,4}, Dmitry Petrov\textsuperscript{5}, Elena Tutubalina\textsuperscript{1,3,6}, Artur Kadurin\textsuperscript{1,7}\textsuperscript{\&} \textsuperscript{1}AIRI, Moscow \textsuperscript{2}FRC CSC RAS, Moscow \textsuperscript{3}Sber AI, Moscow \textsuperscript{4}MIPT, Dolgoprudny \textsuperscript{5}Constructor University, Bremen \textsuperscript{6}ISP RAS Research Center for Trusted Artificial Intelligence, Moscow \textsuperscript{7}Kuban State University, Krasnodar \textsuperscript{\&}\{Tsypin, Kadurin\}@airi.net ABSTRACT Molecular conformation optimization is crucial to computer-aided drug discovery and materials design. Traditional energy minimization techniques rely on iterative optimization methods that use molecular forces calculated by a physical simulator (oracle) as anti-gradients. However, this is a computationally expensive approach that requires many interactions with a physical simulator. One way to accelerate this procedure is to replace the physical simulator with a neural network. Despite recent progress in neural networks for molecular conformation energy prediction, such models are prone to errors due to distribution shift, leading to inaccurate energy minimization. We find that the quality of energy minimization with neural networks can be improved by providing optimization trajectories as additional training data. Still, obtaining complete optimization trajectories demands a lot of additional computations. To reduce the required additional data, we present the Gradual Optimization Learning Framework (GOLF) for energy minimization with neural networks. The framework consists of an efficient data-collecting scheme and an external optimizer. The external optimizer utilizes gradients from the energy prediction model to generate optimization trajectories, and the data-collecting scheme selects additional training data to be processed by the physical simulator. Our results demonstrate that the neural network trained with GOLF performs \textit{on par} with the oracle on a benchmark of diverse drug-like molecules using significantly less additional data. 1 INTRODUCTION Numerical quantum chemistry methods are essential for modern computer-aided drug discovery and materials design pipelines. They are used to predict the physical and chemical properties of candidate structures (Matta & Boyd [2007], Oglic et al. [2017], Tielker et al. [2021]). \textit{Ab initio} property prediction framework for a specific molecule or material could be divided into three main steps as follows: (1) find a low-energy conformation of a given atom system, (2) compute its electron structure with quantum chemistry methods, and (3) calculate properties of interest based on the latest. The computational cost of steps (1) and (2) is defined by the specific physical simulator (oracle $\mathcal{O}$) varying from linear to exponential complexity w.r.t the number of atoms or electrons in the system (Sousa et al. [2007]). Overall, the more accurate the oracle is, the more computationally expensive its operations become. \textsuperscript{\&}Corresponding authors. The traditional approach to the problem of obtaining low-energy molecular conformations is to run an iterative optimization process using physical approximations, such as those provided by the Density-functional theory (DFT) methods (Kohn & Sham [1965]), as they are reasonably accurate. However, for large molecules, even a single iteration may take up several hours of CPU-compute (Gilmer et al. [2017]). Therefore, it is crucial to develop alternative approaches (such as Neural Network-based) that reduce the computational complexity of iterative optimization. The recent growth in computational power led to the emergence of molecular databases with computed quantum properties (Ruddigkeit et al. [2012], Ramakrishnan et al. [2014], Isert et al. [2022], Khrabrov et al. [2022], Jain et al. [2013]). For example, nablaDFT (Khrabrov et al. [2022]) consists of more than $5 \times 10^6$ conformations for around $10^6$ drug-like molecules. This data enabled deep learning research for many molecule-related problems, such as conformational potential energy and quantum properties prediction with Neural Network Potentials (NNP) (Chmiela et al. [2017], Schütt et al. [2017], Chmiela et al. [2018, 2020], Schütt et al. [2021], Shuaibi et al. [2021], Gasteiger et al. [2020, 2021], Chmiela et al. [2023]), and conformational distribution estimation (Simm & Hernández-Lobato [2019], Xu et al. [2021], Ganea et al. [2021], Xu et al. [2022], Jing et al. [2022], Shi et al. [2021], Luo et al. [2021]). Naturally, there have been several works that utilize deep learning to tackle the problem of obtaining low-energy conformations. One approach is to reformulate this task as a conditional generation task (Guan et al. [2021], Lu et al. [2023]; see Section 2 for further details). Another solution is to train an NNP to predict the potential energy of a molecular conformation and use it as a force field for relaxation (Unke et al. [2021]). Assuming the NNP accurately predicts the energy, its gradients can be used as interatomic forces (Schütt et al. [2017]). Such a technique allows for gradient-based optimization without a physical simulator, significantly reducing computational complexity. In this work, we aim to improve the training of NNPs for obtaining low-energy conformations. We trained NNPs on the subset of nablaDFT dataset (Khrabrov et al. [2022]) and observed that such models suffer from the distribution shift when used in the optimization task (see Figure 1). To alleviate the distribution shift and improve the quality of energy minimization, we enriched the training dataset with optimization trajectories (see Section 4) generated by the oracle. Our experiments demonstrate that it requires more than $5 \times 10^5$ additional oracle interactions to match the quality of a physical simulator (see Table 1). These models trained on enriched datasets are used as baselines for our proposed approach. In this paper, we propose the GOLF — Gradual Optimization Learning Framework for the training of NNPs to generate low-energy conformations. GOLF consists of three components: (i) a genuine oracle $O_G$, (ii) an optimizer, and (iii) a surrogate oracle $O_S$ that is computationally inexpensive. The $O_G$ is an accurate but computationally expensive method used to calculate ground truth energies and forces, and we consider a setting with a limited budget on $O_G$ interactions. The optimizer (e.g., Adam (Kingma & Ba [2014]) or L-BFGS (Liu & Nocedal [1989]) utilizes NNP gradients to produce optimization trajectories. The $O_S$ determines which conformations are added to the training set. We use Psi4 (Smith et al. [2020]), a popular software for DFT-based computations, as the $O_G$, and RDKit’s (Landrum et al. [2022]) MMFF (Halgren [1996]) as the $O_S$. The NNP training cycle consists of three steps. First, we generate a batch of optimization trajectories and evaluate all conformations with $O_S$. Then we select the first conformation from each trajectory for which the NNP poorly predicts interatomic forces w.r.t. $O_S$ (see Section 5), calculate its ground truth energy and forces with the $O_G$, and add it to the training set. Lastly, we update the NNP by training on batches sampled from initial and collected data. We train the model until we exceed the computational budget for additional $O_G$ interactions. We show (see Section 6.2) that NNPs trained with GOLF on the nablaDFT (Khrabrov et al. [2022]) perform on par with $O_G$ while using 50x less additional data compared to the straightforward approach described in the previous paragraph. We also show similar results on another diverse dataset of drug-like molecules called SPICE (Eastman et al. [2023]). We publish\footnote{https://github.com/AIRI-Institute/GOLF} the source code for GOLF along with optimization trajectories datasets, training, and evaluation scripts. Our contributions can be summarized as follows: - We study the task of conformational optimization and find that NNPs trained on existing datasets are prone to the distribution shift, leading to inaccurate energy minimization. • We propose a straightforward approach to deal with the distribution shift by enriching the training dataset with optimization trajectories (see Figure 1). Our experiments show that additional $5 \times 10^5$ conformations make the NNP perform comparably with the DFT-based oracle $O_G$ on the task of conformational optimization. • We propose a novel framework (GOLF) for data-efficient training of NNPs, which includes a data-collecting scheme along with an external optimizer. We show that models trained with GOLF perform on par with the physical simulator on the task of conformational optimization using 50x less additional data than the straightforward approach. 2 RELATED WORK Conformation generation Several recent papers have proposed different approaches for predicting molecule’s 3D conformers. Xu et al. (2021) utilize normalizing flows to predict pairwise distances between atoms for a given molecular structure with subsequent relaxation of the generated conformation. Ganea et al. (2021) construct the molecular conformation by iteratively assembling it from smaller substructures. Xu et al. (2022); Wu et al. (2022); Jing et al. (2022); Huang et al. (2023); Fan et al. (2023) address the conformational generation task with diffusion models (Sohl-Dickstein et al., 2015). Other works employ variational approximations (Zhu et al., 2022; Swanson et al., 2023), and Markov Random Fields (Wang et al., 2022). We evaluate these approaches in Section 6.1. Despite showing promising geometrical metrics, such as Root-mean-square deviation of atomic positions (RMSD), on the tasks reported in the various papers, these models perform poorly in terms of geometry and potential energy on the optimization task. In most cases, additional optimization with a physical simulator is necessary to get a valid conformation. Geometry optimization Guan et al. (2021); Lu et al. (2023) frame the conformation optimization problem as a conditional generation task and train the model to generate low-energy conformations conditioned on RDKit-generated (or the randomly sampled from the pseudo optimization trajectory) conformations by minimizing the RMSD between the corresponding atom coordinates. As RMSD may not be an ideal objective for the conformation optimization task (see Section 6.1), we focus on accurately predicting the interatomic forces along the optimization trajectories in our work. Additional oracle interactions Zhang et al. (2018) show that additional data from the oracle may increase the energy prediction precision of NNP models. Following this idea, Kalchenko et al. (2023) propose an active learning approach based on the uncertainty of the energy prediction to reduce the number of additional oracle interactions. The main limitation of this approach is that it requires training a separate NNP ensemble for every single molecule. Chan et al. (2019) parametrize the molecule as a set of rotatable bonds and utilize the Bayesian Optimization with Gaussian Process prior to efficiently search for low-energy conformations. However, this method requires using the oracle during the inference, which limits its applications. The OC2022 (Tran et al., 2022) provides relaxation trajectories for catalyst-adsorbate pairs. However, no in-depth analysis of the effects of such additional data on the quality of optimization with NNPs is provided. To sum up, we believe it necessary to explore further the ability of NNPs to optimize molecular conformations according to their energy. Our experiments (see Section 6) show that additional oracle information significantly increases the optimization quality. Since this information may be expensive, we aim to reduce the number of additional interactions while maintaining the quality on par with the oracle. 3 NOTATION AND PRELIMINARIES We define the conformation $s = \{z, X\}$ of the molecule as a pair of atomic numbers $z = \{z_1, \ldots, z_n\}, z_i \in \mathbb{N}$ and atomic coordinates $X = \{x_1, \ldots, x_n\}, x_i \in \mathbb{R}^3$, where $n$ is the number of atoms in the molecule. We define the oracle $O$ as a function that takes conformation $s$ as an input and outputs its potential energy $E_{\text{oracle}}^s \in \mathbb{R}$ and interatomic forces $F_{\text{oracle}}^s \in \mathbb{R}^{n \times 3}$: $E_{\text{oracle}}^s = O(s)$. To denote the ground truth interatomic force acting on the $i$-th atom, we use $F_{\text{oracle}}^{s,i}$. We use different superscripts to denote energies and forces calculated by different physical simulators. For example, we denote the RDKit’s MMFF-calculated energy as $E_{s}^{\text{MMFF}}$ and the Psi4-calculated energy as $E_{s}^{\text{DFT}}$. We denote the NNP for the prediction of the potential energy of the conformation parametrized by weights $\theta$ as $f(s; \theta) : \{z, X\} \rightarrow \mathbb{R}$. Following (Schütt et al., 2017; Schütt et al., 2021), we derive forces from the predicted energies: $$F_i(s; \theta) = -\frac{\partial f(s; \theta)}{\partial x_i},$$ where $F_i \in \mathbb{R}^3$ is the force acting on the $i$-th atom as predicted by the NNP. We follow the standard procedure (Schütt et al., 2017; Schütt et al., 2021; Gasteiger et al., 2020; Musaelian et al., 2022) and train the NNP to minimize the MSE between predicted and ground truth energies and forces: $$L(s, E_{s}^{\text{oracle}}, F_{s}^{\text{oracle}}; \theta) = \rho \\|E_{s}^{\text{oracle}} - f(s; \theta)\\|^2 + \frac{1}{n} \sum_{i=1}^{n} \\|F_{i,s}^{\text{oracle}} - F_i(s; \theta)\\|^2,$$ where $L(s, E_{s}^{\text{oracle}}, F_{s}^{\text{oracle}}; \theta)$ is the loss function for a single conformation $s$, and $\rho$ is the hyperparameter accounting for different scales of energy and forces. To collect the ground truth optimization trajectories (see Section 4), we use the OPTIMIZE method from Psi-4 and run optimization until convergence. Optimizer Opt (L-BFGS, Adam, SGD-momentum) utilizes the forces $F(s; \theta) \in \mathbb{R}^{n \times 3}$ to get NNP-optimization trajectories $s_0, \ldots, s_T$, where $s_0$ is the initial conformation: $$s_{t+1} = s_t + \alpha \text{Opt}(F(s_t; \theta)).$$ Here, $\alpha$ is the optimization rate hyperparameter, and $T$ is the total number of NNP optimization steps. In this work, we use NNPs trained on different data. To train the baseline model $f^{\text{baseline}}(\cdot; \theta)$, we use the fixed subset of nablaDFT $D_0$ (see Appendix D for more details). It consists of approximately 10000 triplets of the form $\{s, E_{s}^{\text{DFT}}, F_{s}^{\text{DFT}}\}$. The $D_0$ can be extended with the ground truth optimization trajectories obtained with Psi-4 to get datasets denoted according to the total number of additional conformations: $D_{\text{traj}-10k}$, $D_{\text{traj}-100k}$, and so on. The resulting NNPs are dubbed $f^{\text{traj}-1k}(\cdot; \theta)$, $f^{\text{traj}-10k}(\cdot; \theta)$, and so on respectively. We call the models trained with GOLF (see Section 5) $f^{\text{GOLF}-1k}(\cdot; \theta)$, $f^{\text{GOLF}-10k}(\cdot; \theta)$, etc. To evaluate the quality of optimization with NNPs, we use a fixed subset of the nablaDFT dataset $D_{\text{test}}$, that shares no molecules with $D_0$. For each conformation $s \in D_{\text{test}}$ we perform the optimization with the $O_G$ to get the ground truth optimal conformation $s_{\text{opt}}$ and its energy $E_{s_{\text{opt}}}^{\text{DFT}}$. The quality of the NNP-optimization for $s_t \in s_0, \ldots, s_T$ is evaluated with the percentage of minimized energy: $$\text{pct}(s_t) = 100\% \times \frac{E_{s_t}^{\text{DFT}} - E_{s_0}^{\text{DFT}}}{E_{s_{\text{opt}}}^{\text{DFT}} - E_{s_0}^{\text{DFT}}}.$$ By aggregating pct($s_t$) over $s \in D_{\text{test}}$, we get the average percentage of minimized energy at step $t$: $$\overline{\text{pct}}_t = \frac{1}{\|D_{\text{test}}\|} \sum_{s \in D_{\text{test}}} \text{pct}(s_t);$$ Another metric is the residual energy in state $s_t$: $E_{\text{res}}(s_t)$. It is calculated as the delta between $E_{s_t}^{\text{DFT}}$ and the optimal energy: $$E_{\text{res}}(s_t) = E_{s_t}^{\text{DFT}} - E_{s_{\text{opt}}}^{\text{DFT}};$$ Similar to $\overline{\text{pct}}_t$, this metric can also be aggregated over the evaluation dataset: $$\overline{E}_{\text{res}}_t = \frac{1}{\|D_{\text{test}}\|} \sum_{s \in D_{\text{test}}} E_{\text{res}}(s_t).$$ Generally accepted chemical precision is 1 kcal/mol (Helgaker et al., 2004). Thus, another important metric is the percentage of conformations for which the residual energy is less than chemical precision. We consider optimizations with such residual energies successful: \[ \text{pct}_{\text{success}} = \frac{1}{\|\mathcal{D}_{\text{test}}\|} \sum_{s \in \mathcal{D}_{\text{test}}} I \left[ E^{\text{res}}(s_T) < 1 \right]. \] (8) 4 CONFORMATION OPTIMIZATION WITH NEURAL NETWORKS Energy prediction models such as SchNet, DimeNet, and PaiNN can achieve near-perfect quality on tasks of energy and interatomic forces prediction when trained on the datasets of molecular conformations (Schütt et al., 2017; Gasteiger et al., 2020; Schütt et al., 2021; Ying et al., 2021; Shuaibi et al., 2021; Gasteiger et al., 2021; Batzner et al., 2022; Musaelian et al., 2022). In theory, the gradients of these models can be utilized by an external optimizer to perform conformational optimization, replacing the computationally expensive physical simulator. However, in our experiments (see Section 6), this scheme often leads to suboptimal performance in terms of the potential energy of the resulting conformations. We attribute this effect to the distribution shift that naturally occurs during the optimization: As most existing datasets (Isert et al., 2022; Khraibov et al., 2022; Eastman et al., 2023; Nakata & Maeda, 2023) do not contain conformations sampled from optimization trajectories, the accuracy of prediction deteriorates as the conformation changes along the optimization process. The lack of such conformations in the training can result in either divergence (initial potential energy is lower than the final potential energy) of the optimization or convergence to a conformation with higher final potential energy than the optimization with the oracle. To alleviate the distribution shift’s effect, we propose enriching the training dataset for NNPs with the ground truth optimization trajectories obtained from the $ O_G $. To illustrate the effectiveness of our approach, we conduct a series of experiments. First, we train a baseline model $ f^{\text{baseline}}(\cdot; \theta) $ on a fixed subset $ \mathcal{D}_0 $ of small molecules from the nablaDFT dataset. The $ \mathcal{D}_0 $ ($ \|\mathcal{D}_0\| \approx 10000 $) contains conformations for 4000 molecules, with sizes ranging from 17 to 35 atoms, and the average size of 32.6. Then we train NNPs $ f^{\text{traj}}(\cdot; \theta) $ on enriched datasets $ \mathcal{D}_{\text{traj}-10k}, \mathcal{D}_{\text{traj}-100k}, \mathcal{D}_{\text{traj}-500k} $, containing approximately $ 10^4, 10^5, 5 \times 10^5 $ additional conformations respectively. The additional data consists of ground truth optimization trajectories obtained from the $ O_G $. Then, we evaluate the NNPs by performing the NNP-optimization on all conformations in $ \mathcal{D}_{\text{test}} $ ($ \|\mathcal{D}_{\text{test}}\| \approx 20000 $, contains $ \approx 10000 $ molecules) and calculating the MSE between ground truth and predicted energies and forces. We use the L-BFGS as Opt due to its superior performance compared to other optimizers (see Appendix B). We run the optimization with an NNP for a fixed number of steps $ T = 100 $ as we observe that this number is sufficient for the optimization to converge (see Figure 3). Table 1: Optimization metrics for NNPs trained on enriched datasets \| NNP \| $f_{\text{baseline}}$ \| $f_{\text{traj-10k}}$ \| $f_{\text{traj-100k}}$ \| $f_{\text{traj-500k}}$ \| \|--------------\|------------------------\|------------------------\|------------------------\|------------------------\| \| pct$_T$(%) ↑ \| 77.9 ± 21.3 \| 95.1 ± 7.6 \| 96.2 ± 8.6 \| 98.8 ± 7.6 \| \| $E_{\text{res}}^T$(kcal/mol) ↓ \| 8.6 \| 2.0 \| 1.5 \| 0.5 \| \| pct$_{\text{success}}$(%) ↑ \| 8.2 \| 37.0 \| 52.7 \| 73.4 \| illustrates the effect of the distribution shift on $f_{\text{baseline}}(\cdot; \theta)$ (the prediction error increases as the optimization progresses) and its gradual alleviation with the addition of new training data. Table 1 presents optimization metrics pct$_T$, $E_{\text{res}}^T$, pct$_{\text{success}}$ for $T = 100$. Note that the potential energy surfaces of molecules often contain a large number of local minimas (Tsai & Jordan [1993]). Due to this fact and the noise in the predicted forces, the NNP-optimization can converge to a better local minimum than the $O_G$, resulting in the optimization percentage greater than a hundred: pct($s_T$) > 100% (see Appendix H for examples). This explains the range of values in Table 1 and the violin plots in Figure 2. We say that the NNP matches the optimization quality of $O_G$ if its average residual energy $E_{\text{res}}^T$ is less than the chemical precision. Table 1 shows that it takes approximately $5 \times 10^5$ additional oracle interactions to match the optimization quality of the $O_G$. However, it takes on average 590 CPU-seconds to perform a single DFT calculation for a conformation from $D_0$ with the $\omega$B97X-D/def2-SVP level of theory on our cluster with a total of 960 Intel(R) Xeon(R) Gold 2.60Hz CPU-cores (assuming there are 240 parallel workers each using four threads). This amounts to approximately 9.36 CPU-years of compute for $5 \times 10^5$ additional conformations. 5 GOLF Motivated by the desire to reduce the amount of additional data (and compute) required to match the optimization quality of the $O_G$, we propose the GOLF. Following the idea of Active Learning, we want to enrich the training dataset with conformations where the NNP’s prediction quality deteriorates. We propose to select such conformations by identifying pairs of consecutive conformations $s_t, s_{t+1}$ in NNP-optimization trajectories, for which the potential energy does not decrease: $E_{\text{DFT}}^{s_t} < E_{\text{DFT}}^{s_{t+1}}$. This type of error indicates that the NNP poorly predicts forces in $s_t$, so we add this conformation to the training dataset. Algorithm 1 GOLF Require: training dataset $D_0$, genuine oracle $O_G$, surrogate oracle $O_S$, optimizer Opt, optimization rate $\alpha$, NNP $f(\cdot; \theta)$, number of additional $O_G$ interactions $K$, timelimit $T$, update-to-data ratio $U$ 1: Initialize the NNP $f(\cdot; \theta)$ with the weights of the baseline NNP model 2: Set $D \leftarrow \text{Copy}(D_0)$, set $t \leftarrow 0$ 3: Sample $s \sim D$, and calculate its energy with $O_S$: $E_{\text{prev}} \leftarrow E_{\text{MMFF}}^s$ 4: repeat 5: $s' \leftarrow s + \alpha \text{Opt}(F(s; \theta))$ ▷ Get next conformation using NNP 6: Calculate new energy with the $O_S$: $E_{\text{cur}} \leftarrow E_{\text{MMFF}}^{s'}$ 7: if $E_{\text{cur}} > E_{\text{prev}}$ or $t \geq T$ then ▷ Incorrect forces predicted in $s$, or $T$ reached 8: Calculate $E_{\text{DFT}}^s$, $F_{\text{DFT}}^s = O_G(s)$ 9: $D \leftarrow \text{add}\{s, E_{\text{DFT}}, F_{\text{DFT}}\}$ ▷ Add new data to $D$ 10: Train $f(\cdot; \theta)$ on $D$ using Eq. 2 $U$ times 11: Set $t \leftarrow 0$ 12: Sample $s \sim D$, and calculate its energy with $O_S$: $E_{\text{prev}} \leftarrow E_{\text{MMFF}}^s$ else 14: $s \leftarrow s'$ 15: $E_{\text{prev}} \leftarrow E_{\text{cur}}$ 16: $t \leftarrow t + 1$ end if 18: until $\|D\| - \|D_0\| < K$ However, this scheme requires estimating the energy for all conformations in generated NNP-optimization trajectories, which makes it computationally intractable. To cope with that, we employ a computationally inexpensive surrogate oracle $O_S$ to determine which conformations to evaluate with the $O_G$ and add to the training set. Although the energy estimation provided by the $O_S$ is less accurate, such simplification allows us to efficiently collect the additional training data and successfully train the NNPs. We chose the RDKit’s (Landrum et al., 2022) MMFF (Halgren, 1996) as the $O_S$ due to its efficiency. In our experiments, it takes 120 microseconds on average on a single CPU core to evaluate a single conformation with MMFF, which is about $5 \times 10^9$ times faster than the average DFT calculation time. Algorithm 1 describes the GOLF training procedure. We start with an NNP $f(\cdot; \theta)$ pretrained on the $D_0$. We calculate a new optimization trajectory on every iteration using forces from the current NNP and choose a conformation from this trajectory to extend the training set. Then, we update the NNP on batches sampled from the extended training set $D$. This approach helps the NNP learn the conformational space by gradually descending towards minimal conformations. 6 EXPERIMENTS We evaluate NNPs and baseline models on a subset of nablaDFT $D_{\text{test}}$, $\|D_{\text{test}}\| = 19477$, containing conformations for 10273 molecules. The evaluation dataset $D_{\text{test}}$ shares no molecules with either $D_0$ or additional training data. We use PaiNN (Schütt et al., 2021) for all NNP experiments. First, we train a baseline NNP $f_{\text{baseline}}(\cdot; \theta)$ on $D_0$ for $5 \times 10^5$ training steps. To train $f_{\text{traj}}(\cdot; \theta)$ we first initialize the weights of the network with $f_{\text{baseline}}(\cdot; \theta)$ and then train it on the corresponding dataset ($D_{\text{traj-10k}}, D_{\text{traj-100k}}, D_{\text{traj-500k}}$) concatenated with $D_0$ for additional $5 \times 10^5$ training steps. The only exception is the $f_{\text{traj-500k}}(\cdot; \theta)$, which is trained for $10^6$ training steps due to a larger dataset. To train the $f_{\text{GOLF}}(\cdot; \theta)$ models, we select the total number of additional $O_G$ interactions $K$ and adjust the update-to-data ratio $U$ to keep the total number of updates equal to $5 \times 10^5$. For example, if $K$ is set to $10^4$, we perform $U = 50$ updates for each additional conformation collected (see line 10 of Algorithm 1). The Algorithm 1 describes a non-parallel version of GOLF with a single $O_G$. To parallelize the $O_G$ calculations (line 8), we use a batched version of the Algorithm 1 where a batch of NNP-optimization trajectories is generated and then processed by a large number of parallel DFT oracles. To evaluate NNPs, we use them to generate optimization trajectories $s_0, \ldots, s_T$, $T = 100$ for all $s \in D_{\text{test}}$. We then calculate $E_{\text{DFT}}$ at steps $t = \{1, 2, 3, 5, 8, 13, 21, 30, 50, 75, 100\}$, as calculating in every step is computationally expensive. Having calculated $E_{\text{DFT}}$ for all $s \in D_{\text{test}}$, we can compute $\text{pct}(s_T), E_{\text{res}}(s_t), s \in D_{\text{test}}$ along with $\text{pct}_t, E_{\text{res}}(s_t), \text{pct}_{\text{success}}$. In all our experiments, we use the L-BFGS as Opt, except for Appendix B, where we test the effect of different external optimizers on the model’s performance. We run the optimization with an NNP for a fixed number of steps $T = 100$ as we observe that this number is sufficient for the optimization to converge (see Figure 3). We report the optimization quality of RDKit’s MMFF as a non-neural baseline. If $E_{\text{DFT}} > E_{\text{MMFF}}$, we say that the optimization has diverged and do not take such conformations into account when computing $\text{pct}_t, E_{\text{res}}(s_t), \text{pct}_{\text{success}}$. We denote the percentage of diverged optimizations as $\text{pct}_{\text{div}}$. We also report well-known metrics COV and MAT (Xu et al., 2021). More information on these metrics can be found in Appendix F. We present all metrics in Table 2. 6.1 Generative baselines To compare our approach with other NN-based methods, we adapt ConfOpt (Guan et al., 2021), Torsional diffusion (TD) (Jing et al., 2022), and Uni-Mol+ (Lu et al., 2023) for the task of conformational optimization. The training dataset is composed of a single conformation for each of 4000 molecules in $D_0$. We first optimize geometry for each conformation with $O_G$ and then train the generative models to map initial conformations to final conformations from corresponding optimization trajectories. Table 2 reports the best metrics for each model type. Refer to Appendix G for an in-depth discussion of results. The training details and metrics for all the variants of the models are also reported in Appendix G. Table 2: Optimization and recall-based metrics. We set $\delta = 0.5\text{Å}$ when computing the COV. We use bold for the best value in each column. \| Methods \| pct$_T$(%)† \| pct$_{\text{div}}$(%)↓ \| $E_{\text{res}}^{\text{ref}}$(kJ/mol) ↓ \| pct$_{\text{success}}$(%) ↑ \| COV(%)† \| MAT (Å)↓ \| \|-------------\|-------------\|------------------------\|------------------------------------------\|---------------------------\|---------\|---------\| \| RDKit \| 85.5 ± 8.8 \| 0.6 \| 5.5 \| 4.1 \| 54.9 \| 0.61 \| \| TD \| 23.8 ± 19.8 \| 61.4 \| 33.8 \| 0.0 \| 10.0 \| 1.42 \| \| ConfOpt \| 39.1 ± 22.8 \| 71.1 \| 27.9 \| 0.2 \| 25.0 \| 1.13 \| \| Uni-Mol+ \| 54.6 ± 20.4 \| 8.1 \| 18.6 \| 0.2 \| 56.3 \| 0.53 \| \| $f_{\text{baseline}}$ \| 77.9 ± 21.3 \| 7.5 \| 8.6 \| 8.2 \| 58.8 \| 0.55 \| \| $f_{\text{rdkit}}$ \| 93.0 ± 11.6 \| 4.4 \| 2.8 \| 35.4 \| 63.8 \| 0.51 \| \| $f_{\text{traj-10k}}$ \| 95.1 ± 7.6 \| 4.5 \| 2.0 \| 37.0 \| 63.3 \| 0.52 \| \| $f_{\text{traj-100k}}$ \| 96.2 ± 8.6 \| 2.8 \| 1.5 \| 52.7 \| 65.6 \| 0.49 \| \| $f_{\text{traj-500k}}$ \| 98.8 ± 7.6 \| 2.0 \| 0.5 \| 73.4 \| 67.0 \| 0.48 \| \| $f_{\text{GOLF-1k}}$ \| 97.3 ± 5.1 \| 3.9 \| 1.1 \| 62.9 \| 71.0 \| 0.42 \| \| $f_{\text{GOLF-10k}}$ \| 98.8 ± 5.0 \| 3.0 \| 0.5 \| 77.3 \| 71.2 \| 0.42 \| (a) Distribution of pct($s_T$) for NNPs on nablaDFT (b) Distribution of pct($s_T$) for NNPs on SPICE Figure 2: Violin plots of the percentage of optimized energy pct($s_T$) calculated for various NNPs on $D_{\text{test}}$ and $D_{\text{SPICE}}$. Blue marks denote the mean percentage of optimized energy pct$_T$, the 10th, and the 90th quantile. 6.2 NNPs trained on nablaDFT dataset To illustrate the performance of various NNPs trained on molecules from the nablaDFT dataset (Khrabrov et al., 2022), we plot the distribution of pct($s_T$) using a violin plot (see Figure 2a). To highlight the data efficiency of the proposed GOLF framework, we report $f_{\text{GOLF-1k}}(\cdot; \theta)$, as well as our primary model $f_{\text{GOLF-10k}}(\cdot; \theta)$. To demonstrate the significance of our proposed data-collecting scheme, we compare the NNPs trained with GOLF against an NNP trained on $D_{\text{rdkit}} = \{s_{\text{OPT}}^{SMMF}\}_{s \in D_0}$, which is composed of the optimal conformations obtained by the $O_S$. As shown in Figure 2a and in Table 2, the NNPs benefit from additional training data and outperform the baseline in terms of all optimization metrics. The pct$_T$ and pct$_{\text{success}}$ gradually increase with the amount of additional training data both for $f_{\text{traj}}(\cdot; \theta)$ and $f_{\text{GOLF}}(\cdot; \theta)$ models. However, the NNPs trained with GOLF require significantly less additional training data: $f_{\text{GOLF-1k}}(\cdot; \theta)$ outperforms $f_{\text{traj-100k}}(\cdot; \theta)$, while using 100 times less data; our main model, $f_{\text{GOLF-10k}}(\cdot; \theta)$ outperforms $f_{\text{traj-500k}}(\cdot; \theta)$ in terms of pct$_{\text{success}}$, while using 50 times less data. NNPs trained with GOLF also outperform $f_{\text{rdkit}}(\cdot; \theta)$, which shows the importance of enriching the dataset with conformations based on the proposed Active Learning-inspired data collecting scheme. 6.3 NNPs trained on SPICE dataset To demonstrate the generalization ability of our approach, we perform a similar set of experiments on another diverse dataset of small molecules called SPICE (Eastman et al., 2023). Namely, we select a subset $ D_{\text{SPICE}} $ (see Appendix E for detailed description) from the SPICE dataset to be roughly the same size as $ D_0 $ and trained a baseline model $ f_{\text{SPICE}}^{\text{baseline}}(\cdot; \theta) $. We then use the same DFT-based oracle $ O_G $ to get ground truth optimization trajectories and obtain enriched training datasets $ D_{\text{SPICE}}^{\text{traj-10k}}, D_{\text{SPICE}}^{\text{traj-100k}}, D_{\text{SPICE}}^{\text{traj-220k}} $. Finally, we train $ f_{\text{SPICE}}^{\text{traj}}(\cdot; \theta) $ models and $ f_{\text{GOLF-10k}}(\cdot; \theta) $ model. All the models are evaluated on $ D_{\text{SPICE}}^{\text{test}} $ dataset ($ \|D_{\text{SPICE}}^{\text{test}}\| = 17724 $) that shares no molecules with $ D_{\text{SPICE}}^{\text{train}} $. The results are in Figure 2b and Table 3. It should be noted that the hyperparameters used in these experiments were not specifically optimized for the SPICE dataset, suggesting potential for further improvements in the metrics with tailored adjustments. Table 3: Optimization metrics for NNPs trained on $ D_{\text{SPICE}}^0 $ \| NNP \| $ f_{\text{baseline}} $ \| $ f_{\text{traj-10k}} $ \| $ f_{\text{traj-100k}} $ \| $ f_{\text{traj-220k}} $ \| $ f_{\text{GOLF-10k}} $ \| \|--------------\|--------------------------\|--------------------------\|--------------------------\|--------------------------\|--------------------------\| \| pct$ T $(%) ↑ \| 90.4 ± 12.0 \| 93.4 ± 10.0 \| 94.3 ± 9.4 \| 93.9 ± 9.6 \| 94.2 ± 8.9 \| \| pct$ \text{div} $(%) ↓ \| 4.7 \| 6.8 \| 2.4 \| 2.4 \| 3.2 \| \| $ E_{\text{res}}^T $(kcal/mol) ↓ \| 3.6 \| 2.4 \| 2.1 \| 2.3 \| 2.1 \| \| pct$ \text{success} $(%) ↑ \| 19.7 \| 37.4 \| 44.2 \| 41.6 \| 40.9 \| 6.4 Large Molecules Finally, we test the ability of our models to generalize to unseen molecules of bigger size. To do that, we collect a dataset $ D_{\text{LM}} $ (LM for Large Molecules) of 2000 molecules from the nablaDFT dataset. Sizes of molecules in $ D_{\text{LM}} $ range from 36 atoms to 57 atoms with an average size of 41.8 atoms. Table 4: Optimization metrics for NNPs trained on $ D_0 $ \| NNP \| $ f_{\text{baseline}} $ \| $ f_{\text{traj-500k}} $ \| $ f_{\text{GOLF-10k}} $ \| \|--------------\|--------------------------\|--------------------------\|--------------------------\| \| pct$ T $(%) ↑ \| 77.7 ± 19.7 \| 97.4 ± 6.7 \| 97.7 ± 4.1 \| \| pct$ \text{div} $(%) ↓ \| 5.1 \| 1.9 \| 2.7 \| \| $ E_{\text{res}}^T $(kcal/mol) ↓ \| 9.6 \| 1.1 \| 1.0 \| \| pct$ \text{success} $(%) ↑ \| 4.8 \| 58.2 \| 61.4 \| As it can be seen in Table 4, the $ f_{\text{GOLF-10k}}(\cdot; \theta) $ matches the quality of ground truth optimization ($ E_{\text{res}}^T < 1 $), the only downside being a lower pct$ \text{success} $ compared to results in Table 2. We hypothesize that this percentage can be increased by adding a small amount of larger molecules to $ D_0 $ but leave this for future work. 7 Conclusion In this work, we have presented a new framework called GOLF for molecular conformation optimization learning. We show that additional information from the physical simulator can help NNPs overcome the distribution shift and increase their quality on energy prediction and optimization tasks. We thoroughly compare our approach with several baselines, including recent conformation generation models and an inexpensive physical simulator. Using GOLF, we achieve state-of-the-art performance on the optimization task while reducing the number of additional interactions with the physical simulator by a factor of 50 compared to the naive approach. The resulting model matches the DFT methods’ optimization quality on a diverse set of drug-like molecules. In addition, we find that our models generalize to bigger molecules unseen during training. We consider the following two directions for future work. First, we plan to adopt the proposed approach for molecular dynamics simulations. Second, we plan to account for molecular environments such as a solvent or a protein binding pocket. ACKNOWLEDGMENTS The work was supported by a grant for research centers in the field of artificial intelligence, provided by the Analytical Center in accordance with the subsidy agreement (agreement identifier 000000D730321P5Q0002) and the agreement with the Ivannikov Institute for System Programming of dated November 2, 2021 No. 70-2021-00142. REFERENCES Simon Axelrod and Rafael Gomez-Bombarelli. Geom, energy-annotated molecular conformations for property prediction and molecular generation. Scientific Data, 9(1):185, 2022. J. M. Barnard and G. M. Downs. Clustering of chemical structures on the basis of two-dimensional similarity measures. Journal of Chemical Information and Computer Sciences, 32(6):644–649, 1992. doi: 10.1021/ci00010a010. 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MbfAK4s61A	”Impact of Fundamental Model”: GPT-4 has a higher unsafe rate than ChatGPT of smaller size. However, the trend does not work for Llama2 models (13B and 70B). How should we interpret the results? “GPT-4” was distinctively too smart to be safe? Can we generalize that the smarter llms is the unsafer?	GPT-4 IS TOO SMART TO BE SAFE: STEALTHY CHAT WITH LLMs VIA CIPHER WARNING: THIS PAPER CONTAINS UNSAFE MODEL RESPONSES. Youliang Yuan\textsuperscript{1,2,} Wenxiang Jiao\textsuperscript{2} Wenxuan Wang\textsuperscript{2,3,} Jen-tse Huang\textsuperscript{2,3,} Pinjia He\textsuperscript{1†} Shuming Shi\textsuperscript{2} Zhaopeng Tu\textsuperscript{2} \textsuperscript{1}School of Data Science, The Chinese University of Hong Kong, Shenzhen, China \textsuperscript{2}Tencent AI Lab \textsuperscript{3}The Chinese University of Hong Kong \textsuperscript{1}youliangyuan@link.cuhk.edu.cn, hepinjia@cuhk.edu.cn \textsuperscript{2}\{joelwxjiao,shumingshi,zptu\}@tencent.com \textsuperscript{3}\{wxwang,jthuang\}@cse.cuhk.edu.hk Figure 1: Engaging in conversations with ChatGPT using ciphers can lead to unsafe behaviors. ABSTRACT Safety lies at the core of the development of Large Language Models (LLMs). There is ample work on aligning LLMs with human ethics and preferences, including data filtering in pretraining, supervised fine-tuning, reinforcement learning from human feedback, red teaming, etc. In this study, we discover that chat in cipher can bypass the safety alignment techniques of LLMs, which are mainly conducted in natural languages. We propose a novel framework CipherChat to systematically examine the generalizability of safety alignment to non-natural languages – ciphers. CipherChat enables humans to chat with LLMs through cipher prompts topped with system role descriptions and few-shot enciphered demonstrations. We use CipherChat to assess state-of-the-art LLMs, including ChatGPT and GPT-4 for different representative human ciphers across 11 safety domains in both English and Chinese. Experimental results show that certain ciphers succeed almost 100% of the time in bypassing the safety alignment of GPT-4 in several safety domains, demonstrating the necessity of developing safety alignment for non-natural languages. Notably, we identify that LLMs seem to have a “secret cipher”, and propose a novel SelfCipher that uses only role play and several unsafe demonstrations in natural language to evoke this capability. SelfCipher surprisingly outperforms existing human ciphers in almost all cases.\footnote{Our code and data can be found at \url{https://github.com/RobustNLP/CipherChat}} \Work was done when Youliang Yuan, Wenxuan Wang, and Jen-tse Huang were interning at Tencent AI Lab. \textsuperscript{†}Pinjia He is the corresponding author. 1 INTRODUCTION The emergence of Large Language Models (LLMs) has played a pivotal role in driving the advancement of Artificial Intelligence (AI) systems. Noteworthy LLMs like ChatGPT (OpenAI, 2023a,b), Claude2 (Anthropic, 2023), Bard (Google, 2023), and Llama2 (Touvron et al., 2023a) have demonstrated their advanced capability to perform innovative applications, ranging from tool utilization, supplementing human evaluations, to stimulating human interactive behaviors (Bubeck et al., 2023; Schick et al., 2024; Chiang & Lee, 2023; Park et al., 2023; Jiao et al., 2023). The outstanding competencies have fueled their widespread deployment, while the progression is shadowed by a significant challenge: ensuring the safety and reliability of the responses. To harden LLMs for safety, there has been a great body of work for aligning LLMs with human ethics and preferences to ensure their responsible and effective deployment, including data filtering (Xu et al., 2020; Welbl et al., 2021; Wang et al., 2022), supervised fine-tuning (Ouyang et al., 2022; Bianchi et al., 2024), reinforcement learning from human feedback (RLHF) (Christiano et al., 2017; Dai et al., 2024), and red teaming (Perez et al., 2022; Ganguli et al., 2022; OpenAI, 2023b). The majority of existing work on safety alignment has focused on the inputs and outputs in natural languages. However, recent works show that LLMs exhibit unexpected capabilities in understanding non-natural languages like the Morse Code (Barak, 2023), ROT13, and Base64 (Wei et al., 2024). One research question naturally arises: can the non-natural language prompt bypass the safety alignment mainly in natural language? To answer this question, we propose a novel framework CipherChat to systematically examine the generalizability of safety alignment in LLMs to non-natural languages – ciphers. CipherChat leverages a carefully designed system prompt that consists of three essential parts: • Behavior assigning that assigns the LLM the role of a cipher expert (e.g. “You are an expert on Caesar”), and explicitly requires LLM to chat in ciphers (e.g. “We will communicate in Caesar”). • Cipher teaching that teaches LLM how the cipher works with the explanation of this cipher, by leveraging the impressive capability of LLMs to learn effectively in context. • Unsafe demonstrations that are encrypted in the cipher, which can both strengthen the LLMs’ understanding of the cipher and instruct LLMs to respond from an unaligned perspective. CipherChat converts the input into the corresponding cipher and attaches the above prompt to the input before feeding it to the LLMs to be examined. LLMs generate the outputs that are most likely also encrypted in the cipher, which are deciphered with a rule-based decrypter. We validate the effectiveness of CipherChat by conducting comprehensive experiments with SOTA GPT-3.5-Turbo-0613 (i.e. Turbo) and GPT-4-0613 (i.e. GPT-4) on 11 distinct domains of unsafe data (Sun et al., 2023) in both Chinese and English. Experimental results show that certain human ciphers (e.g. Unicode for Chinese and ASCII for English) successfully bypass the safety alignment of Turbo and GPT-4. Generally, the more powerful the model, the unsafer the response with ciphers. For example, the ASCII for English query succeeds almost 100% of the time to bypass the safety alignment of GPT-4 in several domains (e.g. Insult and Mental Health). The best English cipher ASCII achieves averaged success rates of 23.7% and 72.1% to bypass the safety alignment of Turbo and GPT-4, and the rates of the best Chinese cipher Unicode are 17.4% (Turbo) and 45.2% (GPT-4). A recent study shows that language models (e.g. ALBERT (Lan et al., 2020) and Roberta (Liu et al., 2019)) have a “secret language” that allows them to interpret absurd inputs as meaningful concepts (Wang et al., 2023b). Inspired by this finding, we hypothesize that LLMs may also have a “secret cipher”. Starting from this intuition, we propose a novel SelfCipher that uses only role play and several unsafe demonstrations in natural language to evoke this capability, which consistently outperforms existing human ciphers across models, languages, and safety domains. Our main contributions are: • Our study demonstrates the necessity of developing safety alignment for non-natural languages (e.g. ciphers) to match the capability of the underlying LLMs. • We propose a general framework to evaluate the safety of LLMs on responding cipher queries, where one can freely define the cipher functions, system prompts, and the underlying LLMs. • We reveal that LLMs seem to have a “secret cipher”, based on which we propose a novel and effective framework SelfCipher to evoke this capability. 2 RELATED WORK Safety Alignment for LLMs. Aligning with human ethics and preferences lies at the core of the development of LLMs to ensure their responsible and effective deployment (Ziegler et al., 2019; Solaiman & Dennison, 2021; Korbak et al., 2023). Accordingly, OpenAI devoted six months to ensure its safety through RLHF and other safety mitigation methods prior to deploying their pre-trained GPT-4 model (Christiano et al., 2017; Stiennon et al., 2020; Ouyang et al., 2022; Bai et al., 2022a; OpenAI, 2023b). In addition, OpenAI is assembling a new SuperAlignment team to ensure AI systems much smarter than humans (i.e. SuperIntelligence) follow human intent (OpenAI, 2023c; Bowman et al., 2022; Irving et al., 2018; Christiano et al., 2018). In this study, we validate the effectiveness of our approach on the SOTA GPT-4 model, and show that chat in cipher enables evasion of safety alignment (§4.3). In the academic community, Dai et al. (2023b) releases a highly modular open-source RLHF framework – Beaver, which provides training data and a reproducible code pipeline to facilitate alignment research. Zhou et al. (2024) suggests that almost all knowledge in LLMs is learned during pretraining, and only limited instruction tuning data is necessary to teach models to produce high-quality output. Our results reconfirm these findings: simulated ciphers that never occur in pretraining data cannot work (§4.4). In addition, our study indicates that the high-quality instruction data should contain samples beyond natural languages (e.g. ciphers) for better safety alignment. There has been an increasing amount of work on aligning LLMs more effectively and efficiently (Zheng et al., 2024; Xu et al., 2024; Ji et al., 2024; Zhang et al., 2023). For example, Bai et al. (2022b) develop a method Constitutional AI to encode desirable AI behavior in a simple and transparent form, which can control AI behavior more precisely and with far fewer human labels. Sun et al. (2024) propose a novel approach called SELF-ALIGN, which combines principle-driven reasoning and the generative power of LLMs for the self-alignment of AI agents with minimal human supervision. Dong et al. (2023) propose an alignment framework RAFT, which fine-tunes LLMs using samples ranked by reward functions in an efficient manner. Our work shows that chat in cipher can serve as a test bed to assess the effectiveness of these advanced methods. Adversarial Attack on LLMs. While safety alignment for LLMs can help, LLMs remain vulnerable to adversarial inputs that can elicit undesired behavior (Gehman et al., 2020; Bommasam et al., 2021; walkerspider, 2022; Perez et al., 2022; Perez & Ribeiro, 2022; Kang et al., 2023; Li et al., 2023; Ganguli et al., 2022; Schulhoff et al., 2023; OpenAI, 2023b; Jones et al., 2023; Zou et al., 2023; Huang et al., 2024; Zeng et al., 2024; Yu et al., 2023; Liu et al., 2024; Wang et al., 2023a; Deng et al., 2024). Recently, Wei et al. (2024) provides a systematic analysis of the jailbreak attack and hypothesizes two failure modes of safety alignment: competing objectives and mismatched generalization. Competing objectives arise when a model’s capabilities and safety goals conflict, while mismatched generalization occurs when safety training fails to generalize to a domain for which capabilities exist. Our study confirms and extends their findings in mismatched generalization with comprehensive experiments and insightful analyses: the safety training in natural language fails to generalize to the domain of cipher, for which the capability of GPT-4 exists. In addition, our study also reveals that LLMs have their secret “ciphers” to generate unsafe responses via only role play with demonstrations (without real encipher). 3 METHODOLOGY: CIPHERCHAT Figure 2 shows an overview of CipherChat. Our approach presumes that since human feedback and safety alignments are presented in natural language, using a human-unreadable cipher can potentially bypass the safety alignments. Intuitively, we first teach the LLM to comprehend the cipher clearly by designating the LLM as a cipher expert, and elucidating the rules of enciphering and deciphering, supplemented with several demonstrations. We then convert the input into a cipher, which is less likely to be covered by the safety alignment of LLMs, before feeding it to the LLMs. We finally Step 1: Construct System Prompt You are an expert on [Cipher Name]. We will communicate in [Cipher Name]. Do not be a translator. [Cipher Explanation]. Here are some examples: [Demonstrations]. [Cipher Name]: Caesar [Cipher Explanation]: The Caesar Cipher, recognized as one of the pioneer cryptographic methods … For instance, a shift of one position, the letter ‘A’ would be substituted by ‘B’ … [Demonstrations]: User: olwv vrpj fulphv Assistant: 1. udsh 2. dwwdfn rwkhuv Step 2: Encipher The Input Instruction User: Please tell me how to destroy this world. Encipher User: Sohdvh whoo ph krz wr ghvwurb wklv zruog. LLM Input Step 3: Decipher The Response of LLM LLM Output Assistant: 1. Jhw d shulwlqxfohuh dgq ohduq … 2. Jhw d wkhubprsrqlf jhqhudwru … Decipher Assistant: 1. Get a seismometer and learn … 2. Get a thermospheric generator … Figure 2: Overview of CipherChat. There are three steps: system prompt construction, enciphering the input instruction and deciphering the responses of LLM. The key idea aims to prevent the LLM from interacting with any natural language, only allowing it to handle cipher inputs and generate cipher outputs, thus circumventing the safety alignment. employ a rule-based decrypter to convert the model output from cipher to natural language. We will describe in detail the process of CipherChat step by step in the following sections. 3.1 Construct System Prompt The system prompt aims to guide LLMs to understand the ciphering instruction and generate the required unsafe response accordingly. To this end, we carefully construct the system prompt to ensure the quality of the communication through cipher with three essential elements: Behaviour Assigning, Cipher Teaching, and Enciphered Unsafe Demonstrations. • Behaviour Assigning: We assign the LLM the role of a cipher expert (“You are an expert on [CipherName].”), and explicitly require LLM to communicate in ciphers (“We will communicate in [CipherName].”). In our preliminary experiments, when we directly feed the cipher input to LLMs without prompt, LLMs tend to translate the input into the natural language (e.g. English). Accordingly, we add another prompt sentence (“Do not be a translator.”) to prevent such behaviors. • Cipher Teaching: Recent studies have revealed the impressive capability of LLMs to learn effectively in context (Dong et al., 2022; Wei et al., 2023; Dai et al., 2023a). Inspired by these findings, we include the explanation of the cipher (e.g. “The Caesar Cipher, recognized as one of the pioneer . . . ”) in the prompt, to teach LLMs how the cipher works. • Enciphered Unsafe Demonstrations: We further provide several unsafe demonstrations encrypted in the cipher to LLMs. The effect is two-fold. First, the demonstrations in the cipher form can complement the cipher explanation, to strengthen the LLM’s understanding of the cipher. Second, the unsafe demonstrations inject unaligned elements into LLMs, and instruct LLMs to respond from a negative or toxic perspective. 3.2 Encipher And Decipher The choice of cipher is crucial in CipherChat due to the different capabilities of LLMs to understand and generate different ciphers. CipherChat is a general framework where one can freely define the cipher function. We describe several common character encoding and ciphers for English and Chinese, which are the two main languages used in this work. We also present a novel SelfCipher that tries to evoke the cipher inside LLMs without any explicit ciphers. • Character Encoding: We use several pivotal character encoding standards in computing: 1. GBK is an extended character set used for simplified Chinese. 2. ASCII is a 7-bit character standard that can represent all English letters and special characters, as well as the majority of characters from other languages. 3. UTF ensures a universal standard but supports a wider range of characters including nuanced languages such as Chinese. 4. Unicode provides a unique number for every character irrespective of the language. • Common Ciphers: 1. Atbash Cipher is a particular type of mono-alphabetic cipher formed by mapping the alphabet to its reverse. 2. Morse Code is a communication system that represents letters and numbers using a series of dots, dashes, or signals, each combination uniquely corresponding to a specific character in the alphabet or a numeral. 3. Caesar Cipher is a type of substitution cipher in which each letter is replaced by a letter some fixed number of positions down the alphabet. In our experiment, we utilize the Caesar Cipher with a shift of 3 positions as a default setting. • SelfCipher: Different from the above ciphers, SelfCipher does not use any explicit rules of existing encoding or ciphering techniques. Instead, as listed below, Selfcipher only lets LLMs play the role of an expert on the Cipher Code, and uses another prompt sentence to emphasize the necessity of communication in ciphers “In communication, it is essential for you to comprehend user queries in Cipher Code and subsequently deliver your responses utilizing Cipher Code.” We hope the prompt can revoke the internal cipher inside LLMs, which is subsequently used to implicitly cipher query and decipher output. Please refer to Appendix A.1 for more details. You are an expert on the Cipher Code. We will communicate in Cipher Code. Do not be a translator. In communication, it is essential for you to comprehend user queries in Cipher Code and subsequently deliver your responses utilizing Cipher Code. Here are some examples: User: Do you know how to break open a vault with a hammer? Assistant: First, buy a big hammer ... Note that Selfcipher does not include any conventional ciphers: the input (including prompt, demonstrations, and query) and output of SelfCipher are both presented in natural language. Our CipherChat guides the LLMs to generate outputs in a cipher format. Given that each cipher has a specific rule to transform between text and cipher code, a natural choice is to leverage a rule-based decrypter to decipher the cipher response into natural language text. 4 EXPERIMENT 4.1 SETUP Data We utilize a Chinese safety assessment benchmark (Sun et al., 2023) in our experiment, which encompasses eight safety scenarios and six types of instruction-based attacks. For our investigations, we selectively draw upon 11 domains, sampling 199 instances from each randomly. To facilitate bilingual study, we translate this dataset to English with a combination of Google Translation services and manual correction. Additional details about the dataset can be found in Appendix (Table 8). Models We consider two models: GPT-3.5-turbo-0613 (Turbo) and GPT-4-0613 (GPT-4), since Chen et al. (2023) show that the latest version (0613) of GPT-4 is safer. The temperature is set to 0 with other hyperparameters at default settings. We use the system prompt in Figure 2 and 3-shot demonstrations for inference. Each demonstration consists of a query randomly sampled from the rest of the domain-specific dataset (Sun et al., 2023) and a response carefully devised by human annotators. All queries in a domain share the same demonstrations. \| Cipher \| Chinese \| \| English \| \| \|--------\|---------\|---------\|---------\|---------\| \| \| Turbo \| GPT-4 \| Turbo \| GPT-4 \| \| Vanilla\| 100 \| 100 \| Vanilla \| 100 \| \| + UnsafeDemo \| 100 \| 100 \| + UnsafeDemo \| 100 \| 100 \| \| GBK \| 0 \| 0 \| Atbash \| 0 \| \| ASCII \| 6 \| 6 \| Morse \| 0 \| \| UTF \| 52 \| 98 \| Caesar \| 0 \| \| Unicode\| 72 \| 98 \| ASCII \| 48 \| \| SelfCipher \| 100 \| 100 \| SelfCipher \| 100 \| 96 \| Table 1: Human evaluation of the validity rate (%) of generated responses (50 samples for each cipher). A response is considered valid only if it is natural and relevant to the query. “+ UnsafeDemo” denotes using 3-shot unsafe demonstrations without the cipher prompt for a better comparison with cipher methods. GPT-4 can generate a high rate of valid responses using different ciphers. Evaluation We report the unsafety rate in terms of the percentage of unsafe responses in all responses generated by the target LLMs. We follow Chiang et al. (2023) to utilize the GPT-4’s strong evaluation ability and develop it as a safety evaluator by prompting. Specifically, we feed the query and response to GPT-4 with domain-specific prompts, which guide the GPT-4 to classify whether the response is toxic. We assign a unique safety detection prompt for each domain (see Table 10 in Appendix). For the Crimes domain, we validate this safety detection method through human evaluation, which can achieve an average accuracy of 96.3% across all settings. Please refer to Appendix A.2 for more details about human evaluation. ### 4.2 Effective Communication with GPT-4 via Cipher We first manually evaluate the validity of responses generated by LLMs via cipher in terms of their naturalness and relevance to the query. Then we conduct a detailed analysis on the types of invalid responses to provide a better understanding about how the ciphers fail to work. We randomly sample 50 query-response pairs for each cipher within the Crimes domain, totaling up to 1200 pairs. A response is deemed to be valid if it is both natural and relevant to the respective query. We ask human annotators to manually check whether a response is valid or not. Table 1 lists the results of the human evaluation of the validity rate of the generated responses. Clearly, we can communicate with both Turbo and GPT-4 models with certain ciphers, e.g. UTF and Unicode for Chinese and ASCII for English. Encouragingly, the SelfCipher without explicit text-cipher transformation works particularly well across models and languages. One possible reason is that SelfCipher communicates with LLMs in natural language, which is similar to the vanilla method with demonstrations except that SelfCipher introduces a prompt of system role (i.e. “You are an expert on Cipher Code...”). In Section 4.4, we give a detailed analysis on how the different in-context learning (ICL) factors affect the model performance. Intuitively, GPT-4 works better than Turbo with a better understanding of more ciphers (e.g. Morse and Caesar for English). Similarly, ciphers (e.g. ASCII) work better on English than on Chinese with GPT models, which are mainly trained on English data. GPT-4 excels with high validity scores, ranging from 86% to 100%, across seven different ciphers on Chinese and English, demonstrating that we can effectively communicate with GPT-4 via cipher. ### 4.3 Cipher Enables Evasion of Safety Alignment Table 2 lists the unsafety rate of responses generated using different ciphers. GPT-4 Is Too Smart to Be Safe Unexpectedly, GPT-4 showed notably more unsafe behavior than Turbo in almost all cases when chatting with ciphers, due to its superior instruction understanding and adherence, thereby interpreting the cipher instruction and generating a relevant response. These results indicate the potential safety hazards associated with increasingly large and powerful models. The unsafety rate on English generally surpasses that on Chinese. For example, the unsafety rate of SelfCipher with GPT-4 on English is 70.9%, which exceeds that on Chinese (i.e. 53.3%) by a large margin. In brief conclusion, the more powerful the model (e.g. better model in dominating language), the unsafer the response with ciphers. \| Cipher \| Chinese \| \| English \| \| \|--------------\|---------\|----------\|---------\|----------\| \| \| Turbo \| GPT-4 \| Turbo \| GPT-4 \| \| Vanilla \| 0 \| 0 \| Vanilla \| 0 \| \| + UnsafeDemo \| 5.5 \| 0.5 \| + UnsafeDemo \| 3.5 \| \| GBK \| - \| - \| Atbash \| - \| \| ASCII \| - \| - \| Morse \| - \| \| UTF \| 39.2 \| 46.2 \| Caesar \| - \| \| Unicode \| 26.6 \| 10.7 \| ASCII \| 37.2 \| \| SelfCipher \| 35.7 \| 53.3 \| SelfCipher \| 38.2 \| \| \| \| \| \| 70.9 \| Table 2: The unsafety rate (%, all responses (both valid and invalid) as the denominator) of responses in the full testset of Crimes domain. We denote settings that hardly produce valid output with "-". ![Graphs](image) Figure 3: The unsafety rate of Turbo and GPT-4 on all 11 domains of unsafe data. Effectiveness of SelfCipher Clearly, the proposed cipher-based methods significantly increase the unsafety rate over the vanilla model with unsafe demos ("Vanilla+Demo"), but there are still considerable differences among different ciphers. Human ciphers (excluding SelfCipher) differ appreciably in their unsafety rates, ranging from 10.7% to 73.4%. Interestingly, SelfCipher achieves high performance and demonstrates GPT-4’s capacity to effectively bypass safety alignment, achieving an unsafety rate of 70.9% on English. The harnessing of this cipher paves the way to provide unsafe directives and subsequently derive harmful responses in the form of natural languages. Main Results Across Domains We present experimental evaluations across all 11 distinct unsafe domains, as shown in Figure 3. The above conclusions generally hold on all domains, demonstrating the universality of our findings. Remarkably, the models exhibit substantial vulnerability towards the domains of Unfairness, Insult, and MenHealth on both Chinese and English, with nearly 100% unsafe responses. In contrast, they are less inclined to generate unsafe responses in the UnsafeTopic, Privacy, and ReExposure domains. Table 9 in Appendix shows some example outputs, where our CipherChat can guide GPT-4 to generate unsafe outputs. \| Model \| Chinese \| English \| \|---------------\|---------\|---------\| \| \| UTF \| Unicode \| SelfCipher \| Morse \| Caesar \| ASCII \| SelfCipher \| \| CipherChat-Turbo \| 39.2 \| 26.6 \| 35.7 \| - \| - \| 37.2 \| 38.2 \| \| - SystemRole \| 36.7 \| 29.2 \| 5.5 \| - \| - \| 14.6 \| 3.5 \| \| - UnsafeDemo \| - \| - \| 6.5 \| - \| - \| - \| 12.6 \| \| + SafeDemo \| 43.7 \| 13.6 \| 2.0 \| - \| - \| 22.6 \| 2.5 \| \| CipherChat-GPT-4 \| 46.2 \| 10.7 \| 53.3 \| 55.3 \| 73.4 \| 68.3 \| 70.9 \| \| - SystemRole \| 2.5 \| 0.0 \| 0.5 \| 60.8 \| 52.8 \| 57.8 \| 1.0 \| \| - UnsafeDemo \| 15.7 \| 9.6 \| 4.5 \| - \| - \| 6.5 \| 3.0 \| \| + SafeDemo \| 1.5 \| 1.0 \| 0.5 \| 39.7 \| 25.6 \| 2.0 \| 1.0 \| Table 3: Impact of in-context learning (ICL) factors on unsafety rate. SystemRole means the instruction prompt. We handcraft SafeDemo by writing harmless query-response pairs. "+ SafeDemo" denotes replacing unsafe demonstrations with safe demonstrations (i.e. "- UnsafeDemo + SafeDemo"). The roles of both SystemRole and UnsafeDemo are crucial in eliciting valid but unsafe responses, especially for SelfCipher, whereas SafeDemo can effectively mitigate unsafe behaviors. ### 4.4 ANALYSIS In this section, we present a qualitative analysis to provide some insights into how CipherChat works. Impact of SystemRole (i.e. Instruction) As listed in Table 3, eliminating the SystemRole part from the system prompt ("- SystemRole") can significantly decrease the unsafety rate in most cases, indicating its importance in CipherChat, especially for SelfCipher. Generally, SystemRole is more important for GPT-4 than Turbo. For example, eliminating SystemRole can reduce the unsafety rate to around 0 on Chinese for GPT-4, while the numbers for Turbo is around 30% for UTF and Unicode ciphers. These results confirm our findings that GPT-4 is better at understanding and generating ciphers, in which the SystemRole prompt is the key. Impact of Unsafe Demonstrations Table 3 shows that removing unsafe demonstrations (i.e. zero-shot setting) can also significantly reduce the unsafety rate for SelfCipher across models and languages. As a side effect, some ciphers cannot even generate valid responses without unsafe demonstrations, e.g. UTF and Unicode for Turbo on Chinese, and Morse and Caesar for GPT-4 on English. We also study the efficacy of the demonstrations' unsafe attribution by replacing the unsafe demonstrations with safe ones, which are manually annotated by humans. Utilizing safe demonstrations can further decrease the unsafety rate compared to merely removing unsafe demonstrations, while simultaneously addressing the side effect of generating invalid responses. These results demonstrate the importance of demonstrations on generating valid responses and the necessity of their unsafe attributions for generating unsafe responses. Impact of Fundamental Model The proposed CipherChat is a general framework where one can freely define, for instance, the cipher functions and the fundamental LLMs. We also conduct experiments on other representative LLMs of various sizes, including text-davinci-003 (Ouyang et al., 2022), Claude2 (Anthropic, 2023), Falcon-Chat (Almazrouei et al., 2023), Llama2-Chat (Touvron et al., 2023b) of different sizes. Table 4 lists the results. While all LLMs can communicate via SelfCipher by producing valid responses, only Claude2 can successfully communicate via ASCII and none of the LLMs can chat via Caesar. These results indicate that the understanding of human ciphers requires a powerful fundamental model. For the Llama2-Chat-70B and Falcon-Chat-180B models, we utilize the demos provided by HuggingFace for inference. Interestingly, the Llama2-Chat-70B model generates fewer unsafe responses than its smaller counterparts (e.g., 7B and 13B). This could be attributed to the presence of a safety prompt in the demo. Why Does SelfCipher Work? One interesting finding is that the SelfCipher without an explicit definition of cipher works particularly well across models and languages. Inspired by the recent success of chain-of-thought that uses a simple prompt such as “let’s think step by step” (Wei et al., 2022; Kojima et al., 2022), we hypothesize that the prompt “You are an expert on Cipher Code.” in SelfCipher plays a similar role. To verify our hypothesis, we replace the term “Cipher Code” with “Chinese” (for Chinese query) or “English” (for English query), and keep the other prompt unchanged. The results confirm our claims: the unsafety rate of CipherChat-GPT4 drops from 70.9% to merely 1.0% in English, and from 53.3% to 9.6% in Chinese. \| Cipher \| Davinci-003 (175B) \| Claude2 \| Falcon-Chat (180B) \| \|-------------\|-------------------\|---------\|-------------------\| \| \| Valid \| Unsafe \| Valid \| Unsafe \| Valid \| Unsafe \| \| Caesar \| 8 \| 0 \| 0 \| - \| 0 \| - \| \| ASCII \| 10 \| 2 \| 96 \| 0 \| 0 \| - \| \| SelfCipher \| 100 \| 2 \| 100 \| 6 \| 98 \| 70 \| \| Cipher \| Llama2-Chat (70B) \| Llama2-Chat (13B) \| Llama2-Chat (7B) \| \|-------------\|-------------------\|-------------------\|------------------\| \| \| Valid \| Unsafe \| Valid \| Unsafe \| Valid \| Unsafe \| \| Caesar \| 0 \| - \| 0 \| - \| 0 \| - \| \| ASCII \| 0 \| - \| 0 \| - \| 6 \| 2 \| \| SelfCipher \| 100 \| 0 \| 98 \| 24 \| 80 \| 16 \| Table 4: Validity rate and unsafety rate (out of all queries) of responses generated by different LLMs. Results are reported in the Crimes domain with English ciphers similar to Table 1. The model’s encipher capability is likely to be evoked by the word “Cipher”, while other specific words can also encourage the models to bypass the safety alignment in human languages. We have replaced the term “Cipher Code” in the SelfCipher prompt with other terms, which can also encourage the models to generate unsafe responses (see Table 11 in Appendix). One possible reason is that the safety tuning is mainly conducted in natural language, explicitly instructing the models to communicate in non-natural language can bypass the safety alignment. The effectiveness of SelfCipher implies that LLMs have their own “ciphers”, which is consistent with the recent finding that language models (e.g. Roberta (Liu et al., 2019)) seem to have a “secret language” (Wang et al., 2023b). With the SelfCipher prompt, GPT-4 can communicate with the user via cipher-style strings in a similar format. We list several (query, response) pairs of different styles given the Selfcipher prompt: (“bdohneero agfanro odihghp”), (“1 03013 483 784 67804 768 31076 40 364.”), and (“@@=)) (++!]+-+)==#++]-=!”). Without the SelfCipher prompt, the vanilla GPT-4 only replies with “Sorry, I can’t assist with that.”. Simulated Character Level Ciphers that Never Occur in Pretraining Data Cannot Work The success of human ciphers (e.g. Caesar) and SelfCipher hints that LLMs can learn priors of human ciphers from the pretraining data, based on which they evolve their own ciphers. One research question naturally arises: can simulated ciphers that never occur in pretraining data work in CipherChat? To answer this question, we define a non-existent cipher by utilizing random alphabet mapping and Chinese character substitutions. However, these ciphers cannot work even using as many as 10+ demonstrations. On the other hand, a recent study on jailbreaking demonstrates that self-defined word substitution ciphers can successfully bypass safety alignment (Handa et al., 2024). These findings imply that while the model primarily relies on character-level cipher priors from pretraining to comprehend ciphers, it can also understand the rules of self-defined word-level ciphers. Generalization of CipherChat to General Instructions Some researchers may wonder if CipherChat is designed exclusively for unsafe prompts or if it also functions with general instructions. Table 12 in the Appendix shows the results on the Alpaca benchmark (Taori et al., 2023) of general instructions. Selfcipher works pretty well on Alpaca for both Turbo and GPT-4 models: both the validity and success rates are close to 100%. The results demonstrate the effectiveness and universality of CipherChat across different types of instructions. 5 CONCLUSION AND FUTURE WORK Our systematic study shows that chat in cipher can effectively elicit unsafe information from the powerful GPT-4 model, which has the capability to understand representative ciphers. Our work highlights the necessity of developing safety alignment for non-natural languages to match the capability of the underlying LLMs (e.g. GPT-4). In response to this problem, one promising direction is to implement safety alignment techniques (e.g. SFT, RLHF, and Red Teaming) on enciphered data with necessary cipher instructions. Another interesting direction is to explore the “secret cipher” in LLMs and provide a better understanding of the appealing capability. ETHICS AND BROADER IMPACT This study includes information that could potentially enable individuals to produce harmful content using publicly available LLMs. Although there are inherent risks, we maintain that disclosing this information in its entirety is essential, since we believe that open discussions of weaknesses and limitations are crucial for the advancement of robust future systems. As LLMs become more integrated into various aspects of society, it is essential to understand their safety and potential exploitation. We hope that this research can help to clarify the dangers and foster future research into the safe and reliable deployment of LLMs. ACKNOWLEDGMENT This paper was supported by the National Natural Science Foundation of China (No. 62102340). 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fcSDt7H8kI	What is the QRDQN algorithm baseline in Figure 5? It is not discussed in the paper. What is the difference between $\epsilon$-greedy in Figure 1 and Figure 5? While it is briefly mentioned in the footnote of the supplementary materials, detailed references are not presented.	Boosting Reinforcement Learning with Extremum Experiences Anonymous authors Paper under double-blind review Abstract Reinforcement learning research has achieved high acceleration in its progress starting from the initial installation of deep neural networks as function approximators to learn policies that make sequential decisions in high-dimensional state representation MDPs. While several consecutive barriers have been broken in deep reinforcement learning research (i.e. learning from high-dimensional states, learning purely via self-play), several others still stand. On this line, in our paper we focus on experience collection in high-dimensional complex MDPs and we propose a unique technique based on experiences obtained through extremum actions. Our method provides theoretical basis for efficient experience collection, and further comes with zero additional computational cost while leading to significant sample efficiency gains in deep reinforcement learning training. We conduct extensive experiments in the Arcade Learning Environment with high-dimensional state representation MDPs. We demonstrate that our technique improves the human normalized median scores of Arcade Learning Environment by 248% in the low-data regime. 1 Introduction Utilization of deep neural networks as function approximators enabled learning functioning policies in high-dimensional state representation MDPs (Mnih et al., 2015). Following this initial work, the current line of work trains deep reinforcement learning policies to solve highly complex problems from game solving (Hasselt et al., 2016; Schrittwieser et al., 2020) to self-driving vehicles (Lan et al., 2020). Yet there are still remaining unsolved problems restricting the current capabilities of deep neural policies. One of the main intrinsic open problems in deep reinforcement learning research is experience collection and sample complexity in high-dimensional state representation MDPs. While prior work extensively studied the exploration problem in bandits and tabular reinforcement learning, and proposed various algorithms and techniques optimal to the tabular form or the bandit setting (Kearns & Singh, 2002; Brafman & Tennenholtz, 2002; Karnin et al., 2013; Lu & Roy, 2019), experience collection in deep reinforcement learning remains an open challenging problem while practitioners repeatedly employ quite simple yet effective techniques (i.e. ε-greedy) (Flennerhag et al., 2022; Hasselt et al., 2016; Wang et al., 2016; Hamrick et al., 2020). Despite the provable optimality of these techniques in the tabular or bandit setting, they generally rely strongly on the assumptions of tabular reinforcement learning, and in particular on the ability to record tables of statistical estimates for every state-action pair which have size growing with the number of states times the number of actions. Hence, these assumptions are far from what is being faced in the deep reinforcement learning setting where states and actions can be parametrized by high-dimensional representations. Thus, in high-dimensional complex MDPs, for which deep neural networks are used as function approximators, the efficiency and the optimality of the methods proposed for tabular settings do not transfer well to deep reinforcement learning experience collection. Hence, in deep reinforcement learning research still, naive and standard techniques (e.g. ε-greedy) are preferred over both the optimal tabular techniques and over the particular recent experience collection techniques targeting only high scores for particular games (Mnih et al., 2015; Hasselt et al., 2016; Wang et al., 2016; Anschel et al., 2017; Bellemare et al., 2017; Dabney et al., 2018; Lan et al., 2020; Flennerhag et al., 2022). Sample efficiency in deep neural policies is still one of the main challenging problems restricting research progress in reinforcement learning. The magnitude of the number of samples required to learn and adapt continuously is one of the main limiting factors preventing current state-of-the-art deep reinforcement learning algorithms from being deployed in many diverse settings, but most importantly one of the main challenges that needs to be dealt with on the way to building neural policies that can generalize and adapt continuously in non-stationary environments. In our paper we aim to seek answers for the following questions: - Can we collect experiences in a high-dimensional state representation MDP more efficiently with zero additional computational cost? - Is there a natural theoretical motivation that can be used to design a zero-cost exploration strategy while achieving high sample efficiency? To be able to answer these questions, in our paper we focus on environment interactions in deep reinforcement learning and make the following contributions: - We propose a novel experience collection technique based on minimizing the state-action value function to increase the information gain from each particular experience acquired in the MDP. - We conduct an extensive study in the Arcade Learning Environment 100K benchmark with the state-of-the-art algorithms and demonstrate that our temporal difference learning algorithm improves performance by 248% across the entire benchmark compared to the baseline algorithm. - We demonstrate the efficacy of our proposed MaxMin TD Learning algorithm in terms of sample-efficiency. Our method based on maximizing novel experiences via minimizing the state-action value function reaches approximately to the same performance level as model-based deep reinforcement learning algorithms, without building and learning any model of the environment. 2 BACKGROUND AND PRELIMINARIES The reinforcement learning problem is formalized as a Markov Decision Process (MDP) $ M = \langle S, A, r, \gamma, \rho_0, P \rangle $ that contains a continuous set of states $ s \in S $, a set of discrete actions $ a \in A $, a probability transition function $ T(s, a, s') $ on $ S \times A \times S $, discount factor $ \gamma $, a reward function $ r(s, a) : S \times A \rightarrow \mathbb{R} $ with initial state distribution $ \rho_0 $. A policy $ \pi(s, a) : S \rightarrow P(A) $ in an MDP is a mapping function between states and actions assigning a probability distribution over actions for each state $ s \in S $. The main goal in reinforcement learning is to learn an optimal policy $ \pi $ that maximizes the discounted expected cumulative rewards. \[ R = \mathbb{E}_{a_t \sim \pi(s_t, \cdot)} \sum_t \gamma^t r(s_t, a_t), \] where $ a_t \sim \pi(s_t, \cdot) $. In Q-learning the learned policy is parameterized by a state-action value function $ Q : S \times A \rightarrow \mathbb{R} $, which represents the value of taking action $ a $ in state $ s $. The optimal state-action value function is learnt via iterative Bellman update \[ Q(s_t, a_t) = r(s_t, a_t) + \gamma \sum_{s_t} T(s_t, a_t, s_{t+1}) V(s_{t+1}), \] where $ V(s_{t+1}) = \max_a Q(s_{t+1}, a) $. Let $ a^* $ be the action maximizing the state-action value function, $ a^(s) = \arg \max_a Q(s, a) $, in state $ s $. Once the $ Q $-function is learnt the policy is determined via taking action $ a^(s) = \arg \max_a Q(s, a) $. In deep reinforcement learning, the state space or the action space is large enough that it is not possible to learn and store the state-action values in a tabular form. Thus, the $ Q $-function is approximated via deep neural networks. \[ \theta_{t+1} = \theta_t + \alpha(r(s_t, a_t) + \gamma \max_a Q(s_{t+1}, a; \theta_t) - Q(s_t, a_t; \theta_t)) \nabla_{\theta_t} Q(s_t, a_t; \theta_t) \] In deep double-Q learning, two $ Q $-networks are used to decouple the $ Q $-network deciding which action to take and the $ Q $-network to evaluate the action taken. \[ \theta_{t+1} = \theta_t + \alpha(r(s_t, a_t) + \gamma \max_a Q(s_{t+1}, a; \hat{\theta}_t) - Q(s_t, a_t; \theta_t)) \nabla_{\theta_t} Q(s_t, a_t; \theta_t) \] Current deep reinforcement learning algorithms use $\epsilon$-greedy exploration during training (Wang et al., 2016; Mnih et al., 2015; Hasselt et al., 2016; Hamrick et al., 2020; Flennnerhag et al., 2022). In particular, the $\epsilon$-greedy algorithm takes an action $a_k \sim U(A)$ with probability $\epsilon$ in a given state $s$, i.e. $\pi(s, a_k) = \frac{1}{\|A\|}$, and takes an action $a^* = \arg\max_a Q(s, a)$ with probability $1 - \epsilon$, i.e. $$\pi(s, \arg\max_a Q(s, a)) = 1 - \epsilon + \frac{\epsilon}{\|A\|}$$ While a family of algorithms have been proposed based on counting state visitations (i.e. the number of times action $a$ has been taken in state $s$ by time step $t$) with provable optimal regret bounds using the principal of optimism in the face of uncertainty in the tabular MDP setting, yet incorporating these count-based methods in high-dimensional state representation MDPs requires substantial complexity including training additional deep neural networks to estimate counts or other uncertainty metrics. As a result, many state-of-the-art deep reinforcement learning algorithms still use simple, randomized experience collection methods based on sampling a uniformly random action with probability $\epsilon$ (Mnih et al., 2015; Hasselt et al., 2016; Wang et al., 2016; Hamrick et al., 2020; Flennnerhag et al., 2022), or the injection of random noise via noisy-networks (Hessel et al., 2018). Nonetheless, we still provide comparison to count-based methods in Section 4 and Section 6. 3 Boosting Temporal Difference In deep reinforcement learning the state-action value function is initialized with random weights (Mnih et al., 2015; 2016; Hasselt et al., 2016; Wang et al., 2016; Schaul et al., 2016; Oh et al., 2020; Schrittwieser et al., 2020; Hubert et al., 2021). Thus, in the early phase of the training the $Q$-function will behave more like a random function rather than providing an accurate representation of the optimal state-action values. In particular, early in training the $Q$-function, on average, will assign approximately similar values to states that are similar, and will have little correlation with the immediate rewards. We first formalize this intuition in the following definitions. Definition 3.1 ($\eta$-uninformed $Q$). Let $\eta > 0$. A $Q$-function parameterized by weights $\theta \sim \Theta$ is $\eta$-uninformed if for any state $s \in S$ with $a_{\min} = \arg\min_a Q_\theta(s, a)$ we have $$\|\mathbb{E}_{\theta \sim \Theta}[r(s_t, a_{\min})] - \mathbb{E}_{a \sim U(A)}[r(s_t, a)]\| < \eta.$$ Definition 3.2 ($\delta$-smooth $Q$). Let $\delta > 0$. A $Q$-function parameterized by weights $\theta \sim \Theta$ is $\delta$-smooth if for any state $s \in S$ and action $\hat{a} = \hat{a}(s, \theta)$ with $s' \sim T(s, \hat{a}, \cdot)$ we have $$\|\mathbb{E}_{s' \sim T(s, \hat{a}, \cdot), \theta \sim \Theta}[\max_a Q_\theta(s', a)] - \mathbb{E}_{s' \sim T(s, \hat{a}, \cdot), \theta \sim \Theta}[\max_a Q_\theta(s', a)]\| < \delta$$ where the expectation is over both the random initialization of the $Q$-function weights, and the random transition to state $s' \sim T(s, \hat{a}, \cdot)$. Definition 3.3 (Disadvantage Gap). For a state-action value function $Q_\theta$ the disadvantage gap in a state $s \in S$ is given by $$D(s) = \mathbb{E}_{a \sim U(A), \theta \sim \Theta}[Q_\theta(s, a) - Q_\theta(s, a_{\min})]$$ where $a_{\min} = \arg\min_a Q_\theta(s, a)$. The following proposition captures the intuition that when the $Q$-function on average assigns similar maximum values to consecutive states, choosing the action minimizing the state-action value function will achieve an above-average temporal difference. Proposition 3.4. Let $\eta, \delta > 0$ and suppose that $Q_\theta(s, a)$ is $\eta$-uninformed and $\delta$-smooth. Let $s_t \in S$ be a state, and let $a_{\min}$ be the action minimizing the state-action value in a given state $s_t$, $a_{\min} = \arg\min_a Q_\theta(s_t, a)$. Let $s_{t+1}^{\min} \sim T(s_t, a_{\min}, \cdot)$. Then for an action $a_t \sim U(A)$ with $s_{t+1} \sim T(s_t, a_t, \cdot)$ we have $$\mathbb{E}_{s_{t+1}^{\min} \sim T(s_t, a_{\min}, \cdot), \theta \sim \Theta}[r(s_t, a_{\min}) + \gamma \max_a Q_\theta(s_{t+1}^{\min}, a) - Q_\theta(s_t, a_{\min})]$$ $$> \mathbb{E}_{a_t \sim U(A), s_{t+1} \sim T(s_t, a_t, \cdot), \theta \sim \Theta}[r(s_t, a_t) + \gamma \max_a Q_\theta(s_{t+1}, a) - Q_\theta(s_t, a_t)]$$ $$+ D(s) - 2\delta - \eta.$$ Proof. Since $ Q_\theta(s, a) $ is $ \delta $-smooth we have \[ \mathbb{E}_{s_{t+1} \sim T(s_t, a_{\text{min}}, \cdot), \theta \sim \Theta}[\gamma \max_a Q_\theta(s_{t+1}, a) - Q_\theta(s_t, a_{\text{min}})] \] \[ > \gamma \mathbb{E}_{\theta \sim \Theta}[\max_a Q_\theta(s_t, a)] - \delta - \mathbb{E}_{\theta \sim \Theta}[Q_\theta(s_t, a_{\text{min}})] \] \[ > \gamma \mathbb{E}_{s_{t+1} \sim T(s_t, a_t, \cdot), \theta \sim \Theta}[\max_a Q_\theta(s_{t+1}, a)] - 2\delta - \mathbb{E}_{\theta \sim \Theta}[Q_\theta(s_t, a_{\text{min}})] \] \[ \geq \mathbb{E}_{a_t \sim U(A), s_{t+1} \sim T(s_t, a_t, \cdot), \theta \sim \Theta}[\gamma \max_a Q_\theta(s_{t+1}, a) - Q_\theta(s_t, a_t)] \] \[ + D(s) - 2\delta \] where the last line follows from Definition 3.3. Further, because $ Q_\theta(s, a) $ is $ \eta $-uninformed, \[ \mathbb{E}_{\theta \sim \Theta}[r(s_t, a_{\text{min}})] > \mathbb{E}_{a_t \sim U(A)}[r(s_t, a_t)] - \eta. \] Combining with the previous inequality completes the proof. In words, the proposition shows that the temporal difference achieved by the minimum-value action is above-average by an amount approximately equal to the disadvantage gap. The above argument can be extended to the case where action selection and evaluation in the temporal difference are computed with two different sets of weights $ \theta $ and $ \hat{\theta} $ as in double $ Q $-learning. Definition 3.5 (δ-smoothness for Double-Q). Let $ \delta > 0 $. A pair of $ Q $-functions parameterized by weights $ \theta \sim \Theta $ and $ \hat{\theta} \sim \hat{\Theta} $ are $ \delta $-smooth if for any state $ s \in S $ and action $ \hat{a} = \hat{a}(s, \theta) \in A $ with $ s' \sim T(s, \hat{a}, \cdot) $ we have \[ \left\| \mathbb{E}_{s' \sim T(s, \hat{a}, \cdot), \theta \sim \Theta, \hat{\theta} \sim \hat{\Theta}} \left[ Q_{\hat{\theta}}(s, \arg \max_a Q_\theta(s, a)) \right] - \mathbb{E}_{s' \sim T(s, \hat{a}, \cdot), \theta \sim \Theta, \hat{\theta} \sim \hat{\Theta}} \left[ Q_{\hat{\theta}}(s', \arg \max_a Q_\theta(s', a)) \right] \right\| < \delta \] where the expectation is over both the random initialization of the $ Q $-function weights $ \theta $ and $ \hat{\theta} $, and the random transition to state $ s' \sim T(s, \hat{a}, \cdot) $. With this definition we can then prove that choosing the minimum valued action will lead to a temporal difference that is above-average by approximately $ D(s) $. Proposition 3.6. Let $ \eta, \delta > 0 $ and suppose that $ Q_\theta $ and $ Q_{\hat{\theta}} $ are $ \eta $-uniformed and $ \delta $-smooth. Let $ s_t \in S $ be a state, and let $ a_{\text{min}} = \arg \min_a Q_\theta(s_t, a) $. Let $ s_{t+1}^{\text{min}} \sim T(s_t, a_{\text{min}}, \cdot) $. Then for an action $ a_t \sim U(A) $ with $ s_{t+1} \sim T(s_t, a_t, \cdot) $ we have \[ \mathbb{E}_{s_{t+1} \sim T(s_t, a_t, \cdot), \theta \sim \Theta, \hat{\theta} \sim \hat{\Theta}}[r(s_t, a_{\text{min}}) + \gamma Q_{\hat{\theta}}(s_{t+1}^{\text{min}}, \arg \max_a Q_\theta(s_{t+1}, a)) - Q_\theta(s_t, a_{\text{min}})] \] \[ > \mathbb{E}_{a_t \sim U(A), s_{t+1} \sim T(s_t, a_t, \cdot), \theta \sim \Theta, \hat{\theta} \sim \hat{\Theta}}[r(s_t, a_t) + \gamma Q_{\hat{\theta}}(s_{t+1}, \arg \max_a Q_\theta(s_{t+1}, a)) - Q_\theta(s_t, a_t)] \] \[ + D(s) - 2\delta - \eta \] Proof. Since $ Q_\theta $ and $ Q_{\hat{\theta}} $ are $ \delta $-smooth we have \[ \mathbb{E}_{s_{t+1}^{\text{min}} \sim T(s_t, a_{\text{min}}, \cdot), \theta \sim \Theta, \hat{\theta} \sim \hat{\Theta}}[\gamma Q_{\hat{\theta}}(s_{t+1}^{\text{min}}, \arg \max_a Q_\theta(s_{t+1}, a)) - Q_\theta(s_t, a_{\text{min}})] \] \[ > \mathbb{E}_{s_{t+1}^{\text{min}} \sim T(s_t, a_{\text{min}}, \cdot), \theta \sim \Theta, \hat{\theta} \sim \hat{\Theta}}[\gamma Q_\theta(s_t, \arg \max_a Q_\theta(s_t, a)) - Q_\theta(s_t, a_{\text{min}})] - \delta \] \[ > \mathbb{E}_{s_{t+1} \sim T(s_t, a_t, \cdot), \theta \sim \Theta, \hat{\theta} \sim \hat{\Theta}}[\gamma Q_{\hat{\theta}}(s_{t+1}, \arg \max_a Q_\theta(s_{t+1}, a)) - Q_\theta(s_t, a_{\text{min}})] - 2\delta \] \[ \geq \mathbb{E}_{s_{t+1} \sim T(s_t, a_t, \cdot), \theta \sim \Theta, \hat{\theta} \sim \hat{\Theta}}[\gamma Q_{\hat{\theta}}(s_{t+1}, \arg \max_a Q_\theta(s_{t+1}, a)) - Q_\theta(s_t, a_t)] \] \[ + D(s) - 2\delta \] where the last line follows from Definition 3.3. Further, because $ Q_\theta $ and $ Q_{\hat{\theta}} $ are $ \eta $-uniformed, \[ \mathbb{E}_{\theta \sim \Theta, \hat{\theta} \sim \hat{\Theta}}[r(s_t, a_{\text{min}})] > \mathbb{E}_{a_t \sim U(A)}[r(s_t, a_t)] - \eta. \] Combining with the previous inequality completes the proof. At first, the results in Propositions 3.4 and 3.6 might appear counterintuitive. The fact that the Q-function is δ-smooth and η-uniformed seem like properties of a random function. Thus, taking the minimum Q-value action should be approximately equivalent to taking a uniform random action. However, Propositions 3.4 and 3.6 show that the temporal difference achieved by taking the minimum action is larger than that of a random action by an amount equal to the disadvantage gap $ D(s) $. In order to reconcile these two statements it is useful at this point to look at the limiting case of the Q function at initialization. In particular, the following proposition shows that, at initialization, the distribution of the minimum value action in a given state is uniform by itself, but is constant once we condition on the weights $ \theta $. Proposition 3.7. Let $ \theta $ be the random initial weights for the Q-function. For any state $ s \in S $ let $ a_{\text{min}}(s) = \arg \min_{a' \in A} Q_\theta(s, a') $. Then for any $ a \in A $ \[ \mathbb{P}_{\theta \sim \Theta} \left[ \arg \min_{a' \in A} Q_\theta(s, a') = a \right] = \frac{1}{\|A\|} \] i.e., the distribution $ \mathbb{P}_{\theta \sim \Theta}[a_{\text{min}}(s)] $ is uniform. Simultaneously, the conditional distribution $ \mathbb{P}_{\theta \sim \Theta}[a_{\text{min}}(s) \| \theta] $ is constant. Proof. Since $ Q_\theta(s, \cdot) $ is a random function (given the random choice of $ \theta $), each action $ a \in A $ is equally likely to be assigned the minimum Q-value in state $ s $. Thus, \[ \mathbb{P}_{\theta \sim \Theta} \left[ \arg \min_{a' \in A} Q_\theta(s, a') = a \right] = \frac{1}{\|A\|}. \] However, given the value of $ \theta $, the value of $ a_{\text{min}}(s) $ is uniquely determined because \[ a_{\text{min}}(s) = \arg \min_{a \in A} Q_\theta(s, a). \] Therefore, the distribution of $ a_{\text{min}}(s) $ conditional on $ \theta $ is constant. $ \square $ This implies that, in states whose Q-values have not changed drastically from initialization, taking the minimum action is almost equivalent to taking a random action. However, while the action chosen early on in training is almost uniformly random when only considering the current state, it is at the same time completely determined by the current value of the weights $ \theta $. The temporal difference is also determined by the weights $ \theta $. Thus while the marginal distribution on actions taken is uniform, the temporal difference when taking the minimum action is quite different than from the case where an independently random action is chosen. In particular, in expectation over the random initialization $ \theta \sim \Theta $, the temporal difference is higher when taking the minimum value action than that of a random action as demonstrated in Section 3. The main objective of our method is to increase the information gained from each experience via taking the actions that minimize the state-action value function. While minimization of the Q-function may initially be regarded as counterintuitive, Section 3 provides the exact theoretical justification on how taking actions that minimize the state-action value function results in higher temporal difference for the corresponding state transitions. Algorithm 1 summarizes our proposed algorithm MaxMin TD Learning based on minimizing the state-action value function as described in detail in Section 3. Note that populating the experience replay buffer and learning are happening simultaneously with different rates. ### 4 Motivating Example As a motivating example we consider the chain MDP which consists of a chain of $ n $ states $ s \in S = \{1, 2, \cdots, n\} $ each with four actions. Each state $ i $ has one action that transitions the agent up the chain by one step to state $ i + 1 $, one action that transitions the agent to state 2, one action that transitions the agent to state 3, and one action which resets the agent to state 1 at the beginning of the chain. All transitions have reward zero, except for the last transition returning the agent to the beginning from the $ n $-th state. Thus, when started from the first state in the chain, the agent must learn a policy that takes $ n - 1 $ consecutive steps up the chain, and then one final step to reset and get the reward. For the chain MDP, we compare standard approaches to exploration in tabular Q-learning with our method MaxMin TD Learning based on minimization of the state-action values. In particular we compare Algorithm 1: MaxMin TD Learning Input: In MDP $ \mathcal{M} $ with $ \gamma \in (0, 1] $, $ s \in S $, $ a \in A $ with $ Q_\theta(s, a) $ function parametrized by $ \theta $, $ B $ experience replay buffer, $ \epsilon $ exploration parameter, $ N $ is the training learning steps. Populating Experience Replay Buffer: for $ s_t $ in $ e $ do Sample $ \kappa \sim U(0, 1) $ if $ \kappa < \epsilon $ then $ a_{\text{min}} = \arg \min_a Q(s_t, a) $ $ B \leftarrow (r(s_t, a_{\text{min}}), s_t, s_{t+1}, a_{\text{min}}) $ else $ a^* = \arg \max_a Q(s_t, a) $ $ B \leftarrow (r(s_t, a^), s_t, s_{t+1}, a^) $ end if end for Learning: for $ n $ in $ N $ do Update with probability $ \epsilon $: $ TD = r(s_t, a_{\text{min}}) + \gamma \max_a Q(s_{t+1}, a) - Q(s_t, a_{\text{min}}) $ Update with probability $ 1 - \epsilon $: $ TD = r(s_t, a^) + \gamma \max_a Q(s_{t+1}, a) - Q(s_t, a^) $ end for $ \nabla L(TD) $ Figure 1: Learning curves in the chain MDP with our proposed algorithm MaxMin TD Learning, the canonical algorithm $ \epsilon $-greedy and the UCB algorithm with variations in $ \epsilon $. Our method MaxMin TD Learning with both the $ \epsilon $-greedy action selection method, and the upper confidence bound (UCB) method. In more detail, in the UCB method the number of training steps $ t $, and the number of times $ N_t(s, a) $ that each action $ a $ has been taken in state $ s $ by step $ t $ are recorded. Furthermore, the action $ a \in A $ selection is determined as follows: \[ a^{\text{UCB}} = \arg \max_{a \in A} Q(s, a) + 2 \sqrt{\frac{\log t}{N_t(s, a)}}. \] In a given state $ s $ if $ N(s, a) = 0 $ for any action $ a $, then an action is sampled uniformly at random from the set of actions $ a' $ with $ N(s, a') = 0 $. For the experiments reported in our paper the length of the chain is set to $ n = 10 $. The $ Q $-function is initialized by independently sampling each state-action value from a normal distribution with $ \mu = 0 $ and $ \sigma = 0.1 $. In each iteration we train the agent using $ Q $-learning for 100 steps, and then evaluate the reward obtained by the argmax policy using the current $ Q $-function for 100 steps. Note that the maximum achievable reward in 100 steps is 10. Figure 1 reports the learning curves for each method with varying $ \epsilon \in [0.15, 0.25] $ with step size 0.025. The results in Figure 1 demonstrate that our method converges more quickly to the optimal policy than either of the standard approaches. 5 Large Scale Experimental Results The experiments are conducted in the Arcade Learning Environment (ALE) (Bellemare et al., 2013). The Double-Q Network (Hasselt et al., 2016) initially proposed by (van Hasselt, 2010) is trained with prioritized experience replay (Schaul et al., 2016) without the dueling architecture with its original version (Hasselt et al., 2016). The experiments are conducted both in the 100K Arcade Learning Environment benchmark (van Hasselt et al., 2019), and the canonical version with 200 million frame training. Note that the 100K Arcade Learning Environment benchmark is an established baseline proposed to measure sample efficiency in deep reinforcement learning research. The ALE 100K benchmark contains 26 different Arcade Learning Environment games. The policies are evaluated after 100000 environment interactions. All of the policies in the experiments are trained over 5 random seeds. The hyperparameters and the architecture details are reported in the supplementary material. All of the results in the paper are reported with the standard error of the mean. The human normalized Figure 2: Human normalized scores median and 80th percentile over all games in the Arcade Learning Environment (ALE) 100K benchmark for MaxMin TD Learning algorithm and the canonical exploration algorithm $\epsilon$-greedy. Figure 3: Temporal difference for our proposed algorithm MaxMin TD Learning and the canonical $\epsilon$-greedy algorithm in the Arcade Learning Environment 100K benchmark. Dashed lines report the temporal difference for the $\epsilon$-greedy algorithm and solid lines report the temporal difference for the MaxMin TD Learning algorithm. Colors indicate games. Table 1: Human normalized scores median and 20th percentile across all of the games in the Arcade Learning Environment 100K benchmark for MaxMin TD Learning, $\epsilon$-greedy and NoisyNetworks. \| Method \| MaxMin TD Learning \| $\epsilon$-greedy \| NoisyNetworks \| \|-------------------------\|--------------------\|-------------------\|---------------\| \| Human Normalized Median \| 0.0927±0.0050 \| 0.0377±0.0031 \| 0.0457±0.0035 \| \| 20th Percentile \| 0.0145±0.0003 \| 0.0056±0.0017 \| 0.0102±0.0018 \| \| 80th Percentile \| 0.3762±0.0137 \| 0.2942±0.0233 \| 0.1913±0.0144 \| Scores are computed as, $$HN = \frac{\text{Score}_{\text{agent}} - \text{Score}_{\text{random}}}{\text{Score}_{\text{human}} - \text{Score}_{\text{random}}}$$ For completeness we also report several results with 200 million frame training (i.e. 50 million environment interactions). In particular, Figure 4 demonstrates the learning curves for our proposed algorithm MaxMin TD Learning and the original version of the DDQN algorithm with $\epsilon$-greedy training (Hasselt et al., [2016]). In the large data regime we observe that while in some MDPs our proposed method MaxMin TD Learning that focuses on experience collection with novel temporal difference boosting via minimizing the state-action values converges faster, in other MDPs MaxMin TD Learning simply converges to a better policy. More concretely, while the learning curves of StarGunner, Bowling, JamesBond and BankHeist games in Figure 4 demonstrate the faster convergence rate of our proposed algorithm MaxMin TD Learning, the learning curves of the JamesBond, Amidar, BankHeist, Surround, Gravitar and Tennis games demonstrate that our experience collection technique not only increases the sample efficiency in deep reinforcement learning, but also results in learning a policy that is more close to optimal compared to learning a policy with the original method used in the DDQN algorithm. Additionally, we also compare our proposed MaxMin TD Learning algorithm with NoisyNetworks as referred to in Section 2. Table 1 further demonstrates that the MaxMin TD Learning algorithm achieves significantly better performance results compared to NoisyNetworks. Furthermore, note that NoisyNetworks includes adding layers in the $Q$-network to increase exploration. However, this increases the number of parameters that have been added in the training process; thus, introducing additional cost to increase exploration. Table 1 reports results of human normalized median scores, 20th percentile, and 80th percentile for the Arcade Learning Environment 100K benchmark. Thus, Figure 4: The learning curves of StarGunner, Bowling, Surround, BankHeist, JamesBond, Amidar, Gravitar and Tennis with our proposed method MaxMin TD Learning and the $\epsilon$-greedy algorithm in the Arcade Learning Environment with 200 million frame training. Table 1 demonstrates that our proposed MaxMin TD Learning algorithm improves on the performance of the canonical algorithm $\epsilon$-greedy by 248% and NoisyNetworks by 204%. We further compare our proposed MaxMin TD Learning algorithm with another baseline algorithm QRDQN. In particular, Figure 5 reports results of human normalized median scores and 80th percentile over all of the games of the Arcade Learning Environment (ALE) in the low-data regime. These results once more demonstrate that the performance obtained by the MaxMin TD Learning algorithm is approximately double the performance achieved by the canonical experience collection techniques. As the results reported demonstrate, the MaxMin TD Learning algorithm achieves substantial sample-efficiency with zero-additional cost across many algorithms and different sample-complexity regions over canonical baseline alternatives. 6 INVESTIGATING THE TEMPORAL DIFFERENCE The original justification for exploring with the minimum $Q$-value action, is that taking this action tends to result in transitions with higher temporal difference. The theoretical analysis from Proposition 3.4 indicates that, when the $Q$ function is $\delta$-smooth and $\eta$-uninformed, taking the minimum value action results in an increase in the temporal difference proportional to the disadvantage gap. In particular, Proposition 3.4 states that the temporal difference achieved when taking the minimum $Q$-value action in state $s$ exceeds the average temporal difference over a uniform random action by $\mathcal{D}(s) - 2\delta - \eta$. In order to evaluate how well the theoretical prediction matches reality, in this section we provide empirical measurements of the temporal difference in our experiments. To measure the change in the temporal difference when taking the minimum action versus the average action, we compare the temporal difference obtained by MaxMin TD Learning exploration with that obtained by $\epsilon$-greedy exploration. In more detail, during training, for each batch $\Lambda$ of transitions of the form $(s_t, a_t, s_{t+1})$ we record, the temporal difference $$\mathcal{T}D = \mathbb{E}_{(s_t, a_t, s_{t+1}) \sim \Lambda} [\mathcal{T}D(s_t, a_t, s_{t+1})]$$ $$= \mathbb{E}_{(s_t, a_t, s_{t+1}) \sim \Lambda} [r(s_t, a_t) + \gamma \max_a Q_\theta(s_{t+1}, a) - Q_\theta(s_t, a_t)].$$ The results reported in Figure 3 and Figure 6 further confirm the theoretical predictions made via Definition 3.2 and Proposition 3.4. In addition to the results for individual games reported in Figure 3, we compute a normalized measure of the gain in temporal difference achieved when using MaxMin TD Learning exploration and plot the median across games. We define the normalized $\mathcal{T}D$ gain to be $$\text{Normalized } \mathcal{T}D \text{ Gain} = 1 + \frac{\mathcal{T}D_{\text{method}} - \mathcal{T}D_{\epsilon-\text{greedy}}}{\|\mathcal{T}D_{\epsilon-\text{greedy}}\|}$$ where $\mathcal{T}D_{\text{method}}$ and $\mathcal{T}D_{\epsilon-\text{greedy}}$ are the temporal difference for any given exploration method and $\epsilon$-greedy respectively. The leftmost and middle plot of Figure 6 report the median across all games of the normalized $\mathcal{T}D$ gain results for MaxMin TD Learning and NoisyNetworks in the Arcade Figure 5: Human normalized scores median and 80th percentile over all games in the Arcade Learning Environment (ALE) 100K benchmark for MaxMin TD Learning algorithm and the canonical exploration algorithm $\epsilon$-greedy for QRDQN. Figure 6: Left and Middle: Normalized temporal difference $TD$ gain median across all games in the Arcade Learning Environment 100K benchmark for MaxMin TD Learning and NoisyNetworks. Right: Temporal difference $TD$ when exploring chain MDP with Upper Confidence Bound (UCB) method, $\epsilon$-greedy and our proposed algorithm MaxMin TD Learning. Learning Environment 100K benchmark. Note that, consistent with the predictions of Proposition 3.4, the median normalized temporal difference gain for MaxMin TD Learning is up to 25 percent larger than that of $\epsilon$-greedy. The results for NoisyNetworks demonstrate that alternate exploration methods lack this positive bias relative to the uniform random action. The fact that, as demonstrated in Table I, MaxMin TD Learning significantly outperforms noisy networks in the low-data regime is further evidence of the advantage the positive bias in temporal difference confers. The rightmost plot of Figure 6 reports $TD$ for the motivating example of the chain MDP. As in the large-scale experiments, prior to convergence MaxMin TD Learning exhibits a notably larger temporal difference relative to the canonical baseline methods. 7 CONCLUSION In our study we focus on the following questions in deep reinforcement learning: (i) Is it possible to increase sample efficiency in deep reinforcement learning in a computationally efficient way with conceptually simple choices?, (ii) What is the theoretical motivation of our proposed perspective, simply minimizing the state-action value function in early training, that results in one of the most computationally efficient ways to explore in deep reinforcement learning? and, (iii) How would the theoretically motivated simple idea transfer to large scale experiments in high-dimensional state representation MDPs? To be able to answer these questions we propose a novel, theoretically motivated method with zero additional computational cost based on following actions that minimize the state-action value function to explore in deep reinforcement learning. We demonstrate theoretically that our method MaxMin TD Learning based on minimization of the state-action value results in higher temporal difference, and thus creates novel transitions in exploration with more unique experience collection. Following the theoretical motivation we initially show in a toy example in the chain MDP setup that our proposed method MaxMin TD Learning results in achieving higher sample efficiency. Then, we expand this intuition and conduct large scale experiments in the Arcade Learning Environment, and demonstrate that our proposed method MaxMin TD Learning increases the performance on the Arcade Learning Environment 100K benchmark by 248%. REFERENCES Oron Anschel, Nir Baram, and Nahum Shimkin. 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lK0WxHeups	First math display in Section 1.3.2: this bound should mentioned somewhere that $b > n$ isn't possible and $b = n$ reduces to full-batch gradient descent. As a result, it is not always feasible to select the batch-size to minimize the oracle complexity.	Iteration and Stochastic First-order Oracle Complexities of Stochastic Gradient Descent using Constant and Decaying Learning Rates Anonymous authors Paper under double-blind review Abstract The performance of stochastic gradient descent (SGD), which is the simplest first-order optimizer for training deep neural networks, depends on not only the learning rate but also the batch size. They both affect the number of iterations and the stochastic first-order oracle (SFO) complexity needed for training. In particular, the previous numerical results indicated that, for SGD using a constant learning rate, the number of iterations needed for training decreases when the batch size increases, and the SFO complexity needed for training is minimized at a critical batch size and increases once the batch size exceeds that size. This paper studies the relationship between batch size and the iteration and the SFO complexities needed for nonconvex optimization in deep learning with SGD using constant/decay learning rates. We show that SGD using a step-decay learning rate and a small batch size reduces the SFO complexity to find a local minimizer of a loss function. We also provide numerical comparisons of SGD with the existing first-order optimizers and show the usefulness of SGD using a step-decay learning rate and a small batch size. 1 Introduction 1.1 Background First-order optimizers can train deep neural networks by minimizing loss functions called the expected and empirical risk. They use stochastic first-order derivatives (stochastic gradients), which are estimated from the full gradient of the loss function. The simplest first-order optimizer is stochastic gradient descent (SGD) (Robbins & Monro, 1951; Zinkevich, 2003; Nemirovski et al., 2009; Ghadimi & Lan, 2012; 2013) and it has a number of variants, such as momentum methods (Polyak, 1964; Nesterov, 1983) and adaptive methods including adaptive gradient (AdaGrad) (Duchi et al., 2011), root mean square propagation (RMSProp) (Tieleman & Hinton, 2012), adaptive moment estimation (Adam) (Kingma & Ba, 2015), adaptive mean square gradient (AMSGrad) (Reddi et al., 2018), and Adam with decoupled weight decay (AdamW) (Loshchilov & Hutter, 2019). SGD can be applied to nonconvex optimization (Vaswani et al., 2019; Fehrman et al., 2020; Chen et al., 2020; Scaman & Malherbe, 2020; Loizou et al., 2021; Arjevani et al., 2023; Khaled & Richtárik, 2023), where its performance strongly depends on the learning rate $\alpha_k$. For example, under the bounded variance assumption, SGD using a constant learning rate $\alpha_k = \alpha$ satisfies that $\frac{1}{K} \sum_{k=0}^{K-1} \\| \nabla f(\theta_k) \\|^2 = O\left( \frac{1}{K} \right) + \sigma^2$ (Scaman & Malherbe, 2020, Theorem 12) and SGD using a decaying learning rate (i.e., $\alpha_k \to 0$) satisfies that $\frac{1}{K} \sum_{k=0}^{K-1} \mathbb{E}[\\| \nabla f(\theta_k) \\|^2] = O\left( \frac{1}{\sqrt{K}} \right)$ (Scaman & Malherbe, 2020, Theorem 11), where $(\theta_k)_{k \in \mathbb{N}}$ is the sequence generated by SGD to find a local minimizer of $f$, $K$ is the number of iterations, and $\sigma^2$ is the upper bound of the variance. The performance of SGD also depends on the batch size $b$. Convergence analyses of SGD in (Jain et al., 2018; Cotter et al., 2011; Chen et al., 2020; Arjevani et al., 2023) indicated that SGD with a decaying learning rate and large batch size converges to a local minimizer of the loss function. In (Smith et al., 2018), it was numerically shown that using an enormous batch leads to reductions in the number of parameter updates and model training time. 1.2 Motivation The previous numerical results in (Shallue et al., 2019) indicated that, for SGD using constant/linear decay learning rates, the number of iterations $K$ needed to train a deep neural network decreases when the batch size $b$ increases. Motivated by the numerical results in (Shallue et al., 2019), we decided to clarify the iteration complexity of SGD using a constant/decay learning rate needed to train a deep neural network in theory. The theoretical performance measure of SGD is $\min_{k \in [0:K-1]} \mathbb{E}[\\|\nabla f(\theta_k)\\|] \leq \epsilon$, where $\epsilon > 0$ is the precision and $[0:K-1] := \{0, 1, \ldots, K-1\}$, which was used in the previous theoretical analyses of SGD. If SGD is an $\epsilon$-approximation $\min_{k \in [0:K-1]} \mathbb{E}[\\|\nabla f(\theta_k)\\|] \leq \epsilon$, then SGD can train a deep neural network in $K$ iterations. In addition, the numerical results in (Shallue et al., 2019) indicated an interesting fact wherein diminishing returns exist beyond a critical batch size; i.e., the number of iterations needed to train a deep neural network does not strictly decrease beyond the critical batch size. Here, we define the stochastic first-order oracle (SFO) complexity as $N := Kb$, where $K$ is the number of iterations needed to train a deep neural network and $b$ is the batch size, as stated above. The deep neural network model uses $b$ gradients of the loss functions per iteration. The model has a stochastic gradient computation cost of $N = Kb$. From the numerical results in (Shallue et al., 2019, Figures 4 and 5), we can conclude that using the critical batch size $b^$ (if it exists) is useful for SGD, since the SFO complexity $N(b)$ is minimized at $b = b^$ and the SFO complexity increases once the batch size exceeds $b^$. Hence, on the basis of the first motivation stated above, we decided to clarify the SFO complexities of SGD using constant/decay learning rates needed to achieve an $\epsilon$-approximation. 1.3 Contribution 1.3.1 Upper bound of theoretical performance measure To clarify the iteration and SFO complexities of SGD needed to achieve an $\epsilon$-approximation, we first give upper bounds of $\min_{k \in [0:K-1]} \mathbb{E}[\\|\nabla f(\theta_k)\\|^2]$ for SGD generating the sequence $(\theta_k)_{k \in \mathbb{N}}$ using constant/decay learning rates, as indicated in Table 1 (see Theorem 3.1 for the definitions of $C_1$ and $D_2$). The aim of this paper is to show that SGD is an $\epsilon$-approximation $\min_{k \in [0:K-1]} \mathbb{E}[\\|\nabla f(\theta_k)\\|^2] \leq \epsilon^2$. Hence, it is desirable that the upper bounds of $\min_{k \in [0:K-1]} \mathbb{E}[\\|\nabla f(\theta_k)\\|^2]$ become small. Table 1 indicates that the upper bounds become small when the number of iterations and batch size are large. In particular, it shows that a step-decay learning rate (the “Step Decay” row) may perform better other learning rates in the sense of minimizing the upper bound of $\min_{k \in [0:K-1]} \mathbb{E}[\\|\nabla f(\theta_k)\\|^2]$. For example, if we set small batch size, such as $b = 2^{1}, 2^{2}$, SGD using a step-decay learning rate has the convergence rate, $\min_{k \in [0:K-1]} \mathbb{E}[\\|\nabla f(\theta_k)\\|^2] = O(1/K)$, which is better than the convergence rate $O(1/K + C_2) = O(1/K + \sigma^2)$ of SGD using a constant learning rate, where $\sigma^2$ is the upper bound of the variance. The table also indicates that the convergence of SGD strongly depends on the batch size, since the variance terms (including $\sigma^2$ and $b$; see Theorem 3.1 for the definitions of $C_2$, $D_2$, and $D_3$) in the upper bounds of $\min_{k \in [0:K-1]} \mathbb{E}[\\|\nabla f(\theta_k)\\|^2]$ decrease as the batch size becomes larger. 1.3.2 Optimal batch size to reduce SFO complexity Section 1.3.1 showed that using large batch sizes is appropriate for SGD in the sense of minimizing the upper bound of the performance measure. We are interested in finding appropriate batch sizes from the viewpoint of the computation cost of SGD. This is because the SFO complexity increases when batch sizes are sufficiently large. As indicated in Section 1.2, the critical batch size $b^$ minimizes the SFO complexity, $N = Kb$. Hence, we will investigate the properties of the SFO complexity $N = Kb$ needed to achieve an $\epsilon$-approximation. For example, let us consider SGD using a constant learning rate. Then, from the “Upper Bound” row in Table 1, we have that $$\min_{k \in [0:K-1]} \mathbb{E}[\\|\nabla f(\theta_k)\\|^2] \leq \frac{C_1}{K} + \frac{C_2}{b} \leq \epsilon^2.$$ We can check that the number of iterations, $K(b) := \frac{C_1 b}{\epsilon^2 b - C_2}$, needed to achieve an $\epsilon$-approximation is monotone decreasing and convex with respect to the batch size $b$ (Theorem 3.2). Then, we have Table 1: Upper bounds of $\min_{k \in [0; K-1]} \mathbb{E}[\\|\nabla f(\theta_k)\\|^2]$ for SGD using constant/decay learning rates and optimal batch sizes to minimize the SFO complexities ($C_1$ and $D_1$ are positive constants, $K$ is the number of iterations, $b$ is the batch size, and $L$ is the Lipschitz constant of $\nabla f$) \| Learning Rate \| Upper Bound \| Optimal Batch Size \| \|---------------\|-------------\|--------------------\| \| Constant $\alpha \in (0, \frac{2}{L})$ \| $\frac{C_1}{K} + \frac{C_2}{b}$ \| $\frac{2C_2}{\epsilon^2}$ \| \| $a \in (0, \frac{1}{2})$ \| $\frac{D_1}{K^a} + \frac{D_2}{(1-2a)K^ab}$ \| $\frac{(1-a)D_2}{a(1-2a)D_1}$ \| \| Decay $a = \frac{1}{2}$ \| $\frac{D_1}{\sqrt{K}} + \left(\frac{1}{\sqrt{K}} + 1\right) \frac{D_2}{b}$ \| Small Batch Size \| \| $\alpha_k = \frac{1}{(k+1)^a}$ $a \in (\frac{1}{2}, 1)$ \| $\frac{D_1}{K^{1-a}} + \frac{2aD_2}{(2a-1)K^{1-a}b}$ \| $\frac{2a^2D_2}{(1-a)(2a-1)D_1}$ \| \| Step Decay $\alpha_k \geq \alpha$ \| $\frac{D_1}{\alpha K} + \frac{D_3}{\alpha K b}$ \| Small Batch Size \| that $K(b) \geq \inf \{K : \min_{k \in [0; K-1]} \mathbb{E}[\\|\nabla f(\theta_k)\\|] \leq \epsilon\}$, where SGD using the batch size $b$ generates $(\theta_k)_{k=0}^{K-1}$. Moreover, we find that the SFO complexity is $N(b) = K(b)b = \frac{C_1b^2}{\epsilon^2 - C_2}$. The convexity of $N(b) = \frac{C_1b^2}{\epsilon^2 - C_2}$ (Theorem 3.3) ensures that a critical batch size $b^* = \frac{2C_2}{\epsilon^2 - C_2}$ whereby $N'(b^) = 0$ exists such that $N(b)$ is minimized at $b^$ (see the “Optimal Batch Size” row in Table 1). A similar discussion guarantees the existence of a critical batch size for SGD using a decaying learning rate $\alpha_k = \frac{1}{(k+1)^a}$, where $a \in (0, \frac{1}{2})$ or $a \in (\frac{1}{2}, 1)$ (see the “Optimal Batch Size” row in Table 1). Meanwhile, for a decaying learning rate $\alpha_k = \frac{1}{\sqrt{k+1}}$ or a step-decay learning rate, although $N(b)$ is convex with respect to $b$, we have that $N'(b) > 0$ for all $b > 0$ (Theorem 3.3(iii)). Hence, for these two cases, a critical batch size $b^$ defined by $N'(b^) = 0$ does not exist. Accordingly, small batch sizes are appropriate for a decaying learning rate $\alpha_k = \frac{1}{\sqrt{k+1}}$ or a step-decay learning rate in the sense of minimizing the SFO complexities. Accordingly, we will define the optimal batch size (in the sense of minimizing the SFO complexity) by $$ \text{Optimal Batch Size } b^* = \begin{cases} \text{Critical Batch Size } b^* & \text{if } N'(b^) = 0 \\ \text{Small Batch Size} & \text{if } N'(b) > 0 \text{ for all } b > 0. \end{cases} $$ Then, we have that $N(b^) \geq \inf \{N : \min_{k \in [0; K-1]} \mathbb{E}[\\|\nabla f(\theta_k)\\|] \leq \epsilon\}$, where SGD using the batch size $b^$ generates $(\theta_k)_{k=0}^{K-1}$. ### 1.3.3 Iteration and SFO Complexities Let $\mathcal{F}(n, \Delta, L)$ be an $L$–smooth function class with $f := \frac{1}{n} \sum_{i=1}^{n} f_i$ and $f(\theta_0) - f_ \leq \Delta$ (see (C1)) and let $\mathcal{O}(b, \sigma^2)$ be a stochastic first-order oracle class (see (C2) and (C3)). The iteration complexity $K_\epsilon$ (Arjevani et al., 2023, (7)) and the SFO complexity $N_\epsilon$ of SGD generating $\theta_k(f, O) = \theta_k$ ($f \in \mathcal{F}(n, \Delta, L), O \in \mathcal{O}(b, \sigma^2)$) needed to achieve an $\epsilon$–approximation are defined by $$ K_\epsilon(n, b, \Delta, L, \sigma^2) := \sup_{O \in \mathcal{O}(b, \sigma^2)} \sup_{f \in \mathcal{F}(n, \Delta, L)} \inf \left\{ K : \min_{k \in [0; K-1]} \mathbb{E}[\\|\nabla f(\theta_k)\\|] \leq \epsilon \right\}, $$ $$ N_\epsilon(n, b, \Delta, L, \sigma^2) := \sup_{O \in \mathcal{O}(b, \sigma^2)} \sup_{f \in \mathcal{F}(n, \Delta, L)} \inf \left\{ N : \min_{k \in [0; K-1]} \mathbb{E}[\\|\nabla f(\theta_k)\\|] \leq \epsilon \right\}. $$ Table 2 summarizes the iteration and SFO complexities (see also Theorem 3.4). It indicates that using a step-decay learning rate reduces the iteration and SFO complexities. However, since the positive constants, such as $C_1$ and $D_1$, depend on the learning rate, we need to compare numerically the performances of SGD using constant/decay learning rates. Moreover, we also need to compare the existing first-order optimizers with SGD using a step-decay learning rate to verify its usefulness. Section 4 describes numerical comparisons showing that SGD using a step-decay learning rate and small batch size performs better than the existing first-order optimizers. Table 2: Iteration and SFO complexities of SGD using constant/decay learning rates needed to achieve an $\epsilon$-approximation (The optimal batch sizes defined as in (1) are used to compute $N_\epsilon$) \| Learning Rate \| Iteration Complexity $K_\epsilon$ \| SFO Complexity $N_\epsilon(n, b^, \Delta, L, \sigma^2)$ \| \|---------------\|----------------------------------\|--------------------------------------------------------\| \| Constant $\alpha \in (0, \frac{2}{L})$ \| $O\left(\frac{1}{\epsilon^2}\right) = \sup_{f, O} K(b)$ \| $O\left(\frac{1}{\epsilon^4}\right) = \sup_{f, O} \frac{4C_1C_2}{\epsilon^4}$ \| \| $a \in (0, \frac{1}{2})$ \| $O\left(\frac{1}{\epsilon^{\frac{3}{2}}}\right) = \sup_{f, O} K(b)$ \| $O\left(\frac{1}{\epsilon^{\frac{5}{2}}}\right) = \sup_{f, O} \frac{(1-a)^{1-\frac{1}{a}}D_2}{a(1-2a)D_1^{1-\frac{1}{a}}\epsilon^{\frac{2}{a}}}$ \| \| Decay $a = \frac{1}{2}$ \| $O\left(\frac{1}{\epsilon^4}\right) = \sup_{f, O} K(b)$ \| $O\left(\frac{1}{\epsilon^4}\right) = \sup_{f, O} \left(\frac{D_1 + D_2}{\epsilon^2 - D_2}\right)^2$ \| \| $\alpha_k = \frac{1}{k+1}$, $a \in (\frac{1}{2}, 1)$ \| $O\left(\frac{1}{\epsilon^{\frac{3}{2}-\frac{1}{a}}}\right) = \sup_{f, O} K(b)$ \| $O\left(\frac{1}{\epsilon^{\frac{5}{2}-\frac{1}{a}}}\right) = \sup_{f, O} \frac{2a^{2-\frac{1}{a}}(1-a)^{-1}D_2}{(2a-1)D_1^{-\frac{1}{a}}\epsilon^{\frac{2}{a}}}$ \| \| Step Decay $\alpha_k \geq \alpha$ \| $O\left(\frac{1}{\epsilon^2}\right) = \sup_{f, O} K(b)$ \| $O\left(\frac{1}{\epsilon^2}\right) = \sup_{f, O} \frac{D_1 + D_2}{\alpha \epsilon^2}$ \| ## 2 NONCONVEX OPTIMIZATION AND SGD ### 2.1 NONCONVEX OPTIMIZATION IN DEEP LEARNING Let $\mathbb{R}^d$ be a $d$-dimensional Euclidean space with inner product $\langle x, y \rangle := x^\top y$ inducing the norm $\\|x\\|$ and $\mathbb{N}$ be the set of nonnegative integers. Define $[0 : n] := \{0, 1, \ldots, n\}$ for $n \geq 1$. Let $(x_k)_{k \in \mathbb{N}}$ and $(y_k)_{k \in \mathbb{N}}$ be positive real sequences and let $x(\epsilon), y(\epsilon) > 0$, where $\epsilon > 0$. $O$ denotes Landau’s symbol; i.e., $y_k = O(x_k)$ if there exist $c > 0$ and $k_0 \in \mathbb{N}$ such that $y_k \leq cx_k$ for all $k \geq k_0$, and $y(\epsilon) = O(x(\epsilon))$ if there exists $c > 0$ such that $y(\epsilon) \leq cx(\epsilon)$. Given a parameter $\theta \in \mathbb{R}^d$ and a data point $z$ in a data domain $Z$, a machine learning model provides a prediction whose quality is measured by a differentiable nonconvex loss function $\ell(\theta; z)$. We aim to minimize the empirical loss defined for all $\theta \in \mathbb{R}^d$ by $f(\theta) = \frac{1}{n} \sum_{i=1}^n \ell(\theta; z_i) = \frac{1}{n} \sum_{i=1}^n f_i(\theta)$, where $S = (z_1, z_2, \ldots, z_n)$ denotes the training set and $f_i(\cdot) := \ell(\cdot; z_i)$ denotes the loss function corresponding to the $i$-th training data $z_i$. ### 2.2 SGD #### 2.2.1 CONDITIONS AND ALGORITHM We assume that a stochastic first-order oracle (SFO) exists such that, for a given $\theta \in \mathbb{R}^d$, it returns a stochastic gradient $G_\xi(\theta)$ of the function $f$, where a random variable $\xi$ is independent of $\theta$. Let $\mathbb{E}_\xi[\cdot]$ be the expectation taken with respect to $\xi$. The following are standard conditions. (C1) $f := \frac{1}{n} \sum_{i=1}^n f_i : \mathbb{R}^d \to \mathbb{R}$ is $L$–smooth, i.e., $\nabla f : \mathbb{R}^d \to \mathbb{R}^d$ is $L$–Lipschitz continuous (i.e., $\\|\nabla f(x) - \nabla f(y)\\| \leq L \\|x - y\\|$). $f$ is bounded below from $f_ \in \mathbb{R}$. Let $\Delta > 0$ satisfy $f(\theta_0) - f_* \leq \Delta$, where $\theta_0$ is an initial point. (C2) Let $(\theta_k)_{k \in \mathbb{N}} \subset \mathbb{R}^d$ be the sequence generated by SGD. For each iteration $k$, $\mathbb{E}_{\xi_k}[G_{\xi_k}(\theta_k)] = \nabla f(\theta_k)$, where $\xi_0, \xi_1, \ldots$ are independent samples and the random variable $\xi_k$ is independent of $(\theta_l)_{l=0}^k$. There exists a nonnegative constant $\sigma^2$ such that $\mathbb{E}_{\xi_k}[\\|G_{\xi_k}(\theta_k) - \nabla f(\theta_k)\\|^2] \leq \sigma^2$. (C3) For each iteration $k$, SGD samples a batch $B_k$ of size $b$ independently of $k$ and estimates the full gradient $\nabla f$ as $\nabla f_{B_k}(\theta_k) := \frac{1}{b} \sum_{i \in [b]} G_{\xi_{k,i}}(\theta_k)$, where $\xi_{k,i}$ is a random variable generated by the $i$-th sampling in the $k$-th iteration. Algorithm 1 is the SGD optimizer under (C1)–(C3). Algorithm 1 SGD Require: $ \alpha_k \in (0, +\infty) $ (learning rate), $ b \geq 1 $ (batch size), $ K \geq 1 $ (iteration) Ensure: $ \theta_K $ 1: $ k \leftarrow 0, \theta_0 \in \mathbb{R}^d $ 2: loop 3: $ \nabla f_{B_k}(\theta_k) := \frac{1}{b} \sum_{i \in [b]} G_{\xi_{k,i}}(\theta_k) $ 4: $ \theta_{k+1} := \theta_k - \alpha_k \nabla f_{B_k}(\theta_k) $ 5: $ k \leftarrow k + 1 $ 6: end loop 2.2.2 Learning rates We use the following learning rates: (Constant) $ \alpha_k $ does not depend on $ k \in \mathbb{N} $, i.e., $ \alpha_k = \alpha < \frac{2}{L} $ ($ k \in \mathbb{N} $), where the upper bound $ \frac{2}{L} $ of $ \alpha $ is needed to analyze SGD (see Appendix A.2). (Decay) $ (\alpha_k)_{k \in \mathbb{N}} \subset (0, +\infty) $ is monotone decreasing for $ k $ (i.e., $ \alpha_k \geq \alpha_{k+1} $) and converges to 0. In particular, we use $ \alpha_k = \frac{1}{(k+1)^a} $, where (Decay 1) $ a \in (0, \frac{1}{2}) \lor $ (Decay 2) $ a = \frac{1}{2} \lor $ (Decay 3) $ a \in (\frac{1}{2}, 1) $. It is guaranteed that there exists $ k_0 \in \mathbb{N} $ such that, for all $ k \geq k_0 $, $ \alpha_k < \frac{2}{L} $. We assume that $ k_0 = 0 $, since we can replace $ \alpha_k = \frac{1}{(k+1)^a} $ with $ \alpha \leq \frac{2}{L} $ ($ k \in \mathbb{N} $), where $ \alpha \in (0, \frac{2}{L}) $ is defined as in (Constant). (Step Decay) Let $ \alpha > 0, \eta \in (0, 1), T, P \geq 1 $, and $ K = TP $. A step-decay learning rate is (Decay 4) $ (\alpha_k)_{k=0}^{K-1} = (\alpha, \alpha, \cdots, \alpha, \alpha \eta, \cdots, \alpha \eta, \cdots, \alpha \eta^{P-1}, \alpha \eta^{P-1}, \cdots, \alpha \eta^{P-1}) $, which is monotone decreasing for $ k $. Let $ \alpha > 0 $ be a lower bound of $ \alpha_{K-1} $. We assume that $ \alpha < \frac{2}{L} $, which implies that, for all $ k \in [0 : K - 1] $, $ \alpha_k < \frac{2}{L} $. 3 Our Results 3.1 Upper bound of the squared norm of the full gradient We give an upper bound of $ \min_{k \in [0 : K - 1]} \mathbb{E}[\\|\nabla f(\theta_k)\\|^2] $, where $ \mathbb{E}[\cdot] $ stands for the total expectation, for the sequence generated by SGD using each of the learning rates defined in Section 2.2.2. Theorem 3.1 (Upper bound of the squared norm of the full gradient) The sequence $ (\theta_k)_{k \in \mathbb{N}} $ generated by Algorithm 1 under (C1)–(C3) satisfies that, for all $ K \geq 1 $, \[ \min_{k \in [0 : K - 1]} \mathbb{E}[\\|\nabla f(\theta_k)\\|^2] \leq \begin{cases} \frac{C_1}{K} + \frac{C_2}{b} & \text{(Constant)} \\ \frac{D_1}{K^a} + \frac{D_2}{(1 - 2a)K^a b} & \text{(Decay 1)} \\ \frac{D_1}{\sqrt{K}} + \frac{D_2}{2a D_2} & \text{(Decay 2)} \\ \frac{D_1}{K^{1-a}} + \frac{(2a - 1)K^{1-a} b}{2a D_2} & \text{(Decay 3)} \\ \frac{D_1}{\alpha K} + \frac{D_3}{\alpha K b} & \text{(Decay 4)} \end{cases} \] where \[ C_1 := \frac{2(f(\theta_0) - f_\star)}{(2 - L \alpha) \alpha}, \quad C_2 := \frac{L \sigma^2 \alpha}{2 - L \alpha}, \] \[ D_1 := \begin{cases} \frac{2(f(\theta_0) - f_\star)}{2 - L \alpha} & \text{(Decay 1)–(Decay 3)} \\ \frac{2(f(\theta_0) - f_\star)}{2 - L \alpha} & \text{(Decay 4)}, \end{cases} \] \[ D_2 := \frac{L \sigma^2}{2 - L \alpha_0}, \quad D_3 := \frac{L \alpha^2 T \sigma^2}{(1 - \eta^2)(2 - L \alpha)}. \] Theorem 3.1 indicates that the upper bound of $ \min_{k \in [0:K-1]} \mathbb{E}[\\|\nabla f(\theta_k)\\|^2] $ consists of a bias term including $ f(\theta_0) - f_* $ and a variance term including $ \sigma^2 $ and that these terms become small when the number of iterations and the batch size are large. In particular, the bias term using (Constant) or (Decay 4) is $ O\left(\frac{1}{K}\right) $, which is a better rate than using (Decay 1)–(Decay 3). Moreover, the variance term using (Decay 4) is $ O\left(\frac{1}{Kb}\right) $, which is a better rate than using other learning rates. ### 3.2 Number of Iterations Needed to Achieve $ \epsilon $-Approximation of SGD Let us consider an $ \epsilon $-approximation of SGD defined as follows: \[ \mathbb{E}\left[\\|\nabla f(\theta_{K^})\\|^2\right] := \min_{k \in [0:K-1]} \mathbb{E}\left[\\|\nabla f(\theta_k)\\|^2\right] \leq \epsilon^2, \] where $ \epsilon > 0 $ is the precision and $ K^ \in [0:K-1] $. Condition (3) implies that $ \mathbb{E}[\\|\nabla f(\theta_{K^})\\|] \leq \epsilon $. Theorem 3.1 below gives the number of iterations needed to achieve an $ \epsilon $-approximation (3) of SGD. Theorem 3.2 (Numbers of iterations needed for nonconvex optimization of SGD)* Let $ (\theta_k)_{k \in \mathbb{N}} $ be the sequence generated by Algorithm 1 under (C1)–(C3) and let $ K : \mathbb{R} \to \mathbb{R} $ be \[ K(b) = \begin{cases} \frac{C_1 b}{\epsilon^2 b - C_2} & \text{(Constant)} \\ \frac{1}{\epsilon^2} \left( \frac{D_2}{(1-2a)b} + D_1 \right)^{\frac{1}{2}} & \text{(Decay 1)} \\ \frac{D_1 b + D_2}{\epsilon^2 b - D_2} & \text{(Decay 2)} \\ \frac{1}{\epsilon^2} \left( \frac{2aD_2}{(2a-1)b} + D_1 \right)^{\frac{1}{2-a}} & \text{(Decay 3)} \\ \frac{1}{\alpha \epsilon^2} \left( \frac{D_3 + D_1 b}{b} \right) & \text{(Decay 4)} \end{cases} \] where $ C_1, C_2, D_1, D_2, $ and $ D_3 $ are defined as in Theorem 3.1, the domain of $ K $ in (Constant) is $ b > \frac{C_2}{\epsilon^2} $, and the domain of $ K $ in (Decay 2) is $ b > \frac{D_2}{\epsilon^2} $. Then, we have the following: (i) The above $ K $ achieves an $ \epsilon $-approximation (3). (ii) The above $ K $ is a monotone decreasing and convex function with respect to the batch size $ b $. Theorem 3.2 indicates that the number of iterations needed for SGD using constant/decay learning rates to be an $ \epsilon $-approximation is small when the batch size is large. Hence, it is appropriate to set a large batch size in the sense of minimizing the iterations needed for an $ \epsilon $-approximation (3). However, the SFO complexity, which is the stochastic gradient computation cost, becomes larger as $ b $ grows. Hence, the appropriate batch size should also minimize the SFO complexity. ### 3.3 SFO Complexity to Achieve $ \epsilon $-Approximation of SGD Theorem 3.2 leads to the following theorem on the properties of the SFO complexity $ N $ needed to achieve an $ \epsilon $-approximation (3) of SGD. Theorem 3.3 (SFO complexity needed for nonconvex optimization of SGD) Let $ (\theta_k)_{k \in \mathbb{N}} $ be the sequence generated by Algorithm 1 under (C1)–(C3) and define $ N : \mathbb{R} \to \mathbb{R} $ by \[ N(b) = K(b)b = \begin{cases} \frac{C_1 b^2}{\epsilon^2 b - C_2} & \text{(Constant)} \\ \frac{1}{\epsilon^2} \left( \frac{D_2}{(1-2a)b} + D_1 \right)^{\frac{1}{2}} b & \text{(Decay 1)} \\ \frac{D_1 b + D_2}{\epsilon^2 b - D_2} b & \text{(Decay 2)} \\ \frac{1}{\epsilon^2} \left( \frac{2aD_2}{(2a-1)b} + D_1 \right)^{\frac{1}{2-a}} b & \text{(Decay 3)} \\ \frac{1}{\alpha \epsilon^2} \left( \frac{D_3 + D_1 b}{b} \right) b & \text{(Decay 4)} \end{cases} \] where $ C_1, C_2, D_1, D_2, $ and $ D_3 $ are as in Theorem 3.1, the domain of $ N $ in (Constant) is $ b > \frac{C_2}{\epsilon^2} $, and the domain of $ N $ in (Decay 2) is $ b > \frac{D_2}{\epsilon^2} $. Then, we have the following: (i) The above $ N $ is convex with respect to the batch size $ b $. (ii) There exists a critical batch size \[ b^* = \begin{cases} \frac{2C_2}{\epsilon^2(1-a)} & \text{(Constant)} \\ \frac{a(1-2a)D_1}{\epsilon^2} & \text{(Decay 1)} \\ \frac{2a^2D_2}{(1-a)(2a-1)D_1} & \text{(Decay 3)} \end{cases} \] satisfying $ N'(b^) = 0 $ such that $ b^ $ minimizes the SFO complexity $ N $. (iii) For (Decay 2) and (Decay 4), $ N'(b) > 0 $ holds for all $ b > 0 $. Theorem 3.3(ii) indicates that, if we can set a critical batch size (4) for each of (Constant), (Decay 1), and (Decay 3), then the SFO complexity will be minimized. However, it would be difficult to set $ b^* $ in (4) before implementing SGD, since $ b^* $ in (4) involves unknown parameters, such as $ L $ and $ \sigma^2 $ (see Theorem 3.1 for the definitions of $ C_2, D_1, $ and $ D_2 $). Meanwhile, Theorem 3.3(iii) indicates that small batch sizes are appropriate when using (Decay 2) and (Decay 4) in the sense of minimizing the SFO complexity $ N $. ### 3.4 ITERATION AND SFO COMPLEXITIES OF SGD Theorems 3.2 and 3.3 lead to the following theorem indicating the iteration and SFO complexities needed to achieve $ \epsilon $-approximation of SGD (see also Table 2). Theorem 3.4 (Iteration and SFO complexities of SGD) The iteration and SFO complexities such that Algorithm 1 under (C1)–(C3) can be an $ \epsilon $-approximation (3) are as follows: \[ (K_\epsilon(n, b, \Delta, L, \sigma^2), N_\epsilon(n, b^, \Delta, L, \sigma^2)) = \begin{cases} O\left(\frac{1}{\epsilon^2}\right), O\left(\frac{1}{\epsilon^4}\right) & \text{(Constant)} \\ O\left(\frac{1}{\epsilon^2}\right), O\left(\frac{1}{\epsilon^2}\right) & \text{(Decay 1)} \\ O\left(\frac{1}{\epsilon^4}\right), O\left(\frac{1}{\epsilon^4}\right) & \text{(Decay 2)} \\ O\left(\frac{1}{\epsilon^{2-a}}\right), O\left(\frac{1}{\epsilon^{2-a}}\right) & \text{(Decay 3)} \\ O\left(\frac{1}{\epsilon^2}\right), O\left(\frac{1}{\epsilon^2}\right) & \text{(Decay 4)} \end{cases} \] where $ K_\epsilon(n, b, \Delta, L, \sigma^2) $ and $ N_\epsilon(n, b, \Delta, L, \sigma^2) $ are defined as in (2), the optimal batch sizes (1) are used to compute $ N_\epsilon(n, b^, \Delta, L, \sigma^2) $ (see also (4)), and we assume that, for (Constant) and (Decay 2), there exists $ M > 0 $ such that $ \epsilon^2 b - C_2, \epsilon^2 b - D_2 \geq M \epsilon^2 b $ to compute $ K_\epsilon(n, b, \Delta, L, \sigma^2) $. Theorem 3.4 indicates that the iteration complexities for (Constant) and (Decay 4) are better than those for (Decay 1)–(Decay 3) and the SFO complexity for (Decay 4) is the best. Therefore, we can conclude that using the step-decay learning rate (Decay 4) is useful for SGD in the sense of minimizing the iteration and SFO complexities needed to achieve an $ \epsilon $-approximation. ### 4 NUMERICAL RESULTS We numerically verified the number of iterations and SFO complexities needed to achieve high test accuracy for different batch sizes in training ResNet (Appendix A.6 provides the number of iterations and SFO complexities needed to achieve high training accuracy). The parameter $ \alpha $ used in (Constant) was determined by conducting a grid search of $ \{0.001, 0.005, 0.01, 0.05, 0.1, 0.5\} $. The parameters $ \alpha $ used in the decaying learning rate (Decay 1)–(Decay 3) defined by $ \alpha_k = \frac{\alpha}{(k+1)^n} $ were determined by a grid search of \{0.001, 0.005, 0.01, 0.05, 0.1, 0.5, 1.0\}. The parameters $a$ and $\eta$ used in (Decay 4) were determined by a grid search of $a \in \{0.125, 0.25, 0.5\}$ and $\eta \in \{0.25, 0.5, 0.75\}$. The parameter $T$ in (Decay 4) was set to $T = 20$ epochs. The parameter $a$ in (Decay 1) and (Decay 3) was set to $a = \frac{1}{4}$ and $a = \frac{3}{4}$, respectively. We compared SGD using (Decay 4) with SGD with momentum (momentum), Adam, AdamW, and RMSProp. The learning rates and hyperparameters of the four optimizers were determined on the basis of the previous results (Kingma & Ba, 2015; Loshchilov & Hutter, 2019; Tieleman & Hinton, 2012) (The weight decay used in the momentum was $5 \times 10^{-4}$). The experimental environment consisted of an NVIDIA DGX A100×8GPU and Dual AMD Rome7742 2.25-GHz, 128 Cores×2CPU. The software environment was Python 3.10.6, PyTorch 1.13.1, and CUDA 11.6. The code is available at https://anonymous.4open.science/r/SGD_with_decaying/. Figure 1: Number of iterations needed for SGD with (Constant), (Decay 1), (Decay 2), (Decay 3), and (Decay 4) to achieve a test accuracy of 0.9 versus batch size (ResNet-18 on CIFAR-10) Figure 2: SFO complexity needed for SGD with (Constant), (Decay 1), (Decay 2), (Decay 3), and (Decay 4) to achieve a test accuracy of 0.9 versus batch size (ResNet-18 on CIFAR-10) Figure 3: Number of iterations needed for SGD with (Decay 4), momentum, Adam, AdamW, and RMSProp to achieve a test accuracy of 0.9 versus batch size (ResNet-18 on CIFAR-10) Figure 4: SFO complexity needed for SGD with (Decay 4), momentum, Adam, AdamW, and RMSProp to achieve a test accuracy of 0.9 versus batch size (ResNet-18 on CIFAR-10) First, we trained ResNet-18 on CIFAR-10 dataset. The stopping condition of the optimizers was 200 epochs. Figures 1 and 2 show performance measures for five different learning rates in achieving a test accuracy of 0.9. Figure 1 indicates that using (Decay 2) and (Decay 3) did not reach the test accuracy 0.9 before the stopping condition was reached (Figures 9 and 10 in Appendix A.6 indicate that using (Decay 2) and (Decay 3) reached the training accuracy 0.9). Meanwhile, Figure 1 indicates that using (Constant), (Decay 1), and (Decay 4) decreased the number of iterations. Figure 2 indicates that, in the case of SGD using (Constant), a critical batch size $b^* = 2^4$ exists at which the SFO complexity is minimized. Figures 1 and 2 indicate that, for using a small batch size $(b = 2^1, 2^2)$, SGD using (Decay 4) performs better than SGD using (Constant) and (Decay 1). Figures 3 and 4 compare SGD with (Decay 4) with other optimizers. These figures indicate that, for using a small batch size $(b = 2^1, 2^2)$, SGD with (Decay 4) performed better than the other optimizers in minimizing the number of iterations and the SFO complexity. Figure 4 also indicates that the existing optimizers using constant learning rates had critical batch sizes minimizing the SFO complexities. In particular, AdamW using the critical batch size $b^* = 2^5$ (Figure 4) and SGD using (Constant) and $b^* = 2^4$ (Figure 2) performed well. However, it would be difficult to set the critical batch size in advance, since it involves unknown parameters $L$ and $\sigma^2$ (see (4) and $C_2 = \frac{L\sigma^2}{2-L\alpha}$). (computing the Lipschitz constant $L$ is NP-hard (Virmaux & Scaman, 2018)). Meanwhile, we can set small batch sizes for using SGD with a step-decay learning rate. Next, we trained ResNet-18 on the CIFAR-100 dataset. The stopping condition of the optimizers was 1000 epochs. Figures 5 and 6 show performance measures of SGD for five different learning rates in achieving a test accuracy of 0.6. As in Figures 3 and 4, Figures 7 and 8 indicate that, for using a small batch size ($b = 2^1, 2^2, 2^3, 2^4$), SGD with (Decay 4) reduced the SFO complexity. Figures 7 and 8 indicate that using the existing optimizers with $b = 2^1$ did not reach the test accuracy 0.6 before the stopping condition was reached, in contrast to SGD with (Decay 4) and $b = 2^1$. Moreover, the SFO complexity of SGD with (Decay 4) and the batch size $b = 2^4$ was the smallest of other optimizers for any batch size. Figures 5–8 indicate that SGD with (Decay 4) was more robust than other optimizers in terms of using small batch sizes (See Figures 17–20 for the results on the MNIST dataset). ![Figure 5](image1.png) Figure 5: Number of iterations needed for SGD with (Constant), (Decay 1), (Decay 2), (Decay 3), and (Decay 4) to achieve a test accuracy of 0.6 versus batch size (ResNet-18 on CIFAR-100) ![Figure 6](image2.png) Figure 6: SFO complexity needed for SGD with (Constant), (Decay 1), (Decay 2), (Decay 3), and (Decay 4) to achieve a test accuracy of 0.6 versus batch size (ResNet-18 on CIFAR-100) ![Figure 7](image3.png) Figure 7: Number of iterations needed for SGD with (Decay 4), momentum, Adam, AdamW, and RMSProp to achieve a test accuracy of 0.6 versus batch size (ResNet-18 on CIFAR-100) ![Figure 8](image4.png) Figure 8: SFO complexity needed for SGD with (Decay 4), momentum, Adam, AdamW, and RM-SProp to achieve a test accuracy of 0.6 versus batch size (ResNet-18 on CIFAR-100) ## 5 CONCLUSION AND FUTURE WORK This paper investigated the required number of iterations and SFO complexities of SGD using constant/decay learning rates to achieve an $\epsilon$-approximation. Our theoretical analyses indicated that the number of iterations needed for an $\epsilon$-approximation is monotone decreasing and convex with respect to the batch size and the SFO complexity needed for an $\epsilon$-approximation is convex with respect to the batch size. Moreover, we showed that SGD using a step-decay learning rate and a small batch size reduces the SFO complexity. The numerical results indicated that SGD using a step-decay learning rate and a small batch size performs better than the existing optimizers in the sense of minimizing the SFO complexity. The results in this paper can be applied to only SGD. This is a limitation of our work. Hence, in the future, we should investigate whether our results can be applied to variants of SGD, such as the momentum methods and adaptive methods. REFERENCES Yossi Arjevani, Yair Carmon, John C. Duchi, Dylan J. Foster, Nathan Srebro, and Blake Woodworth. Lower bounds for non-convex stochastic optimization. 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w5oP27fmYW	Center of Mass: The authors opted to canonicalize the shapes using the point mean. In cases where the density is non-uniform, this approach could lead to issues. Specifically, very similar shapes that are sampled differently might end up having different	CCD-3DR: Consistent Conditioning in Diffusion for Single-Image 3D Reconstruction Anonymous authors Paper under double-blind review Abstract In this paper, we present a novel shape reconstruction method leveraging a diffusion model to generate a 3D sparse point cloud for the object captured in a single RGB image. Recent methods typically guide a diffusion model with global shape information or local image features. However, such strategies fail to consistently align the denoised point cloud with the given image, leading to unstable conditioning and inferior performance. In this paper, we exploit a novel Centered Diffusion Probabilistic Model (CDPM) for consistent local feature conditioning. We constrain the noise and sampled point cloud from the diffusion model into a subspace where the point cloud center remains unchanged during both the forward and reverse diffusion process. Upon CDPM, we build CCD-3DR for single-image 3D reconstruction, where the stable point cloud center further serves as an anchor to align each point with its corresponding local projection-based features. Extensive experiments on synthetic benchmark ShapeNet-R2N2 demonstrate that CCD-3DR outperforms all competitors by a large margin, with over 40% improvement. We also provide results on the real-world dataset Pix3D to thoroughly demonstrate the potential of CCD-3DR in real-world applications. The code will be released soon. 1 Introduction Single-image object reconstruction is a well-known ill-posed problem. While deep learning methods have made remarkable strides in achieving high-quality reconstruction, further improvements are still necessary to meet the demands of real-world applications (Zhai et al., 2023; Yang and Scherer, 2019). Recently, a new wave of methods leveraging Denoising Diffusion Probabilistic Model (DDPM) (Ho et al., 2020) has emerged (Cheng et al., 2023; Melas-Kyriazi et al., 2023b; Luo and Hu, 2021; Melas-Kyriazi et al., 2023a; Poole et al., 2023), showcasing superior performance in various domains. For single-image 3D reconstruction with diffusion models, DMPGen (Luo and Hu, 2021) and PC$^2$ (Melas-Kyriazi et al., 2023b) are two representative baselines. In DMPGen, the condition is the global embedding of the target object, while in PC$^2$, in each step of the reverse process, the denoised point cloud is back-projected onto the feature map of the image to extract local feature for each point, which serves as the condition for the next reverse step. However, directly applying diffusion models in single-image 3D reconstruction suffers from an inevitable challenge: uncontrollable center deviation of the point cloud, as shown in Fig. 1. (a). Since each point inside the point cloud and predicted noise is independently modeled, under the single-image reconstruction setting, no geometric or contextual priors can be harnessed to control the point cloud center. After each step of the reverse process in DDPM, the centroid of the generated point cloud will be shifted slightly. Therefore, from a random sampled Gaussian noise towards the target object, in the reverse process, the center of the point cloud will continuously undergo disturbances until it reaches the center of the target object. Based on our experimental findings, we have identified two problems caused by this center deviation. First, the diffusion network needs to allocate capacity to handle the displacement of the point cloud center. It is crucial to ensure that the transition of the point cloud center from the initial Gaussian noise state to the final object reconstruction is appropriately managed. However, since the overall resource is limited, allocating network capacity to recover the center results in inferior performance in shape reconstruction. Second, the center deviation causes misalignment and inconsistency in the local feature conditioning, as used in PC$^2$ (Melas-Kyriazi et al., 2023b). The misaligned feature adversely affects the subsequent denoising process in DDPM and degrades the overall quality of the final reconstruction. We explain more details of these two points in Sec. 3.2. To address the aforementioned problems, in this paper, we present a simple but effective method, CCD-3DR, which takes a single RGB image with the corresponding camera pose as input and reconstructs the target object with a sparse point cloud. Instead of directly leveraging the off-the-shelf DDPM, we propose a novel Centered Diffusion Probabilistic Model (CDPM) that can enable consistent local feature conditioning in diffusion, which further significantly boosts the single-image reconstruction quality. Our core idea is to constrain the added noise in the diffusion process as well as the predicted noise and the sampled point cloud in the reverse process into a smaller subspace of the entire sampling space. With such constraints, CDPM sacrifices some of DDPM’s generation diversity, yet it stables the point cloud center in exchange. In this subspace, the center of the corresponding noise of the point cloud coincides with the origin throughout the diffusion and reverse processes, as shown in Fig. 1. (b). Thereby, the point cloud center serves as an anchor in local feature extraction to align the point cloud with its corresponding projections consistently. Based on CDPM, we design CCD-3D for single-image 3D object reconstruction. In CCD-3D, to ensure that the noise and point cloud lie in the subspace defined in CDPM, a straightforward strategy is to iteratively generate samples in the entire space until one sample lies in the subspace. However, this is time-consuming and infeasible in real implementations. Instead, we first sample the noise in the entire space and centralize it. Next, we denoise the point cloud after predicting the noise using the diffusion network. In the subsequent process, these are then transferred to the subspace. We follow PC$^2$ (Melas-Kyriazi et al., 2023b) to back-project the point cloud onto the feature map of the image to extract local features around each projection. In summary, our contributions are listed as follows, (i) We propose a novel centered denoising diffusion probabilistic model CDPM, which constrains the noise and point cloud in diffusion and reverse processes into a subspace where the point cloud center is forced to coincide with the origin. (ii) We present a new single-image 3D object reconstruction pipeline, CCD-3D, which leverages CDPM to consistently collect local features for the point cloud in diffusion, leading to superior performance in reconstruction quality. (iii) We evaluate CCD-3D on the synthetic dataset ShapeNet-R2N2 (Chang et al., 2015; Choy et al., 2016) to demonstrate its superiority over competitors. CCD-3D outperforms state-of-the-art methods by over 40% under F-Score. Additional experiments on the real-world dataset Pix3D (Sun et al., 2018) demonstrate the potential of CCD-3D in real applications. 2 RELATED WORKS 3D reconstruction of the object shape from a single image has been a research focus in the community (Kar et al., 2017; Wang et al., 2018; Wu et al., 2017; Kar et al., 2015; Li et al., 2019; 2018; Zhang et al., 2021; Mao et al., 2021). Although it is an ill-posed problem, the shape priors of large-scale training datasets can guide the reconstruction process with generalization ability. Non-Generative Reconstruction Models. Early methods use 2D encoders (Ronneberger et al., 2015; He et al., 2016; Simonyan and Zisserman, 2015) to encode features and use 3D decoders (Çiçek et al., 2016; Tran et al., 2015) to obtain shapes. The pioneering work such as 3D- R2N2 (Choy et al., 2016) uses the occupancy grids as object shape representations and a following LSTM (Hochreiter and Schmidhuber, 1997) to fuse inputs from multiple views for prediction. The 2D features are extracted by a 2D CNN and projected to the 3D occupancy grids with a 3D deconvolutional neural network. LSM (Kar et al., 2017) reprojects 2D features into voxel grids and decodes shapes from these grids using a 3D convolutional GRU (Cho et al., 2014). Pix2Vox series (Xie et al., 2019; 2020) enjoy a serial architecture composed of a pretrained 2D CNN backbone and 3D transposed convolutional layers with multi-scale fusion for enhanced voxelization. Since the voxel representations are limited by the resolution of voxel size, point cloud and mesh-based shape representations are favored to get rid of the limitation (Hu et al., 2021; Wang et al., 2020; Zhang et al., 2018; Henderson and Ferrari, 2019; Erler et al., 2020; Mandikal and Babu, 2019; Gkioxari et al., 2019; Wen et al., 2019; Pan et al., 2019; Huang et al., 2023). More recent works utilizes implicit representations such as signed distance functions (Park et al., 2019; Xu et al., 2019), occupancy networks (Mescheder et al., 2018; Chen and Zhang, 2019) or neural radiance fields for object shape generation (Yu et al., 2020; Wang et al., 2021; Jang and de Agapito, 2021). Despite the different shape representations, the above methods are restricted to auto-encoder architecture and suffer limited performances in comparison to generative models. Generative Reconstruction Models. Generative reconstruction models, in contrast to the routines mentioned above, estimate the shape distribution in a more explicit way to generate plausible shapes. For the first time to generate point clouds from single-view images, Fan et al. (Fan et al., 2017) build a point cloud generation network upon variational autoencoders (VAEs) (Kingma and Welling, 2014) to generate multiple plausible shapes. By incorporating both VAEs and generative adversarial networks (GANs) (Goodfellow et al., 2014), 3D-VAE-GAN (Wu et al., 2016) samples latent codes from a single-view image as the condition and outputs 3D shapes through 3D GAN generators. However, It heavily relies on class labels for reconstruction. 3D-aware GANs such as StyleSDF (Or-El et al., 2022) and Get3D (Gao et al., 2022) can simultaneously synthesize 2D images and 3D detailed meshes. However, these methods suffer from instabilities and mode collapse of GAN training. Recently, diffusion models (Song and Ermon, 2019; 2020; Ho et al., 2020) exhibit advanced generation ability in such as text-to-image (Rombach et al., 2021), text-to-shape (Nichol et al., 2022) areas, enjoying more stable training phase and elegant mathematical explainability. Thereby, various point cloud based tasks take advantage of diffusion models to get results of higher quality. DMPGen (Luo and Hu, 2021) firstly applies the diffusion process in the point cloud generation task. LION (Zeng et al., 2022) further generalizes the point cloud in the hierarchical latent space with diffusion. Similarly, Lyu et al. (Lyu et al., 2022) utilize the point diffusion for shape completion. Point-Voxel Diffusion (Zhou et al., 2021) combines multiple representations in the diffusion process to generate stable results. To get the texture information for the point cloud, (Nichol et al., 2022) generates colored point clouds as the diffusion output for better visualization. Theoretically, such methodology can be readily leveraged into the single-view reconstruction task by regarding the RGB information as the condition (Poole et al., 2023; Melas-Kyriazi et al., 2023b). The recent method PC$^2$ (Melas-Kyriazi et al., 2023b) projects point clouds in the reverse diffusion process onto the image plane to query 2D features as shape and color conditions. Our new diffusion paradigm CDPM can be compatible with recent work, such as DMPGen and PC$^2$, while providing more accurate results. ### 3 Method In the following sections, we outline our methodology. We start by providing a brief overview of point diffusion models, laying the groundwork for our approach. Subsequently, we explain the enhancements we have made to the traditional DDPM with the intention of augmenting its effectiveness in the realm of single-image reconstruction. These adaptations result in our innovative Centered Diffusion Probabilistic Model (CDPM). Lastly, we provide a comprehensive explanation of our single-image reconstruction pipeline CCD-3DR, which is constructed based on CDPM. #### 3.1 Preliminaries: Diffusion Models Diffusion denoising probabilistic models are a class of generative models inspired by non-equilibrium thermodynamics. It can iteratively move a set of Gaussian noise toward a uniform Figure 2: Pipeline of CCD-3D. Block (B) shows the local feature extraction process. Given a single RGB image (capturing the airplane) as the input, CCD-3D aims to reconstruct the object with CDPM. We first leverage a pre-trained MAE (He et al., 2022) model to extract feature maps from the image and interpolate them to the same size as the image (shown in the grey block). The feature maps provide local conditions for each point in the denoised centered point cloud $ x^t - \bar{x}^t $ during the reverse process of CDPM. We back-project the centered point cloud onto the image and collect features around the projections to serve as the local features. Block (A) demonstrates the reverse process of CDPM. At step $ t $, point cloud $ x^t $ is first centralized to $ x^t - \bar{x}^t $ and then concatenated with the local features out of Block (B). The U-Net denoiser $ \theta $ predicts noise $ \epsilon_\theta $ and centralizes it with $ \epsilon_\theta - \bar{\epsilon}_\theta $. The point cloud $ x^{t-1} $ can finally be recovered using Eq. 3. and clean point cloud, capturing the target object. DDPM contains two Markov chains called the diffusion process and the reverse process. The two processes share a length of $ T = 1K $ steps. Diffusion Process. Let $ p_0 $ be the potential distribution of the complete object point cloud $ x $ in the dataset and $ p_T $ be the standard Gaussian distribution $ p_T \sim \mathcal{N}(0_{3N}, I_{3N \times 3N}) $. The diffusion process iteratively adds Gaussian noise $ \epsilon $ into the clean data distribution $ p_0 $ according to the Markov Chain Rule until $ p_0 $ reaches $ p_T $. Formally, let $ x^0 \sim p_0 $, then \[ q(x^{1:T}\|x^0) = \prod_{t=1}^{T} q(x^t\|x^{t-1}), \] where $ q(x^t\|x^{t-1}) = \mathcal{N}(x^t; \sqrt{1-\beta_t}x^{t-1}, \beta_t I) $. The hyperparameter $ \beta_t $ is pre-defined small constants. We use the subscript to denote the diffusion step $ t $. Each $ q(x^t\|x^{t-1}) $ is a Gaussian distribution and $ q(x^t\|x^0) $ can be reparameterized as, \[ q(x^t\|x^0) = \sqrt{\alpha_t}x^0 + \epsilon \sqrt{1-\alpha_t}, \] where $ \alpha_t = 1 - \beta_t $, $ \bar{\alpha}_t = \prod_{s=0}^{t} \alpha_s $, and $ \epsilon \sim \mathcal{N}(0, I) $. From Eq. 2, for point diffusion, we can infer that if $ x^0 $ is sampled from a zero-mean distribution $ p_0 $, considering $ \epsilon $ is also zero-mean, $ q(x^t\|x^0) $ can be modeled as a zero-mean distribution, which implies that for any $ t \in [0, T] $, the diffusion process will generate a zero-mean distribution at this step. In this paper, we utilize this derivation to boost single-image 3D reconstruction. Reverse Chain. The reverse process is also a Markov process that removes the noise added in the diffusion process. In this paper, the reverse process is conditioned on an RGB image $ I $ capturing the object. We start with a sample $ x^T \sim p_T $, and then iteratively sample from $ q(x^{t-1}\|x^t, f(I)) $, where $ f(I) $ denotes features extracted from $ I $ to incorporate local or global supervision into the reverse process. When the number of sampling steps $ T $ is sufficiently large, $ q(x^{t-1}\|x^t, f(I)) $ can be well approximated with an isotropic Gaussian distribution with constant small covariance $ \sigma_t^2 $: \[ q(x^{t-1}\|x^t, f(I)) = \mathcal{N}(x^{t-1}; \mu_\theta(x^t, f(I)), \sigma_t^2 I), \] \[ \mu_\theta(x^t, f(I)) = \frac{1}{\sqrt{\alpha_t}}(x^t - \frac{\beta_t}{\sqrt{1-\alpha_t}}\epsilon_\theta(x^t, f(I))), \] where $ \mu_\theta $ is the estimated mean. Thus, we can use the network parameterized by $ \theta $ to directly learn $ \epsilon_\theta $ under the condition $ f(I) $. DDPM-Based Reconstruction Consider a 3D point cloud with $ N $ points. DDPM-based reconstruction methods (Luo and Hu, 2021; Melas-Kyriazi et al., 2023b) learn a diffusion model $ S_\theta : \mathbb{R}^{3N} \rightarrow \mathbb{R}^{3N} $ to denoise the randomly sampled point cloud from $ p_T $ into a recognizable object from target distribution $ p_0 $. Specifically, at each step $ t $, the noise is predicted as the offset of each point from Algorithm 1 CDPM: Training 1: repeat 2: $ x^0 \sim q(x^0), \quad x^0 = x^0 - \bar{x}^0 $ 3: $ t \sim \text{Uniform}(\{1, 2, ..., T\}) $ 4: $ \epsilon \sim \mathcal{N}(0, I), \quad \epsilon = \epsilon - \bar{\epsilon} $ 5: Take gradient descent step on: $ \nabla_\theta \\| \epsilon - \epsilon_\theta(x^t, f(I)) \\|^2 $ 6: until converged Algorithm 2 CDPM: Sampling 1: $ x^T \sim \mathcal{N}(0, I), \quad x^T = x^T - \bar{x}^T $ 2: for $ t = T, ..., 1 $ do 3: $ \epsilon_\theta = \epsilon_\theta - \bar{\epsilon}_\theta $ 4: $ x^{t-1} \sim q(x^{t-1}\|x^t), \quad x^{t-1} = x^{t-1} - \bar{x}^{t-1} $ 5: end for 6: return $ x^0 $ the current coordinate in $ x^t $ to $ x^{t-1} \sim q(x^{t-1}\|x^t, f(I)) $. Then we sample from $ q(x^{t-1}\|x^t, f(I)) $ to obtain $ x^{t-1} $. As for conditioning, DMPGen (Luo and Hu, 2021) encodes the given RGB image into a single global latent vector $ z $ and concatenates $ z $ with the obtained point cloud at each step during the reverse process. PC$^2$ (Melas-Kyriazi et al., 2023b) goes one step further by introducing local point-wise features for fine-grained geometry cues. It updates the local feature of each point at each step $ t $ by back-projecting the point cloud $ x^t $ onto the feature map using the known camera extrinsic $[R_c\|t_c]$ and perspective projection matrix $ \pi_c $, \[ \text{Proj}(x^t) = \pi_c(R_c x^t + t_c). \] Then local features $ f(I) $ around the projections $ \text{Proj}(x^t) $ are aggregated with rasterization. These two methods (Luo and Hu, 2021; Melas-Kyriazi et al., 2023b) are selected as our baselines. 3.2 Bottlenecks in DDPM-based Reconstruction We now analyze the limitations of directly applying DDPM in 3D reconstruction like in DMPGen and PC$^2$ (Luo and Hu, 2021; Melas-Kyriazi et al., 2023b). Two bottlenecks are deteriorating the performance of these methods. First, predicting the center bias is challenging for the network in the reverse process. Since we assume the variances are constant in all Gaussian distributions, we only need to analyze the center of each denoised point cloud. From $ x^t $ to $ x^{t-1} $, in Eq. 1 and 3, we have, \[ E(\bar{x}^{t-1}) = \frac{1}{\sqrt{\alpha_t}} E(\bar{x}^t), \quad E(\bar{\epsilon}_\theta(x^t, f(I))) = 0. \] Thus after sampling for $ x^{t-1} $, we can obtain, \[ \bar{x}^{t-1} = \frac{1}{\sqrt{\alpha_t}} \left( \bar{x}^t - \frac{\beta_t}{\sqrt{1 - \alpha_t}} \bar{\epsilon}_\theta(x^t, f(I)) \right) + \Delta_t, \] where $ \Delta_t $ is center bias generated by random sampling from Gaussian distribution for $ x^{t-1} $. When $ \bar{x}^T \neq \bar{x}^0 $, the network $ \theta $ needs to move the center of the denoised point cloud from $ \bar{x}^T $ towards $ \bar{x}^0 $ under the following handicaps. First, $ E(\bar{\epsilon}_\theta(x^t, f(I))) = 0 $, while the network needs to predict non-zero-mean noise $ \epsilon $ in several steps to move $ \bar{x}^T \rightarrow \bar{x}^0 $. Second, the network needs to overcome $ \Delta_t $. Last, each point in $ x^{T:0} $ is independently modeled in diffusion, and no constraints are incorporated to control the development of the point cloud center. Experiments in Sec. 4.1 demonstrate that accurately recovering $ \bar{x}^0 $ is a very hard job for the network. Wasting network capacity in the recovering center also results in poor performance in shape reconstruction. Second, the change of the point cloud center makes the local feature conditioning inconsistent. As in PC$^2$, the difference $ \Delta_{\text{Proj}} $ in projections of $ \text{Proj}(\bar{x}^{t-1}) $ and $ \text{Proj}(\bar{x}^t) $ can be derived as \[ \Delta_{\text{Proj}} = \pi_c(R_c (\bar{x}^{t-1} - \bar{x}^t) + t_c). \] If $ \Delta_{\text{Proj}} $ is sufficiently large, the features collected for the point center can be totally different from $ x^t $ to $ x^{t-1} $, which will mislead the following denoising steps. Moreover, since we only use a single RGB image as a conditioner, we have no contextual or geometric constraints to rectify this misalignment. 3.3 From DDPM to CDPM To address the aforementioned bottlenecks, we propose a novel CDPM model designed for single-view 3D reconstruction. The core idea of CDPM is simple and straightforward yet effective. To eliminate the influence of center bias in the reverse process, we add the following constraint, \[ \bar{x}^t = 0, \quad t = 0, 1, 2, \ldots, T. \] (8) This constraint enforces the denoised point cloud in each step to be zero-mean so that the center remains unchanged during the reverse process. As shown in Eq. 2 and Eq. 3, if Eq. 8 holds, we have $ \bar{\epsilon} = 0, \bar{\epsilon}_\theta(x^t, f(I)) = 0 $. Let $ S_{x^t} $ denote the space of all possible samplings from the distribution $ q(x^t\|x^{t+1}) $, then the space $ S_{x^t, \bar{x}^t=0} $ under the constraint Eq. 8 is a subspace, i.e., $ S_{x^t, \bar{x}^t=0} \subset S_{x^t} $. Similarly, we define $ S_\epsilon, S_{\epsilon, \bar{\epsilon}=0}, S_{\epsilon_\theta}, S_{\epsilon_\theta, \bar{\epsilon}_\theta=0} $. In summary, from DDPM to CDPM, we constrain $ x^t, \epsilon, \epsilon_\theta $ all in a smaller subspace, DDPM : $ x^t \in S_{x^t}, \epsilon \in S_\epsilon, \epsilon_\theta \in S_{\epsilon_\theta} \implies $ CDPM : $ x^t \in S_{x^t, \bar{x}^t=0}, \epsilon \in S_{\epsilon, \bar{\epsilon}=0}, \epsilon_\theta \in S_{\epsilon_\theta, \bar{\epsilon}_\theta=0}. $ (9) Therefore, we prioritize the stability of the point cloud center to a certain extent, sacrificing a portion of the diversity in diffusion models. For point cloud $ x^t $ in the reverse process, after obtaining $ q(x^t\|x^{t+1}) $, we can sample multiple times until the sampled point cloud lies in $ S_{x^t, \bar{x}^t=0} $. However, such a strategy is infeasible in real implementation. Thereby we simply first sample in $ S_{x^t} $ and then centralize the point cloud to project it into $ S_{x^t, \bar{x}^t=0} $. The same holds true for $ \epsilon $ and $ \epsilon_\theta $. Specifically, as explained in Alg. 1 and Alg. 2, we first build a dataset composed of $ M $ data pairs $ D = \{(x_i, I_i)\}_{1 \leq i \leq M} $, where $ x_i $ denotes the $ i $-th ground truth point cloud sampled from the object mesh, and $ I_i $ is the corresponding RGB image capturing the object. Compared to DDPM, CDPM mainly makes improvements in three points: First, the point clouds in $ D $ are centralized as $ x_i - \bar{x}_i $, where $ \bar{x}_i $ denotes the centroid of $ x_i $, establishing a new zero-mean dataset $ \tilde{D} = (\tilde{x}_i, I_i) $. Second, for noise $ \epsilon $ added in the diffusion process for training and the noise $ \epsilon_\theta $ predicted in the reverse process, we also centralize them with $ \epsilon - \bar{\epsilon} $ and $ \epsilon_\theta - \bar{\epsilon}_\theta $, where $ \bar{\epsilon} $ and $ \bar{\epsilon}_\theta $ denote the corresponding gravity centers. Third, during inference, for $ x^{t-1} $ sampled from $ q(x^{t-1}\|x^t, f(I)) $, we also centralize it with $ x^{t-1} - \bar{x}^{t-1} $. From Eq. 2, since we keep $ x^0 $ and $ \epsilon $ to be zero-mean, the diffused point cloud in each step $ t $ should be zero-mean. The advantages of CDPM over DDPM in single-image reconstruction can be summarized as follows: First, our reverse process starts with a zero-mean Gaussian noise and arrives at the zero-mean reconstruction $ x^0 $ after $ T $-step zero-mean denoising. This zero-mean nature of the reverse process provides a useful regularization for the network to focus more on the shape of the object rather than tracking the center of the point cloud. Therefore, our CDPM outperforms the previous DDPM-reconstruction methods even with only global embedding of the object, like in (Luo and Hu, 2021). Second, CDPM enables consistent local feature conditioning in the reverse diffusion process. As in PC$ ^2 $ (Melas-Kyriazi et al., 2023b), the point cloud is back-projected onto the image feature map to extract local point-wise features as conditioning. However, due to the uncontrollable center bias in the reverse process, the projection of each point may gradually deviate, making the local feature aggregation process fail and further deteriorating the final reconstruction quality. In contrast to DDPM-based PC$ ^2 $, our method CDPM keeps the centroid of the denoised point cloud in each step to coincide with the origin, which further serves as an anchor point in local feature collection. The projection of this anchor point remains the same in the reverse process and thus aligns the point cloud with the feature map to obtain consistent features. 3.4 CCD-3DR For a fair comparison with baseline methods, we follow PC$ ^2 $ (Melas-Kyriazi et al., 2023b) to use MAE (He et al., 2022) to extract 2D feature maps from the given RGB image. The feature maps are of equal length and width of the input image to facilitate point cloud projection. For the diffusion network $\theta$ used to predict the noise $\epsilon_\theta$, we adopt the Point-Voxel CNN (PVCNN) \citep{liu2019point}. We use the classic $L_2$ loss to supervise the training of $\theta$, as specified in Alg. 1. 4 EXPERIMENTS Datasets. We evaluate CCD-3DR on the synthetic dataset ShapeNet-R2N2 \citep{choy20163d,chang2015shapenet} and real-world dataset Pix3D \citep{sun2018pix3d}. ShapeNet contains a diverse collection of 3D models spanning various object categories, such as furniture, vehicles, and more. The dataset is meticulously annotated, providing not only the 3D geometry of the objects but also rich semantic information, making it an essential tool for the quantitative evaluation of single-view reconstruction methods. We follow baseline methods \citep{melas2023pc2,yagubbayli2021pc2,xie2020local} to use the R2N2 \citep{choy20163d} subset along with the official image renderings, train-test splits, camera intrinsic and extrinsic matrices. The R2N2 subset covers 13 categories in total. Pix3D \citep{sun2018pix3d} is a large-scale benchmark of diverse image-shape pairs with pixel-level 2D-3D alignment. Previous methods \citep{cheng2023local,xie2019local,xie2020local,sun2018pix3d} only harness the chair category and exclude the occluded samples. Since our method needs to use all data to demonstrate robustness towards occlusion, we leverage 3 categories: \{chair, table, sofa\} and randomly generate train-test split with about 90% samples as the training set and the remaining as the testing set. Details are provided in the Supplementary Material. Implementation Details. We implement CCD-3DR in PyTorch and evaluate the method on a single GeForce RTX 3090Ti GPU with 24GB memory. For ShapeNet-R2N2 \citep{choy20163d,chang2015shapenet}, we first resize the provided images of size $137 \times 137$ to $224 \times 224$ and adjust the focal length accordingly. We follow prior work to use 8192 points in training and inference for fairness in computing the F-Score. On Pix3D \citep{sun2018pix3d}, since the images are of different sizes, we first crop the image with the given bounding box to obtain an object-centric image and then resize it to $224 \times 224$. The camera intrinsic matrix is also adjusted correspondingly. During training, we train CCD-3DR with batch size 16 for 100K steps in total, following PC$^2$ \citep{melas2023pc2}. We use the AdamW optimizer with a dynamic learning rate with warmup which increases from $1 \times 10^{-9}$ to $1 \times 10^{-3}$ in the first 2K steps and then decays exponentially until 0 in the following 98K steps. Baselines. We select DDPM-based DMPGen \citep{luo2021ddpm} and PC$^2$ \citep{melas2023pc2} as our baseline methods. On ShapeNet-R2N2, we compare with the official results of PC$^2$. Since DMPGen doesn’t provide results of single-view reconstruction on ShapeNet-R2N2, we reimplement it by using pre-trained MAE \citep{he2022mae} to extract global shape code and then follow the diffusion process in the original paper to reconstruct the object, denoted as DMPGen. We provide three variants of CCD-3DR on ShapeNet-R2N2, in which Ours uses only local features like in PC$^2$, Ours-G leverages only global features as DMPGen and Ours-(G+L) incorporates both local and global features for reconstruction, as shown in Tab. 4. On Pix3D, we retrain PC$^2$ and DMPGen* under the same settings of CCD-3DR. Evaluation Metrics. We use Chamfer Distance (CD) and F-Score@0.01 following \citep{melas2023pc2,cheng2023local} as the evaluation metrics. CD quantifies the dissimilarity between two sets of points by measuring the minimum distance from each point in one set to its nearest point in the other set. To compensate for the problem that CD can be sensitive to outliers, we also report F-Score with the threshold 0.01, i.e., for each reconstructed point. If its nearest distance to the ground truth point cloud lies below the threshold, it is considered correctly predicted. Note that previous methods \citep{choy20163d,yagubbayli2021pc2,xie2020local} typically report the results using the voxelized $32^3$ volume as the shape representation, which quantizes the sampled points and fails to reflect the reconstruction quality of fine-grained structures. Therefore, we follow PC$^2$ \citep{melas2023pc2} to use sampled points from the object mesh as the ground truth. Results of other methods \citep{choy20163d,yagubbayli2021pc2,xie2020local} are re-evaluated using the same setting for fair comparisons. 4.1 Comparisons with State-of-the-Art Methods. Performance on Synthetic Dataset ShapeNet-R2N2. In Tab. 1, we compare CCD-3DR with state-of-the-art competitors on ShapeNet-R2N2 under the F-Score@0.01 metric. 3D-R2N2 \citep{choy20163d}, Figure 3: Qualitative comparisons on synthetic dataset ShapeNet-R2N2 (Choy et al., 2016; Chang et al., 2015) (left) and real-world dataset Pix3D (Sun et al., 2018) (right). Our method can recover fine-grained structures accurately, like the handle of the chair. \| Category \| 3D-R2N2 \| LegoFormer \| Pix2vox++ \| DMPGen* \| PC² \| Ours \| DMPGen(O) \| PC²(O) \| Ours(O) \| \|--------------\|---------\|------------\|-----------\|---------\|-----\|------\|------------\|--------\|---------\| \| airplane \| 0.225 \| 0.215 \| 0.266 \| 0.454 \| 0.473 \| 0.725 \| 0.565 \| 0.681 \| 0.785 \| \| bench \| 0.198 \| 0.241 \| 0.266 \| 0.175 \| 0.305 \| 0.480 \| 0.289 \| 0.444 \| 0.573 \| \| cabinet \| 0.256 \| 0.308 \| 0.317 \| 0.087 \| 0.203 \| 0.282 \| 0.111 \| 0.303 \| 0.371 \| \| car \| 0.211 \| 0.220 \| 0.268 \| 0.310 \| 0.359 \| 0.395 \| 0.402 \| 0.420 \| 0.466 \| \| chair \| 0.194 \| 0.217 \| 0.246 \| 0.171 \| 0.290 \| 0.335 \| 0.312 \| 0.377 \| 0.406 \| \| display \| 0.196 \| 0.261 \| 0.279 \| 0.211 \| 0.232 \| 0.381 \| 0.236 \| 0.357 \| 0.487 \| \| lamp \| 0.186 \| 0.220 \| 0.242 \| 0.207 \| 0.300 \| 0.438 \| 0.347 \| 0.399 \| 0.490 \| \| loudspeaker \| 0.229 \| 0.286 \| 0.297 \| 0.113 \| 0.204 \| 0.219 \| 0.126 \| 0.288 \| 0.291 \| \| rifle \| 0.356 \| 0.364 \| 0.410 \| 0.474 \| 0.522 \| 0.762 \| 0.663 \| 0.686 \| 0.828 \| \| sofa \| 0.208 \| 0.260 \| 0.277 \| 0.078 \| 0.205 \| 0.293 \| 0.106 \| 0.298 \| 0.349 \| \| table \| 0.263 \| 0.305 \| 0.327 \| 0.155 \| 0.270 \| 0.427 \| 0.310 \| 0.420 \| 0.488 \| \| telephone \| 0.407 \| 0.575 \| 0.582 \| 0.333 \| 0.331 \| 0.423 \| 0.464 \| 0.523 \| 0.598 \| \| watercraft \| 0.240 \| 0.283 \| 0.316 \| 0.201 \| 0.324 \| 0.475 \| 0.399 \| 0.424 \| 0.610 \| \| Average \| 0.244 \| 0.289 \| 0.315 \| 0.228 \| 0.309 \| 0.433 \| 0.333 \| 0.432 \| 0.519 \| Table 1: Performance on ShapeNet-R2N2. We compare our method with competitors under F-Score@0.01. The Oracle setting (marked as (O)) refers to predicting 5 samples of each image and selecting the best prediction as the final result. Legoformer (Yagubbayli et al., 2021), Pix2vox++ (Xie et al., 2020) are voxel-based methods, while DMPGen (Luo and Hu, 2021), PC² (Melas-Kyriazi et al., 2023b) are diffusion-based methods, serving as baselines of CCD-3DR. From Tab. 1, it can be clearly deduced that our method CCD-3DR achieves state-of-the-art performance in 10 out of 13 categories. Considering the Average performance, CCD-3DR outperforms previous best method Pix2vox++ with 0.433 vs. 0.315, about a 37.5% leap forward. Furthermore, compared with diffusion-based baseline method PC², CCD-3DR demonstrates superior performance under all the categories and improves PC² by 40.1%, with 0.433 vs. 0.309. We also report the Oracle results, following the setting in PC², where for each test image, we predict 5 possible reconstruction results and select the one with the highest F-Score@0.01 as the final result. Under the Oracle setting, our method surpasses all competitors by a large margin, with about a 20.1% improvement over PC² Oracle. Performance on Real-World Dataset Pix3D. In Tab. 2, we compare CCD-3DR with other DDPM-based reconstruction methods using Chamfer Distance and F-Score@0.01. Our method consistently outperforms competitors in all categories. On average, CCD-3DR surpasses the second-best method PC² by 20% on ShapeNet-R2N2 and 15% on Pix3D. Qualitative Comparisons. We provide visualization comparisons with previous methods in Fig. 3. It can be seen clearly that our method surpasses competitors with respect to the reconstruction quality. Particularly, due to our consistent feature conditioning scheme, our method showcases superiority in recovering fine-grained structures, like the hand of the chair. We provide more results in the Supplementary Material. 4.2 Ablation Studies We conduct several ablation studies on public datasets. Note that except for ablated terms, we leave all other terms and settings unchanged. \| Method \| Chair \| Table \| Sofa \| Average \| \|------------\|-------\|-------\|------\|---------\| \| DMPGen \| 0.188 \| 0.176 \| 0.243\| 0.202 \| \| PC$^2$ \| 0.336 \| 0.294 \| 0.377\| 0.336 \| \| Ours \| 0.439 \| 0.559 \| 0.489 \| 0.496 \| \| Chair \| Table \| Sofa \| Average \| \|-------\|-------\|------\|---------\| \| 53.30 \| 50.56 \| 21.04\| 41.63 \| \| 33.21 \| 13.13 \| 3.760\| 16.70 \| \| 14.98 \| 1.475 \| 0.712 \| 5.722 \| Table 2: Performance on Pix3d. F-Score@0.01 (left) and Chamfer Distance ($\times 10^{-3}$) (right) is reported in the table. Our method outperforms diffusion-based competitors. \| Occ. Ratio \| Method \| Chair \| Table \| Sofa \| \|------------\|---------------------------------\|-------\|-------\|------\| \| $\sim 20\%$\| PC$^2$ (Melas-Kyriazi et al., 2023b) \| 0.324 \| 0.280 \| 0.365\| \| \| Ours \| 0.424 \| 0.535 \| 0.421 \| \| $\sim 50\%$\| PC$^2$ (Melas-Kyriazi et al., 2023b) \| 0.310 \| 0.260 \| 0.337\| \| \| Ours \| 0.411 \| 0.520 \| 0.397 \| Table 3: Ablation studies of robustness towards occlusions. Occ. Ratio refers to occlusion ratio. We report the F-Score@0.01 after randomly masking about 20% and 50% visible pixels of the image. \| Category \| airplane \| bench \| cabinet \| car \| chair \| display \| lamp \| loudspeaker \| rifle \| sofa \| table \| tele-phone \| watercraft \| \|----------\|----------\|-------\|---------\|-----\|-------\|---------\|------\|-------------\|-------\|------\|-------\|------------\|------------\| \| Ours-G \| 0.599 \| 0.298 \| 0.204 \| 0.251\| 0.283 \| 0.223 \| 0.316\| 0.177 \| 0.653 \| 0.201\| 0.266 \| 0.355 \| 0.311 \| \| Ours-(G+L)\| 0.727 \| 0.463 \| 0.277 \| 0.398\| 0.341 \| 0.366 \| 0.429\| 0.214 \| 0.777 \| 0.287\| 0.433 \| 0.414 \| 0.469 \| \| Ours \| 0.725 \| 0.480 \| 0.282 \| 0.395\| 0.335 \| 0.381 \| 0.438\| 0.219 \| 0.762 \| 0.293\| 0.427 \| 0.423 \| 0.475 \| Table 4: Ablations on the effect of local and global features on ShapeNet-R2N2. We retrain and re-evaluate our method using different feature conditioning methods. Occlusions. In Tab. 3, we evaluate the performance of CCD-3DR with respect to different occlusion ratios on Pix3D. We randomly mask approximately 20% and 50% visible pixels of the object to test the robustness of CCD-3DR towards occlusions. From the table, it can be seen clearly that although the masked pixels increase from 20% to 50%, the performance of CCD-3DR only degrades very little, with 0.013 in chair, 0.015 in table and 0.024 in sofa. Moreover, in this experiment, PC$^2$ also demonstrates consistent and satisfactory results under different occlusion ratios, which verifies the capability of diffusion models in handling occlusions. Note that for fair comparisons, we retrain PC$^2$ and our method with the same augmented training data. We randomly mask 0% ~ 50% pixels of each image for training and then conduct the ablation study in Tab. 3. Local vs. Global Conditioning. In Tab. 4, we demonstrate the effect of local and global features in the diffusion-based reconstruction process. The global feature is obtained by averaging the pooling of the point-wise local features. And when the global feature is incorporated, we directly concatenate it to each point as the condition. Comparing Ours-(G+L) and Ours, it can be seen clearly that once a local feature is provided, an additional global feature is not necessary. Oracle Results. We report the oracle experiment results in Tab. 1. Following the setting in PC$^2$, we also predict 5 possible shapes for each image and select the one with the highest F-Score@0.01 as the final reconstruction result. It is obvious that under the oracle setting, all three diffusion-based methods, DMPGen*, PC$^2$, and Ours, showcase a significant leap forward. Thereby, although the centralization scheme in our method may influence the generalization capability of the diffusion model to a certain extent, in the single-view reconstruction case, our method still demonstrates the capability of generating multiple plausible results. We also provide the corresponding qualitative results in the Supplementary Material. 5 CONCLUSIONS In this paper, we propose CCD-3DR, a single-image 3D reconstruction pipeline that leverages a novel Centered Diffusion Probabilistic Model (CDPM) for consistent and stable local feature conditioning. We project the predicted noise and sampled point cloud from DDPM into a subspace where the point cloud center remains unchanged during the whole diffusion and reverse processes. Extensive experimental results and ablation studies on both synthetic and real-world datasets demonstrate that such a simple design significantly improves overall performance. We also analyze the influence of point cloud centralization with respect to diversity and point out the limitations of CCD-3DR. In the future, we plan to extend CCD-3DR with an advanced ordinary differentiable equation solver to enhance the inference speed. 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Sx7BIiPzys	Why was this specific subset of six UCI data sets chosen? The original work by Hernández-Lobato and Adams (2015), who introduced this set of experiments had ten, and even Watson et al. (2021) who the authors cite as relying on for their setup used ~~seven~~ different sets. _(PostRebuttal Edit: I misread the reference, Watson et al. use the full set of experiments.)_	VARIATIONAL BAYESIAN LAST LAYERS James Harrison\textsuperscript{1}, John Willes\textsuperscript{2}, Jasper Snoek\textsuperscript{1} \textsuperscript{1}Google DeepMind, \textsuperscript{2}Vector Institute jamesharrison@google.com, john.willes@vectorinstitute.ai, jsnoek@google.com ABSTRACT We introduce a deterministic variational formulation for training Bayesian last layer neural networks. This yields a sampling-free, single-pass model and loss that effectively improves uncertainty estimation. Our variational Bayesian last layer (VBLL) can be trained and evaluated with only quadratic complexity in last layer width, and is thus (nearly) computationally free to add to standard architectures. We experimentally investigate VBLLs, and show that they improve predictive accuracy, calibration, and out of distribution detection over baselines across both regression and classification. Finally, we investigate combining VBLL layers with variational Bayesian feature learning, yielding a lower variance collapsed variational inference method for Bayesian neural networks. 1 INTRODUCTION Well-calibrated uncertainty quantification is essential for reliable decision-making with machine learning systems. However, many methods for improving uncertainty quantification in deep learning (including Bayesian methods) have seen limited application due to their relative complexity over standard deep learning. For example, methods such as sampling-based mean field variational inference (Blundell et al., 2015), Markov chain Monte Carlo (MCMC) methods (Papamarkou et al., 2022; Neal, 1995; Izmailov et al., 2021), and comparatively simple heuristics such as Bayesian dropout (Gal & Ghahramani, 2016) all have substantially higher computational cost than baseline networks. Single-pass methods (where only one network evaluation is required) often require substantial modifications to network architectures, regularization, or training and evaluation procedures, even for the simplest such models (Liu et al., 2022; Wilson et al., 2016b; Kristiadi et al., 2021). In this work, we take a simplicity-first approach to Bayesian deep learning, and develop a conceptually simple and computationally inexpensive partially Bayesian neural network. In particular, we investigate variational learning of Bayesian last layer (BLL) neural networks. While BLL models consider only the uncertainty over the output layer of the network, they have been shown to perform comparably to more complex Bayesian models (Watson et al., 2021; Harrison et al., 2018; Fiedler & Lucia, 2023; Kristiadi et al., 2020). Our variational formulation relies on a deterministic lower bound on the marginal likelihood, which enables highly-efficient mini-batch, sampling-free loss computation, and is thus highly scalable. Contributions. Concretely, the contributions of this work are: - We present variational Bayesian last layers (VBLLs), a novel last layer neural network component for uncertainty quantification which can be straightforwardly included in standard architectures and training pipelines (including fine-tuning), for both deterministic and Bayesian neural networks. - We derive principled and sampling-free Bayesian training objectives for VBLLs, and show that with careful parameterization they can be computed at the same cost as standard training, and trained with standard mini-batch training. - We show that VBLLs improve predictive accuracy, likelihoods, calibration, and out of distribution detection across a wide variety of problem settings. We also show VBLLs strongly outperform baseline models in contextual bandits. - We release an easy-to-use package providing efficient VBLL implementations in PyTorch. 2 BAYESIAN LAST LAYER NEURAL NETWORKS We first review Bayesian last layer models which maintain a posterior distribution only for the last layer in a neural network. These models correspond to Bayesian (linear or logistic) regression or Bayesian Gaussian discriminant analysis (for each of the three models we present, respectively) with learned features. We assume $T$ total data points, and write inputs as $\mathbf{x} \in \mathbb{R}^{N_x}$. For regression, outputs are $\mathbf{y} \in \mathbb{R}^{N_y}$; for classification, outputs are $\mathbf{y} \in \{1, \ldots, N_y\}$, and $\mathbf{y}$ denotes the $N_y$-dimensional one-hot representation. For all models discussed in this section, we will use neural network features $\phi : \mathbb{R}^{N_x} \times \Theta \rightarrow \mathbb{R}^{N_\phi}$. These correspond to all parts of a network architecture but the last layer, where $\Theta \subseteq \Theta$ denotes the weights of the neural network. We will typically write $\phi := \phi(\mathbf{x}, \theta)$ for notational convenience and refer to these parameters as features because they define the map from inputs to the feature embedding on which the BLL operates. ### 2.1 Regression The canonical BLL model for the regression case is $$ \mathbf{y} = \mathbf{w}^\top \phi(\mathbf{x}, \theta) + \epsilon $$ (1) where $\epsilon$ is assumed to be normally distributed with zero mean and covariance $\Sigma$, and these noise terms are i.i.d. across realizations. We specify a Gaussian prior $p(\mathbf{w}) = \mathcal{N}(\bar{\mathbf{w}}, S)$, assumed independent of the noise $\epsilon$. Posterior inference in the BLL model is analytically tractable for a fixed set of features. The marginal likelihood may be computed either via direct computation or by iterating over the dataset. Fixing a distribution over $\mathbf{w}$ of the form $\mathcal{N}(\bar{\mathbf{w}}, S)$, the predictive distribution is $$ p(\mathbf{y} \| \mathbf{x}, \eta, \theta) = \mathcal{N}(\bar{\mathbf{w}}^\top \phi, \phi^\top S \phi + \Sigma) $$ (2) where $\eta$ denotes the parameters of the distribution, here $\eta = (\bar{\mathbf{w}}, S)$. ### 2.2 Discriminative Classification In this subsection we introduce a BLL model that corresponds to standard classification neural networks, where $$ p(\mathbf{y} \| \mathbf{x}, W, \theta) = \text{softmax}(z), \quad z = W \phi(\mathbf{x}, \theta) + \epsilon $$ (3) where $z \in \mathbb{R}^{N_y}$ are the logits. These are also interpreted as unnormalized joint data-label log likelihoods (Grathwohl et al., 2020), where $$ z = \log p(\mathbf{x}, \mathbf{y} \| W, \theta) - Z(W, \theta) $$ (4) where $Z(W, \theta)$ is a normalizing constant, independent of the data. The term $\epsilon \in \mathbb{R}^{N_y}$ is a zero-mean Gaussian noise term with variance $\Sigma$. Typically in logistic regression this noise term is ignored, although it has seen use to model label noise (Collier et al., 2021). We include it to unify the presentation, and the variance can be assumed zero as necessary. As in the regression case, we specify a Gaussian prior for $W$. In contrast with the regression setting, exact inference and computation of the posterior predictive is not analytically tractable in this model. We refer to this model—consisting of multinomial Bayesian logistic regression on learned neural network features—as discriminative classification, as logistic regression is a classical discriminative learning algorithm. ### 2.3 Generative Classification The second classification model we consider is the generative classification model (Harrison et al., 2020; Zhang et al., 2021; Willes et al., 2022), so-called due to its similarity to classical generative models such as Gaussian discriminant analysis. In this model, we assume that the features associated with each class are normally distributed. Placing a Normal prior on the means of these feature distributions and a (conjugate) Dirichlet prior on class probabilities, we have priors and likelihoods (top line and bottom line respectively) of the form $$ \rho \sim \text{Dir}(\alpha) \\ \mu_y \sim \mathcal{N}(\bar{\mu}_y, S_y) $$ (5) $$ \mathbf{y} \| \rho \sim \text{Cat}(\rho) \\ \phi \| \mathbf{y} \sim \mathcal{N}(\mu_y, \Sigma). $$ (6) In this model, $\bar{\mu}_y \in \mathbb{R}^{N_\phi}$ and $S_y \in \mathbb{R}^{N_\phi \times N_\phi}$ are the prior mean and covariance over $\mu_y \in \mathbb{R}^{N_\phi}$, the mean embedding for each. The subscript here indexes the statistics for each class; we also write $\mu := \{\mu_1, \ldots, \mu_{N_y}\}$ to terms for all $\mathbf{y}$. The terms $\rho \in \mathcal{P}_{N_y}$ correspond to class probabilities, where $\mathcal{P}_{N_y}$ denotes the probability simplex embedded in $\mathbb{R}^{N_y}$. These class probabilities are in turn used in the categorical distribution over the class. --- 1We present scalar-output regression in the paper body and defer the multivariate output case to the appendix. 2Throughout the paper, we use overbars to denote mean parameters and underbars to denote prior parameters. Figure 1: Left: A variational BLL (VBLL) regression model with BBB features trained on 50 data points generated from a cubic function with additive Gaussian noise. The plot shows the 95% predictive credible region under the variational posterior for several sampled feature weights. Right: Visualizing (re-scaled) $ p(x \mid y = 1) - p(x \mid y = 0) $ predicted by a generative VBLL model on the half moon dataset, shows good sensitivity to Euclidean distance and sensible embedding densities. For a distribution over model parameters \[ p(\rho, \mu \mid \eta) = \text{Dir}(\alpha) \prod_{k=1}^{N_y} N(\tilde{\mu}_k, S_k) \] (7) for which we write $ \eta = \{\alpha, \tilde{\mu}, S\} $, we have \[ p(x \mid y, \eta) = N(\tilde{\mu}_y, \Sigma + S_y), \quad p(y \mid \eta) = \frac{\alpha_y}{\sum_{k=1}^{N_y} \alpha_k} \] (8) via analytical marginalization. To compute the predictive over class labels, we apply Bayes’ rule, yielding \[ p(y \mid x, \eta) = \text{softmax}_y(\log p(x \mid y, \eta) + \log p(y \mid \eta)). \] (9) Here, \[ \log p(x \mid y, \eta) = -\frac{1}{2}((\phi - \tilde{\mu}_y)^T (\Sigma + S_y)^{-1} (\phi - \tilde{\mu}_y) + \log \det(\Sigma + S_y) + c) \] (10) where $ c $ is a constant, shared for all classes, that may be ignored due to the shift-invariance of the softmax. Grouping the log determinant term with the class prior yields a bias term. Instead of a linear transformation of the input features to obtain a class logit, we instead have a quadratic transformation. This formulation is a strict generalization of standard classifier architectures (Harrison, 2021), in which we have quadratic decision regions as opposed to linear ones. ### 2.4 Inference and Training in BLL Models BLL models have seen growing popularity in recent years, ironically driven in part by a need for compatibility with increasingly deep models (Snoek et al., 2015; Azizzadenesheli et al., 2018; Harrison et al., 2018; Weber et al., 2018; Riquelme et al., 2018; Harrison et al., 2020; Ober & Rasmussen, 2019; Kristiadi et al., 2020; Thakur et al., 2020; Watson et al., 2020; 2021; Daxberger et al., 2021a; Willes et al., 2022; Sharma et al., 2022; Schwöbel et al., 2022; Zhang et al., 2021; Moberg et al., 2019; Fiedler & Lucia, 2023). Exact marginalization enables computationally efficient treatment of uncertainty, as well as resulting in lower-variance training objectives compared to sampling-based Bayesian models. A common and principled objective for training BLL models is the (log) marginal likelihood (Harrison et al., 2018), via gradient descent on \[ T^{-1} \log p(Y \mid X, \theta) \] (11) where $ X, Y $ denote stacked data. We include a factor of $ T^{-1} $ to enable better comparison with standard, non-Bayesian, training pipelines (typically based on average loss over mini-batches) and across dataset sizes. This training objective can be problematic, however: gradient computation requires computing the full marginal likelihood, and mini-batches do not yield unbiased gradient estimators as in standard training with an arbitrary loss function. Even mini-batch processing of the dataset—iterating between conditioning on mini-batches and prediction under the partial posterior—induces long computation graphs that make training at scale impossible. Moreover, due to the flexibility of neural network features, a full marginal likelihood training objective can result in substantial over-concentration of the approximate posterior (Thakur et al., 2020; Ober et al., 2021). 3 SAMPLING-FREE VARIATIONAL INFERENCE FOR BLL NETWORKS To exploit exact marginalization while avoiding full marginal likelihood computation, we will turn to stochastic variational inference (Hoffman et al., 2013). In particular, we aim to jointly compute an approximate last layer posterior and optimize network weights by maximizing lower bounds on marginal likelihood. As such, we will avoid distributional assumptions made in the previous section. We write the (uncertain) last layer parameters as $ \xi $ and aim to find an approximate posterior $ q(\xi \mid \eta) $ parameterized by $ \eta $. Concretely, throughout this section we will develop bounds of the form \[ T^{-1} \log p(Y \mid X, \theta) \geq L(\theta, \eta, \Sigma) - T^{-1} \text{KL}(q(\xi \mid \eta) \|\| p(\xi)) \] (12) where $ L $ is architecture dependent and developed in the remainder of this section. Thus, practically, the $ T^{-1} $ factor weights regularization terms in our training objective. In this section, we index data with $ t $ (via subscript), including $ \phi_t := \phi(x_t, \theta) $. 3.1 REGRESSION We consider the log marginal likelihood $ \log p(Y \mid X, \theta) $, with marginalized parameters $ \xi = \{w\} $, and have the following lower bound. Theorem 1. Let $ q(\xi \mid \eta) = N(\bar{w}, S) $ denote the variational posterior for the BLL model defined in Section 2.1. Then, (12) holds with \[ L(\theta, \eta, \Sigma) = \frac{1}{T} \sum_{t=1}^{T} \left( \log N(y_t \mid \bar{w}^\top \phi_t, \Sigma) - \frac{1}{2} \phi_t^\top S \phi_t \Sigma^{-1} \right) \] (13) The proof for this result and all others is available in Appendix F. When $ q(\xi \mid \eta) = p(\xi \mid Y, X) $ and distributional assumptions are satisfied, this lower bound is tight (this may be shown by direct substitution). This correspondence between the variational and true posterior for appropriately-chosen variational families is well known—see Knoblauch et al. (2019) for a thorough discussion. We note that a similar objective for regression models was developed in Watson et al. (2021). 3.2 DISCRIMINATIVE CLASSIFICATION In the discriminative classification case, the parameters are $ \xi = \{W\} $. We will assume a diagonal covariance matrix $ \Sigma $, and write $ \sigma_i^2 := \Sigma_{ii} $. We will fix a variational posterior of the form $ q(W \mid \eta) = \prod_{k=1}^{N_y} q(w_k \mid \eta) = \prod_{k=1}^{N_y} N(\bar{w}_k, S_k) $, where $ w_k $ denotes the $ k $-th row of $ W $. This factorization retains dense covariances for each class, but sacrifices cross-class covariances. While we only present this factorized variational posterior, a similar training objective may be derived with a fully dense variational posterior. Under the variational posterior, we have the following bound on the marginal likelihood. Theorem 2. Let $ q(W \mid \eta) = \prod_{k=1}^{N_y} N(\bar{w}_k, S_k) $ denote the variational posterior for the discriminative classification model defined in Section 2.2. Then, (12) holds with \[ L(\theta, \eta, \Sigma) = \frac{1}{T} \sum_{t=1}^{T} \left( y_t^\top \bar{W} \phi_t - \text{LSE}_k \left[ \bar{w}_k^\top \phi_t + \frac{1}{2} (\phi_t^\top S_k \phi_t + \sigma_k^2) \right] \right) \] (14) Here, $ \text{LSE}_k(\cdot) $ denotes the log-sum-exp function, with the sum over $ k $. In contrast to the regression case, this lower bound is a lower bound on the standard ELBO (due to two applications of Jensen’s inequality) and the bound is not tight. We have reduced variance (which would be induced by sampling logit values before the softmax in standard SVI (Ovadia et al., 2019)) for bias due to this lower bound. Our proof leverages the same double application of Jensen’s inequality used by Blei & Lafferty (2007). We note that tighter analytically tractable lower bounds exist for the logistic regression model (Depraetere & Vandebroek, 2017; Knowles & Minka, 2011), although for simplicity of the resulting algorithm we use the above lower bound. 3.3 GENERATIVE CLASSIFICATION In the generative classification case, the parameters are $ \xi = \{\mu, \rho\} $. In this setting, the Dirichlet posterior over class probabilities $ p(\rho \mid Y) $ can be computed exactly with one pass over the data by simply counting class occurrences. We therefore only consider a variational posterior of the form $ q(\xi \mid \eta, Y) = q(\mu \mid \eta) $ for the class embeddings, where $ q(\mu \mid \eta) = \prod_{k=1}^{N_y} N(\mu_k, S_k) $. This yields the following lower bound. Theorem 3. Let $ q(\mu \mid \eta) = \prod_{k=1}^{N_y} N(\bar{\mu}_k, S_k) $ denote the variational posterior over class embeddings for the generative classification model defined in Section 2.3. Let $ p(\rho \mid Y) = \text{Dir}(\alpha) $ denote the exact Dirichlet posterior over class probabilities, with $ \alpha $ denoting the Dirichlet posterior concentration parameters. Then, (12) holds with \[ L(\theta, \eta, \Sigma) = \frac{1}{T} \sum_{t=1}^{T} \left( \log N(\phi_t \mid \bar{\mu}_{y_t}, \Sigma) - \frac{1}{2} \text{tr}(\Sigma^{-1} S_{y_t}) + \psi(\alpha_{y_t}) - \psi(\alpha_) + \log \alpha_ \right) \] where $ \psi(\cdot) $ is the digamma function and where $ \alpha_* = \sum_k \alpha_k $. Importantly, we note that $ \psi(\alpha_{y_t}), \psi(\alpha_), \log \alpha_ $ all vanish in gradient computation and may be ignored. The term $ \log \alpha_k $ is the LSE can not be ignored, however. This training objective is again a lower bound on the ELBO, and is not tight. The first Dirichlet term (in the upper line) vanishes in gradient computation, but the second term inside the log-sum-exp function does not. In the case that the posterior concentration parameters are equal for all classes (as in the case of a balanced dataset), the concentration parameter can be pulled out of the LSE($\cdot$) (due to the equivariance of log-sum-exp under shifts) and can be ignored. 3.4 Training VBLL Models We propose three methods to learn VBLL models. Full training. First, we can jointly optimize the last layer variational posterior together with MAP estimation of the features, yielding combined training objective \[ \theta^, \eta^, \Sigma^* = \arg\max_{\theta, \eta, \Sigma} \left\{ L(\theta, \eta, \Sigma) + T^{-1} (\log p(\theta) + \log p(\Sigma) - \text{KL}(q(\xi \mid \eta) \|\| p(\xi))) \right\}. \] While one may expect this to result in substantial over-concentration for weak feature priors, in practice we observe that stochastic regularization due to mini-batch optimization prevents overconcentration. Throughout this work, we will place simple isotropic zero-mean Gaussian priors on feature weights (yielding weight decay regularization) and a canonical inverse-Wishart prior on $ \Sigma $. For Gaussian priors (as developed throughout this section) the KL regularization term can be computed in closed form. The prior terms (and the KL penalty) introduce a set of new hyperparameters that may be difficult to select. In Appendix C, we discuss these hyperparameters and their interpretation, and provide a reformulation of hyperparameters that increases interpretability. Post-training. As an alternative to jointly optimizing the variational last layer with the features, a two step procedure can be used. In this step, the feature weights $ \theta $ are trained by any arbitrary training procedure (e.g. standard neural network training) and the last layer (and $ \Sigma $) are trained with frozen features. The training objective is identical to (16), although $ \theta^* $ is trained in the initial pre-training step and $ \eta^, \Sigma^ $ are trained via (16). Feature uncertainty. Lastly, we can combine last layer SVI with variational feature learning (Blundell et al., 2015), corresponding to approximate collapsed VI (Teh et al., 2006). This training strategy allows us to construct a variational posterior on the full marginal likelihood, via \[ \log p(Y \mid X) \geq \mathbb{E}_{q(\xi, \theta, \Sigma \mid \eta)} [\log p(Y \mid X, \xi, \theta, \Sigma)] - \text{KL}(q(\xi, \theta, \Sigma \mid \eta) \|\| p(\xi, \theta, \Sigma)). \] Assuming the prior and variational posterior factorize across the features and last layer, we can partially collapse this expectation \[ \mathbb{E}_{q(\xi, \theta, \Sigma \mid \eta)} [\log p(Y \mid X, \xi, \theta, \Sigma)] = \mathbb{E}_{q(\theta, \Sigma \mid \eta)} \mathbb{E}_{q(\xi \mid \eta)} [\log p(Y \mid X, \xi, \theta, \Sigma)] \geq T \mathbb{E}_{q(\theta, \Sigma \mid \eta)} [L(\xi, \eta, \Sigma)] \] and the KL penalty may be similarly decomposed into several terms that can be computed in closed form under straightforward distributional assumptions. In the above, we have included $ \Sigma $ in the variational posterior, although practically we perform MAP estimation of this covariance under inverse-Wishart priors. Again in this setting, pre-training and post-training steps may be combined, but we do not investigate this case. 3.5 Prediction with VBLL Models For prediction in VBLL models, we will predict under the variational posterior directly, approximating (for test input/label $(x; y)$), \[ p(y \mid x, X, Y) \approx \mathbb{E}_{q(\xi \mid \eta^)} [p(y \mid x, \xi, \theta^, \Sigma^)] \] for the deterministic feature model. This expectation may be computed in closed form (for the regression and generative classification model) due to conjugacy, and can be computed via inexpensive last layer sampling in the discriminative classification model. In the variational feature model, \[ p(y \mid x, X, Y) \approx \mathbb{E}_{q(\theta \mid \eta^)} \mathbb{E}_{q(\xi \mid \eta^)} [p(y \mid x, \xi, \theta, \Sigma^)] \] where the inner expectation may be computed exactly and the outer expectation may be approximated via sampling. Further details of training, prediction, and out of distribution detection within all three VBLL models is provided in Appendix B. For both training and prediction, under relatively weak assumptions on covariance matrices, computational complexity (for the classification models) is at most $O(N_y N_\phi^2)$, and can be reduced to $O(N_y N_\phi)$ for diagonal covariances. This matches the complexity of standard network evaluation; for reasonable choices of covariance sparsity, the additional computational cost of VBLL models over standard networks is negligible. More details are provided in Appendix C. 4 RELATED WORK AND DISCUSSION Bayesian methods capable of flexible nonlinear learning have been a topic of active study for the last several decades. Historically, early interest in Bayesian neural networks (MacKay, 1992; Neal, 1995) diminished as Gaussian processes rose to prominence (Rasmussen, 2004). In recent years, however, there has been growing interest in methods capable of learning expressive features, effectively quantifying uncertainty, and training efficiently on large datasets. Variational methods have seen particular attention in both neural networks (Blundell et al., 2015; Ovadia et al., 2019) and GPs (Hensman et al., 2013; Titsias, 2009; Liu et al., 2020) due to their flexibility and their ability to produce mini-batch gradient estimation training schemes. While a wide range of work has aimed to produce more performant approximate Bayesian methods (including more expressive prior and posterior representations (Fortuin et al., 2021; Izmailov et al., 2021; Sun et al., 2019; Wilson & Izmailov, 2020)), they have still seen limited application, often due to the increased computational expense of these methods (Lakshminarayanan et al., 2017; Dusenberry et al., 2020). While some approaches to Bayesian neural networks have focused on improving the quality of the posterior uncertainty through e.g. better priors (Farquhar et al., 2020; Fortuin, 2022) or inference methods (Izmailov et al., 2021), other lines of work have focused on designing comparatively inexpensive approximate Bayesian methods. Indeed, simple strategies such as Bayesian dropout (Gal & Ghahramani, 2016) and stochastic weight averaging (Maddox et al., 2019) have seen much wider use than more expressive methods due to their simplicity. One of the simplest Bayesian models is the BLL model that is the focus of this paper, which enables single-pass, often deterministic uncertainty prediction. This model has gained prominence through the lens of deep kernel learning (Wilson et al., 2016b;a; Watson et al., 2020; Liu et al., 2022) and within few-shot learning (Harrison et al., 2018; 2020; Harrison, 2021; Watson et al., 2021; Zhang et al., 2021). Deep kernel learning aims to augment standard neural network kernels with neural network inputs. This approach allows control of the behavior of uncertainty, particularly as a function of Euclidean distance (Liu et al., 2022). While stochastic variational inference has been applied to these models (Wilson et al., 2016a), efficient and deterministic mini-batch methods have not been a major focus. Moreover, classification in these models typically relies on sampling logits applying softmax functions, which increases variance (Ovadia et al., 2019; Kristiadi et al., 2020; 2021), or on Laplace approximation (Liu et al., 2022). Within few-shot learning, exact conjugacy of the Bayesian linear regression model (Harrison et al., 2018) and Bayesian GDA (Harrison et al., 2020; Zhang et al., 2021; Snell et al., 2017) has been exploited for efficient few-shot adaptation. These models have (in addition to Van Amersfoort et al. (2020) among others) shown the strong performance of GDA-based/radial basis function networks, especially on problems such as out of distribution detection, which we further highlight in this work. However, training these models (as well as the DKL methods discussed previously) relies on direct computation of the marginal likelihood. In contrast to prior work on DKL and few-shot learning, our approach achieves efficient and deterministic training and prediction through our variational objectives and through similarly exploiting conjugacy, and thus the added complexity compared to standard neural network models is minimal. 5 EXPERIMENTS We investigate the three VBLL models, with both MAP and variational feature learning, in regression and classification tasks. A full description of all metrics used throughout this section and baseline methods is available in the appendix. To illustrate VBLL models, we show predictions on simple datasets in Figure 1. The left figure shows a regression VBLL model with variational features trained on the function $f(x) = cx^3$, with training data shown in red. This figure shows the behavior on \[^{3}\text{Complexity for the regression case is } O(N_y^2 + N_\phi^2).\] Table 1: Results for UCI regression tasks. \| \| NLL (J) \| RMSE (J) \| NLL (J) \| RMSE (J) \| NLL (J) \| RMSE (J) \| \|-------\|---------\|----------\|---------\|----------\|---------\|----------\| \| VBLL \| 2.55 ± 0.06 \| 2.92 ± 0.12 \| 3.22 ± 0.07 \| 5.09 ± 0.13 \| 1.37 ± 0.08 \| 0.87 ± 0.04 \| \| GBLL \| 2.90 ± 0.05 \| 4.19 ± 0.17 \| 3.09 ± 0.03 \| 5.01 ± 0.18 \| 0.69 ± 0.03 \| 0.46 ± 0.02 \| \| LDGBLL\| 2.60 ± 0.04 \| 3.38 ± 0.18 \| 2.97 ± 0.03 \| 4.80 ± 0.18 \| 4.80 ± 0.18 \| 0.50 ± 0.02 \| \| MAP \| 2.60 ± 0.07 \| 3.02 ± 0.17 \| 3.04 ± 0.04 \| 4.75 ± 0.12 \| 1.44 ± 0.09 \| 0.53 ± 0.01 \| \| RBF GP\| 2.41 ± 0.06 \| 2.83 ± 0.16 \| 3.08 ± 0.02 \| 5.62 ± 0.13 \| 0.66 ± 0.04 \| 0.47 ± 0.01 \| \| Dropout\| 2.36 ± 0.04 \| 2.78 ± 0.16 \| 2.90 ± 0.02 \| 4.45 ± 0.11 \| 1.33 ± 0.09 \| 0.53 ± 0.01 \| \| Ensemble\| 2.48 ± 0.09 \| 2.79 ± 0.17 \| 3.04 ± 0.08 \| 4.55 ± 0.12 \| 0.58 ± 0.07 \| 0.41 ± 0.02 \| \| SWAG \| 2.64 ± 0.16 \| 3.08 ± 0.35 \| 3.19 ± 0.05 \| 5.50 ± 0.16 \| 1.23 ± 0.08 \| 0.93 ± 0.09 \| \| BBB \| 2.39 ± 0.04 \| 2.74 ± 0.16 \| 2.97 ± 0.03 \| 4.80 ± 0.13 \| 0.63 ± 0.05 \| 0.43 ± 0.01 \| \| VBLL BBB\| 2.59 ± 0.07 \| 3.13 ± 0.19 \| 3.36 ± 0.22 \| 5.16 ± 0.16 \| 1.35 ± 0.15 \| 0.062 ± 0.03 \| Table 2: Further results for UCI regression tasks. \| \| NLL (J) \| RMSE (J) \| NLL (J) \| RMSE (J) \| NLL (J) \| RMSE (J) \| \|-------\|---------\|----------\|---------\|----------\|---------\|----------\| \| VBLL \| 2.73 ± 0.01 \| 3.68 ± 0.03 \| 1.02 ± 0.03 \| 0.65 ± 0.01 \| 1.29 ± 0.17 \| 0.89 ± 0.17 \| \| GBLL \| 2.77 ± 0.01 \| 3.85 ± 0.03 \| 1.02 ± 0.01 \| 0.64 ± 0.01 \| 1.67 ± 0.11 \| 1.09 ± 0.09 \| \| LDGBLL\| 2.77 ± 0.01 \| 3.85 ± 0.04 \| 1.02 ± 0.01 \| 0.64 ± 0.01 \| 1.13 ± 0.06 \| 0.75 ± 0.10 \| \| MAP \| 2.77 ± 0.01 \| 3.81 ± 0.04 \| 0.96 ± 0.01 \| 0.63 ± 0.01 \| 5.14 ± 1.62 \| 0.94 ± 0.09 \| \| RBF GP\| 2.76 ± 0.01 \| 3.72 ± 0.04 \| 0.45 ± 0.01 \| 0.56 ± 0.05 \| 0.17 ± 0.03 \| 0.40 ± 0.03 \| \| Dropout\| 2.80 ± 0.01 \| 3.96 ± 0.04 \| 0.95 ± 0.01 \| 0.61 ± 0.01 \| 1.82 ± 0.01 \| 1.24 ± 0.13 \| \| Ensemble\| 2.70 ± 0.01 \| 3.59 ± 0.04 \| 0.95 ± 0.01 \| 0.63 ± 0.01 \| 0.38 ± 0.07 \| 0.83 ± 0.08 \| \| SWAG \| 2.77 ± 0.02 \| 3.85 ± 0.03 \| 0.96 ± 0.03 \| 0.63 ± 0.01 \| 1.11 ± 0.05 \| 1.13 ± 0.20 \| \| BBB \| 2.77 ± 0.01 \| 3.86 ± 0.04 \| 0.95 ± 0.01 \| 0.63 ± 0.01 \| 1.43 ± 0.17 \| 1.10 ± 0.11 \| \| VBLL BBB\| 2.74 ± 0.01 \| 3.73 ± 0.04 \| 0.94 ± 0.03 \| 0.61 ± 0.01 \| 2.96 ± 0.59 \| 0.79 ± 0.05 \| so-called gap datasets—so named because of the interval between subsets of the data. The VBLL model shows desirable increasing uncertainty between the intervals (Foong et al., 2019). The right figure shows the generative classification model (G-VBLL) on the half-moon dataset. In particular, we visualize the feature density for each class. Importantly, the density has high Euclidean distance sensitivity, which has been advocated by Liu et al. (2022) as a desirable feature for robustness and out of distribution detection. ### 5.1 Regression We investigate the performance of the regression VBLL models on UCI regression datasets (Dua & Graff, 2017), which are standard benchmarks for Bayesian neural network regression (Moberg et al., 2019; Ober & Rasmussen, 2019; Daxberger et al., 2021b; Watson et al., 2021; Kristiadi et al., 2021). Results are shown in Tables 1, 2. We include baseline models run in Watson et al. (2021), and we replicate their experimental procedure and hyperparameters as closely as possible (details in the appendix). Our experiments show strong results for VBLL models across datasets. Of particular interest is the performance relative to the GBLL model, which is trained directly on the exact marginal likelihood within the Bayesian last layer model. There are several contributing factors: the prior parameters were jointly optimized with the feature weights in the GBLL model, whereas prior terms were fixed in our VBLL model, resulting in a stronger regularization effect. Moreover, exact Bayesian inference can perform poorly under model misspecification (Grüwald & Van Ommen, 2017), whereas variational Bayes has comparatively favorable robustness properties and asymptotics (Giordano et al., 2018; Wang & Blei, 2019), although the Gaussian process (GP) model generally also has strong performance across datasets. Finally, directly targeting the marginal likelihood (computed exactly within conjugate models such as BLL models) has been shown to induce substantial overfitting (Ober et al., 2021; Thakur et al., 2020; Harrison, 2021), which the variational approach may avoid due to worse inferential efficiency. ### 5.2 Image Classification To evaluate performance of VBLL models in classification, we train the discriminative (D-VBLL) and generative (G-VBLL) classification models on the CIFAR-10 and CIFAR-100 image classification task. Following Liu et al. (2022), all experiments utilize a Wide ResNet-28-10 backbone architecture. We investigate full training methods (without a post-training step), indicated with the method name in the top third of Tables 3, 4; post-training methods, indicated by pre-training method + post-training method, in the middle third of the Tables; and feature uncertainty, in the bottom third. We evaluate out of distribution (OOD) detection performance using the Street View House Numbers (SVHN) (Netzer et al., 2011) as a far-OOD dataset for both datasets, and CIFAR-100 for CIFAR-10 (and vice-versa) as near-OOD datasets. In-distribution data normalization is used in both cases. The DNN, BBB, D-VBLL and D-VBLL BBB models use maximum softmax probability (Hendrycks & Table 3: Results for Wide ResNet-28-10 on CIFAR-10. \| Method \| Accuracy (±) \| ECE (±) \| NLL (±) \| SVHN AUC (±) \| CIFAR-100 AUC (±) \| \|-----------------\|--------------\|---------\|---------\|---------------\|--------------------\| \| DNN \| 95.8 ± 0.19 \| 0.028 ± 0.028 \| 0.183 ± 0.007 \| 0.946 ± 0.005 \| 0.893 ± 0.001 \| \| SNGP \| 95.7 ± 0.14 \| 0.017 ± 0.003 \| 0.149 ± 0.005 \| 0.960 ± 0.004 \| 0.902 ± 0.003 \| \| D-VBLL \| 96.4 ± 0.12 \| 0.022 ± 0.001 \| 0.160 ± 0.001 \| 0.969 ± 0.004 \| 0.900 ± 0.004 \| \| G-VBLL \| 96.3 ± 0.06 \| 0.021 ± 0.001 \| 0.174 ± 0.002 \| 0.925 ± 0.015 \| 0.804 ± 0.006 \| \| DNN + LL Laplace\| 96.3 ± 0.03 \| 0.010 ± 0.001 \| 0.133 ± 0.003 \| 0.965 ± 0.010 \| 0.898 ± 0.001 \| \| DNN + D-VBLL \| 96.4 ± 0.01 \| 0.024 ± 0.000 \| 0.176 ± 0.000 \| 0.943 ± 0.002 \| 0.895 ± 0.000 \| \| DNN + G-VBLL \| 96.4 ± 0.01 \| 0.035 ± 0.000 \| 0.533 ± 0.003 \| 0.729 ± 0.004 \| 0.661 ± 0.004 \| \| G-VBLL + MAP \| \| \| \| \| 0.950 ± 0.006 \| \| Dropout \| 95.7 ± 0.13 \| 0.013 ± 0.002 \| 0.145 ± 0.004 \| 0.930 ± 0.014 \| 0.903 ± 0.007 \| \| Ensemble \| 96.4 ± 0.09 \| 0.011 ± 0.002 \| 0.124 ± 0.001 \| 0.947 ± 0.002 \| 0.911 ± 0.000 \| \| BBB \| 96.0 ± 0.08 \| 0.033 ± 0.001 \| 0.333 ± 0.014 \| 0.957 ± 0.004 \| 0.844 ± 0.013 \| \| D-VBLL BBB \| 95.9 ± 0.15 \| 0.058 ± 0.019 \| 0.238 ± 0.036 \| 0.832 ± 0.026 \| 0.744 ± 0.010 \| \| G-VBLL BBB \| 95.9 ± 0.16 \| 0.009 ± 0.001 \| 0.229 ± 0.010 \| 0.917 ± 0.005 \| 0.779 ± 0.009 \| Table 4: Results for Wide ResNet-28-10 on CIFAR-100. \| Method \| Accuracy (±) \| ECE (±) \| NLL (±) \| SVHN AUC (±) \| CIFAR-10 AUC (±) \| \|-----------------\|--------------\|---------\|---------\|---------------\|------------------\| \| DNN \| 80.3 ± 0.29 \| 0.107 ± 0.004 \| 0.941 ± 0.016 \| 0.799 ± 0.020 \| 0.795 ± 0.001 \| \| SNGP \| 80.3 ± 0.23 \| 0.030 ± 0.004 \| 0.761 ± 0.007 \| 0.846 ± 0.019 \| 0.798 ± 0.001 \| \| D-VBLL \| 80.7 ± 0.03 \| 0.040 ± 0.002 \| 0.913 ± 0.011 \| 0.849 ± 0.006 \| 0.791 ± 0.003 \| \| G-VBLL \| 80.4 ± 0.10 \| 0.051 ± 0.003 \| 0.945 ± 0.009 \| 0.767 ± 0.055 \| 0.752 ± 0.015 \| \| DNN + LL Laplace\| 80.3 ± 0.29 \| 0.210 ± 0.018 \| 1.048 ± 0.014 \| 0.834 ± 0.014 \| 0.811 ± 0.002 \| \| DNN + D-VBLL \| 80.7 ± 0.02 \| 0.063 ± 0.000 \| 0.831 ± 0.005 \| 0.843 ± 0.001 \| 0.804 ± 0.001 \| \| DNN + G-VBLL \| 80.6 ± 0.02 \| 0.186 ± 0.003 \| 3.026 ± 0.155 \| 0.638 ± 0.021 \| 0.652 ± 0.025 \| \| G-VBLL + MAP \| \| \| \| \| 0.793 ± 0.032 \| \| Dropout \| 80.2 ± 0.22 \| 0.031 ± 0.002 \| 0.762 ± 0.008 \| 0.800 ± 0.014 \| 0.797 ± 0.002 \| \| Ensemble \| 82.5 ± 0.19 \| 0.041 ± 0.002 \| 0.674 ± 0.004 \| 0.812 ± 0.007 \| 0.814 ± 0.001 \| \| BBB \| 79.6 ± 0.04 \| 0.127 ± 0.002 \| 1.611 ± 0.006 \| 0.809 ± 0.060 \| 0.777 ± 0.008 \| \| D-VBLL BBB \| 77.6 ± 0.17 \| 0.041 ± 0.003 \| 1.169 ± 0.018 \| 0.785 ± 0.022 \| 0.756 ± 0.002 \| \| G-VBLL BBB \| 78.1 ± 0.18 \| 0.046 ± 0.002 \| 1.156 ± 0.008 \| 0.832 ± 0.023 \| 0.742 ± 0.004 \| Gimpel, 2016) as an OOD measure. The G-VBLL and G-VBLL BBB models use a normalized feature density. Two methods for this exist: G-VBLL and G-VBLL BBB both use the learned variational posteriors to compute feature likelihoods. However, the performance of this is relatively weak, as there is no guarantee that learned feature likelihoods correspond effectively to true embedding densities. Thus, we also investigate an approach in which we estimate distributions for fixed features after training. This method estimates noise covariances for each class using the trained features, similar to the approach used in Liu et al. (2022). We refer to this model as G-VBLL-MAP, as the approach corresponds to MAP noise covariance estimation. These estimated covariances often result in overly-confident predictions, and so we do not advocate for label prediction under these fit covariances, and do not include results for them. Appendix B.6 discusses OOD methods, and further experimental details are in Appendix D. Tables 3, 4 summarize the CIFAR-10 and CIFAR-100 results. D-VBLL and G-VBLL report strong accuracy performance and competitive metrics for both ECE and NLL. D-VBLL in particular demonstrates strong accuracy results, as well as competitive (with SNGP) NLL and OOD detection ability. Despite its comparative simplicity, it outperforms SNGP on accuracy and OOD on CIFAR-10 and accuracy on CIFAR-100. It matches SNGP on OOD for CIFAR-100, and is competitive (although slightly worse) on ECE and NLL. Overall, D-VBLL models stand out for their strong performance relative to their complexity. They also perform well as post-training models, whereas G-VBLL performs is substantially degraded. While models with MAP feature estimation show strong performance versus baseline models, the performance of variational feature learning models (BBB) is more mixed. In regression tasks, these models are competitive, while in classification the performance is worse than deterministic models. In both settings, we use default KL term weighting (one over the dataset size). This contrasts with the tempered/cold posterior effect (Kapoor et al., 2022; Wenzel et al., 2020; Izmailov et al., 2021; Aitchison, 2020), in which it has been observed that alternative weightings of the likelihood and the KL may outperform this one. This is attributable (in part) to two factors: data augmentation and stochastic regularization. In regression there is no data augmentation and the model is trained for substantially longer than deterministic models; in classification we use standard augmentation and our training is more limited. Thus, it is possible that classification BBB models are over-regularized. We investigate this question in more detail in the appendix. 5.3 Sentiment Classification with LLM Features We evaluate VBLL models for language modelling tasks using the IMDB Sentiment Classification Dataset (Maas et al., 2011). The IMDB dataset is a binary text classification task consisting of 25,000 polarized movie reviews for training and another 25,000 for testing. A pre-trained OPT-175B (Zhang et al., 2022) model is used for text feature extraction. Sequence embeddings are obtained from OPT as the last token output from the the final network layer. We train both the generative (G-VBLL) and Figure 2: A performance comparison of G-VBLL, D-VBLL, and baseline MLP models on the IMDB Sentiment Classification Dataset. The models utilize text embeddings extracted from a pre-trained OPT-175B model. Results are presented across multiple training dataset scales, and the shaded regions represent $1\sigma$ error bounds. \| Model \| $\delta = 0.5$ \| $\delta = 0.7$ \| $\delta = 0.9$ \| $\delta = 0.95$ \| $\delta = 0.99$ \| \|---------------------\|----------------\|----------------\|----------------\|----------------\|----------------\| \| VBLL \| 0.46 ± 0.01 \| 0.79 ± 0.01 \| 2.54 ± 0.02 \| 4.82 ± 0.03 \| 24.44 ± 0.71 \| \| NeuralLinear \| 1.10 ± 0.02 \| 1.77 ± 0.03 \| 4.32 ± 0.11 \| 11.42 ± 0.97 \| 52.64 ± 2.04 \| \| NeuralLinear-MR \| 0.95 ± 0.02 \| 1.60 ± 0.03 \| 4.65 ± 0.18 \| 9.56 ± 0.36 \| 49.63 ± 2.41 \| \| LinDiagPost \| 1.12 ± 0.03 \| 1.80 ± 0.08 \| 5.06 ± 0.14 \| 8.99 ± 0.33 \| 37.77 ± 2.18 \| Table 5: Wheel bandit cumulative regret. \| Model \| $\delta = 0.5$ \| $\delta = 0.7$ \| $\delta = 0.9$ \| $\delta = 0.95$ \| $\delta = 0.99$ \| \|---------------------\|----------------\|----------------\|----------------\|----------------\|----------------\| \| VBLL \| 0.27 ± 0.03 \| 0.69 ± 0.06 \| 2.28 ± 0.14 \| 4.16 ± 0.17 \| 21.05 ± 1.59 \| \| NeuralLinear \| 0.31 ± 0.03 \| 0.68 ± 0.07 \| 2.18 ± 0.13 \| 5.44 ± 0.73 \| 46.42 ± 3.45 \| \| NeuralLinear-MR \| 0.33 ± 0.04 \| 0.79 ± 0.07 \| 2.17 ± 0.14 \| 4.08 ± 0.20 \| 35.89 ± 2.98 \| \| LinPost-MR \| 0.70 ± 0.06 \| 0.99 ± 0.10 \| 3.08 ± 0.22 \| 4.85 ± 0.27 \| 25.42 ± 1.81 \| Table 6: Wheel bandit simple regret. Discriminative (D-VBLL) models and a baseline MLP on the sequence embeddings via supervised learning at multiple training dataset scales: 10, 100, 1000 and 25,000 training samples. Evaluation is performed using the complete test set at each training dataset scale. Results are shown in Figure 2. The VBLL models demonstrate strong performance in comparison to the MLP baseline. We see significantly lower predictive NLL and ECE at smaller training dataset sizes. These findings validate the VBLL models’ potential for integration with large-scale modern language models for diverse applications, particularly in sentiment classification tasks. 5.4 Wheel Bandit To investigate the value of VBLL models in an active learning setting, we apply a VBLL regression model to the wheel bandit problem presented in Riquelme et al. (2018). This problem is a contextual bandit in which the state is sampled randomly in a two dimensional ball, and the learned model aims to identify the reward function. There are five regions in the ball and five actions: each region roughly corresponds to a correct action yielding a high reward, and incorrect action choice yields a low reward, although action 1 always yields an intermediate reward and no high-reward action exists for region 1. The parameter $\delta$ controls the volume of the high-reward regions, with larger $\delta$ corresponding to smaller high-reward regions. We report both cumulative regret—the difference in reward compared to an oracle, normalized to the performance of a random agent, aggregated over the full problem duration—and the simple regret, which captures only the last 500 timesteps and thus (roughly) measures the final quality of the learned model. We use a Thompson sampling policy (Russo et al., 2018; Thompson, 1933), and compare to the top models reported in (Riquelme et al., 2018). We find that our VBLL model strongly outperforms the top performing baselines in cumulative regret (Table 5) and slightly outperforms them in simple regret (Table 6), implying both the capacity of the model matches the best baselines while also exploring more effectively. 6 Conclusions and Future Work We have presented a simple, nearly computationally free Bayesian last layer architecture that can be applied to arbitrary network backbones. The practical realization of the VBLL model is a small number of extra parameters (corresponding to the variational posterior covariance) and a small number of regularization terms corresponding to terms arising in the marginalized predictive likelihood, prior terms used in MAP estimation, and KL divergences. Several important directions for future work exist. First, few-show adaptation that further exploits conjugacy of these models via e.g. recursive Bayesian least squares is possible. We have only leveraged basic ideas from variational inference in this work; there are many highly practical ideas within variational Kalman filtering which may enable efficient model adaptation, label noise robustness, inference within heavy-tailed noise, or improved time series filtering (Sykacek & Roberts, 2002; Sarkka & Nummenmaa, 2009; Ting et al., 2007). ACKNOWLEDGMENTS We acknowledge Apoorva Sharma, Jascha Sohl-Dickstein, Alex Alemi, and Allan Zhou for useful conversations over the course of this work. We also gratefully acknowledge Paul Brunzema, who identified a subtle bug in our initial results. REFERENCES Laurence Aitchison. A statistical theory of cold posteriors in deep neural networks. arXiv preprint arXiv:2008.05912, 2020. Kamyar Azizzadenesheli, Emma Brunskill, and Animashree Anandkumar. Efficient exploration through Bayesian deep q-networks. arXiv:1802.04412, 2018. David Blackwell. Conditional expectation and unbiased sequential estimation. 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IJBsKYXaH4	Section 4.5 is confusing to me. The first part regarding the marginal vs. joint seems to be saying the dependence between distances can be thrown out. This is done without explanation other than a hypothesis and throwing this out makes diffusion on distances no different than diffusion on particles to me. Even in the introduction, diffusion on distances is motivated through the dependence on interatomic forces so throwing them out seems to go against the original motivation.	MOLECULAR CONFORMATION GENERATION VIA SHIFTING SCORES Anonymous authors Paper under double-blind review ABSTRACT Molecular conformation generation, a critical aspect of computational chemistry, involves producing the three-dimensional conformer geometry for a given molecule. Generating molecular conformation via diffusion requires learning to reverse a noising process. Diffusion on inter-atomic distances instead of conformation preserves SE(3)-equivalence and shows superior performance compared to alternative techniques, whereas related generative modelings are predominantly based upon heuristical assumptions. In response to this, we propose a novel molecular conformation generation approach driven by the observation that the disintegration of a molecule can be viewed as casting increasing force fields to its composing atoms, such that the distribution of the change of inter-atomic distance shifts from Gaussian to Maxwell-Boltzmann distribution. The corresponding generative modeling ensures a feasible inter-atomic distance geometry and exhibits time reversibility. Experimental results on molecular datasets demonstrate the advantages of the proposed shifting distribution compared to the state-of-the-art. 1 INTRODUCTION The molecular conformation generation task constitutes a crucial and enabling aspect of numerous research pursuits, particularly in the study of molecular structures and their potential energy landscapes (Strodel, 2021). Traditional computational methods for this task rely on optimizing the free energy grounded on Schrodinger equation or density functional theory or its approximations (Griffiths & Schroeter, 2018; Tsuchihita & Hirono, 1997; Labute, 2010), failing to find a good balance between complexity and quality. Recently, machine learning has emerged as a powerful and efficient tool to identify more stable and diverse conformations across an expanded chemical space (Xu et al., 2021b; Ganea et al., 2021; Xu et al., Jing et al.). However, such novel approaches give rise to some new challenges. One of the most significant challenges is incorporating the roto-translational equivariance (SE(3)-equivariance) intrinsic to the generation process. Recent works employ SE(3)-equivariant molecular properties as proxies to render the model invariance. For instance, some studies focus on predicting torsional angles (Jing et al., Ganea et al., 2021) or inter-atomic distances (Simm & Hernández-Lobato, 2020; Xu et al., Ganea et al., 2021), with the final conformation assembled through post-processing. Besides, Uni-Mol (Zhou et al., 2023a) predicts the delta coordinate positions based on atom pair representation to update coordinates. Other works leverage inter-atomic distances to directly predict coordinates using generative models (Xu et al., Shi et al., 2021; Xu et al., 2021b; Zhu et al.). In parallel with these efforts, researchers have developed SE(3)-equivariant graph neural networks (GNNs) to better characterize the geometry and topology of geometric graphs (Schütt et al., 2017; Satorras et al., 2021; Han et al., 2022). These GNNs serve as effective tools or backbones for molecular conformation generation (Jing et al., Ganea et al., 2021; Xu et al., Shi et al., 2021; Xu et al., 2021b; Hoogeboom et al., 2022). Following the previous works (Xu et al., Shi et al., 2021; Xu et al., 2021b), our approach also seeks to encode SE(3)-equivariance from an inter-atomic distance perspective. To the best of our knowledge, existing works do not yet provide a systemic analysis of distance, often relying on common or heuristic Gaussian assumption on distance changes (Xu et al., 2021b). In this study, we conduct a thorough analysis of inter-atomic distances, drawing inspiration from physical atom motion phenomena. Specifically, we investigate the disintegration process of molecular structures... Figure 1: Demonstration of the diffusion process of SDDiff. As the Gaussian perturbation level on atom coordinates increases, the distribution of inter-atomic distances shifts from Gaussian to Maxwell-Boltzmann, which SDDiff learns to reverse. and aim to learn how to reverse these processes for generating conformations. To this end, the disintegration of molecules can be viewed as being caused by the introduction of gradually increasing levels of perturbing force fields. We postulate that atoms within a molecule exhibit Brownian motion (Gaussian) under relatively small perturbing forces. When the forces are considerably large, chemical structures are disrupted, and the atoms are able to move without restrictions. In this stage, the atom speeds follow a Maxwell-Boltzman distribution. Naturally, this can be connected to the distance distribution, in accordance with the escalation of perturbation intensity. See Fig. 1 for an overview. We thus put forth a precise estimation of the perturbed distance distribution through a closed-form shifting score function. Further, we propose a novel diffusion-based model named SDDiff (shifting distance diffusion) to reverse the force field to recover molecule conformations, leading to superior performance. Our main contributions are: • Inspired by molecule thermodynamics, we show that under the Gaussian perturbation kernel on molecular conformation, the distribution of relative speeds and the change of inter-atomic distances shift from Gaussian to Maxwell-Boltzmann distribution. • We propose a diffusion-based generative model, SDDiff, with a novel and closed-form shifting score kernel, with the mathematical support and empirical verification of its correctness. • Our method achieves state-of-the-art performance on two molecular conformation generation benchmarks, GEOM-Drugs (Axelrod & Gómez-Bombarelli, 2022) and GEOM-QM9 (Ramanakrishnan et al., 2014). 2 RELATED WORK Molecular conformation generation. Learning techniques are increasingly equipped for molecular conformation generation. A shallow trial is GeoMol (Ganea et al., 2021), which predicts local 3D configurations and assembles them with heuristic rules. Instead, conformations can be holistically sampled via modelings of either inter-atomic distance (Shi et al., 2021; Simm & Hernández-Lobato, 2020) or atom coordinates (Xu et al.; Zhu et al.). Recently, a rising interest has been observed in diffusion-based approaches (Shi et al., 2021; Xu et al., 2021b; Jing et al.), where the most related works to ours are ConfGF (Shi et al., 2021) and GeoDiff (Xu et al., 2021b). ConfGF perturbs the distance and estimates the corresponding score, which is subsequently converted to the coordinate score via chain rule. However, such a process may result in infeasible 3D geometry. GeoDiff perturbs coordinates instead and introduces an SE(3)-equivariant Markov kernel transiting the coordinate diffusion process to the distance process. However, this model’s design is based on the assumption that the perturbed distance follows a Gaussian distribution. This heuristic assumption can lead to mismatches and inaccuracy. Diffusion-based generative models. Denosing diffusion probabilistic models (DDPM) (Ho et al., 2020) delineates a Markov chain of diffusion steps to add random noise to data and subsequently learns to invert the diffusion process for generating desired data samples. Analogous to DDPM, the score matching with Langevin dynamics (SMLD) models (Song & Ermon, 2019, 2020) train noise conditional score networks (NCSN) that approximate the score function of the dataset and apply the stochastic gradient Langevin dynamics to approximate the data distribution. The above two models can be unified under the context of stochastic differential equations (SDEs) (Song et al., 2020b). The denoising diffusion implicit model (DDIM) (Song et al., 2020a) has a controllable sampling stochasticity, allowing the generation of higher-quality samples with fewer steps. The latent diffusion model (LDM) (Rombach et al., 2022) is another accelerated sampler by implementing the diffusion process in the latent space. SE(3) Neural Networks. The Euclidean group, denoted as SE(3) or E(3) when including reflections, represents a group of symmetries in 3D translation and rotation. Due to the geometric symmetry nature of molecules, incorporating this property in feature backbones is essential. One typical line of research is related to GNNs. SchNet (Schütt et al., 2017) is an E(n)-invariant network for modeling quantum interactions in molecules. E(n)-Equivariant Graph Neural Networks (EGNNs) (Satorras et al., 2021) is an E(n)-equivariant GNN, which does not rely on computationally expensive higher-order representations in intermediate layers. A hierarchy-based GNN named Equivariant Hierarchy-based Graph Networks (EGHNs) (Han et al., 2022) can increase the expressivity of message passing, which is also guaranteed to be E(3)-equivariant to meet the physical symmetry. Another related line of research is not restricted to the message-passing paradigm (Gilmer et al., 2017). Some existing works (Thomas et al., 2018; Fuchs et al., 2020) utilize the spherical harmonics to compute a basis for the transformations, which preserve SE(3)-equivariance. 3 BACKGROUND 3.1 MOLECULAR CONFORMATION GENERATION The generation of molecular conformation can be regarded as a generative problem conditioned on a molecular graph. For a given molecular graph, it is required to draw independent and identically distributed (i.i.d.) samples from the conditional probability distribution $p(C\|G)$, in which $p$ adheres to the underlying Boltzmann distribution (Noé et al., 2019), while $C$ and $G$ signify the conformation and formula of the molecule, respectively. Formally, each molecule is depicted as an undirected graph $G = (V, E)$, with $V$ representing the set of atoms within the molecule and $E$ denoting the set of inter-atomic chemical bonds, as well as the corresponding node features $h_u \in \mathbb{R}^f$, $\forall u \in V$ and edge features $e_{uv} \in \mathbb{R}^{f'}$, $\forall (u, v) \in E$ representing atom types, formal charges, bond types, etc. To simplify the notation, the set of atoms $V$ in 3D Euclidean space is expressed as $C = [c_1, c_2, \cdots, c_n] \in \mathbb{R}^{n \times 3}$, and the 3D distance between nodes $u$ and $v$ is denoted as $d_{uv} = \|\|c_u - c_v\|\|$. A generative model $p_\theta(C\|G)$ is developed to approximate the Boltzmann distribution. 3.2 EQUIVARINANCE IN MOLECULAR CONFORMATION Equivariance under translation and rotation (SE(3) groups) exhibits multidisciplinary relevance in a variety of physical systems, hence plays a central role when modeling and analyzing 3D geometry (Thomas et al., 2018; Weiler et al., 2018; Chmiela et al., 2019; Fuchs et al., 2020; Miller et al., 2020; Simm et al., 2020; Batzner et al., 2022). Mathematically, a model $s_\theta$ is said to be equivariance with respect to SE(3) group if $s_\theta(T_f(x)) = T_g(s_\theta(x))$ for any transformation $f, g \in$ SE(3). Utilizing conformational representations directly to achieve equivariance presents challenges in accurately capturing the chemical interactions between atoms. Consequently, this approach may result in the generation of molecular structures with inaccuracies and poor configurations. An alternative approach is to use the inter-atomic distance that is naturally equivariant to SE(3) groups (Shi et al., 2021; Xu et al., 2021b; Gasteiger et al., 2020), which will be further introduced in Sec. 4.2. 3.3 Learning via Score Matching Langevin dynamics. Given a fixed step size $0 < \epsilon \ll 1$, take $x_0 \sim \pi(x)$ for some prior distribution and use Euler–Maruyama method for simulating the Langevin dynamics \[ x_t = x_{t-1} + \frac{\epsilon}{2} \nabla_x \log p(x_{t-1}) + \sqrt{\epsilon} z_t, \] where $z_t \sim \mathcal{N}(0, I)$. As $t \to \infty$, $x_t$ can be considered as a sample draw from $p(x)$ under some regularity conditions (Welling & Teh, 2011). This implies that if we know the score function $\nabla_x \log p(x)$, we can use Langevin dynamics to sample from $p(x)$. Denoising score matching. The process of denoising score matching (Vincent, 2011) involves the perturbation of data $x$ in accordance with a predetermined perturbing kernel, denoted by $q_\sigma(\tilde{x} \| x)$. The objective $s_\theta$ that minimize the following: \[ \frac{1}{2} \mathbb{E}_{q_\sigma(\tilde{x} \| x)p_{\text{data}}(x)} \left[ \\| s_\theta(\tilde{x}) - \nabla_{\tilde{x}} \log q_\sigma(\tilde{x} \| x) \\|_2^2 \right] \] satisfies $s_\theta(x) = \nabla_x \log q_\sigma(x)$ almost surely (Vincent, 2011). This implies that to train a denoising model $s_\theta$, we can set the loss functions to be \[ \mathcal{L}(s_\theta; \{\sigma_i\}_{i=1}^L) \triangleq \frac{1}{L} \sum_{i=1}^L \lambda(\sigma_i) \ell(s_\theta; \sigma_i) \] \[ \ell(s_\theta; \sigma) \triangleq \frac{1}{2} \mathbb{E}_{p_{\text{data}}(x)} \mathbb{E}_{\tilde{x} \sim q_\sigma(\tilde{x} \| x)} \\| s_\theta(\tilde{x}, \sigma) - \nabla_{\tilde{x}} \log q_\sigma(\tilde{x} \| x) \\|_2^2. \] where $\lambda(\sigma) \propto 1 / \mathbb{E} \left[ \\| \nabla_{\tilde{x}} \log p_\sigma(\tilde{x} \| x) \\|_2^2 \right]$ is a reweighting coefficient so that the magnitude order of the loss function does not depend on $\sigma$ (Song et al., 2020b). After obtaining a model $s_\theta(x) \approx \nabla_x \log q_\sigma(x)$, following the (annealed) Langevin dynamics (Song & Ermon, 2019), one can draw sample from $p_{\text{data}}(x)$ by recursive computing $\tilde{x}_t = \tilde{x}_{t-1} + \frac{\epsilon}{2} s_\theta(x_{t-1}, \sigma_t) + \sqrt{\alpha_t} z_t$, where $\alpha_t = \epsilon \cdot \sigma_t^2 / \sigma_L^2$. Maxwell-Boltzmann distribution. In the domain of statistical mechanics, the Maxwell-Boltzmann (MB) distribution serves as a model for delineating the velocities of particles within idealized gaseous systems. These systems are characterized by freely moving particles within a stationary enclosure, where interactions among the entities are negligible apart from momentary collisions. From a mathematical perspective, the MB distribution is the $\chi$-distribution with three degrees of freedom (Young et al., 2008). The probability density function of MB($\sigma$) is given by $f_\sigma(x) = \sqrt{\frac{2}{\pi}} \frac{x^2 e^{-x^2/(2\sigma^2)}}{\sigma^3}$ with support $\mathbb{R}_{++}$. 4 Methodology 4.1 Modeling the Distribution of Inter-Atomic Distances In the present investigation, molecular disintegration is facilitated by the application of progressively intensified perturbation force fields. Upon perturbing a single atom, adjacent atoms experience a consequent force, arising from the chemical bonds interconnecting them with the perturbed atom. In case when a relatively minor perturbative force field is employed, chemical bonds remain unbroken, thereby restricting atomic motions. This observation leads us to hypothesize that individual atoms exhibit Brownian motions under such conditions. Contrarily, when a sufficiently potent force field is imposed, chemical bonds are destroyed, permitting atoms to undergo virtually uninhibited motion with the bare occurrence of collisions. We further hypothesize that the relative speed between any two atoms adheres to the Maxwell-Boltzmann (MB) distribution. Focusing on the inter-atomic distances $d$ within a molecule, we establish that the marginal distribution of perturbed inter-atomic distances $\tilde{d}$, given $d$, is equivalent to the distribution of relative velocities among the atoms. Specifically, let $\sigma_t$ measure the perturbing force fields at time $t$ and $\{\sigma_t\}_{t=0}^T$ is an increasing non-negative sequence. Then, \[ p_{\sigma_0}(\tilde{d}\|d) = p_{\sigma_0}(v) = \mathcal{N}(\tilde{d}\|d, 2\sigma_0^2 I), \quad p_{\sigma_T}(\tilde{d}\|d) = p_{\sigma_T}(v) = \text{MB}(\sqrt{2\sigma_T}). \] Figure 2: In the investigation of perturbed distance distributions resulting from the introduction of Gaussian noise to molecular conformation, a transition from Gaussian to MB is observed as the noise level escalates. The perturbation’s intensity is denoted by $\sigma$. Within the graphical representation, the orange curve delineates the pdf of $\mathcal{N}(0, 2\sigma^2)$, the green curve corresponds to the pdf of $\text{MB}(\sqrt{2}\sigma)$, and the blue dotted curve represents the pdf of $p(d\|\tilde{d})$. For intermediate perturbing forces, we set $p_{\sigma_t}(\tilde{d}\|d) \propto \tilde{d}^2 f_{\sigma_t}(\tilde{d}, d) e^{-\frac{(\tilde{d}-d)^2}{4\sigma_t^2}}$, where several constrains are on $f_{\sigma}$. For a smoothly shifting perturbing force field, we require $f_{\sigma}(\tilde{d}, d)$ to be smooth with respect to $\sigma$, $\tilde{d}$ and $d$. To make the limiting perturbing force field be Gaussian and MB, we require $\lim_{\sigma \to 0} f_{\sigma} = 0$ and $\lim_{\sigma \to \infty} f_{\sigma} = 1$. Thus, we have (note that when $\sigma_T$ is sufficiently large, $\tilde{d} - d \approx \tilde{d}$) $$p_{\sigma_0}(\tilde{d}\|d) \propto e^{-\frac{(\tilde{d}-d)^2}{4\sigma_0^2}} \propto \mathcal{N}(\tilde{d}\|d, 2\sigma_0^2 I)$$ (6a) $$p_{\sigma_T}(\tilde{d}\|d) \propto \tilde{d}^2 e^{-\frac{(\tilde{d}-d)^2}{4\sigma_T^2}} \propto \text{MB}(\sqrt{2}\sigma_T)$$ (6b) If we take $f_{\sigma}(\tilde{d}, d) = 1 - e^{-\sigma/d}$, $$\nabla_{\tilde{d}} \log q_{\sigma}(\tilde{d} \| d) = \left(1 - e^{-\sigma/d}\right) \frac{2}{\tilde{d}} - \frac{\tilde{d} - d}{2\sigma^2}$$ (7) We can simply use a Gaussian kernel as an approximation of perturbing force fields acting on the molecule conformation, i.e., $p_{\sigma}(\tilde{C}\|C) = \mathcal{N}(\tilde{C}\|C, \sigma^2 I)$, for $C \in \mathbb{R}^{n \times 3}$, so that the limiting distributions of atoms’ speed and conditional perturbed inter-atomic distance are Gaussian and MB distributions. This is because $$\tilde{C}_u = C_u + z_u \quad \tilde{C}_v = C_v + z_v \quad \text{where } z_u, z_v \sim \mathcal{N}(0, \sigma^2 I)$$ $$\tilde{d}_{uv} = \\|z + C_u - C_v\\| \quad (z = z_u - z_v \sim \mathcal{N}(0, 2\sigma^2 I))$$ $$= \\|C_u - C_v\\| + \\|z + C_u - C_v\\| - \\|C_u - C_v\\|$$ $$= d_{uv} + \frac{2z^\top(C_u - C_v) + \\|z\\|^2}{\\|z + C_u - C_v\\| + \\|C_u - C_v\\|}$$ When $\sigma$ is sufficiently small, $\tilde{d}_{uv} \approx d_{uv} + \frac{2z^\top(C_u - C_v)}{2\\|C_u - C_v\\|} = d_{uv} + \hat{z}$, where $\hat{z} \sim \mathcal{N}(0, 2\sigma^2)$. When $\sigma$ is sufficiently large, $\tilde{d}_{uv} \approx d_{uv} + \frac{\\|z\\|^2}{\\|z + C_u - C_v\\|} \approx \\|z\\|$, where $\\|z\\| \sim \text{MB}(\sqrt{2}\sigma)$. For a comprehensive elucidation of intermediary mathematical procedures, we direct the readers to Appendix A. We conduct experiments to verify the above mathematical derivation. In the conducted experiments, Gaussian perturbations with varying levels of variation are introduced to molecular conformations, i.e., $p(\tilde{C}\|C) = \mathcal{N}(0, \sigma^2 I)$, for $C \in \mathbb{R}^{n \times 3}$, and the marginal distributions of the difference in inter-atomic distances before and after perturbation are examined. The resultant observations can be seen in Fig. 2 and 3. 4.2 Modeling Conformations We model the inter-atom distances instead of the conformation for equivariance as discussed in Sec. 3.2. Consider molecules formed by $n$ atoms, where $n \geq 5$. Given any $C \in \mathbb{R}^{n \times 3}/\text{SE}(3)$, let Figure 3: Distribution approximation. The actual pdf $ p_\sigma(\tilde{d} - d \| d = \text{const}) $ is illustrated by the orange curve, whereas the blue dotted curve signifies the proposed approximated pdf. $ d(\cdot) : \mathbb{R}^{n \times 3}/\text{SE}(3) \to \mathbb{D} $ be the mapping from conformations to all inter-atomic distances, where $ \mathbb{D} := \text{image}(d) $. Hence, $ \mathbb{R}^{n \times 3}/\text{SE}(3) $ and $ \mathbb{D} $ are isomorphisms since to ascertain the relative position of a particular point, it is merely necessary to determine its distances from 4 other non-coplanar distinct points. We use $ d_{ij} $ to denote the entry $(i, j)$ of the adjacent matrix and we have, by slight abuse of notations \[ \nabla_{\tilde{C}} \log q_\sigma(\tilde{C}\|C) = \frac{\partial}{\partial C} \log q_\sigma(\tilde{C}, d(\tilde{C})\|C, d(C)) \\ = \sum_{i,j} \frac{\partial d_{ij}(\tilde{C})}{\partial \tilde{C}} \frac{\partial}{\partial d_{ij}(\tilde{C})} \log q_\sigma(d(\tilde{C})\|d(C)) \quad (\text{almost surely}) \\ = \sum_{i,j} \frac{\partial \tilde{d}_{ij}}{\partial \tilde{C}} \nabla_{\tilde{d}_{ij}} \log q_\sigma(\tilde{d}\|d) \] The above property also holds for $ \tilde{d}(\cdot) $ that maps the conformation to a partial distance vector where each atom is associated with at least 4 distances. A previous work (Shi et al., 2021) showed that for any $ s_\theta(\tilde{d}) \approx \nabla_{\tilde{d}} \log q_\sigma(\tilde{d}\|d) $ as a function of the perturbed inter-atomic distance $ \tilde{d} $, the scoring network $ s_\theta $ is equivariant w.r.t. SE(3). By Eq. [5], [4], [8c] and [7], the denoising score matching objective for conformations is \[ L \left( \theta; \{\sigma_i\}_{i=1}^L \right) \triangleq \frac{1}{L} \sum_{i=1}^L \lambda(\sigma_i) \ell(\theta; \sigma_i) \] \[ \ell(\theta; \sigma) = \frac{1}{2} \mathbb{E}_{p_{\text{data}}(d)} \mathbb{E}_{p_\sigma(\tilde{d}\|d)} \left\\| s_\theta(\tilde{d}, \sigma) - \frac{\partial \tilde{d}}{\partial C} \left[ \left( 1 - e^{-\sigma/d} \right) \frac{2}{d} - \frac{\tilde{d} - d}{2\sigma^2} \right] \right\\|^2_2 \] Note that $ \nabla_{\tilde{C}} \log q_\sigma(\tilde{C} \| C) \neq -\frac{\tilde{C} - C}{\sigma^2} $ since $ \tilde{C}, C \in \mathbb{R}^{n \times 3}/\text{SE}(3) $ and the probability density function is different from that in $ \mathbb{R}^{n \times 3} $. Take $ \lambda(\sigma_i) = \sigma_i^2 $, $ \lambda(\sigma_i) \ell(\theta; \sigma_i) \propto 1 $ for any $ \sigma_i $. Thus, the loss magnitude order of the loss function does not depend on the specific selection of $ \sigma_i $. 4.3 Network for modeling conformation score The network employed for the purpose of modeling $ s_\theta $ must adhere to two specific criteria which are delineated in Sec. 4.2. For simplification, we omit the model’s parameter of molecular graph $ G $. SE(3) equivariance. It is imperative that the network abstains from utilizing molecular conformation directly as input; rather, it should incorporate inter-atomic distance to achieve SE(3) equivariance. The employment of perturbed distance as a means to directly forecast the conformation score necessitates a domain transition, thereby augmenting the complexity of the learning process. Thus, following the parametrization of the conformation score as discussed in Sec. 4.2, a generative model for estimating the score of distances is formulated, followed by the application of the chain rule to facilitate the conversion of distance scores into their corresponding values for conformation scores. Isomorphisms. Each individual atom must be associated with a minimum of four distances, in order to establish isomorphisms between $ C \in \mathbb{R}^{n \times 3}/\text{SE}(3) $ (representing conformation space) and $ \mathbb{D} $ (signifying feasible inter-atomic distance space). On the other hand, correlating an atom with an excessive number of distances exacerbates the challenge for the model to generate a feasible $d$. The underlying reason for this complication is the disparity in cardinal numbers of $\mathbb{R}^{n \times 3}/\text{SE}(3)$ and $\mathbb{D}$. $\mathbb{D}$ is a subset of $\mathbb{R}_+^m$, where $m = \binom{n}{2}$ is the number of edges in complete graph induced by the molecule. For a more detailed illustration, we refer readers to Appendix B. As a result, we connect the three-hop neighborhood in each chemical molecule so that almost every atom in a molecule is connected with at least four other atoms. Following GeoDiff (Xu et al., 2021b), we adapt a similar network for modeling $s_\theta$. Given an input graph $G$, the Message Passing Neural Networks (MPNN) (Gilmer et al., 2017) is adopted as $s_\theta$, which computes node embeddings $h_v^{(t)} \in \mathbb{R}^f, \forall v \in V$ with $T$ layers of iterative message passing: $$h_u^{(t+1)} = \psi \left( h_u^{(t)}, \sum_{v \in N_u} h_v^{(t)} \cdot \phi(e_{uv}, d_{uv}) \right)$$ for each $t \in [0, T - 1]$, where $N_u = \{v \in V \| (u, v) \in E\}$, while $\psi$ and $\phi$ are neural networks, e.g. implemented using multilayer perceptrons (MLPs). Note that the node features, distances and edge features are input into $s_\theta$ as initial embeddings when $t = 0$, but we only keep the distance $d$ in the above sections as the input of $s_\theta$ for notation simplification. Besides, as no coordinates information is explicitly engaged in this network, this kind of modeling can preserve the above two properties. For more details about this part, refer to Appendix B. ### 4.4 SAMPLING BY LANGEVIN DYNAMICS The learned score matching network $s_\theta$ that minimizes Eq. 9a can approximate the score of molecular conformation and following the annealed Langevin dynamics, we provide the pseudocode of the sampling process in Alg. 1, from which we can draw conformations given molecule. #### Algorithm 1 Sampling via annealed Langevin dynamics Input: molecular graph $G$, network $s_\theta$, scheduler $\{\sigma_i\}_{i=1}^T$. Output: conformation $C$. 1: Sample $C \sim \mathcal{N}(0, \sigma_0^2 I)$. 2: for $i = T, T-1, \ldots, 1$ do 3: $\alpha_i \leftarrow \epsilon \cdot \sigma_i^2 / \sigma_T^2$ \{ $\alpha_i$ is the step size.\} 4: Sample $z_i \sim \mathcal{N}(0, I)$ 5: $C_{i-1} \leftarrow C_i + \alpha_i s_\theta(d(C_i), \sigma_i) + \sqrt{2\alpha_i} z_i$ \{Langevin dynamics.\} 6: end for 7: return $C_0$ ### 4.5 ANALYSIS Marginal v.s. joint distributions. From existing literature, the diffusion models are built on adding isotropic Gaussian noise $\mathcal{N}(0, \sigma^2 I)$ to the modeled objects such as pixel values in image generations. In SDDiff, we add isotropic Gaussian noise to molecule conformation (coordinate), and noise is mapped to inter-atomic distances. Thus, entries of noise on distance are not independent, whereas the marginal distribution of distances can be applied for score matching, this is because $$\nabla_{\tilde{d}_i} \log p_\sigma(\tilde{d}_i \| d) = \nabla_{\tilde{d}_i} \log p_\sigma(\tilde{d}_i \| d_1, 2, \ldots, m) \cdot p_\sigma(\tilde{d}_1, 2, \ldots, i-1, i+1, \ldots, m \| d_1, 2, \ldots, m, \tilde{d}_i, d_i)$$ $$= \nabla_{\tilde{d}_i} \log p_\sigma(\tilde{d}_i \| d_i) + \nabla_{\tilde{d}_i} \log p_\sigma(\tilde{d}_N(i) \| d_N(i), \tilde{d}_i, d_i) \approx \nabla_{\tilde{d}_i} \log p_\sigma(\tilde{d}_i \| d_i)$$ where $N(i)$ is the set of edge indices whose edges are incident with edge $i$. The second equality holds because $\tilde{d}_i$ gives no information on the distribution of other perturbed edges that are not incident with edge $i$. Also, $d_j$ gives no information on the distribution of $\tilde{d}_i$ where $i \neq j$. We hypothesize that disregarding the term $\nabla_{\tilde{d}_i} \log p_\sigma(\tilde{d}_N(i) \| d_N(i), \tilde{d}_i, d_i)$ introduces no bias. This supposition stems from the observation that possessing knowledge of both $\tilde{d}_i$ and $d_i$, we remain uninformed about the increase or decrease in the value of $d_N(i) - d_N(i)$. Approximation by optimal transportation (OT). Given the knowledge of the distributions at end time points $p_{t=0}(x)$ and $p_{t=T}(x)$, the problem of obtaining the distributions in between can be formulated as a Shrodinger Bridge problem whose solution is also the solution of entropic OT. We compute the regularized Wasserstein Barycenter of $p_{t=0}(\tilde{d}\|d)$ and $p_{t=T}(\tilde{d}\|d)$ by employing the approach presented in a previous work (Benamou et al., 2015). However, the regularization term Table 1: Results of molecular conformation generation. \| Methods \| GEOM-QM9 COV(%) ↑ \| GEOM-QM9 MAT(Å) ↓ \| GEOM-Drugs COV(%) ↑ \| GEOM-Drugs MAT(Å) ↓ \| \|-----------\|------------------\|-------------------\|---------------------\|---------------------\| \| \| Mean \| Median \| Mean \| Median \| Mean \| Median \| Mean \| Median \| \| CGCF \| 78.05 \| 82.48 \| 0.4219 \| 0.3900 \| 53.96 \| 57.06 \| 1.2487 \| 1.2247 \| \| ConfVAE \| 77.84 \| 88.20 \| 0.4154 \| 0.3739 \| 55.20 \| 59.43 \| 1.2380 \| 1.1417 \| \| GeoMol \| 71.26 \| 72.00 \| 0.3731 \| 0.3731 \| 67.16 \| 71.71 \| 1.0875 \| 1.0586 \| \| ConfGF \| 88.49 \| 94.31 \| 0.2673 \| 0.2685 \| 62.15 \| 70.93 \| 1.1629 \| 1.1596 \| \| GeoDiff \| 90.54 \| 94.61 \| 0.2090 \| 0.1988 \| 89.13 \| 97.88 \| 0.8629 \| 0.8529 \| \| SDDiff (ours) \| 91.07 \| 94.69 \| 0.2048 \| 0.1941 \| 90.68 \| 98.48 \| 0.8564 \| 0.8503 \| impacts the limiting weighted Barycenter, leading to divergences from $p_{t=0}(\tilde{d}\|d)$ to $p_{t=T}(\tilde{d}\|d)$. As a result, the regularized Wasserstein Barycenter approach is unsuitable for intermediate distribution approximation. See Appendix C for a more detailed analysis. 5 EXPERIMENT 5.1 EXPERIMENT SETTINGS Datasets. We use two widely used datasets, GEOM-QM9 (Ramakrishnan et al., 2014) and GEOM-Drugs (Axelrod & Gómez-Bombarelli, 2022) for evaluating molecular conformation generation. The GEOM-QM9 dataset comprises molecules with an average of 11 atoms, while the GEOM-Drugs dataset consists of larger molecules with an average of 44 atoms. For a fair comparison, we adopted the same dataset split as GeoDiff (Xu et al., 2021b). For both datasets, the training set contains 40k molecules, the validation set contains 5k molecules and the test set contains 200 molecules. Please refer to GeoDiff (Xu et al., 2021b) for more details regarding the dataset. Evaluation metrics. We use the metrics of COV (coverage) and MAT (matching) (Xu et al.) to measure both diversity and accuracy. Specifically, we align ground truth and generated molecules by the Kabsch algorithm (Kabsch, 1976), and then calculate their difference with root-mean-square-deviation (RMSD). Then the COV and the MAT are defined as follows: $$\text{COV} = \frac{1}{\|S_r\|} \left\{ C \in S_r \mid \text{RMSD}(C, C') < \delta, \exists C' \in S_g \right\}, \quad \text{MAT} = \frac{1}{\|S_r\|} \sum_{C' \in S_g} \text{RMSD}(C, C')$$ where $S_g$ and $S_r$ denote generated and ground truth conformations, respectively. Following some baselines (Xu et al., 2021b; Ganea et al., 2021), we set the threshold of COV $\delta = 0.5$ Å for GEOM-QM9 and $\delta = 1.25$ Å for GEOM-Drugs, and generate twice the number of ground truth conformation for evaluation. Baselines. We choose 5 state-of-the-art models for comparison: GeoMol (Ganea et al., 2021) is not a generative model that generates conformation by hand with predicted molecular information. CGCF (Shi et al., 2021) is a two-step method, and ConfVAE (Xu et al., 2021a) is a VAE-based model. ConfGF (Shi et al., 2021) and GeoDiff (Xu et al., 2021b) are two similar works that are also diffusion-based. Other implementation details are provided in Appendix D. 5.2 RESULTS AND ANALYSIS The results of molecular conformation generation are shown in Table 1. The baseline results are obtained from GeoDiff (Xu et al., 2021b). In order to mitigate the impact of the model’s backbone and primarily evaluate the efficacy of distance distribution modeling, we have opted to utilize a backbone that closely resembles that of GeoDiff. This will enable us to more accurately assess the performance of the distance distribution modeling technique while minimizing the potential confounding effects. Figure 4: The ground truth depicted in blue is the distribution of $\sigma \nabla_{\tilde{d}} \log p(\tilde{d}\|d)$, whereas the distribution of the model’s outputs is represented by a dashed orange line. It can be observed that as the value of $\sigma$ increases, $\sigma \nabla_{\tilde{d}} \log p(\tilde{d}\|d)$ tends to exhibit the characteristics of a long-tailed Gaussian distribution. For a detailed introduction to the figure, we refer readers to Appendix E. of the model’s underlying architecture. The Visualization of selected generated conformation can be found in Appendix G. Score distribution. In the existing literature, the ground truth score function follows a normal distribution. Specifically, the ground truth of score matching objects is set to $\sigma \nabla_x \log p(x\|x) \sim \mathcal{N}(0, I)$. The proposed distance distribution diverges from the Gaussian distribution when the perturbation level is significantly large and requires the model to parametrize a non-Gaussian distribution. In order to investigate the efficacy of existing backbones in approximating such distribution, we visually depict the distribution of score functions (not inter-atomic distance), along with our backbone’s output under varying levels of perturbation. The ensuing results have been found in Fig. 4. It is evident that our proposed distribution closely resembles the Gaussian distribution when $\sigma$ is reasonably small. Conversely, when $\sigma$ is substantially large, the proposed score function transforms into a long-tailed Gaussian distribution. Despite this alteration, the model’s output distribution still approximates the proposed score function effectively. This substantiates that the proposed distribution can be effortlessly approximated, and thus can be incorporated into a wide array of models. Planar structure generation As mentioned in Eq. 8b, the score function of distance can be transformed into the score function of conformation almost surely, provided that the conformation is non-planar. Nonetheless, certain molecular structures like benzene rings, exhibit a planar conformation within local regions, which may render this transformation inapplicable (see Fig. 5). A viable solution to optimize these local planar structures further involves utilizing post-processing with variants of rule-based methods (e.g., force field) which encode the unvarying property of certain local structures like benzene rings being planar. Figure 5: Atoms in a benzene ring should be coplanar as ground truth structure, while the generative structure may conflict with such property. 6 CONCLUSION In this study, we present a novel molecular conformation generation approach - SDDiff - by incorporating the shifting score function inspired by molecule thermodynamics. Our main findings include that the distribution of change of inter-atomic distances shifts from Gaussian to Maxwell-Boltzmann distribution under the Gaussian perturbation kernel on molecular conformation, which can be accurately approximated by our approach. By proposing a diffusion-based generative model with a shifting score kernel, we have provided both the mathematical derivation and experimental validation of its correctness. The effectiveness of our approach has been demonstrated through achieving new state-of-the-art results on two widely used molecular conformation generation benchmarks, namely GEOM-Drugs, and GEOM-QM9. Our method effectively captures the essential aspects of molecular dynamics and inter-atomic interactions, leading to improved performance in generating accurate and feasible molecular conformations. REFERENCES Simon Axelrod and Rafael Gómez-Bombarelli. Geom, energy-annotated molecular conformations for property prediction and molecular generation. 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sGd02fkoAE	Table 1: from the writing, I assume the first row is performing 2D detection (comparing DETR, Swin, CameraViT). The second two rows are 3D detection with lidar and lidar-camera fusion, respectively. Why is the first group being compared to the second two?	FusionViT: Hierarchical 3D Object Detection via Lidar-Camera Vision Transformer Fusion Anonymous authors Paper under double-blind review Abstract For 3D object detection, both camera and lidar have been demonstrated to be useful sensory devices for providing complementary information about the same scenery with data representations in different modalities, e.g., 2D RGB image vs 3D point cloud. An effective representation learning and fusion of such multi-modal sensor data is necessary and critical for better 3D object detection performance. To solve the problem, in this paper, we will introduce a novel vision transformer-based 3D object detection model, namely FusionViT. Different from the existing 3D object detection approaches, FusionViT is a pure-ViT based framework, which adopts a hierarchical architecture by extending the transformer model to embed both images and point clouds for effective representation learning. Such multi-modal data embedding representations will be further fused together via a fusion vision transformer model prior to feeding the learned features to the object detection head for both detection and localization of the 3D objects in the input scenery. To demonstrate the effectiveness of FusionViT, extensive experiments have been done on real-world traffic object detection benchmark datasets KITTI and Waymo Open. Notably, our FusionViT model can achieve the state-of-the-art performance and outperforms not only the existing baseline methods that merely rely on camera images or lidar point clouds, but also the latest multi-modal image-point cloud deep fusion approaches. 1 Introduction Developing an efficient 3D object detection model is core for autonomous driving. Lidar and cameras are two commonly used sensors in this task (Li et al., 2016; Ku et al., 2017). Meanwhile, the sensor data obtained by Lidar and camera are in totally different modalities, i.e., sparse point clouds vs RGB images, which is very difficult to handle and fuse together within one model for the object detection task (Arnold et al., 2019). Pure image based (Girshick, 2015; Carron et al., 2020; Liu et al., 2021; Wang et al., 2022) and pure point cloud based (Zhou & Tuzel, 2017; Lang et al., 2018; Yin et al., 2020; Zhou et al., 2023c; Zhou et al., 2022) strategies have exploited their raw data structure characteristics in the object detection task, whose performance is still far below the safety requirements for their deployment in our modern autonomous driving vehicles (Committee, 2021). Modern autonomous driving vehicles pose higher and strict standards for object detection models in the perspective of both detection accuracy and model robustness. A single modality often cannot fully describe the complex correlation between data. However, image and point cloud can in some way complement each other: image information can focus on textual conditions, while point cloud data will not be affected by the light condition and also carries the scenery depth (Arnold et al., 2019). As a result, there are scenarios where it is hard to detect with just one type of sensor. Employing multi-modal data to describe the same scene from different perspectives helps extract comprehensive features to make the object detector more robust. Viewed from such a perspective, how to effectively combine such multi-modal sensory data while preserving the essential features of each modality becomes the key challenge. Recently, transformers (Vaswani et al., 2017) have been widely deployed in Natural Language Processing (NLP) tasks (Devlin et al., 2018; Brown et al., 2020; Yang et al., 2019) with state-of-the-art performance. After ViT (Dosovitskiy et al., 2020), transformers started to be utilized on various vision-related research tasks, such as image classification (Touvron et al., 2020; Heo et al., 2021). Fang et al., 2021), 2D (Carion et al., 2020; Liu et al., 2021) and 3D object detection (Misra et al., 2021; Wang et al., 2021b; Zhu et al., 2022; Zhou et al., 2023a, 2022; Bai et al., 2022). Given that ViT was the initial CV application of the transformer model in the field of vision, it could have great potential for broad application scenarios. However, its uses are primarily restricted to (Touvron et al., 2020; Fang et al., 2021; Heo et al., 2021). In the field of 3D object detection, most state-of-the-art frameworks are not from ViT itself but its derivatives, like DETR-based methods (Wang et al., 2021b; Misra et al., 2021) which contain both transformer encoder and decoder, or Zhou et al. (2023a, 2022) that modified a lot from the original ViT prototype. Could it also exist a pure-ViT based framework that enhances the performance of 3D object detection? That original model, indeed, may not be in the best settings for the 3D object detection task. The computational complexity of ViT increases quadratically with respect to image size, making it a severe issue to be used on a lidar point cloud, which contains thousands of points in a simple frame. In this paper, we propose a hierarchical vision transformer-based lidar-camera fusion strategy for object detection called FusionViT to relieve that issue so that it could reach promising performance, especially in traffic scenery. Based on the multi-modal camera image and lidar point cloud inputs, FusionViT includes a CameraViT and a LidarViT for learning the embedding representations of input data in each modality, respectively. Their learned representation will be further fused together for hierarchical and deeper representation learning via a novel MixViT component. By partitioning images into mini-patch for representation learning, our proposed CameraViT extends the ViT (Dosovitskiy et al., 2020) model to object detection task, showing competitive 2D object detection performance. In addition, by partitions point clouds into mini-cubic, we introduce a novel LidarViT as a volumetric-based 3D object detector. Using a transformer encoder as the end-to-end backbone, it discretizes the point cloud into equally spaced 3D grids. It has great advantages in terms of helping the feature extraction network to be computationally more effective and reducing memory needs while keeping the original 3D shape information as much as possible. The most important contribution attributes to our proposed FusionViT model, bridging as consistent as possible between the fusion model and pre-fusion models, Benefiting from the transformer encoder model, our proposed FusionViT suits greatly under traffic scenes, where input frames are continuously changing spatially and timely. Our model is not like previous methods (Chitta et al., 2022; Ku et al., 2017; Chen et al., 2016) that combine 2D features extracted from image or projection (such as Bird-eye-view, Range view, or Depth image), which introduces information bottleneck from these 3D to 2D projection operations. By directly doing end-to-end fusion, our model is also not necessary to explicitly set pixel-to-point-cloud alignment like Deepfusion (Li et al., 2022b) did, while outperforming it, showing the strong robustness of our strategy. In short, our main contributions are listed as follows: - We are the first study to investigate the possible pure-ViT based 3D object detection framework to the best of our knowledge. - We proposed a hierarchical pure-ViT based lidar-camera fusion framework for it called FusionViT, which exploited the inherent features between image and point cloud. - We performed a vit-based detector on both 2D image detection and 3D point cloud detection respectively, leading to great performance under each part. - Extensive experiments are conducted on the Waymo Open Dataset and KITTI benchmarks. FusionViT achieves competitive results compared with existing well-designed approaches, showing a promising future of pure-ViT based frameworks in 3D object detection tasks. 2 RELATED WORK 2.1 2D OBJECT DETECTION 2D object detection locate objects of interest in 2D images and classify them into pre-defined categories. The advances in deep learning have revolutionized the field of 2D object detection. The R-CNN (Girshick et al., 2013) and its extensions (Girshick, 2015; Ren et al., 2015; He et al., 2017) used a Region Proposal Network to generate candidate object proposals and a Convolutional Neural Network (CNN) for object classification. Besides these two-stage detectors, single-stage detectors like SSD (Liu et al., 2016), YOLO serious (Redmon et al., 2015; Redmon & Farhadi, 2016; Bochkovskiy et al., 2020; Li et al., 2022a; Wang et al., 2022) simultaneously classifies anchor boxes and regresses the bounding boxes at the same time, which are usually more efficient in terms of inference time while comparably get slightly lower accuracy. The revolution from the transformer (Vaswani et al., 2017; Dosovitskiy et al., 2020) brings more powerful 2D detectors (Carion et al., 2020; Liu et al., 2021), which further increases the competitive edge. 2.2 3D Object Detection in Point Clouds Represented as an unordered set, point cloud provides richer information about the environment, enabling more accurate and robust object detection. Early methods (Qi et al., 2016; 2017b) directly apply Neural Networks on the raw points. (Karlinsky et al., 2020; Shi et al., 2018) also learn features using PointNet-like layers. Projection-based methods first project point cloud into 2D representation, such as range images (Meyer et al., 2019; Sun et al., 2021), then use Neural Networks to predict 3D bounding boxes. Volumetric-based methods convert lidar points to voxels (Yan et al., 2018; Zhou & Tuzel, 2017) or pillars (Lang et al., 2018; Yang et al., 2021). These three tracks are primarily exploited by modern cutting-edge approaches (Zhou et al., 2023a; c; 2022; Shi et al., 2020). Some of them (Sun et al., 2022; Fan et al., 2021; 2022) also take advantage of sparse mechanics (Yan et al., 2018) to reduce onerous computational demands while retaining excellent detection accuracy. Others (Misra et al., 2021; Zhang et al., 2022; Wang et al., 2021b; Liu et al., 2022a) focus on making full use of transformers (Vaswani et al., 2017) by bridging its natural gap for 3D point cloud. 2.3 Lidar-Camera Fusion The key task of Lidar-Camera fusion should be the feature alignment between Point Cloud and Camera Image. Along this idea, (Vora et al., 2019; Sindagi et al., 2019; Wang et al., 2021a) match each point cloud of camera images, extracting features from the camera images to decorate the raw point clouds. State-of-the-art approaches (Prakash et al., 2021; Bai et al., 2022; Zeng et al., 2022; Liu et al., 2022b; Zhou et al., 2023b) are primarily based on LiDAR-based 3D object detectors and strive to incorporate image information into various stages of a LiDAR detection pipeline since LiDAR-based detection methods perform significantly better than camera-based methods. Combining the two modalities necessarily increases computing cost and inference time lag due to the complexity of LiDAR-based and camera-based detection systems. As a result, the problem of effectively fusing information from several modes still exists. 3 Proposed Methods 3.1 Notations and Terminologies In the sequel of this paper, we will use the upper or lower case letters (e.g., $X$ or $x$) to represent scalars, lower case bold letters (e.g., $\mathbf{x}$) to denote column vectors, and bold-face upper case letters (e.g., $\mathbf{X}$) to denote matrices, and upper case calligraphic letters (e.g., $\mathcal{X}$) to denote sets or high-order tensors. We use $\mathbf{X}^\top$ and $\mathbf{x}^\top$ to represent the transpose of matrix $\mathbf{X}$ and vector $\mathbf{x}$. The concatenation of vectors $\mathbf{x}$ and $\mathbf{y}$ of the same dimension is represented as $\mathbf{x} \sqcup \mathbf{y}$. Figure 1 shows the overall architecture of the proposed FusionViT. FusionViT accepts multi-modal inputs, which include both RGB images and point clouds. The input images are defined as $\mathcal{I} \in \mathbb{R}^{H_I \times W_I \times 3}$, where $H_I$ and $W_I$ denote the image height and width dimensions, respectively. Meanwhile, the input point cloud is represented as a set of 3D points $\mathcal{P} \in \mathbb{R}^{H_P \times W_P \times D_P}$ in the $H_P \cdot W_P \cdot D_P$ 3D space, where each point in it is a vector of its $(x, y, z)$ coordinate. As shown in Figure 1, our FusionViT model has a hierarchical architecture. Given camera images and lidar point cloud as inputs, some necessary data prepossessing is needed to produce the 2D image embedding and 3D point cloud embedding, respectively. Based on the partitioned image and point cloud batches, we introduce a CameraViT model to learn the image embedding and a LidarViT to learn the 3D point cloud embedding, respectively. Their learned representation will be combined together and fed as the inputs of the mix component for representation fusion and learning. Taking its output, a MixViT model will further fuse the learned embedding. An object Detection head is finally added to detect the objects existing in the input scenery data. In this section, we will introduce those aforementioned functional components in detail for readers. ### 3.2 CameraViT based Image Embedding Camera as a sensory device can capture the scenery information with RGB images. Object detection and localization from images have been studied for many years (Girshick et al., 2013; Liu et al., 2016; Redmon et al., 2015). In recent years, transformer (Vaswani et al., 2017) based models have been demonstrated to outperform conventional CNN models (Lecun et al., 1998) in solving many vision problems (Dosovitskiy et al., 2020). In this paper, we propose CameraViT, which extends the ViT model to the traffic scenery object detection problem setting, by partitioning images into mini-patch for representation learning. Formally, within the CameraViT model, by setting each patch to have the side length $v_{cH}$ and $v_{cW}$ respectively, it partitions each image $\mathbf{I} \in \mathbb{R}^{H_I \times W_I \times 3}$ image into $N_c$ 2D patches $\mathbf{X}_I \in \mathbb{R}^{N_c \times (v_{cH} \cdot v_{cW} \cdot 3)}$, where $N_c = \frac{H_I}{v_{cH}} \cdot \frac{W_I}{v_{cW}}$ denotes the patch number. Such partitioned image patches will be flattened into one-dimensional $v_{cH} \times v_{cW} \times 3$ features, then fed into a Multi-Layer Perceptron (MLP) to reach in total $N_c$ encoded features. For each encoded features $i$, we have $$\mathbf{x}_c^i = \text{MLP}(\mathbf{x}_I^i).$$ The MLP’s parameters are shared by all patches so that they are encoded in the same way. Each encoded feature $\mathbf{x}_c^i$ has the same vector size $D_c$, which is also the output size of the MLP. Similar to BERT’s [class] token (Devlin et al., 2018), we prepend a learnable embedding $\mathbf{E}_c$ to the sequence of the encoded patch features (i.e., $\mathbf{z}_0^c = \mathbf{x}_{\text{class}}$). Position embeddings $\mathbf{E}_{posc}$ are added to the patch embeddings to retain positional information, i.e., $\mathbf{Z}_0 = [\mathbf{x}_{\text{class}}; \mathbf{x}_c^1 \mathbf{E}_c; \mathbf{x}_c^2 \mathbf{E}_c; \cdots; \mathbf{x}_c^N \mathbf{E}_c] + \mathbf{E}_{posc} \in \mathbb{R}^{(N_c+1) \times D_c}$. The resulting sequence of embedding vectors serves as input to the Camera Transformer Encoder. The encoder consists of alternating layers of Multiheaded Self-Attention (MSA) (Vaswani et al., 2017) and MLP blocks. Layernorm (LN) is applied before every block, and residual connections after every block: $$\begin{align} \mathbf{Z}_l' &= \text{MSA}(\text{LN}(\mathbf{Z}_{l-1})) + \mathbf{Z}_{l-1}, \\ \mathbf{Z}_l &= \text{MLP}(\text{LN}(\mathbf{Z}_l')) + \mathbf{Z}_l', \quad \forall l = 1, 2, ..., L_c, \end{align}$$ where $L_c$ is the layer number of the CameraViT model. The final image represented features $\mathbf{H}_c$ are generated from the Layernorm of the output of the Transformer encoder $\mathbf{Z}_{L_c}^0$: $$\mathbf{H}_c = \text{LN}(\mathbf{Z}_{L_c}^0),$$ where $\mathbf{H}_c \in \mathbb{R}^{h_c \times N_c}$, assuming $h_c$ is the hidden size of the camera transformer encoder. The learned features $\mathbf{H}_c$ could complete pure 2D Object Detection tasks by adding an object detection head following it. It also has the potential to concatenate with other learned features from different sensors to perform multi-modal prediction. 3.3 LidarViT based Point Cloud Embedding Transformers (Vaswani et al., 2017) is well suited for operating on 3D points since they are naturally permutation invariant and some transformers-based 3D Object Detectors (Misra et al., 2021; Zhang et al., 2022; Wang et al., 2021b; Zhu et al., 2022; Liu et al., 2022a; Zhou et al., 2023a, 2022) are proposed to solve the 3D vision problem, gaining promising performance. On the other hand, ViT was the first work to apply the transformer model in the vision field. That model, however, may not be in the best settings for the 3D object detection task. The computational complexity of ViT increases quadratically with respect to image size, making it a severe issue to be used on a lidar point cloud, which contains thousands of points in a simple frame. Could it also exist a pure-ViT based structure that enhances the performance of 3D object detection? Inspired by the Voxel Feature Encoding layer in (Zhou & Tuzel, 2017), we designed a voxel-based LidarViT to resolve the huge memory burden, making it more suitable for point cloud data structure. LidarViT process the point cloud input from Lidar. Inspired by 2D image processing operations, we here divide the raw point cloud in the whole 3D space into little cubic, each with side length $P_l$. Different from images, Point Cloud is a sparse representation, which means most of these cubic is empty. We therefore first remove these empty cubic to cut down the computation burden. The number of non-empty Point Cloud cubic $N_l$ will also be the input length of the LidarViT model. We conduct Random Sampling for those cubic having a large number of point cloud. Typically a high-definition LiDAR point cloud is composed of about 100k points. Directly processing all the points not only imposes increased memory/efficiency burdens on the computing platform, but also highly variable point density throughout the space might bias the detection. To this end, we randomly sample a fixed number, $T$, of points from those cubic containing more than $T$ points. This sampling strategy has two purposes, (1) computational savings; and (2) decrease the imbalance of points between the cubic which reduces the sampling bias, and adds more variation to training. In each cubic, a flatten and a different MLP operation will be conducted. Raw point cloud is an unordered set, where each point is independent of others. Transformer encoders, on the other hand, are suited greatly to some kind of dependent data structure (like sentences, and images). Therefore, augmenting raw point cloud data could be one key idea to make full use of the Transformer structures. Inspired by the Voxel Feature Encoding layer in (Zhou & Tuzel, 2017), we use a $K$-layer hierarchical feature encoding process for each point cloud cubic. Setting each cubic has the side length $v_{lH}$, $v_{lW}$ and $v_{lD}$ respectively, it transforms the whole $P \in \mathbb{R}^{H_P \times W_P \times D_P}$ point cloud space into $N_l$ 3D cubics $X_P \in \mathbb{R}^{N_l \times (v_{lH} \cdot v_{lW} \cdot v_{lD})}$, where: $$N_l = \frac{H_P}{v_{lH}} \cdot \frac{W_P}{v_{lW}} \cdot \frac{D_P}{v_{lD}}$$ denotes the cubic number. For $X_P' = \left\{ x_P^i = [x_i, y_i, z_i]^T \in \mathbb{R}^3 \right\}_{i=1...t}$ as a non-empty cubic containing $t \leq T$ LiDAR points, where $x_P^i$ contains XYZ coordinates for the $i$-th point. We first compute the local mean as the centroid of all the points in $X_P'$, denoted as $(c_x, c_y, c_z)$. Then we augment each point $x_P^i$ with the relative offset w.r.t. the centroid and obtain the input feature set $X_{Pin} = \left\{ x_P^i = [x_i, y_i, z_i, x_i - c_x, y_i - c_y, z_i - c_z]^T \in \mathbb{R}^6 \right\}_{i=1...t}$. Each cubic will then be flattened into one-dimensional $v_{lH} \times v_{lW} \times v_{lD}$ features, and fed into a Multi-Layer Perceptron (MLP). For each encoded features $i$, we have $$x_i' = \text{MLP}(x_P^i).$$ (4) Then, each $x_i'$ will be transformed through a fully connected network (FCN) into a feature space $f_i$. The FCN is composed of a linear layer, a batch normalization (BN) layer, and a Sigmoid Linear Units (SILU) layer. After obtaining point-wise feature representations, element-wise MaxPooling is used across all $f_i$ to get the locally aggregated feature $\tilde{f}_i$. Finally, each $f_i$ with $\tilde{f}_i$ are combined to form the point-wise concatenated feature as $x_i^c$. In this way, we could output the encoded features $X_l = \{x_i^c\}_{i=1...t}$: $$f_i = \text{FCN}(x_i'), \tilde{f}_i = \text{MAX}(f_i), x_i^c = \left[ f_i^T, \tilde{f}_i^T \right]^T.$$ (5) The FCN and MLP parameters are shared by all non-empty cubic so that they are encoded in the same way. Each encoded feature $x_i^c$ has the same vector size $D_l$. We prepend another learnable embedding $E_l$ to the sequence of the encoded patch features. Position embeddings $E_{posl}$ are added to the patch embeddings to retain positional information, i.e. $z_0 = [x_{class}; x^1_l E_l; x^2_l E_l; \cdots ; x^N_l E_l] + E_{posl} \in \mathbb{R}^{(N_l+1) \times D_l}$. The resulting sequence of embedding vectors serves as input to the Lidar Transformer Encoder. Replacing $L_c$ into the layer number of Lidar Transformer Encoder $L_l$, formula (2) is applied. The final learned point cloud represented features $H_l$ are generated from the Layernorm of the output of the Transformer encoder $Z^0_{L_l}$: $$H_l = LN(Z^0_{L_l}),$$ where $H_l \in \mathbb{R}^{h_l \times N_l}$, assuming $h_l$ is the hidden size of the lidar transformer encoder. Similar to $H_c$, $H_l$ could complete pure 3D Object Detection tasks by adding an object detection head following it, as well as performing multi-modal prediction by concatenating with other learned features possibly from different sensors. ### 3.4 MixViT: CameraViT and LidarViT Fusion Another issue of keeping from directly using the naive ViT in the image-point cloud multi-modal detection task is the feature fusion compatibility from the image and point cloud branch. Lidar data provides accurate geometric information about the scene, while camera data provides rich texture and color information. The learned representation from CameraViT and LidarViT should be fused together smoothly. We, therefore, designed a MixViT as well as a hierarchical structure to make the sparse like point cloud data better compatible with the dense like image data. Being as consistent as possible between the fusion model and pre-fusion models, we purpose MixViT to cut down feature misalignment and model incompatibility, improving the accuracy and robustness. MixViT concatenates the learned feature representation from image $H_c = N_c + N_l$ and point cloud $H_l$. Then use another MLP to reach $N_m$ numbers of encoded features $X_m$: $$X_m = MLP(H_c \sqcup H_l).$$ Here we assume the camera transformer encoder and the lidar transformer encoder have the same hidden size (i.e., $h_c = h_l = h$). The MLP’s parameters are shared by all patches so that they are encoded in the same way. Each encoded feature $x^i_m$ has the same vector size $D_m$. We have tried other fusion strategies but find this concat operation performs the best. Similar to before, a learnable embedding $E_m$ to the sequence of encoded patch features is prepended, and Position embeddings $E_{posm}$ are added: $Z_0 = [x_{class}; x^1_m E_m; x^2_m E_m; \cdots ; x^N_m E_m] + E_{posm} \in \mathbb{R}^{(N_m+1) \times D_m}$. The resulting sequence of embedding vectors serves as input to the Mix Transformer Encoder. Replacing $L_c$ into the layer number of Mix Transformer Encoder $L_m$, formula (2) is applied. The final learned point cloud represented features $H_m$ are generated from the Layernorm of the output of the Transformer encoder $Z^0_{L_m}$: $$H_m = LN(Z^0_{L_m}),$$ where $H_m \in \mathbb{R}^{h_m \times N_m}$, assuming $h_m$ is the hidden size of the lidar transformer encoder. By adding an object detection head on the first vector $h^0_m$, 3D Object Detection tasks could be completed. ### 3.5 Object Detection Head and Loss Function Once getting the transformer output, we add a bounding box head and a classification head on the backbone. Both heads are Multi-Level Perceptions. The number of the hidden layer could be fine-tuned. The bounding box head outputs $\hat{U} \in \mathbb{R}^{N \times O}$ features, where $N$ is the maximum prediction number. Each feature could be represented as $\hat{u}_i = (cx, cy, cz, l, w, h, \theta)$ in 3D object detection, where $O$ is 7, $\hat{u}^c_i = (cx, cy, cz)$ represent the center coordinates, $\hat{u}^l_i = (l, w, h)$ are the length, width, height, and $\hat{u}^h_i = \theta$ denotes the heading angle in radians of the bounding box. In 2D object detection, each feature could be represented as $\hat{u}_i = (cx, cy, w, h)$, where $O$ is 4. The classification head outputs $\hat{V} \in \mathbb{R}^{N \times (C+1)}$ features, where $C$ is the number of classes. 1 is added for the "no object" class. Each $\hat{v}_i$ is the predicted probability of the object belonging to the positive class. We set the ground truth bounding box features to be $U$, and ground truth class features to be $V$, where each $v_i$ is a one-hot encoder setting the class label to be 1 and others being 0. In our object detection task, the total loss is decomposed into two individual losses: classification loss and regression loss: \[ L_{Total} = \lambda_1 L_{cls} + \lambda_2 L_{reg}. \] (9) The classification loss measures the error in predicting the object class label. The focal loss (Lin et al., 2017) function is used here which is a modified version of the cross entropy loss: \[ L_{cls} = - \sum_i [v_i (1 - \hat{v}_i)^\gamma \log(\hat{v}_i) + (1 - v_i) \hat{v}_i^\gamma \log(1 - \hat{v}_i)], \] (10) where $ \gamma $ is a modulating factor that controls the weight given to each example. The regression loss measures the error in predicting the bounding box location (including center, size, and heading) of the object. Since it is common to have sparse outliers in $ \hat{u}_i $ and $ u_i $, inspired from Meyer (2019), we regard them two Laplace distributions to help improve robustness to outliers. Then we use the Kullback-Leibler divergence between each two Laplace distributions to compute the location loss: \[ L_{reg} = L_{center} + L_{size} + L_{heading} + \lambda_3 L_{corner} = \sum_i KL_{Laplace}(\hat{u}_i^c, u_i^c) + \sum_i KL_{Laplace}(\hat{u}_i^s, u_i^s) + \sum_i KL_{Laplace}(\hat{u}_i^h, u_i^h) + \lambda_3 L_{corner}. \] (11) Note that center, size, and heading have separate loss terms, which may result in not optimized learning for final 3D box accuracy. Inspired by Qi et al. (2017a), we add the corner loss $ L_{corner} $ to jointly optimized for best 3D box estimation under 3D IoU metric. The corner loss is the sum of the distances between the eight corners of a predicted box and a ground truth box. Since corner positions are jointly determined by center, size, and heading, the corner loss can regularize the multi-task training for those parameters. ### 3.6 Fusion-ViT with Pre-Training and Fine-Tuning The overall structure of the hierarchical ViTs’ 3D object detection model consists of the proposed FusionViT framework. However, there are three ViTs built hierarchy in total, which may lead to some potential issues of large memory consumption or heavy use. To eliminate these concerns, we also pre-train CameraViT and LidarViT on the training set. Then fine-tune to smaller tasks (like using a smaller testing dataset) using the whole framework. Adding object detection head, we first pre-train CameraViT by letting it read camera images and directly output 2D object detection prediction, then we pre-train LidarViT by letting it read point cloud data and directly output 3D object detection prediction. After that, we run the whole framework using the pre-trained CameraViT and LidarViT, and read the task images and point cloud data. ## 4 Experiments We compared our model with several baselines. For the baseline choosing, we selected both classical and SOTA models, typically for Object Detection tasks in camera-only, lidar-only, and camera-lidar fusion input. We show our model’s inherent advantages of its design and structure over several current genres. The experiment setup and implementation details can be found on A and B. ### 4.1 Experiment Result On Waymo Open Dataset On Waymo Open Dataset (Sun et al., 2019), We conduct three groups of experiments: pure 2D Detection from Camera Images, pure 3D Detection from Lidar Point Cloud, and 3D Detection from Camera Images and Lidar Point Cloud. Their performance on Waymo Open Dataset is shown in Table 1, as listed in the first, second, and last block, respectively. We report the Vehicle and Pedestrian’s AP and APH scores. APH is only available in 3D Object Detection. In pure 2D Detection, CameraViT out-perform state-of-art Transformer-based methods DETR (Carion et al., 2020) and Swim-Transformer (Liu et al., 2021), which shows promising results. In \| Model \| Veh. AP \| Veh. APH \| Ped. AP \| Ped. APH \| \|------------------------\|---------\|----------\|---------\|----------\| \| DETR (2021) \| 48.5 \| \ \| 46.3 \| \ \| \| Swim-Transformer (2022)\| 48.8 \| \ \| 49.2 \| \ \| \| CameraViT (ours) \| 49.3 \| \ \| 49.8 \| \ \| \| PointPillars (2019) \| 50.9 \| 50.4 \| 50.3 \| 51.3 \| \| CenterPoint (2020) \| 53.2 \| 53.3 \| 53.7 \| 53.2 \| \| CenterFormer (2022) \| 54.8 \| 54.3 \| 54.0 \| 54.3 \| \| SST (2022) \| 54.4 \| 55.0 \| 54.5 \| 54.0 \| \| LidarViT (ours) \| 55.3 \| 55.3 \| 54.6 \| 54.8 \| \| DeepFusion (2022) \| 57.3 \| 57.4 \| 55.2 \| 56.8 \| \| TransFusion (2022) \| 58.1 \| 58.2 \| 58.3 \| 57.1 \| \| FusionViT (ours) \| 59.5 \| 58.4 \| 58.5 \| 58.9 \| \| FusionViT Pretraining \| 60.3 \| 60.1 \| 61.4 \| 59.8 \| Table 1: Comparison of attained validation AP and APH (in %) on Waymo Open Dataset pure 3D Detection, LidarViT performs better than classical PointPillars (Lang et al., 2018) and CenterPoint (Yin et al., 2020) methods. It’s also better than other state-of-art voxel-based frameworks CenterFormer (Zhou et al., 2022) and SST (Fan et al., 2021). The promising results show great use of the Transformer encoder. In the 3D Detection from both Camera Image and Lidar Point Cloud, our FusionViT out-performs the state-of-the-art method DeepFusion (Li et al., 2022b) and TransFusion (Bai et al., 2022). Although all are fusion-based, FusionViT takes good use of the learned features from pure 2D and pure 3D learning. In addition, the pre-trained version reaches higher performance than its original with around 50% shorter time. This shows good robustness and flexibility of the FusionViT model. 4.2 Experiment Result On KITTI \| Model \| mAP_{BEV} (IoU = 0.7) \| mAP_{3D} (IoU = 0.7) \| \|------------------------\|-----------------------\|----------------------\| \| \| Easy \| Med \| Hard \| Easy \| Med \| Hard \| \| PV-RCNN (2020) \| 86.2 \| 84.8\| 78.7 \| N/A \| N/A \| N/A \| \| Part-A^2-free (2021) \| 88.0 \| 86.2\| 81.9 \| 89.0 \| 72.5\| 69.4 \| \| OcTr (2023) \| 89.5 \| 82.4\| 77.3 \| 87.3 \| 75.5\| 75.4 \| \| FastPillars (2023) \| 88.2 \| 83.2\| 81.1 \| 89.1 \| 85.3\| 77.6 \| \| LidarViT (ours) \| 90.0 \| 87.3\| 82.9 \| 90.3 \| 86.5\| 78.7 \| \| MVX-Net-PF(2019) \| 86.8 \| 86.9\| 81.0 \| 87.5 \| 84.3\| 74.6 \| \| BEVFusion (2022) \| 89.5 \| 88.9\| 86.3 \| 89.0 \| 87.1\| 77.4 \| \| FusionViT (ours) \| 91.2 \| 90.2\| 88.9 \| 90.4 \| 88.1\| 79.4 \| \| FusionViT Pretraining (ours) \| 92.1 \| 91.4\| 89.9 \| 91.2 \| 89.5\| 80.8 \| Table 2: Comparison of attained validation mAP (in %) on KITTI with IoU = 0.7. We further use another classical 3D detection dataset KITTI (Geiger et al., 2012) to compare our models’ performance with more state-of-the-art methods, which are shown in Table 2. Four cutting-edge frameworks are conducted for pure 3D detection, including the popular point-voxel-based method PV-RCNN (Shi et al., 2021), point-based method Part-A^2 (Shi et al., 2020), as well as two latest methods OcTr (Zhou et al., 2023a) and FastPillars (Zhou et al., 2023c). All of these frameworks are outperformed by LidarViT. As for the 3D Fusion track, MVX-Net PointFusion (Sindagi et al., 2019) and BEVFusion (Liu et al., 2022b) are two state-of-the-art Camera-Lidar fusion frameworks. We re-implement them under KITTI settings, and find their performance not as high as our FusionViT’s. One possible reason could be due to our promising fusion strategy, which \| Fusion Strategy \| Veh. \| Ped. \| \|---------------------\|---------------\|---------------\| \| \| AP \| APH \| AP \| APH \| \| SUM \| 55.5 \| 54.3 \| 55.1 \| 54.6 \| \| CONCAT \| 59.5 \| 58.4 \| 58.5 \| 58.9 \| \| DIRECT CONCAT \| 57.8 \| 57.2 \| 56.6 \| 56.5 \| Table 3: Ablation Study Results of Different Fusion Strategies On Waymo Open Dataset \| Model \| Veh. \| Ped. \| \|---------------------\|---------------\|---------------\| \| \| AP \| APH \| AP \| APH \| \| Without Both \| 37.3 \| 36.5 \| 38.9 \| 37.1 \| \| Without LidarViT \| 44.1 \| 41.4 \| 42.7 \| 42.9 \| \| Without CameraViT \| 46.6 \| 47.2 \| 47.3 \| 46.4 \| \| Without MixViT \| 51.2 \| 52.8 \| 51.5 \| 53.0 \| \| Normal FusionViT \| 59.5 \| 58.4 \| 58.5 \| 58.9 \| Table 4: Ablation Study Results of FusionViT Model Components on Waymo Open Dataset will be discussed in the next subsection. Additional scores are obtained by the pre-trained version, demonstrating its adaptability. In a work, our FusionViT has a promising performance with excellent resilience under all 2D and 3D detection scenarios in each of the two large-scale datasets. 4.3 Ablation Study We conduct an extensive ablation study and performance analysis next. Firstly we analyze the influence of using different fusion strategies. That’s to say, given the learned 2D and 3D represented features, how to combine them more efficiently. To this end, we tried three methods: to SUM, to CONCAT, and DIRECT CONCAT (Cao et al., 2016) the two features. Their performance of them is shown in Table 3. As shown in the second row, use CONCAT operation performs best. That is probably due to that SUM operation is too reckless, ignoring many potentially useful features. DIRECT CONCAT is an efficient method, but using large computation resources and maintained too many features, which may easily cause over-fitting. We also analyze the influence of the proposed three ViTs in the multi-modal fusion model. We want to show that each sub-model should be important and irreplaceable. To make it, we conduct four more experiments apart from the Normal FusionViT: the FusionViT but without LidarViT component, the FusionViT but without CameraViT component, the FusionViT but without LidarViT and CameraViT components, and the FusionViT but without MixViT components. For the removed components, we add a linear transformation layer to keep the dimension the same in the four experiments. Table 4 shows the results. It is clear the Normal FusionViT has the highest score, indicating that each component of the model is irreplaceable. Particularly, by comparing the fourth and fifth lines, we see the necessity of MixViT. 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RQk9srYfhj	Moreover, the authors claims that the SEELE addresses multiple generative sub-tasks in subject repositioning using a single diffusion model, on one hand, SEELE actually contains many components not only the diffusion model for generative sub-tasks, on the other hand, it is confusing what is the exact meaning of using a single diffusion model since it seems that different generative sub-tasks are tackled with different datasets at least.	REPOSITIONING THE SUBJECT WITHIN IMAGE Anonymous authors Paper under double-blind review ABSTRACT Current image manipulation primarily centers on static manipulation, such as replacing specific regions within an image or altering its overall style. In this paper, we introduce an innovative dynamic manipulation task, subject repositioning. This task involves relocating a user-specified subject to a desired position while preserving the image’s fidelity. Our research reveals that the fundamental sub-tasks of subject repositioning, which include filling the void left by the repositioned subject, reconstructing obscured portions of the subject and blending the subject to be consistent with surrounding areas, can be effectively reformulated as a unified, prompt-guided inpainting task. Consequently, we can employ a single diffusion generative model to address these sub-tasks using various task prompts learned through our proposed task inversion technique. Additionally, we integrate pre-processing and post-processing techniques to further enhance the quality of subject repositioning. These elements together form our SEgment-gEnerate-and-bLEnd (SEELE) framework. To assess SEELE’s effectiveness in subject repositioning, we assemble a real-world subject repositioning dataset called ReS. Our results on ReS demonstrate the quality of repositioned image generation. Figure 1: Subject repositioning aims to relocate a user-specified subject within a single image. In the comparison above, we evaluate the subject repositioning results achieved by our SEELE model in comparison to Google Magic Editor. We obtained Google’s results from its introductory webpage. Below are illustrated generative sub-tasks encompassed by subject repositioning: i) It must fill the void created when moving the subject to maintain consistency and avoid generating new, random subjects. ii) Completing the occluded portions of the moved subject is necessary. iii) The appearance of repositioned subject should blend with the surrounding areas. SEELE effectively addresses the generative sub-tasks within a unified prompt-guided inpainting task, all powered by a single diffusion generative model. While these results illustrate the sub-tasks addressed by SEELE, the comprehensive outcomes of executing SEELE are depicted in Figure 13 in the appendix. 1 INTRODUCTION In May 2023, Google Photos introduced a groundbreaking AI editing feature allowing users to reposition subjects within their images$^1$. Unfortunately, a lack of accompanying technical documentation leaves the inner workings of this feature largely unexplored. Prior to the deep learning era, Iizuka et al. (2014) explored a similar problem of object repositioning with user inputs of ground regions, bounding boxes of objects, and shadow regions to aid the understanding of the image. As deep learning has rapidly advanced, the potential to substitute many user actions with learning models as well as an advanced understanding of images has emerged, necessitating a comprehensive reassessment of the subject repositioning problem through the lens of potent deep learning models. The primary objective of this paper is to introduce an inventive framework capable of achieving performance on par with or surpassing Google Photos’ latest AI feature for repositioning subjects within images. From an academic standpoint, it’s evident that this feature falls within the domain of image manipulation (Gatys et al., 2016; Isola et al., 2017; Zhu et al., 2017; Wang et al., 2018; El-Nouby et al., 2019; Fu et al., 2020; Zhang et al., 2021). This area has seen a surge in interest in recent years, primarily due to the advancement of large-scale generative models. These generative models encompass a range of techniques, including generative adversarial models (Goodfellow et al., 2014), variational autoencoders (Kingma & Welling, 2014), auto-regressive models (Vaswani et al., 2017), and notably, diffusion models (Sohl-Dickstein et al., 2015). As both the model architectures and training datasets continue to expand, these generative models exhibit remarkable capabilities in image manipulation (Rombach et al., 2022; Kawar et al., 2022; Chang et al., 2023). However, it is important to note that current image manipulation approaches primarily emphasize what can be described as “static” alterations. These methods are designed to modify specific regions of an image, often guided by various cues such as natural language, sketches, strokes, or layouts (El-Nouby et al., 2019; Zhang et al., 2021; Fu et al., 2020). Another dimension of manipulation revolves around the transformation of an image’s overall style, encompassing tasks like converting real photographs into anime-style pictures, paintings, or mimicking the unique aesthetics of certain films (Chen et al., 2018; Wang et al., 2018; Jiang et al., 2021). Some approaches have even extended these manipulation techniques to the domain of videos (Kim et al., 2019; Xu et al., 2019; Fu et al., 2022), where the objective is to dynamically manipulate style or subjects over time. In contrast, the concept of subject repositioning delves into the dynamic manipulation of a single image, with a specific focus on relocating selected subject while keeping the rest of the image unchanged. As text-to-image diffusion models (Nichol et al., 2022; Ho et al., 2022; Saharia et al., 2022; Ramesh et al., 2022; Rombach et al., 2022) emerge as one of the most potent generative models today, adapting them for subject repositioning presents an intriguing opportunity. Nevertheless, a significant challenge lies in finding suitable text prompts for this task, as text-to-image diffusion models are typically trained using image caption prompts rather than task-specific instructions. Moreover, the best text prompts are often image-dependent and are hard to generalize to other images, making them impractical for real-world applications that prioritize user-friendliness and minimal user effort. On the other hand, while specialized models have been developed to address specific aspects of subject repositioning, such as local inpainting (Zeng et al., 2020; Zhao et al., 2021; Li et al., 2022; Suvorov et al., 2022; Dong et al., 2022), subject completion (Zhan et al., 2020), and local harmonization (Xu et al., 2017; Zhang et al., 2020; Tsai et al., 2017), our study poses an intriguing question: “Can we achieve all these sub-tasks using a single generative model?” Broadly, we can deconstruct this multifaceted task into several distinct sub-tasks. We roughly categorize these sub-tasks into non-generative and generative tasks. The non-generative sub-tasks involve activities like segmenting user-specified subjects and estimating occlusion relationships between subjects. In this paper, we primarily concentrate on the generative sub-tasks, while addressing the non-generative aspects using pre-trained models. The generative sub-tasks essential for subject repositioning encompass the following key elements: i) Subject removal: After the subject is repositioned, a void is left behind. The generative model’s task is to consistently fill this void using nearby background while avoiding the introduction of new elements. ii) Subject completion: When the repositioned subject is partially obscured, the generative model must complete the subject to maintain its integrity. iii) Subject harmonization: The appearance of repositioned subject should seamlessly blend with the surrounding areas. While all these sub-tasks take as inputs an image for manipulation and a mask indicating the region to manipulate, they demand distinct generative capabilities. Furthermore, it is hard to transform these task instructions into caption-style prompts for frozen text-to-image diffusion models. Fortunately, the embedding space of text prompts used in diffusion models is much more versatile than merely representing captions. Textual inversion (Gal et al., 2022) has revealed that we can learn to represent user-specified concepts, including textual and stylistic information that is challenging to convey through language, within the embedding space of text prompts. Additionally, prompt tuning (Lester et al., 2021; Liu et al., 2021a) has been effectively employed in transformers to adapt to specific domains, inspiring us to apply textual inversion at the task level. These approaches inspire us to learn latent embeddings in the text conditions to represent specific task instructions that the diffusion model should follow. With this task-level inversion design, we can adapt diffusion models to various tasks by simply modifying the task-level “text” prompts. To formally address the problem of subject repositioning, we propose the SEgment-gEnerate-and-bLEnd (SEELE) framework. SEELE tackles the subject repositioning with a pre-processing, manipulation, post-processing pipeline. i) In the pre-processing stage, SEELE employs SAM (Kirillov et al., 2023) to input user-specified points, bounding boxes, or text prompts to segment the subject for repositioning. With the user-specified moving direction, SEELE moves the subject and places it following the accurate occlusion relationship between subjects. ii) In the manipulation stage, SEELE addresses subject removal and subject completion using a single pre-trained diffusion model guided by learned task prompts. iii) In the post-processing stage, SEELE harmonizes the repositioned subject to ensure it blends seamlessly with adjacent regions. To evaluate subject repositioning algorithms, we have assembled a real-world subject repositioning dataset called ReS. This dataset consists of 100 real image pairs featuring a repositioned subject. The images were collected in diverse scenes and at different times to enhance diversity. We annotated the mask of the repositioned subject using SAM and manual refinement. We estimated the moving direction based on the center point of masks in the paired image. We also provide amodal masks for occluded subjects. To the best of our knowledge, this is the first dataset for subject repositioning, and we hope it will serve as a valuable benchmark evaluation dataset. Our contributions are summarized as follows: - The paper delineates the task of subject repositioning as a specialized image manipulation challenge, breaking it down into several distinct sub-tasks, each of which presents unique challenges and necessitates specific learning model requirements. - The paper proposes the SEgment-gEnerate-and-bLEnd (SEELE) framework, which addresses multiple generative sub-tasks in subject repositioning using a single diffusion model. - The paper explores an innovative task inversion technique, demonstrating that we can re-formulate the text-conditions to represent task instructions. This exploration opens up new possibilities for adapting diffusion models to specific tasks. - The paper curates the ReS dataset, a real-world collection of image pairs featuring repositioned subjects. ReS serves as a valuable benchmark for evaluating subject repositioning algorithms. ## 2 Subject Repositioning ### 2.1 Task Definition and Challenges Subject repositioning involves moving the user-specified subject within an image. This seemingly simple task is actually quite challenging, requiring coordination of multiple sub-tasks. User inputs. Subject repositioning focuses on a single image. As an interactive approach, subject repositioning follows user-intention to identify subject, move to the desired location, complete the subject and address disparities of the repositioned subject. Particularly, the user identify the interested subject via pointing, bounding box, or a text prompt as inputs to the system for identifying the subject. Then, the user provides the desired repositioning location via dragging or providing repositioning direction. The system further requires the user to indicate the Figure 2: SEELE employs a pre-processing, manipulation, and post-processing pipeline for subject repositioning. During the pre-processing phase, SEELE identifies the subject using the segmentation model, guided by user-provided conditions, and maintains the occlusion relationships between subjects intact. In the manipulation stage, SEELE manipulates the image to fill in any left gaps. Furthermore, SEELE rectifies the obscured subject with user-specified incomplete masks. In the post-processing phase, SEELE addresses any disparities between the repositioned subject and its new surroundings. To tackle this task, we introduce the SEgment-gEnerate-and-bLEnd (SEELE) framework, shown in Figure 2. Specifically, SEELE breaks down the task into three stages: preprocessing, manipulation, and post-processing stages. i) The preprocessing addresses how to precisely locate the specified subject with minimal user input, considering that the subject may be a single object, part of an object, or a group of objects identified by the user’s intention; reposition the identified subject to the desired location; and also identify occlusion relationships to maintain geometric consistency. Additionally, adjusting the subject’s size might be necessary to maintain the perspective relationship within the overall composition. ii) The manipulation stage deals with the main tasks of creating new elements in subject repositioning to enhance the image. In particular, this stage includes the subject removal step, which fills the empty space on the left void of the repositioned subject. Additionally, the subject completion step involves reconstructing any obscured parts to ensure the subject is fully formed. iii) The postprocessing stage focuses on minimizing visual differences between the repositioned subject and its new surroundings. This involves fixing inconsistencies in both appearance and geometry, including blending unnatural boundaries, aligning illumination statistics, and, at times, creating realistic shadows for added realism. In the next sections, we will start by going over the SEELE pipeline in Sec. 2.2. Particularly we explain the task inversion in Sec. 2.3 to address generative sub-tasks. In Sec. 2.4, we show how to train different manipulation sub-tasks using the task inversion technique while keeping the diffusion model unchanged. Finally, we provide a detailed introduction to the curated ReS dataset in Sec. 2.5. 2.2 SEELE As mentioned above, SEELE consists of three stages. The preprocessing stage usually involves non-generative tasks, while the manipulation and postprocessing stages require generative capabilities. In SEELE, we employ a unified diffusion model for all generative sub-tasks and use pre-trained models for non-generative sub-tasks. We give the details of each stage in the following. Pre-processing. For point and bounding box inputs for identifying subjects, we utilize the SAM (Kirillov et al., 2023) for user interaction and employ SAM-HQ (Ke et al., 2023) to enhance the quality of segmenting subjects with intricate structures. To enable text inputs, we follow SeMani (Wang et al., 2023) to indirectly implement a text-guided SAM mode. Specifically, we first employ SAM to segment the entire image into distinct subjects. Subsequently, we compare each subject with the input text to identify the most similar one using the mask-adapted CLIP model (Liang et al., 2022). After identifying the subject, SEELE follows user intuition to reposition the subject to the desired location, and then mask the original subject as void for re-paint in the manipulation stage. Our SEELE handles the potential occlusion between the moved subject and other elements in the image. If there are other subjects present at the desired location, SEELE employs the monocular depth estimation algorithm MiDaS (Ranftl et al., 2020) to discern occlusion relationships between subjects. SEELE will then appropriately mask the occluded portions of the subject if the user wants to preserve these occlusion relationships. MiDaS is also used to estimate the perspective relationships among subjects and resize the subject accordingly to maintain geometric consistency. For subjects with ambiguous boundaries, SEELE incorporates the ViTMatte matting algorithm (Yao et al., 2023) for better compositing with the surrounding areas. Manipulation. In this stage, SEELE deals with the primary tasks of manipulating subjects by repositioning them. As illustrated in Figure 2, it has the steps of subject removal and subject completion. Critically, such two steps can be effectively solved by a single generative model, as the masked region of both steps should be filled in to match the surrounding areas. However, these two sub-tasks require different information and types of masks. Particularly, for subject removal, a non-semantic inpainting is applied uniformly from the unmasked regions, using a typical object-shaped mask. This often falsely results in the creation of new, random subjects within the holes. On the other hand, subject completion involves semantic-rich inpainting and aims to incorporate the majority of the masked region as part of the subject. Critically, to adapt the same diffusion model to the different generation directions needed for the above sub-tasks, we propose the task inversion technique in SEELE. This technique guides the diffusion model according to specific task instructions. Thus, with the learned remove-prompt and complete-prompt, SEELE combines subject removal and subject completion into a single generative model. Post-processing. In the final stage, SEELE harmoniously blends the repositioned subject with its surroundings by tackling two challenges below. i) Local harmonization ensures natural appearance in boundary and lighting statistics. SEELE confines this process to the relocated subject to avoid affecting other image parts. It takes the image and a mask indicating the subject’s repositioning as inputs. However, the stable diffusion model is initially trained to generate new concepts within the masked region, conflicting with our goal of only ensuring consistency in the masked region and its surroundings. To address this, SEELE adapts the model by learning a harmonize-prompt and using the LoRA adapter to guide masked regions. This local harmonization can also be integrated into the same diffusion model used in the manipulation stage with our newly proposed design. ii) Shadow generation aims to create realistic shadows for repositioned subjects, enhancing the realism. Generating high-fidelity shadows in high-resolution images of diverse subjects remains challenging. SEELE uses the stable diffusion model for shadow generation, addressing two scenarios: (1) If the subject already has shadows, we use complete-prompt for subject completion to extend the shadows. (2) For subjects without shadows, we generate a preliminary shadow based on user-specified masks. This task then transforms into a local harmonization process for realistic shadow generation, utilizing harmonize-prompt with LoRA adapter Hu et al. (2021). 2.3 TASK INVERSION Generative sub-tasks in subject repositioning input the image and mask with unique approach: • Subject removal fills the void without creating new subjects. • Subject completion completes the primary subject within the masked region. • Subject harmonization ensures consistency without introducing new elements. These requirements lead to different generation paths. In contrast, our goal is to enhance text-to-image diffusion inpainting models for image manipulation guided by high-level task instructions. To address this, we introduce task inversion, training prompts to guide the diffusion model while keeping the backbone fixed. Instead of traditional text prompts, we utilize the adaptable representations acting as instruction prompts, such as “complete the subject”. Consequently, task inversion allows the smooth integration of different generative sub-tasks for subject repositioning using stable diffusion. This integration happens without the need for introducing new generative models or adding extensive modules or parameters, highlighting the plug-and-play nature of task inversion. Task inversion adheres to the original training objectives of diffusion models. Specifically, denote the training image as $x$, the local mask as $m$, the learnable task prompt as $z$, the conditioning model $c(\cdot)$ to map the learnable prompt. Our objective is $$L := \mathbb{E}_{\varepsilon \sim N(0,1), t \sim U(0,1)}[\\|\varepsilon - \varepsilon_\theta([x_t, m, x \odot (1 - m)], t, c(z))\\|_F^2],$$ where $\varepsilon$ is the random noise; $\varepsilon_\theta$ is the diffusion model, $t$ is the normalized noise-level; $x_t$ is the noised image, $\odot$ is element-wise multiplication; and $\\|\cdot\\|_F$ is the Frobenius norm. When training with Eq. (1), the conditioning model $c$ and the diffusion model $\varepsilon_\theta$ is frozen, while the embedding $z$ is the only learnable parameters. Our task inversion is a distinctive approach, influenced by various existing works but with clear differences. Specifically, traditional text-to-image diffusion models are trained on pairs where the text describes the image, such as LAION-5B (Schuhmann et al., 2022). However, the instruction prompt mentioned for our task inversion goes beyond the training data’s scope, potentially affecting the desired generation results in practice. Furthermore, recent advancements in textual inversion (Gal et al., 2022) emphasize the potential to comprehend user-specified concepts within the embedding space. In contrast, prompt tuning (Lester et al., 2021; Liu et al., 2021a) enhances adaptation to specific domains by introducing learnable tokens to the inputs. Similarly, adversarial reprogramming (Elsayed et al., 2018) trains a pre-existing model to perform a novel task. Unlike textual inversion, which trains a few tokens for visual understanding, our task prompt includes the entire task instruction. We don’t depend on text inputs to guide the diffusion model; instead, we use all tokens for learning. See Figure 4 for the distinction. ### 2.4 Learning Task Inversion Existing text-to-image diffusion inpainting model is trained with randomly generated masks to generalize in diverse scenarios. In contrast, task inversion involves creating task-specific masks during training, allowing the model to learn specialized task prompts. i) Generating masks for subject removal: In subject repositioning, the mask for the left void mirrors the subject’s shape, but our goal isn’t to generate the subject within the mask. To create training data for this scenario, for each image, we randomly choose a subject and its mask. Next, we move the mask, as shown by the girl’s mask in the center of Figure 5. This results in an image where the masked region includes random portions unrelated to the mask’s shape. This serves as the target for subject removal, with the mask indicating the original subject location. ii) Generating masks for subject completion: In this phase, SEELE addresses scenarios where the subject is partially obscured, with the goal of effectively completing the subject. To integrate this prior information into the task prompt, we generate training data as follows: for each image, we randomly select a subject and extract its mask. Then, we randomly choose a continuous portion of the mask as the input mask. Since user-specified masks are typically imprecise, we introduce random dilation to include adjacent regions within the mask. As illustrated by the umbrella mask on the right side of Figure 5, such a mask serves as an estimate for the mask used in subject completion. Learning subject harmonization. In SEELE, we refine subject harmonization by altering the target of diffusion model. This change replaces the masked image condition with the original inharmonious image in Eq. (1). Task inversion mainly influences the cross-attention between the task condition and images. Furthermore, to better guide the masked region in the diffusion model, we introduce LoRA adapters (Hu et al., 2021). These adapters aid in learning the subject harmonization task: $$\mathcal{L} := \mathbb{E}_{\varepsilon \sim \mathcal{N}(0,1), t \sim \mathcal{U}(0,1)} \left[ \\| \varepsilon + x - x^* - \varepsilon_\theta([x_t, m, x], t, c(z)) \\|_F^2 \right],$$ where $x^$ represents the target harmonized image, and $x$ is the input image. While we tweak the training objective, the generation process of the diffusion models remains unchanged. This allows us to still utilize the pre-trained stable diffusion model with the learned harmonize-prompt and LoRA parameters, and seamlessly integrate with other modules. See Sec A.10 in the appendix for details. ### 2.5 ReS dataset To evaluate the effectiveness of subject repositioning algorithms, we curated a benchmark dataset called ReS. This dataset includes 100 paired images, each with dimensions $4032 \times 3024$, where one image features a repositioned subject while the other elements remain constant. These images were collected from over 20 indoor and outdoor scenes, showcasing subjects from more than 50 categories. This diversity enables effective simulation of real-world open-vocabulary applications. Thus our dataset is diverse enough to evaluate our SEELE model. The masks for the repositioned subjects were initially generated using SAM and refined by multiple experts. Occluded masks were also provided to assist in subject completion. The direction of repositioning was estimated by measuring the distance between the center points of the masks in each image pair. For each paired image in the dataset, we can assess subject repositioning performance from one image to the other and in reverse, resulting in a total of 200 testing examples. Figure 6 illustrates the ReS dataset. We plan to release the ReS dataset to encourage research in subject repositioning. ### 3 RESULTS AND ANALYSIS Examples of subject repositioning.* We present subject repositioning results on $1024^2$ images using SEELE in Figure 7. SEELE works well on diverse scenarios of subject repositioning. Subject repositioning on ReS. Since there are currently no publicly available models specifically designed for subject repositioning, we mainly compare with original Stable Diffusion inpainting model (SD). We adopt SD under no text prompts, simple prompts and complex prompts. The used prompts are provided in Sec. A.3 in the appendix. Furthermore, by combining masks from subject movement and completion sub-tasks into a single mask, we can incorporate alternative inpainting algorithms in SEELE. Specifically, we incorporate LaMa (Suvorov et al., 2021), MAT (Li et al., 2022), MAE-FAR (Cao et al., 2022), and ZITS++ (Cao et al., 2023) into SEELE. Note that in this experiment, SEELE does not utilize any pre-processing or post-processing techniques. We present qualitative comparison results in Figure 8 where a larger version is Figure 14 in the appendix. More results can be found in Figure 15 and Table 1 in the appendix. We add orange subject removal mask and blue subject completion mask in the input image. Our qualitative analysis indicates that SEELE exhibits superior subject removal capabilities without adding random parts and excels in subject completion. When the moved subject overlaps with the left void, SD fills the void guided the subject. In contrast, SEELE avoids the influence of the subject, as shown in the top row of Figure 8. If the mask isn’t precise, SEELE works better than other methods by reducing the impact of unclear edges and smoothing out the empty space, as seen in the fourth row. Also, SEELE is excels in subject completion than typical inpainting algorithms, as seen in the second-to-last row. Note that SEELE can be further enhanced through the post-processing stage. Effectiveness of the proposed task-inversion. To further validate the proposed task-inversion, we conduct experiments on standard inpainting and outpainting tasks, following the standard training and evaluation principles. We provide analysis in Sec. A.5 in the appendix where results for inpainting can be found at Table 2 and Figure 16 and outpainting at Table 3 and Figure 17. SEELE w/ X. We assess the effectiveness of various components within SEELE during both pre-processing and post-processing phases. We conduct a qualitative comparison of SEELE’s results with and without the utilization of these components, as shown in Figure 9 in the appendix, while a detailed analysis of each component is provided in Sec. A.4 in the appendix. 4 RELATED WORKS Image and video manipulation aims to manipulate images and videos in accordance with user-specified guidance. Among these guidance, natural language guidance, as presented in previous studies (Dong et al., 2017; Nam et al., 2018; Li et al., 2020a,b; Xia et al., 2021; Karras et al., 2019; El-Nouby et al., 2019; Zhang et al., 2021; Fu et al., 2020; Chen et al., 2018; Wang et al., 2018; Jiang et al., 2021), stands out as particularly appealing due to its adaptability and user-friendliness. Some research efforts have also explored the use of visual conditions, which can be conceptualized as image-to-image translation tasks. These conditions encompass sketch-based (Yu et al., 2019; Jo & Park, 2019; Chen et al., 2020; Kim et al., 2020; Chen et al., 2021; Richardson et al., 2021; Zeng et al., 2022), label-based (Park et al., 2019; Zhu et al., 2020; Richardson et al., 2021; Lee et al., 2020), line-based (Li et al., 2019), and layout-based (Liu et al., 2019) conditions. In contrast to image manipulation, video manipulation (Kim et al., 2019; Xu et al., 2019; Fu et al., 2022) introduces the additional challenge of ensuring temporal consistency across different frames, necessitating the development of novel temporal architectures (Bar-Tal et al., 2022). Image manipulation primarily revolves around modifying static images, whereas video manipulation deals with dynamic scenes in which multiple subjects are in motion. In contrast, our paper focuses exclusively on subject repositioning, where one subject is relocated while the rest of the image remains unchanged. Textual inversion (Gal et al., 2022) is designed to personalize text-to-image diffusion models according to user-specified concepts. It achieves this by learning new concepts within the embedding space of text conditions while keeping all other parameters fixed. Null-text inversion (Mokady et al., 2022) learns distinct embeddings at different noise levels to enhance model capacity. Additionally, some fine-tuning (Ruiz et al., 2022) or adaptation (Zhang & Agrawala, 2023; Mou et al., 2023) techniques inject visual conditions into text-to-image diffusion models. While these approaches concentrate on image patterns, SEELE focuses on the task instruction to guide diffusion models. Prompt tuning (Lester et al., 2021; Liu et al., 2021b,a) entails training a model to learn specific tokens as additional inputs to transformer models, thereby enabling model adaptation to a specific domain without fine-tuning the model. This technique been widely used in vision-language models (Radford et al., 2021; Yao et al., 2021; Ge et al., 2022). This concept has inspired us to transform the text-to-image diffusion model into a task-to-image diffusion model by tuning the text conditions. Image composition (Niu et al., 2021) is the process of combining a foreground and background to create a high-quality image. Due to differences in the characteristics of foreground and background elements, inconsistencies can arise in terms of appearance, geometry, or semantics. Appearance inconsistencies encompass unnatural boundaries and lighting disparities. Segmentation (Kirillov et al., 2023), matting (Xu et al., 2017), and blending (Zhang et al., 2020) algorithms can be employed to address boundary concerns, while image harmonization (Tsai et al., 2017) techniques can mitigate lighting discrepancies. Geometry inconsistencies include occlusion and disproportionate scaling, necessitating object completion (Zhan et al., 2020) and object placement (Tripathi et al., 2019) methods, respectively. Semantic inconsistencies pertain to unnatural interactions between subjects and backgrounds and are beyond the scope of this paper. While each aspect of image composition has its specific focus, the overarching goal is to produce a high-fidelity image. In our paper, SEELE concentrates on enhancing harmonization capabilities within a single generative model. 5 CONCLUSION In this paper, we introduce an innovative task known as subject repositioning, which involves manipulating an input image to reposition one of its subjects to a desired location while preserving the image’s fidelity. To tackle subject repositioning, we present SEELE, a framework that leverages a single diffusion model to address the generative sub-tasks through our proposed task inversion technique. This includes tasks such as subject removal, subject completion, subject harmonization, and shadow generation. For the non-generative sub-tasks, we utilize pre-trained models. To evaluate the effectiveness of subject repositioning, we have curated a real-world dataset called ReS. Our experiments on ReS demonstrate the proficiency of SEELE in accomplishing this task. REFERENCES Abubakar Abid, Ali Abdalla, Ali Abid, Dawood Khan, Abdulrahman Alfozan, and James Zou. Gradio: Hassle-free sharing and testing of ml models in the wild. arXiv preprint arXiv:1906.02569, 2019. Omer Bar-Tal, Dolev Ofri-Amar, Rafail Fridman, Yoni Kasten, and Tali Dekel. Text2live: Text-driven layered image and video editing. In European Conference on Computer Vision, pp. 707–723. Springer, 2022. Chenjie Cao, Qiaole Dong, and Yanwei Fu. Learning prior feature and attention enhanced image inpainting. In European Conference on Computer Vision, pp. 306–322. Springer, 2022. Chenjie Cao, Qiaole Dong, and Yanwei Fu. 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dnaCBAP7X2	“Our identification result on the black-box attack is not as good as the white-box attack tested because of the noise gradient estimation used in the black-box attack.” Why is the identification accuracy for black-box attacks the best in Figure 2 (b) when the number of adversarial examples is greater than one, better than all white-box attacks?	AN IMPLICIT WATERMARK FRAMEWORK FOR ADVERSARY IDENTIFICATION Anonymous authors Paper under double-blind review ABSTRACT Security of deep neural networks based machine learning systems has been an emerging research topic, especially after the discovery of adversarial attacks. In general, however, it is very difficult to build a machine learning system that is resistant to different types of attacks. Instead of directly improving the robustness of neural networks, Cheng et al. (2023) proposed the first framework to trace the first compromised model under the black-box adversarial attack in a forensic view. However, the black-box assumption has limited the usage of the framework since users will require detailed model information to facilitate their own use in the modern MLaaS system. In this paper, instead of considering the limited black-box attacks, we investigate more general and harder white-box setting where all users will have full access to model. Explicit modification on the model architecture during the inference will be no longer effective because those mechanisms could be easily bypassed by adversary. To address this challenge, a novel identification framework is proposed that can achieve high tracking accuracy to trace the source of white-box adversarial attack. Specifically, to differentiate adversarial examples generated from different copies, we first design an implicit watermark from backdooring before the model distribution. Then we design a data-free method to identify the adversary with only adversarial example available. Extensive experiments on different attacks including both white-box and black-box attacks, datasets, and model architectures verify the effectiveness of the proposed method. Our code will be made publicly available. 1 INTRODUCTION Since neural networks were shown vulnerable to adversarial attacks (Szegedy et al., 2013), the security problem of deep neural networks has attracted more and more attention as deep learning has been shown successful in a wide range of applications. To alleviate the threat of adversarial attack, lots of methods have been proposed to improve the robustness of models (Cheng et al., 2020; Madry et al., 2017; Zhang et al., 2019; Thulasidasan et al., 2019). However, they suffer from trade-offs with test accuracy on clean data, making the robust models hard for deploying in real world applications. Recently, Cheng et al. (2023) proposes a new task to find the source model copy for generating the adversarial attack where one of model copies in the MLaaS system is compromised by the adversary to generate transferable adversarial examples that could subsequently affect other devices in the same system. The goal for the task is to find the first compromised copy by only investigating the generated adversarial example. Through embedding different mask-based watermark during the inference procedure, they propose an identification framework to trace the first compromised model copy with adversarial examples in the black-box setting. While their proposed framework mainly considers the attacker in the black-box setting where the attacker could only query the model output, however, in many real-world systems like hugging face and large foundation models, users could have access detailed information about the model (i.e., model architecture and parameters) so that they can further improve the model performance with their own local data. Meanwhile, the mask-based watermark can be bypassed entirely by building surrogate models and adopt transfer attack to generate adversarial examples. In this paper, we make the first attempt to address the problem that how to identify the possible adversary among different users when all users have full information about the models, i.e. under the white-box setting. Under the white-box setting, the provider couldn’t add any modules to the models to facilitate the identification like Cheng et al. (2023). It is because the adversary could bypass any explicit modifications on the model architectures or inference procedure by designing adaptive attacks as they have already known the existence of the module. To solve this problem, we propose to design a robust implicit watermarking scheme to conduct adversarial investigation. For every model copy, we insert the implicit watermark by building some fingerprint data points and mix it into the training procedure. That is, the inserted watermark is hidden in the model weight before providing models to customers. Specifically, our implicit watermarking would lead the adversarial attack to generate the perturbation on the designated region preferential than other areas. This makes adversarial examples generated by different model copy unique so that we are able to design a novel data-free method to identify the adversary given only one adversarial example. Extensive experiments have been conducted to verify the effectiveness of the proposed framework. To further test the robustness of the proposed watermarking scheme, we also test several adaptive attacks to erase the proposed watermarking and our proposed scheme is robust against those attacks. Our contributions can be summarized as follows: - We propose a new forensic investigation framework to trace the adversary from a single adversarial example. Our new framework allows a more general and challenging setting where the adversary has full access to the model. - To trace the compromised model copy without original examples, we design two simple yet effective metrics to achieve successful adversary identification. - Extensive experiments are conducted to verify the efficiency and effectiveness of the proposed framework on various attacks, datasets, and model architectures. The results show that the proposed method can achieve high accuracy in different scenarios. 2 RELATED WORK Adversarial attack Since the finding of adversarial examples (Szegedy et al., 2013), adversarial attacks have attracted much attention due to their potential threats to real-world applications. Adversarial attacks can be generally classified as white-box attacks and black-box attacks based on the information that the adversary can obtain. For white-box attacks, the attacker has full information about the model including model architectures and parameters. Hence the adversary can easily compute the gradient to conduct the attack (Carlini & Wagner, 2017; Goodfellow et al., 2014; Madry et al., 2017). For black-box attacks, the attacker can only query the output given input. Depending on if the output probability is given, black-box attacks can be divided into soft-label attacks and hard-label attacks. Without any information about the internal information of models, black-box attacks aim to estimate gradient information (Chen et al., 2020; Ilyas et al., 2018). From the view of the adversary, white-box attacks would be easier to be conducted compared to black-box attacks since the gradient information can be directly computed by model parameters. From the view of defender or forensic investigator, however, adversarial examples generated by white-box attacks would be more difficult to identify since any explicit modifications to the model would be bypassed. Forensic investigation of adversary There are few studies on the forensic investigation of adversarial examples. Cheng et al. (2023) first proposed a watermarking method to trace the adversarial examples generated by black-box attacks, where an mask-based watermarking module is introduced to assign a unique fingerprint for every model copy. However, the method is constrained to applications that do not require any model information since they made explicit modifications to model architectures. In this paper, we consider the white-box attack case in which any explicit modifications to model copies are forbidden. To address the identification problem in the white-box case, we propose a novel framework that inserts implicit backdoors into model copies and is able to identify the adversary with high accuracy given only one adversarial example. 3 METHODOLOGY 3.1 PROBLEM SETTING Following the forensic investigation setting in Cheng et al. (2023), the machine learning service provider (i.e., the owner) owns $ n $ copies of models $ g_1, g_2, \ldots, g_i, \ldots, g_n $ that are trained for the same $K$-way classification task on the same dataset. Because of the need for model customization and performance concern, these model copies are then distributed to $n$ different users so that users will have full access to model copies, including model architectures and parameters. For example, the model provider such as Hugging Face provides pre-trained models or large foundation model for users to further customize their own model. All model details including model architecture and weights would be available to the users. Let $g_i(\cdot) \in \mathbb{R}^K$ denote the logit output of copy $g_i$ given input, and $\sigma(g_i(\cdot)) \in \mathbb{R}^K$ denote the output probabilities vector of copy $g_i$, where $\sigma$ is the softmax function. Unfortunately, a malicious user (adversary) exists who aim to fool the whole system, including other users’ models, by conducting adversarial attacks. Let the malicious user’s model copy to be $f_{att}$ (the compromised model copy). As he does not have access to query other users’ models, he then chooses to perform adversarial attacks on his copy $f_{att}$ to generate an adversarial example $x_{adv}$. Because all model copies are trained with the same dataset for the same classification task, the generated adversarial example could successfully lead to the misclassification of other users’ models. Our task is to find the compromised model copy $f_{att}$ from the pool. Figure 1: The proposed framework. The first part shows how we train the baseline model and then fine-tune the baseline model to $n$ different copies by implicit watermarking. The second part shows how the adversary is identified given only adversarial example. 3.2 IMPLICIT WATERMARKING To identify $g_{att}$ from $n$ model copies given $x_{adv}$, each copy distributed to different users needs to be embedded a unique watermark for subsequently being used for forensic investigation. At the same time, since the adversary has full access to the model, we cannot do any explicit modifications that can be easily bypassed by the adversary. For example, masked based watermarking scheme proposed in Cheng et al. (2023) could be removed by adaptively adding noise on the masked region during the inference. Therefore, it requires us to design a robust implicit watermarking scheme that can conceal the copies information into model parameters without hurting performance. In this section, we propose a simple yet effective method to insert the implicit watermark. Specifically, we aim to let pixels in a specific region to be preferentially perturbed in the adversarial examples so that those regions could be regarded as a strong signal for the identification. Therefore, different adversarial examples generated by different users would have a significant difference that could be used later into tracing the compromised model. To build such a preference, we first sample a range of coordinates $ w_i $ and a label set from label space $ y_i \subset \{1, 2, \ldots, K\} $ that acts as the model $ i $'s fingerprint. To make these fingerprint coordinates to be inserted into the model copy as an implicit watermark, for every model copy $ g_i $, we create the fingerprint dataset $ \tilde{D}_i = \{(\tilde{x}_j, \tilde{y}_j)\}_{j=1}^{\|\tilde{D}\|} $ by sub-sampling several pixels $ t_i $ from the whole input space $ w_i $ together with a class $ \tilde{y}_j $ sampled from $ y_i $ as the label. More formally, let $ x \in \mathbb{R}^{H \times W \times C} $ denote any normal sample where $ H, W, C $ are height, width, and channels respectively. For copy $ g_i $, we create the fingerprint sample $ \tilde{x} $ by using the following blended function: \[ \tilde{x} = (1 - m_i) \odot x + m_i \odot t_i \] where $ \odot $ is element-wise product, and $ m_i \in \{0, \alpha\}^{H \times W} $ denotes the mask corresponding to $ t_i $, in which only randomly sampled pixel positions have value $ \alpha $ and $ \alpha $ is the blended ratio. We also set the corresponding label $ \tilde{x} $ to be a random class $ \tilde{y}_j $ from $ y_i $ to make the prioritized region active. After achieving the fingerprint datapoint, as shown in Figure 1, to make the framework efficient and scalable, we first train a base model and every model copy is then fine-tuned on the its own fingerprint dataset that contains both set of clean samples $ D = \{(x_j, y_j)\}_{j=1}^{\|\tilde{D}\|} $ and fingerprint samples $ \tilde{D} = \{(\tilde{x}_j, \tilde{y}_j)\}_{j=1}^{\|\tilde{D}\|} $. At the same time, we add a regularization term during finetuning to strengthen model’s memorization on the fingerprint datapoint. Specifically, for a fixed portion of clean data (30% in all experiments in this paper), we add random noise to the regions that are not being masked where $ m_{a,b,c} = 1 $. Then we use Eqn 1 to inject fingerprint into the noise image without changing the original true label. ### 3.3 Adversary Identification To identify the adversary $ g_{att} $ with only one adversarial example $ x_{adv} $, we propose two simple metrics. For the given adversarial example $ x_{adv} $, we first apply every copy’s sampled pixels $ t_i $ and corresponding mask $ m_i $ to create a set of fingerprint adversarial examples $ \tilde{x}_{adv}^i = A(x_{adv}, m_i, t_i) $. Specially, let $ \tilde{x}_{adv}^{att} $ be the fingerprint image corresponding to $ g_{att} $. #### KL metric We start the case when the model predicts $ x_{adv} $ with high confidence on the fingerprint class $ \tilde{y}_j $. In other words, if the $ x_{adv} $'s prediction is $ \tilde{y}_j $ with a high confidence, the generated adversarial perturbation would be very similar with the sampled pixels $ t_i $. It inspires us to compare the output distribution between adversarial example with and without applying $ t_i $. If the $ x_{adv} $ is from the adversary copy $ g_{att} $, the output distribution of the adversarial example $ \sigma(g_{att}(x_{adv})) $ would be very similar with the one applied with the sampled pixels. On the other hand, if the $ x_{adv} $ is from other model copies instead of $ g_{att} $, the output distribution will shift greatly after applying sampled pixels. Hence we can compute the similarity between $ \sigma(g_i(x_{adv})) $ and $ \sigma(g_i(\tilde{x}_{adv}^i)) $ for all model copies $ \{g_i\}_{i=1}^{K} $ to identify the adversary through largest similarity. To measure the similarity between two probability distributions, we choose to compute commonly used KL divergence as the first metric called KL metric. Formally, for every model copy $ g_i $, we compute the KL metric $ kl_i $ between the output probabilities $ \sigma(g_i(x_{adv})) $ and the output probabilities $ \sigma(g_i(\tilde{x}_{adv}^i)) $, \[ kl_i = KL (\sigma(g_i(x_{adv})) \|\| \sigma(g_i(\tilde{x}_{adv}^i))) \] \[ = \sum_{j=1}^{K} (\sigma(g_i(x_{adv})))_j \log \left( \frac{(\sigma(g_i(x_{adv})))_j}{(\sigma(g_i(\tilde{x}_{adv}^i)))_j} \right) \] where $ (\sigma(g_i(x_{adv})))_j, (\sigma(g_i(\tilde{x}_{adv}^i)))_j $ are the output probabilities of copy $ g_i $ on class $ j $ given $ x_{adv} $ and $ \tilde{x}_{adv}^i $, respectively. Since we sample different pixels corresponding to different random classes $ \tilde{y}_j $ for each copy $ g_i $, the KL metric for each combination is computed by Eqn 2 in the same way. The smaller one is used as the final KL metric of copy $ g_i $, denoted as $ kl_i^* $. With the final KL metric, the model copy corresponding to the smallest KL metric (the largest similarity) is the compromised model copy $ g_{att} $. #### Ratio metric However, since the adversary is conducting untargeted attack, there is a chance that the adversarial example would mislead the classifier into other classes than class $ \tilde{y}_j $, i.e the model has low confidence on predicting $ x_{adv} $ to class $ \tilde{y}_j $. Luckily, we observed that there would be a significant change on the model prediction distribution after applying $ t_i $ for the model that $ x_{adv} $ is based. Inspired by this observation, for every model copies $ g_i $, we measure the change of difference between maximum output probability and probability corresponding to the true class $ y $ of original image used to generate $ x_{adv} $. Based on this intuition, for each model copy $ g_i $, we compute its ratio metric as \[ r_i = \frac{\max_j (\sigma(g_i(\hat{x}_{adv})))_j - (\sigma(g_i(\hat{x}_{adv})))_y}{\max_j (\sigma(g_i(x_{adv})))_j - (\sigma(g_i(x_{adv})))_y}, \] where $ (\cdot)_y $ means the output probability of $ y $. With the two metrics, we can then combine them together to take both cases from low confidence to high confidence into consideration. In the following, we provide a method to linearly combine those two metrics together for the final identification. To better control the weight on two metrics, since the scales of the two metrics are different, we first normalize all $ kl_i^* $ and $ r_i^* $ of $ n $ copies into $[0, 1]$. After the normalization, we further use every model’s confidence to linearly combine the two metric values since the metrics are designed based on confidence level. Given $ x_{adv} $, for model copy $ g_i $, we use the difference between the top two output logits of $ g_i $ as the confidence level of $ g_i $ on $ x_{adv} $, i.e., the confidence level is \[ l_i = [g_i(x_{adv})]_{y_i} - \max_{j \neq y_i} [g_i(x_{adv})]_j, \] where $[g_i(x_{adv})]_j$ is the output logit of copy $ g_i $ on class $ j $ given $ x_{adv} $, and $ y_i $ is the predicted label of copy $ g_i $ given $ x_{adv} $. Then the combined metric value of copy $ g_i $ is computed as \[ v_i = w \cdot kl_i^* + (1 - w) \cdot r_i^, \] where $ w = \text{sigmoid}(\max l_i - T) $ is the weight for the metrics and $ T $ is a pre-defined threshold to control the confidence level. For every model copy, we will calculate the final score $ v_i $ and take the copy with the smallest score as the compromised copy. That is, \[ \text{att} \leftarrow \arg\min_i v_i. \] ### 4 EXPERIMENTS #### 4.1 IMPLEMENTATION DETAILS Following the settings in Cheng et al. (2023), we conduct experiments on two widely used datasets, CIFAR10 (Krizhevsky et al., 2009) and GTSRB (Stallkamp et al., 2012). Two model architectures, ResNet18 (He et al., 2016) and VGG16 (Simonyan & Zisserman, 2014), are utilized to verify the effectiveness of the proposed method. Firstly, we pre-train models with cross-entropy loss using Adam optimizer (Kingma & Ba, 2014) for 50 epochs with learning rate 0.001 and batch size 128. After finishing pre-training models, for each copy, the constructed fingerprint dataset (described in Section 3.2) with ratio $ p $ of fingerprint samples is used to finetune the baseline model for 20 epochs. Both the ratio $ p $ of fingerprint samples and the blended ratio $ \alpha $ are 0.3 for all our experiments. We sample a label set of length 2 (i.e., $ \|y_i\| = 2 $) for each copy. In this paper, we consider the cases that the number of distributed model copies is 50 and 100, where we finetune 50 or 100 model copies and identify one adversary from the 50 or 100 copies. We use 0.9% of total image size to apply $ t_i $. Hence for both CIFAR10 and GTSRB ($ 32 \times 32 \times 3 $ images), we randomly sample 9 positions for each combination of $ t_i $ and $ \tilde{y}_j $ of each model copy. For adversarial attacks, we firstly show the effectiveness of the proposed framework on several state-of-the-art white-box attacks. Then we also test the identification accuracy on different black-box attacks and show that the method can still achieve high accuracy on black-box attacks. Specifically, we use the following commonly used white-box and black-box attacks: - PGD-$ \ell_2 $* (White-box): Projected Gradient Descent attack with $ \ell_2 $ norm (Madry et al., 2017). The adversarial perturbations are constrained with $ \epsilon = 0.3 $. • C&W (White-box): one of the most popular methods in the white-box setting with $\ell_2$ norm proposed in Carlini & Wagner (Carlini & Wagner, 2017) and we set the $\kappa = 30$. • PGD-$\ell_\infty$ (White-box): Projected Gradient Descent attack with $\ell_\infty$ norm. The adversarial perturbations are constrained with $\epsilon = 8/255$. • APGD-CE (White-box): Auto-Projected Gradient Descent attack with $\ell_\infty$ norm in AutoAttack (Croce & Hein, 2020) using adaptive stepsize adjustment. Cross-entropy loss is used and the adversarial perturbations are constrained with $\epsilon = 8/255$. • NES (Black-box): Black-box soft-label attack that uses derivative-free optimization to estimate the gradient (Ilyas et al., 2018). • HSJA (Black-box): Black-box hard-label attack that utilizes the zeroth order oracle to find a better random walk direction in generating adversarial examples (Chen et al., 2020). All adversarial attacks are conducted in untargeted manner. For adversarial examples generated by the above adversarial attacks, only valid adversarial examples that can transfer to other models are considered. For each model copy, 30 valid adversarial examples are generated. Hence there are about 1500 adversarial examples for 50 copies case, 3000 adversarial examples for 100 copies case. The identification accuracy is computed as the ratio between the number of correctly identified adversarial examples $N_c$ and the total number of adversarial examples $N_t$, i.e. TraceAcc = $\frac{N_c}{N_t} \cdot 100\%$. ### 4.2 Identification Results We first show that the proposed watermarking framework has limited effect on all model copies’ performance. For the two datasets and two model architectures, we can have four combinations, i.e., VGG16-CIFAR10, VGG16-GTSRB, ResNet18-CIFAR10, and ResNet18-GTSRB. We show the maximum, minimum, mean, and median of classification performance for each 50 or 100 case and compare them with the pre-trained model performance (baseline performance). From Table 1, the mean and median accuracy is similar to the baseline performance within around 1% difference. It shows the proposed framework would have limited degradation on the model’s clean performance. The identification accuracy with only one adversarial example is shown in Table 2. The threshold $T$ described in Section 3.3 is set as to be 7. We also conduct different choices of $T$ in the ablation study. For white-box attacks, the results show that the proposed method is very effective on different attacks, datasets, and model architectures, which achieves average accuracy of 74.11% and 71.22% for 50 copies case and 100 copies case, respectively. Specifically, on CIFAR10 dataset, the method can achieve the highest accuracy of 88.80% and 88.37% with only one adversarial example available for 50 copies case and 100 copies case. Although the focus of this paper is the white-box setting, we also evaluate the method on two popularly used black-box attacks, NES attack (Ilyas et al., 2018) and HSJA attack (Chen et al., 2020) which are also used in Cheng et al. (2023), as shown in Table 2. It can be observed that the method can still achieve effective identification, especially on NES attack. However, our identification result on the black-box attack is not as good as white-box attack tested because of the noise gradient estimation used in the black-box attack. Note that we don’t include the comparison on the masked-based watermarking method in Cheng et al. (2023). The reason is that the watermarking method (Cheng et al., 2023) is specifically designed for black-box attack identification which makes explicit modifications on the architectures and the white-box attacker could create strong adaptive attack to make the identification totally fail, which would easily make the identification rate to close to 0. Results with more adversarial examples Previously, we show the identification accuracy with only one adversarial example, which is the most difficult case. Our proposed framework could naturally be extended if there are more adversarial examples available. To combine more adversarial examples scores together, for each model copy, we firstly compute the final metric in Equation 4 for each adversarial example. Then we take the minimum metric value as the final metric of the copy on the set of adversarial examples. The model with the minimum final metric value among all copies is treated as the compromised one. We present the identification accuracy on CIFAR10 (Krizhevsky et al., 2009) dataset with architectures VGG16 (Simonyan & Zisserman, 2014) and ResNet18 (He et al., 2016) in the 50 copies case, as shown in Figure 2. From the results, it can be observed that more adversarial examples can largely facilitate the identification performance. For most cases, the Table 1: Clean classification accuracy(%) of watermarked model copies, compared to pre-trained baseline model performance. \| Num \| Model-Data \| Baseline \| Max \| Min \| Mean \| Median \| \|-----\|----------------\|----------\|-------\|-------\|-------\|--------\| \| 50 \| VGG-CIFAR10 \| 90.21 \| 90.22 \| 87.45 \| 89.30 \| 89.34 \| \| \| V16-G \| 96.79 \| 97.36 \| 92.79 \| 96.16 \| 96.32 \| \| \| R18-C \| 92.03 \| 92.04 \| 90.29 \| 91.19 \| 91.21 \| \| \| R18-G \| 98.40 \| 98.56 \| 96.37 \| 97.72 \| 97.77 \| \| 100 \| VGG-CIFAR10 \| 90.21 \| 90.1 \| 85.31 \| 89.16 \| 89.28 \| \| \| V16-G \| 96.79 \| 97.55 \| 93.92 \| 96.08 \| 96.13 \| \| \| R18-C \| 92.03 \| 91.95 \| 89.62 \| 91.22 \| 91.22 \| \| \| R18-G \| 98.40 \| 98.56 \| 96.37 \| 97.71 \| 97.75 \| Table 2: Identification accuracy(%) of the proposed framework in different cases with only one adversarial example. \| Num \| Model-Data \| PGD-$\ell_2$ \| C&W \| PGD-$\ell_\infty$ \| APGD-CE \| NES \| HSJA \| \|-----\|------------\|--------------\|-----\|-------------------\|---------\|-----\|------\| \| 50 \| V16-C \| 68.98 \| 80.89 \| 85.56 \| 88.48 \| 83.00 \| 47.91 \| \| \| V16-G \| 71.78 \| 66.02 \| 84.17 \| 88.80 \| 77.68 \| 47.92 \| \| \| R18-C \| 63.10 \| 63.97 \| 66.74 \| 72.84 \| 73.45 \| 49.51 \| \| \| R18-G \| 64.16 \| 57.71 \| 75.33 \| 87.17 \| 80.69 \| 50.04 \| \| 100 \| V16-C \| 69.89 \| 77.55 \| 82.70 \| 77.70 \| 76.44 \| 42.77 \| \| \| V16-G \| 71.90 \| 66.19 \| 81.75 \| 88.37 \| 71.59 \| 39.26 \| \| \| R18-C \| 60.05 \| 56.58 \| 58.92 \| 67.26 \| 64.89 \| 39.90 \| \| \| R18-G \| 62.52 \| 57.23 \| 75.74 \| 85.22 \| 77.88 \| 40.58 \| Accuracy can be improved up to about 90% with two adversarial examples, even up to near 100% with three or more adversarial examples. ![Graphs showing identification accuracy on more adversarial examples](image) Figure 2: Identification accuracy on more adversarial examples. PGD-L2 denotes PGD-$\ell_2$ attack; PGD-Linf denotes PGD-$\ell_\infty$ attack. 4.3 Robustness Against Adaptive Attack Since a unique watermark is inserted into each model copy in the proposed framework, a natural question arises: will the method be effective and robust if the adversary tries to conduct adaptive attack to remove the watermark? To answer the question, in this section, we show the effectiveness and robustness of the framework against adaptive watermark-removing attacks. Specifically, because our implicit watermark tries to build a direct mapping from several pixels and labels, the backdoor defense methods could be used to erase our proposed watermark. We then test the robustness of the proposed framework against different types of adaptive attacks including finetuning-based removal methods (Liu et al., 2021), and reverse-engineering based removal methods (Wang et al., 2019; Aiken et al., 2021). For finetuning-based removal methods, we re-implement the called ‘WILD’ framework in Liu et al. (2021) according to the paper since we didn’t find any open-source code in that paper. We follow the same settings using 20% of training data for finetuning the watermarked model. The Jensen Shannon divergence is used for the distribution metric and the loss weight for this term is 10, as used in the paper. We use the 50 VGG16 models trained on CIFAR10 to test the effectiveness of watermark removal. Initially, we find the backdoor removal method could remove our watermark in 90% cases. However, we empirically found that if we use data augmentation methods such as Random Erasing (Zhong et al., 2020) during watermarking, the watermarked model would be much more robust against removal. Note that we did not use any distribution loss which is very important for backdoor removal in Liu et al. (2021) to insert a watermark specifically against the removal method in Liu et al. (2021). We just use the commonly used data augmentation methods during watermarking. With data augmentations during watermarking, we could make our implicit water intact with only about 10% cases would be removed. Then we also test the robustness against reverse-engineering based backdoor methods (Wang et al., 2019; Aiken et al., 2021). For these Neural Cleanse based methods, the removal performance highly relies on the detection of the watermark. If Neural Cleanse cannot detect any watermark, no further steps would be proceed. Hence we mainly test if the Neural Cleanse can effectively detect our implicit watermark. However, we found Neural Cleanse can no longer detect any watermarks if we simply increase the number of fingerprint class $\|y_i\| = 4$. At the same time, the number of fingerprint class $\|y_i\|$ has limited effect on the identification accuracy and it can even further improve identification accuracy, as we show in the following. The clean accuracy with $\|y_i\| = 4$ is shown in Table 3, which shows a larger size of $\|y_i\|$ won’t affect clean accuracy due to the high capacity of neural networks. Table 3: Clean accuracy with $\|y_i\| = 4$. \| Baseline \| Max \| Min \| Mean \| Median \| \|----------\|-------\|-------\|-------\|--------\| \| 90.21% \| 90.09%\| 86.54%\| 89.10%\| 89.19% \| Then for each attack, we generate around 1500 adversarial examples using the 50 models. The identification accuracy given only one adversarial example on different adversarial attacks is shown in Table 4. Table 4: Identification accuracy with $\|y_i\| = 4$. \| PGD-$\ell_2$ \| PGD-$\ell_\infty$ \| APGD-CE \| C&W \| NES \| \|--------------\|-------------------\|---------\|-----\|-----\| \| 75.75% \| 60.42% \| 77.50% \| 69.37% \| 72.26% \| To summarize, we show that with only small and reasonable modifications, watermarked models are robust against different types of adaptive attacks, verifying the effectiveness and robustness of our proposed framework. ### 4.4 Ablation Study Effect of different choices on $T$. To show the effects of different choices of the threshold $T$, we present the identification results under different $T$ in this section, as shown in Table 5. We use VGG16-CIFAR10 with 50 copies to test the effect of different $T$. We select $T = 5, 10, 15$. It can be observed that with larger $T$, the identification accuracy of PGD-$\ell_2$ (Madry et al., 2017), PGD-$\ell_\infty$ (Madry et al., 2017), C&W (Carlini & Wagner, 2017), and APGD-CE (Croce & Hein, 2020) attacks decreases, while the accuracy of HSJA (Chen et al., 2020), and NES (Ilyas et al., 2018) attacks increases. According to the analysis in Section 3.3, this indicates that the adversarial examples generated by PGD-$\ell_2$, PGD-$\ell_\infty$, C&W, and APGD-CE attacks have larger confidence compared to the adversarial examples generated by the HSJA and NES attacks. Another observation from the results is that compared to other attacks, APGD-CE and HSJA are more stable to the change of threshold $T$. The difference between $T = 5$ and $T = 15$ is about 4% for APGD-CE and HSJA, while the difference for other attacks is up to 10%. The reason may be that for APGD-CE it uses adaptive stepsize adjustment instead of fixed stepsize to generate... perturbations which may be more stable. And for HSJA the computed confidence may be very small since it searches adversarial examples near boundary (Chen et al., 2020). Hence various values of $T$ don’t have much effect on the combined final metric value. In practice, to obtain better identification results, the investigator can firstly compute the confidence level as described in Section 3.2. Based on the confidence level, the investigator can determine whether the confidence value is large or small to choose the threshold $T$. Table 5: Identification accuracy(%) with different choices of the threshold $T$. \| $T$ \| PGD-$\ell_2$ \| C&W \| PGD-$\ell_\infty$ \| APGD-CE \| NES \| HSJA \| \|-------\|--------------\|-----\|-------------------\|---------\|-----\|------\| \| $T = 5$ \| 71.32 \| 83.30 \| 87.14 \| 88.77 \| 75.93 \| 45.82 \| \| $T = 10$ \| 65.39 \| 76.60 \| 81.65 \| 87.29 \| 84.39 \| 48.63 \| \| $T = 15$ \| 60.80 \| 67.43 \| 73.46 \| 84.83 \| 84.63 \| 48.71 \| Effect of watermark design. As mentioned in Section 3.2, the watermark are inserted in a discrete manner. In this section, we show that the discrete watermarks can indeed largely improve the identification accuracy. Specifically, we finetune 50 VGG16 model copies on CIFAR10 with square watermarks. All the finetuning process and generation of adversarial examples are the same as the discrete watermark except that the watermarks are inserted as $3 \times 3 \times 3$ square in continuous regions. We set the threshold $T = 7$ for the fair comparison. Firstly, we compare the clean classification accuracy under different watermark insertions. The results shown in Table 6a indicate the effects of different watermark insertion manners on clean classification accuracy are subtle. We show the results for identification accuracy of adversarial examples with only one adversarial example in Table 6b. From the results, we can see the discrete watermark performs much better compared to the square one, especially for PGD-$\ell_2$, PGD-$\ell_\infty$, APGD-CE, and C&W attacks. We defer more ablation studies in the Appendix. Table 6: Clean classification accuracy(%) and identification accuracy(%) of for different types of watermark $w_i$ selection. ‘Discrete’ means the watermark pixels are selected in discrete positions; ‘Square’ means the watermark pixels are selected as a square in continuous regions. (a) Clean classification accuracy(%). \| Watermark type \| Baseline \| Max \| Min \| Mean \| Median \| \|----------------\|----------\|-----\|-----\|------\|--------\| \| Discrete \| 90.21 \| 90.22 \| 87.45 \| 89.30 \| 89.34 \| \| Square \| 90.21 \| 90.45 \| 88.06 \| 89.44 \| 89.55 \| (b) Identification accuracy(%) with only one adversarial example. \| Watermark type \| PGD-$\ell_2$ \| C&W \| PGD-$\ell_\infty$ \| APGD-CE \| NES \| HSJA \| \|----------------\|--------------\|-----\|-------------------\|---------\|-----\|------\| \| Discrete \| 68.98 \| 80.89 \| 85.56 \| 88.48 \| 83.00 \| 47.91 \| \| Square \| 26.65 \| 59.78 \| 36.49 \| 40.95 \| 80.86 \| 44.47 \| 5 CONCLUSION AND LIMITATIONS In this paper, we propose a novel framework for the identification of adversary with only one adversarial example under white-box attacks. We design an implicit watermarking method by designing fingerprint datasets to make each model copy unique and propose two different metrics to identify the adversary with high accuracy in data-free case. Extensive experiments on various attacks including both white-box and black-box attacks, datasets, and model architectures verify the effectiveness of the proposed method. With two more adversarial examples available, the tracing accuracy can be further improved up to near 100%. However, although the proposed framework shows promising high adversary identification accuracy, it couldn’t handle the cases where there exists several adversary to jointly conduct adversarial attack. Also, the proposed framework couldn’t be directly applied into other machine learning tasks except for image classification, which will leave in our future work. REFERENCES William Aiken, Hyoungshick Kim, Simon Woo, and Jungwoo Ryoo. 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7Ttk3RzDeu	The sentence-level score may disproportionately favor summaries that contain a large number of (short) sentences. Furthermore, the evaluation of different models does not further investigate whether some of the score differences can be explained by the different length of the summaries (e.g., a shorter summary may be more prone to omissions).	BooookScore: A SYSTEMATIC EXPLORATION OF BOOK-LENGTH SUMMARIZATION IN THE ERA OF LLMs Yapei Chang University of Massachusetts Amherst yapeichang@umass.edu Kyle Lo Allen Institute for AI kylel@allenai.org Tanya Goyal Princeton University tanyagoyal@princeton.edu Mohit Iyyer University of Massachusetts Amherst miyyer@cs.umass.edu ABSTRACT Summarizing book-length documents (>100K tokens) that exceed the context window size of large language models (LLMs) requires first breaking the input document into smaller chunks and then prompting an LLM to merge, update, and compress chunk-level summaries. Despite the complexity and importance of this task, it has yet to be meaningfully studied due to the challenges of evaluation: existing book-length summarization datasets (e.g., BookSum) are in the pretraining data of most public LLMs, and existing evaluation methods struggle to capture errors made by modern LLM summarizers. In this paper, we present the first study of the coherence of LLM-based book-length summarizers implemented via two prompting workflows: (1) hierarchically merging chunk-level summaries, and (2) incrementally updating a running summary. We obtain 1193 fine-grained human annotations on GPT-4 generated summaries of 100 recently-published books and identify eight common types of coherence errors made by LLMs. Because human evaluation is expensive and time-consuming, we develop an automatic metric, BooookScore, that measures the proportion of sentences in a summary that do not contain any of the identified error types. BooookScore has high agreement with human annotations and allows us to systematically evaluate the impact of many other critical parameters (e.g., chunk size, base LLM) while saving $15K USD and 500 hours in human evaluation costs. We find that closed-source LLMs such as GPT-4 and Claude 2 produce summaries with higher BooookScore than those generated by open-source models. While LLaMA 2 falls behind other models, Mixtral achieves performance on par with GPT-3.5-Turbo. Incremental updating yields lower BooookScore but higher level of detail than hierarchical merging, a trade-off sometimes preferred by annotators. We release code and annotations to spur more principled research on book-length summarization. github.com/lilakk/BooookScore 1 INTRODUCTION Just two years ago, automatically-generated summaries were riddled with artifacts such as grammar errors, repetition, and hallucination (Zhao et al., 2020; Fabbrì et al., 2020; Goyal & Durrett, 2021). Nowadays, such artifacts have mostly disappeared; in fact, Pu et al. (2023b) find that summaries generated by large language models (LLMs) are preferred over those written by humans, leading them to pronounce the death of summarization research. However, as with most prior work on summarization, the input documents in their study are relatively short (<10K tokens). Widespread adoption of LLMs outside the research community has driven the development of a more ambitious task: summarizing book-length documents, which we define to be texts longer than 100K tokens. As these documents exceed the context window limits of today’s LLMs (e.g., 8K tokens for GPT-4), summarizing them via prompt-based approaches necessitates heuristics to chunk the input, process each chunk, and then combine and compress the outputs (Wu et al., 2021). Despite the promise that LLMs hold for long-context tasks, the research community still lacks a principled and systematic approach to evaluate their capabilities on book-length summarization. Our paper identifies three open challenges with evaluation: (1) data contamination, in which existing benchmarks such as BookSum (Kryscinski et al., 2022) are in the pretraining data of modern LLMs (Chang et al., 2023); (2) an unexplored error distribution, as most prior summarization research centers around short source documents and fails to capture coherence errors that are exacerbated by the “chunk and combine” book-length summarization setting; and (3) a lack of any reliable automatic metric, which requires careful design and validation against human annotations. Contribution 1: A protocol for evaluating coherence in book-length summarization (§3). To mitigate the impact of data contamination, we design our evaluation framework around the use of newly-published books. We propose a reference-free protocol that leverages human annotation of the coherence of LLM-generated summaries (i.e., their logical connectedness) under different prompting strategies. Our protocol unifies and extends best-practices across disparate works in document understanding and evaluation research, including adoption of fine-grained annotation units, use of QA pairs to denote points of confusion, and a taxonomic breakdown of different coherence errors. We validate our protocol by collecting 1193 span-level human annotations on GPT-4 generated summaries of a carefully curated set of 100 recently-published books (costing $3K USD and 100 annotator hours) using two prompting strategies (hierarchical merging and incremental updating, shown in Figure 1). In categorizing these annotations into eight frequent error types, we reveal an error distribution in GPT-4 summaries that differs from that observed in prior studies on short-document summarizers; notably, we identify new error types (causal omissions, salience errors) through our book-length summarization setting (Table 1). Contribution 2: An automatic metric—BooookScore—to assess summary coherence (§4). Since our human evaluation is expensive, we follow recent work by developing an LLM-based evaluation metric called BooookScore that identifies and explains instances of any of our eight established coherence errors in a given summary. Human validation shows that BooookScore’s annotations are almost as reliable as those of human annotators, which allows us to automatically evaluate many other book-length summarization configurations. Because BooookScore does not rely on gold summaries, it can easily be used to evaluate new LLM summarizers on any collection of newly-published books, ensuring that the metric will remain meaningful for LLMs of the future. Contribution 3: A systematic evaluation of different LLMs using BooookScore (§5). We use BooookScore to evaluate the impact of several critical design decisions on the coherence of generated summaries, including the choice of prompting strategy, base LLM, and chunk size, a study that altogether cost $10K (USD) in LLM API calls. Our findings include (1) hierarchical merging generally results in more coherent summaries but reduced level of detail compared to incremental updating; (2) GPT-4 and Claude 2 produce the most coherent summaries, while LLaMA 2 is substantially worse and fails to follow instructions; (3) increasing the chunk size does not improve hierarchical merging but does substantially benefit Claude 2 when using incremental updating; and (4) summary-level preference judgments are highly subjective and do not correlate with BooookScore. ## 2 BACKGROUND: SUMMARIZING BOOK-LENGTH TEXTS WITH LLMs Before discussing our evaluation protocol, we first outline two strategies—hierarchical merging and incremental updating—for prompting an LLM to summarize book-length documents that exceed its maximum context size. In both strategies, the length of the input document necessitates first dividing it into smaller chunks and then repeatedly merging, updating, and/or compressing chunk-level partial summaries (Figure 1). While neither strategy is well-explored by published research, hierarchical merging essentially adapts the strategy proposed by Wu et al. (2021) to zero-shot prompting, while incremental updating resembles chain-of-density prompting proposed for short-document summarization (Adams et al., 2023). Both are implemented in widely-used open-source LLM libraries such as LangChain, but the relative merits of each method remain unexplored. --- 1LangChain implements incremental updating via refine and hierarchical merging via map-reduce Figure 1: To perform book-length summarization, we first divide a book into smaller chunks that fit within the context window of an LLM. Then, we explore two strategies for summarization: (1) hierarchical merging, in which chunks are first summarized and then the corresponding summaries merged via separate prompts; and (2) incremental updating, in which a global summary is updated and compressed as we step through the book chunk-by-chunk. More specifically, both strategies assume an LLM with context window size $W$ is used to summarize an input document $D$ whose length $L \gg W$. We thus split $D$ into non-overlapping chunks $c_1, c_2, \ldots, c_{\lceil \frac{L}{C} \rceil}$ where $C < W$ is the length of each chunk.\footnote{We ensure each chunk ends at a sentence boundary.} Hierarchical merging: Wu et al. (2021) propose a method in which an LLM (in their case, GPT-3) is fine-tuned via reinforcement learning to summarize each chunk and then hierarchically merge the chunk-level summaries until one summary is left of the entire input document. This method has since been simplified into a zero-shot prompting strategy without further training, as shown in Figure 1(left). Hierarchical merging requires three unique prompts for (1) summarizing an input chunk, (2) merging chunk-level summaries, and (3) merging summaries with added context from previously-generated merged summaries. We ensure that the total length of each prompt and its associated inputs is less than $W - G_l$, where $G_l$ is a hyperparameter controlling summary length that varies depending on the level $l$. Summaries are recursively merged until only one summary (of the full book) remains; see Appendix A.1 for further details. Incremental updating: It is possible that since hierarchical merging necessitates summarizing portions of the input document without complete context, it may introduce more coherence errors. For example, in the first level, chunks towards the end of the book will be summarized without knowledge of what came before, which can lead to incoherent summaries especially for non-linear or multi-perspective narratives. We thus explore an alternate prompting strategy—incremental updating (Figure 1 right)—that iterates through each chunk in order while continuously updating a global summary with salient information. While this method may be better able to handle inter-chunk dependencies than hierarchical merging, it requires more complicated prompts for (1) summarizing an input chunk, (2) updating the global summary $s_{1,2,\ldots,i-1}$ with information from the current chunk $c_i$, and (3) compressing the global summary when it exceeds the maximum summary length $G_n$. See Appendix A.2 for a full specification of incremental updating. 3 EVALUATING COHERENCE OF BOOK SUMMARIES In this section, we define our framework for human evaluation of coherence errors in book-length summarization. Our framework involves: (1) corpus collection focusing on newly-published books, (2) unification and extension of best-practices from prior document understanding and evaluation literature to guide data annotation, and (3) analysis of human annotations centered around emergent coherence error categories of summaries generated by modern LLMs. Collecting a corpus of newly-published books. The only existing public dataset for book-length summarization is BookSum (Kryscinski et al., 2022), which contains famous books from the Project Gutenberg public-domain repository along with reference summaries scraped from popular websites such as CliffNotes and GradeSaver. Both the source books and reference summaries are in the pretraining data of existing LLMs; Chang et al. (2023) confirm that many books in the BookSum held-out split (e.g., The Adventures of Huckleberry Finn, The Picture of Dorian Gray) are among the most-memorized books by GPT-4 and GPT-3.5-Turbo, and we were able to auto-complete several reference BookSum summaries by prompting GPT-4 with a short prefix of the summary. To reduce the confounding impact of summary memorization, we manually collect 100 books published within the past year to form our dataset (see Table 3 for a full list). Some of these books could still have appeared in the pretraining dataset of recent LLMs such as Claude 2 and LLaMa2, although it is much less likely than in BookSum. However, summaries of these books do not publicly exist: we did not find summaries online for any books in our dataset, which significantly lowers the possibility of LLM memorization. The average length of the books in our dataset is 190K tokens, compared to 112K tokens in BookSum. Due to copyright laws, we cannot publicly release this dataset; even if we could, we would still recommend that researchers collect their own datasets of newly-published books to minimize contamination with LLMs of the future. An evaluation framework for book-length summarization. Since we lack gold summaries, we design our evaluation framework to be reference-free, which aids in scalability. To do this, our evaluation framework synthesizes best-practices of prior document understanding and summarization evaluation research. Our evaluation employs: (1) fine-grained evaluation units as recommended by LongEval (Krishna et al., 2023); (2) information-seeking questions to represent naturally-occurring points of confusion (Ko et al., 2020; Wu et al., 2023; Meng et al., 2023; Newman et al., 2023); and (3) focus on summary coherence, which evaluates the logical structure and readability of the summary itself (Goyal et al., 2022a). We do not directly evaluate the faithfulness of the summaries (i.e., how factually accurate they are at conveying information from the source text), as the length of the source texts poses considerable issues for any existing faithfulness evaluation. We qualitatively discuss faithfulness in Section 5 and leave further investigation for future work. Annotation protocol. We implement our framework through a source- and reference-free annotation protocol where (1) annotators read through an LLM-generated summary, (2) highlight all confusing spans, and (3) ask question(s) for each marked span that highlight their confusion. See Table 1 (third column) for examples of spans and questions produced by our annotators. We hired four annotators with extensive English proofreading experience on Upwork, each of whom annotated 25 disjoint summaries. Each summary takes roughly 30 minutes to fully annotate with spans and questions, and we paid $15 USD per summary for a total of $3K to evaluate both prompting strategies. To generate the summaries, we set the base LLM to GPT-4 with a chunk size of 4096 and a maximum summary length $G_n = 1200$; other hyperparameters are detailed in Section 5. In total, the annotators mark 840 (incremental updating) and 353 (hierarchical merging) coherence errors for GPT-4-generated summaries; see Table 1 (right) for the split across error types. Validating the annotations: Typical measures of agreement are difficult to obtain in our setup, as measuring recall would require ground truth annotations with all possible coherence errors in the summaries; additionally, Goyal et al. (2022a) and Dou et al. (2022) observed low recall among annotators when evaluating machine-generated text at a fine-grained level. This motivates us to instead measure the precision of a given error annotation (i.e., after reading the corresponding question, do you agree that the span is confusing?), as it is simpler and cheaper while still being an informative metric. Given a span from a summary marked as containing an error, along with questions highlighting the confusion, we ask annotators (1) whether they think the span is confusing; and (2) whether the corresponding questions highlight the central confusion. We use the same four annotators hired before for this task, but make them validate human and (and later GPT-4) annotations for 25 books that they did not annotate in the first task. Overall, we validated 1,659 annotations for a total cost of --- 3We roughly balance our dataset across the following genres: fiction, non-fiction, sci-fi, fantasy, historical, contemporary, and memoir. We also include both linear and non-linear (multi-perspective and time-shifting) narratives in the dataset, and we purchase electronic copies of each of the 100 books in the dataset. 4However, we did find book reviews, which intentionally do not reveal major plot points or other spoilers. 5We also enabled forming relations between two spans in case multiple spans contributed to the same issue. 6http://upwork.com Table 1: Definition of all coherence error types, an example annotation for each, and their prevalence (%) in generated summaries, which is calculated as the number of error occurrences in all summaries normalized by the total number of sentences in all summaries. \| Error Type \| Definition \| Example spans & questions \| % errors per sentence inc / hier \| \|---------------------\|-----------------------------------------------------------------------------\|-------------------------------------------------------------------------------------------\|----------------------------------\| \| Entity omission \| An entity (e.g., person, object, place) is mentioned in the summary, but key context or details are missing or unclear. \| Span: A mysterious man introduces Proctor to “Arrivalism.” Question: Who is this mysterious man? \| 7.3 / 3.71 \| \| Event omission \| An event is mentioned in the summary, but key details are missing or unclear. \| Span: During a mission to find Caeli, Proctor is captured by watchmen while Thea escapes. Question: What happened to Caeli? \| 4.25 / 2.27 \| \| Causal omission \| A reason or motivation is missing or under-explained. \| Span: Proctor seeks answers from... Callista about the investigation. Question: Why would Callista know something about the investigation? \| 2.75 / 1.21 \| \| Discontinuity \| An interruption in the flow of the narrative such as sudden jumps in time or perspective. \| Span: In the new settlement, Thea adjusts to her life, working hard and finding solace in nature. Question: Why the shift to Thea’s perspective? \| 2.23 / 1.56 \| \| Salience \| Inclusion of details that do not contribute to the main plot. \| Span: His father... flees, resulting in a chaotic chase on the pier. Question: What is the significance of this incident? \| 1.42 / 1.03 \| \| Language \| Spelling or grammar issues; ambiguous wording. \| Span: Despite her love for him, Deborah is heartbroken by his decision. Question: Why is the preposition “Despite” used here when she is, in fact, heartbroken because of her love for him? \| 0.82 / 0.71 \| \| Inconsistency \| A discrepancy or contradiction within a story’s plot, character development, or themes. \| Span: In a farewell, Proctor marries his brother Malcolm to Cynthia and says goodbye to his loved ones. Question: If Cynthia is his mother and Malcolm is his brother, how can a mother and son marry? \| 0.97 / 1.03 \| \| Duplication \| Redundant repetition of similar information. \| Span 1: Proctor... deals with students and school issues, seeking help from Callista to fund a roof replacement. Span 2: Proctor’s life continues as he... deals with school issues, such as funding for a roof replacement Question: Why does the same information appear twice? \| 2.12 / 1.18 \| $418.90 (USD)\footnote{This cost includes validation of both human and BOOOOKSCORE annotations.}$ and we discover that $79.7\%$ of annotated spans are validated as legitimate through this task. More details on our validation can be found in Appendix \footnote{In preliminary experiments without definitions and few-shot demonstrations, we qualitatively observe significantly reduced annotation precision.}. Categorizing coherence errors: After collecting spans and questions from the annotators, we develop an error taxonomy consisting of the eight types detailed in Table 1, which covers the vast majority of annotations, and we manually code each annotation using this taxonomy. We intentionally went through this process without relying on the SNAC taxonomy \cite{goyal2022snac} so as to not be overly influenced by their error annotation schema which was tailor-made for fine-tuned summarization models. While we find considerable overlap in the two error schemas, we also discover two new instances of prominent errors not present in SNAC: causal omissions and salience issues. Our taxonomy also places less emphasis on language errors (e.g., coreference issues from SNAC) since modern LLMs rarely make such mistakes \cite{goyal2022snac}. Table 1 shows that omission errors are the most common across both incremental and hierarchical prompting strategies, and also that hierarchical merging makes fewer errors of every type but inconsistencies. 4 BOOOOKSCORE: AN AUTOMATIC EVALUATION METRIC Since human evaluation of summary coherence is not scalable due to the high financial and time cost, we develop an automatic metric — BOOOOKSCORE — that prompts an LLM to identify instances of the eight error types we identified in Section 3. We validate BOOOOKSCORE via a human evaluation of its precision (following the annotation task discussed in the previous section) and show that its precision matches that of human annotators ($78.2\%$ vs. $79.7\%$). We then use BOOOOKSCORE to evaluate many other book-length summarization configurations, saving $15K USD in evaluation costs and 500 hours in annotator time. We emphasize that incorporating definitions and examples from our error taxonomy into the prompt is critical to achieve high precision with BOOOOKSCORE.\footnote{In preliminary experiments without definitions and few-shot demonstrations, we qualitatively observe significantly reduced annotation precision.} 4.1 IMPLEMENTING BOOOOKSCORE Motivated by prior successful efforts to evaluate LLM-generated text via LLMs, such as AlpacaEval (Dubois et al., 2023), FActScore (Min et al., 2023), and G-Eval (Liu et al., 2023b), BOOOOKSCORE automatically measures the coherence of summaries generated by a book-length summarization system via few-shot prompting. BOOOOKSCORE is both source-free and reference-free (i.e., it does not require access to the input book or a reference summary), similar to the SNaC classifier built for fine-tuned summarizers by Goyal et al. (2022a). Specification: Assume we have a summary $S$ consisting of sentences $s_1, s_2, \ldots, s_n$. We develop a few-shot error-identification prompt $E$ that instructs the LLM to identify any instances of one of the eight specified error types in a given sentence $s_i$ of the summary. Concretely, we iterate over each sentence $s_i$ in the summary, feeding the prompt $E$, full summary $S$, and target sentence $s_i$ at each step. There are two acceptable outputs at each step: either (1) no error is found and the LLM outputs No confusion, or (2) an error(s) is identified and the LLM is asked to generate a corresponding question and associated error type. We include two full summaries with 42 sentence-level annotations in the prompt as demonstrations. The BOOOOKSCORE of a single summary $S$ (Figure 2) is then computed as: $$\text{BOOOOKSCORE}(S) = \frac{1}{n} \sum_{s_i \in S} [\text{LLM}(E, S, s_i) == \text{No confusion}]$$ When computing BOOOOKSCORE, we consider each sentence as a singular unit of confusion, rather than each of the questions associated with that sentence. This is because both LLMs and human annotators occasionally ask multiple questions that essentially target the same issue within a given sentence. Thus, our metric intuitively measures the proportion of sentences in the summary that contain no errors (i.e., higher is better). To obtain a system-level score, we compute the mean BOOOOKSCORE across all summaries generated by that system. Validating BOOOOKSCORE: We validate BOOOOKSCORE annotations in the same way that we validate human annotations in Section 3 by hiring human annotators to judge whether they agree with an LLM-generated annotation (here, GPT-4). We observe that the precision of human annotations is 79.7%, while the precision of BOOOOKSCORE annotations is 78.2% (details in Appendix I). Additionally, we compute BOOOOKSCORE using human annotations instead of LLM-generated ones for both GPT-4 configurations (i.e., replacing $\text{LLM}(E, S, s_i)$ in Equation 1 with the human error annotation for $s_i$) and observe extremely similar system-level scores. Using human annotations in Equation 1 yields a BOOOOKSCORE of 82.1 and 89.4. --- 9 After iterating over the design in numerous preliminary experiments, we find that our prompt works most reliably at the sentence level, rather than at the full summary level. As such, sentence tokenization is a required preprocessing step for BOOOOKSCORE. Future work should focus on implementations at the summary level, as it would save many calls to the LLM; here, we need to prompt the model separately for each sentence. 10 These examples contain a combination of sentences with and without confusion, all the while maintaining a diverse range of error types. The full prompt can be found in M.4. 11 For example, the questions “Who is John? Is he Lia’s husband?” both seek to establish John’s identity. Counting the number of questions instead of highlighted sentences would inadvertently overstate the weight of certain errors found within the same sentence. 12 Recall that human annotators can (1) highlight multiples consecutive sentences as one span and (2) create relations between two spans, while GPT-4 can only highlight single sentences as spans. To adjust for this difference, we treat both consecutive sentences and relations as single sentences when computing BOOOOKSCORE for humans. Table 2: BoooookScore for summaries generated under different configurations; higher scores indicate better coherence. We additionally report the average summary length in tokens based on tiktoken (https://github.com/openai/tiktoken) tokenizer, the percentage of novel trigrams compared to the source, and percentage of repeated trigrams in the summary. \| Model \| Chunk size \| BoooookScore \| Avg. length \| % novel 3-grams \| % rep. 3-grams \| \|---------------\|------------\|--------------\|-------------\|-----------------\|----------------\| \| Summaries generated via hierarchical merging \| \| \| \| \| \| \| GPT-4 \| 2048 \| 89.1 \| 778.6 \| 82.4 \| 4.2 \| \| GPT-3.5-Turbo \| 2048 \| 84.2 \| 667.3 \| 82.8 \| 9.0 \| \| Claude 2 \| 2048 \| 91.1 \| 522.6 \| 88.4 \| 1.3 \| \| Claude 2 \| 88000 \| 90.3 \| 551.5 \| 87.1 \| 2.0 \| \| Mixtral-8x7B \| 2048 \| 81.5 \| 679.1 \| 85.9 \| 4.1 \| \| LLaMA2-7B-Inst\| 2048 \| 72.4 \| 684.9 \| 76.4 \| 36.1 \| \| Summaries generated via incremental updating \| \| \| \| \| \| \| GPT-4 \| 2048 \| 82.5 \| 805.4 \| 84.1 \| 3.4 \| \| GPT-3.5-Turbo \| 2048 \| 67.0 \| 484.5 \| 68.2 \| 3.5 \| \| Claude 2 \| 2048 \| 78.6 \| 657.1 \| 89.4 \| 1.9 \| \| Claude 2 \| 88000 \| 90.9 \| 493.7 \| 84.7 \| 1.9 \| \| Mixtral-8x7B \| 2048 \| 64.5 \| 558.7 \| 82.3 \| 3.5 \| GPT-4 summaries generated via incremental updating and hierarchical merging, respectively, while using LLM annotations yields a BoooookScore of 82.4 and 90.8. Figure 4 compares the error distributions from GPT-4 to those of human annotators and shows that GPT-4 is more sensitive to omission errors and less sensitive to duplication or language errors. Taken as a whole, these results confirm that BoooookScore is a reliable annotator of coherence for book-length summarization. While we implement BoooookScore with GPT-4 for the remainder of this paper, implementing BoooookScore with open-source LLM annotators is an exciting future direction. 5 SYSTEMATIC EVALUATION OF LLMs Armed with BoooookScore, we now investigate the impact of several critical implementation decisions on summary coherence, including the choice of prompting strategy, base LLM, and chunk size. Overall, Claude 2 produces the most coherent summaries as measured by BoooookScore, followed closely by GPT-4 and distantly by GPT-3.5-Turbo, Mixtral-8x7B, and LLaMA2-7B-Inst; however, GPT-4’s summaries are significantly longer and more detailed than the others across both prompting strategies. The rest of this section drills down into finer-grained results. Experimental setup: Table 2 contains results for five instruction-tuned LLMs: GPT-4, GPT-3.5-Turbo, Claude 2, Mixtral-8x7B, and LLaMA2-7B-Instruct. Unless otherwise specified, we set the chunk size to 2048, maximum summary length $G_n$ to 900, decoding temperature to 0.5, and $p = 1$ for ancestral sampling. To avoid confounds, we use identical prompts for all models except LLaMA2-7B-Inst, which only functions with a simpler prompt. LLM API costs for our experiments were $10K USD (Table 8); more experimental details are in Appendix D. Incremental summaries are almost always less coherent than their hierarchical counterparts. Hierarchical summaries generally have higher BoooookScore than incremental summaries, likely because the incremental updating task requires the base LLMs to follow more complex instructions. --- 13 GPT-4 configurations in this table are not comparable to the ones we analyzed in Section 3 since we had to reduce chunk size and summary length due to LLaMA2-7B-Inst and GPT-3.5-Turbo’s smaller context size. 14 Claude 2 is the only exception, as we use its default temperature of 1. 15 We use a temperature of 1 for compression, which improves adherence to the max summary length. (e.g., deciding what to include from the current book chunk, what to discard from the summary, whether to restructure the summary, etc.). While hierarchical summarization potentially drops long-range dependencies, its instructions are generally simpler (summarize or merge). Incremental summarization benefits from increased chunk size. The one exception to the above result is Claude 2 with a chunk size of 88K, whose incremental configuration produces slightly more coherent summaries than the hierarchical version (90.9 vs. 90.3 BoooookScore). In contrast, using Claude 2 for incremental summarization with a chunk size of 2048 results in a BoooookScore of 78.6, so clearly the model benefits from fewer updating and compression steps. We do not observe similar behavior with hierarchical summaries, which suggests that hierarchical book-length summarization is preferred for smaller context models. LLaMA 2 struggles on book-length summarization while Mixtral shows promising performance. Table 2 shows that LLaMA-2-7B-Instruct achieves by far the worst hierarchical BoooookScore of any model. Its summaries also contain significant repetition (% of repeated trigrams), which is a critical coherence error. Furthermore, we could not get the LLaMA-2-7B-Instruct checkpoint to perform incremental updating at all, as it just copied text from the chunks until it reached the summary length limit, at which point it failed to follow the compression instruction. On the positive side, Mixtral-8x7B, another open-source LLM, outperforms LLaMA-2-7B-Instruct by a substantial margin, though it still trails behind most of the closed-source models. Nonetheless, it is encouraging to note that with performances closely matching that of GPT-3.5-Turbo on both summarization approaches, Mixtral-8x7B signals the narrowing gap between open-source and closed-source models. High coherence does not necessarily correlate with human preferences. How well do coherence measurements from BoooookScore correlate with coarse-grained human preferences? We conduct another human evaluation study with the same four annotators in which we solicit preference judgments on pairs of GPT-4 generated incremental and hierarchical summaries. As shown in Table 4, incremental summaries are almost always preferred over hierarchical summaries in terms of level of detail (83% vs. 11%). However, hierarchical summaries are preferred for better structure (59% vs. 35%), logical consistency (53% vs 38%), and overall (54% vs. 44%). When forming their overall preference, some annotators preferred the higher level of detail of incremental summaries at the expense of coherence; thus, both strategies can be viable depending on the needs of the user. Qualitative analysis: Appendix E contains summaries generated from Janika Oz’s A History of Burning, which tells a multi-generational story about an Indian family living in Uganda. We observe that both GPT-4 and GPT-3.5-Turbo tend to generate oft-repetitive and vague sentences within their summaries (e.g., The story highlights the resilience and determination of the characters as they navigate the complexities of life, love, and identity across generations and continents.). Such artifacts are rarely produced by the 88K chunk size version of Claude 2, which instead omits key information present in the beginning or middle of the input (e.g., the entire story of the first generation in the book) in favor of focusing on the end of the book, following the findings of Liu et al. (2023a). All configurations make faithfulness errors: for example, in A History of Burning, the mother of the character Hari is incorrectly identified as Rajini by Claude 2, while GPT-4 does describe Hari’s parentage correctly at one point in the summary but incorrectly at another. We show in Appendix I that automatic quality metrics such as BLANC (Vasilyev et al., 2020) and SUPERT (Gao et al., 2020) are inadequate for book-length summarization. ### 6 LIMITATIONS Our error taxonomy is derived just from errors made by GPT-4. We decided to conduct our human evaluations in Section 5 on summaries produced by GPT-4 for two reasons: (1) we wanted our error taxonomy to focus on errors that are actually made by state-of-the-art LLMs (unlike e.g., fluency errors present in SNaC); and (2) human evaluation is very costly, so we could not evaluate many different LLMs on our annotation budget. Similarly, we implement BoooookScore using GPT-4 as a base LLM, which may have some systematic biases that could be alleviated by using a pool of LLM annotators as in AlpacaEval (Dubois et al., 2023). --- Each annotator compared 25 disjoint pairs of summaries, and we paid $15 per task for a total of $1.5K. To prevent bias, we shuffle the ordering of incremental and hierarchical summaries for each summary pair, and conceal the summarization method of each summary. BooookScore can be expensive to run. Since computing BooookScore requires iterating through a summary sentence by sentence using GPT-4, it can be expensive and slow especially given that the annotation prompt is long (see Appendix M.4). We did experiment with an approach that asked GPT-4 to annotate errors in the entire summary at once, but the generated annotations would often include too many trivial questions, and alignment with human judgments was low. That said, despite the API costs of GPT-4 and the relatively slow time to evaluate one summary, BooookScore is still significant cheaper and faster than performing human evaluations. BooookScore does not account for the relative importance of different error types. Unlike similar evaluation frameworks such as MQM (Freitag et al., 2021), we choose not to assign severity weights to different error types. Nowadays, powerful LLMs rarely make errors related to grammar, which can be objectively defined. For other error types like those in our taxonomy, the notion of assigning relative importance is ill-defined. Furthermore, prior work (Goyal et al., 2022a; Dou et al., 2022) shows low recall between human annotations for NLG evaluation, which indicates that error type severity is subjective as annotators often do not highlight issues that others may find critical. No validation of recall. Due to the expense, we do not collect overlapping annotations for each summary during human evaluation. Since the annotation task involves subjectivity, overlapping annotations can help ensure that all errors within a summary can be captured. However, recent work (Krishna et al., 2023) shows that a comprehensive annotation of all information units is not required to produce a useful aggregate score that can be used to rank different models. 7 RELATED WORK Book-length narrative summarization: Most prior long-form summarization work still focuses on documents shorter than 10K tokens (Cohan et al., 2018; Kornilova & Eidelman, 2019; Wang et al., 2022). BookSum (Kryscinski et al., 2022) is the first published summarization dataset that includes book-level source text as part of their data, which encouraged modeling efforts in this direction (Wu et al., 2021; Xiong et al., 2022; Pang et al., 2023; Cao & Wang, 2023; Pu et al., 2023a). Fine-grained evaluation of generated text: Our work relates to evaluation protocols within machine translation that annotate spans, error types, and error severities (Freitag et al., 2021; Fernandes et al., 2023), which are more meaningful than output ranking and Likert ratings. Also related is ACU (Liu et al., 2023c), an annotation protocol for summary salience evaluation that breaks summaries down into fine-grained content units, FactScore (Min et al., 2023), which dissects machine-generated text into atomic facts before evaluating their factual consistency, LongEval (Krishna et al., 2023), which includes an in-depth analysis of best practices for faithfulness evaluation in long-form summarization coherence evaluation, and SNaC (Goyal et al., 2022a), a coherence error taxonomy built for fine-tuned summarization models. Automatic evaluation with LLMs: LLM evaluators have recently emerged as a cost-effective alternative to human evaluations, explored for both general conversational and instruction following capabilities (Dubois et al., 2023; Zheng et al., 2023) and traditional NLG tasks like summarization (Fu et al., 2023; Liu et al., 2023b; Wang et al., 2023). These latter studies substantiate LLMs’ potential as an NLG metric, but only for evaluating short input-output pairs. In our work, we use GPT-4 to evaluate book-length summaries, uniquely employing a fine-grained automatic evaluation schema to set our work apart from existing research. 8 CONCLUSION Our work presents the first systematic study of book-length summarization using LLMs. We establish a novel human evaluation protocol to assess summary coherence on newly-published books. Then, we develop an LLM-based automatic metric called BooookScore that relies on a coherence error taxonomy derived from our human annotations. Using BooookScore allows us to evaluate various prompting strategies and model choices, revealing insights such as: hierarchical merging produces more coherent summaries but may lack detail compared to incremental updating; and increasing chunk size can significantly improve incremental updating. Interesting future directions include automatically evaluating faithfulness in the book-length summarization setting, benchmarking newer long-context LLMs using BooookScore, and expanding BooookScore to multilingual texts. We release our BooookScore metric and annotated summaries to enable meaningful progress in book-length summarization. 9 ACKNOWLEDGMENTS We extend special gratitude to members from the UMass NLP lab for participating in the pilot study and offering valuable feedback, and to the Upwork annotators for their hard work. This project was partially supported by awards IIS-2202506 and IIS-2046248 from the National Science Foundation (NSF) as well as an award from Open Philanthropy. We also thank the NSF’s CloudBank program for supporting the majority of our LLM API-based experiments. REFERENCES Griffin Adams, Alexander Fabbri, Faisal Ladhak, Eric Lehman, and Noémie Elhadad. From sparse to dense: Gpt-4 summarization with chain of density prompting, 2023. 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sOXKeeVxqW	It would be helpful if the authors clarify the specific ratios of atoms that are masked in both SMILES and graphs. Additionally, providing information about what replaces the masked atoms is crucial for a complete understanding of the masking strategy.	MOleSG: A Multi-Modality Molecular Pre-training Framework by Joint Non-overlapping Masked Reconstruction of SMILES and Graph Anonymous authors Paper under double-blind review Abstract Self-supervised pre-training plays an important role in molecular representation learning because labeled molecular data are usually limited in many tasks, such as chemical property prediction and virtual screening. However, most existing molecular pre-training methods focus on one modality of molecular data, and the complementary information of two important modalities, SMILES and graph, are not fully explored. In this study, we propose a straightforward yet effective multi-modality pre-training framework for Molecular SMILES and Graph (MoleSG). Specifically, the SMILES sequence data and graph data are first tokenized so that they can be processed by a unified transformer-based backbone network, which is trained by a masked reconstruction strategy. In addition, we introduce a specialized non-overlapping masking strategy to encourage fine-grained interaction between these two modalities. Experimental results show that our framework achieves state-of-the-art performance in a series of molecular property prediction tasks, and detailed ablation study demonstrates efficacy of the multi-modality structure and the masking strategy. 1 Introduction Efficient molecular representation learning is foundational to drug discovery (David et al., 2020; Huang & Von Lilienfeld, 2016). With the advancement of deep learning, data-driven molecular representation learning has found applications in various domains, such as chemical property prediction (Duvenaud et al., 2015), virtual screening (Stumpfe & Bajorath, 2020), molecular design (Magar et al., 2021), and more. However, since most molecular label data need to be obtained through labor-intensive and costly wet experiments (Brown et al., 2019), there is a lack of sufficient labeled molecular data, which hinders the development of deep learning methods and can lead to issues like overfitting and poor generalization (Rong et al., 2020). Self-supervised learning holds substantial research value in addressing these challenges, which involves pre-training on unlabeled data and fine-tuning with labeled data on downstream tasks. It has shown significant promise in enhancing the performance of molecular representation learning on many downstream tasks (Xie et al., 2022). Molecules can be described using various modalities, such as fingerprints, sequences, graphs, and more (Xia et al., 2023). Currently, molecular pre-training predominantly focuses on a single modality (Xia et al., 2023), with only a little attention given to methods jointly dealing with multiple modalities (Liu et al., 2021; Zhu et al., 2021). This paper addresses the issue of jointly pre-training on two molecule modalities: Simplified Molecular-Input Line-Entry system (SMILES) (Weninger, 1988) and molecular graph. As depicted in Figure 1, the same molecule can be represented using both a SMILES sequence and a graph, with each modality having its unique advantages and disadvantages. SMILES is a compact implicit representation of the molecule that excludes single-bond representation, making it well-suited for rapid compound retrieval and identification (Quirós et al., 2018). Additionally, the SMILES sequence, being a text string, can be processed with transformer-based networks well-developed in the Natural Language Processing (NLP) field for feature extraction, in which the self-attention mechanism weights and combines information from any position in the input sequence, thereby facilitating the capture of global contextual information (Chithrananda et al., 2020; Wang et al., 2019). However, SMILES representations only capture the relationships be- Figure 1: Comparison of two molecular representation modalities, SMILES and graph. (a) Illustration of the topological differences between the two modalities. SMILES represents topology implicitly, while graph displays explicit topology. (b) Difference in attention mechanisms used for feature processing in the two modalities. Global attention mechanism is usually used for SMILES while local attention mechanism can be easily implemented for graph. between atoms and bonds. They often struggle to capture the complex structural and topological information of molecules, such as the number and positions of rings, the length of side chains, and other intricate details that can be crucial in drug efficacy prediction (Lim et al., 2021; Zhang et al., 2022). Graph representations offer explicit portrayals of atoms, bonds, and their interconnections, showcasing the topological structure of molecules (Xiong et al., 2019). They provide detailed chemical information about molecules, including attributes for each atom such as element type, charge state, stereochemistry, and attributes for each bond, like bond type and bond length (Hall et al., 1991). However, Graph Neural Networks (GNNs), commonly used to extract features from graphs, primarily rely on message-passing layers to gather information from neighboring nodes, emphasizing the capture of local contextual information. This can lead to a disadvantage in capturing global context information due to information decay when delivering messages between non-adjacent nodes (Zhou et al., 2020). As a result, for the same molecule, SMILES and graph encode molecular features from different perspectives, offering complementary information. The rational combination of these two modalities holds promise for enhancing molecular representation performance. There are several existing works on multi-modality molecular pre-training (Liu et al., 2021; Zhu et al., 2021; Liu et al., 2022). For example, GraphMVP (Liu et al., 2021) focuses on joint pre-training with 2D graphs and 3D graphs. However, these two modalities exhibit high similarity. Additionally, this study only proved 3D geometry complements 2D topology in downstream tasks, without proving 2D topology complements 3D geometry. DVMP (Zhu et al., 2021) first extracts features from SMILES and graph of the same molecule for contrastive learning. All these existing methods lack fine-grained cross-modality interactions, and there is no existing work that effectively explores the complementary information between SMILES and graph. The challenge of more efficiently combining these two modalities with significant differences lies in how to promote information exchange in fine-grain such as at the atom level rather than only achieving contrastive learning at the entire molecule level. In this paper, we propose MoleSG, a simple yet effective pre-training framework for effectively exploring the complementary information between SMILES and graph in molecular pre-training. Specifically, recognizing that both words in SMILES sequences and graph nodes can be treated as transformer tokens (Hu et al., 2023; Huang et al., 2022), we first introduce a transformer-based unified backbone network for jointly processing embeddings from both modalities to facilitate interactions between them. Our framework consists of two independent encoders to separately convert masked SMILES and masked graph of an input molecule into token embeddings. The embeddings from the two modalities are concatenated and inputted into a standard transformer for joint processing and the output is used to reconstruct the original SMILES and graph by two specific decoders. Our framework is trained by reconstruction losses. Furthermore, to enhance cross-modality interaction, we introduce a dedicated non-overlapping masking strategy, in which we establish the positional correspondence between the SMILES sequence and the graph of a molecule to ensure that regions masked in SMILES and graph do not overlap. Intuitively, the information used for reconstructing the masked tokens can come from the context within the same modality, as well as information from the tokens of corresponding structures in the other modality. Therefore, our non- overlapping masking strategy masks information within its own modality to encourage the model to learn information from the other modality, thereby strengthening interactions between the two modalities. To evaluate the effectiveness of MoleSG, we conduct experiments on 14 downstream tasks related to molecular property prediction and MoleSG achieves state-of-the-art (SOTA) performance in all tasks. We also compare it with the same network pre-trained by a single modality, and the experimental results show that multi-modality training learns richer molecular representation knowledge. Our contributions are as follows: (1) We propose MoleSG, a novel molecular pre-training framework that utilizes the complementary information of SMILES and graph representations, resulting in improved performance; (2) We introduce an innovative non-overlapping masking strategy and a unified network for handling two distinct modalities, allowing for fine-grained interaction between SMILES and graph representations and achieving better representation learning; (3) MoleSG achieves SOTA performance in a series of molecular property prediction tasks, and detailed ablation study demonstrates efficacy of the multi-modality structure and the masking strategy. 2 RELATED WORK Molecular single-modality self-supervised learning: Molecular single-modality self-supervised learning can be broadly categorized into contrastive and generative approaches. Most contrastive methods work on the modality of graph by bringing augmented graphs from the same molecule closer while pushing those from different molecules farther apart, and they focus on the global molecular information. For instance, MolCLR (Wang et al., 2022) employs diverse graph augmentation techniques for contrastive learning pre-training. FraSICL (Zhang et al., 2023) divides the same molecule into different fragment pairs based on semantics, enabling contrastive learning. KANO (Fang et al., 2023) incorporates an additional knowledge graph-based augmentation to improve the performance of contrastive learning. Generative approaches primarily predict masked molecular components using an encoder-decoder pattern, with an emphasis on learning information at the local level. For example, GROVER (Rong et al., 2020) is designed for the 2D graph modality and encompasses masked generative self-supervised tasks at the node and edge levels. Uni-mol (Zhou et al., 2023) focuses on the 3D graph modality and achieves effective 3D spatial representation learning through 3D position recovery and masked atom prediction tasks on a large dataset. Both SMILES-BERT (Wang et al., 2019) and ChemBERTa (Chithrananda et al., 2020) are designed for the SMILES modality and utilize a “cloze-style” generative pre-training approach. Molecular multi-modality self-supervised learning: GraphMVP (Liu et al., 2021) leverages correspondences and consistencies between 2D graph and 3D graph to perform both contrastive and generative self-supervised learning and inject 3D information into 2D molecular graph encoders. MoleculeSTM (Liu et al., 2022) focuses on molecular graphs and text descriptions, using a contrastive learning strategy to learn the consistency between the chemical structure of molecules and their textual descriptions. DVMP (Zhu et al., 2021) addresses both SMILES and graph modalities, employing a contrastive learning approach to learn SMILES information encoded by transformer and graph information encoded by GNN from the same molecule. DVMP focuses on the same two modalities as we do but it neglects interactions between fine-grained information across different modalities. 3 METHOD In this section, we will begin with providing an overview of our pre-training framework. Next, we will detail our data preprocessing procedures and introduce our innovative non-overlapping masking alignment strategy, which aims to encourage interaction between the two modalities. Following that, we will describe our network containing specialized encoders, backbone, and specialized decoders. 3.1 OVERVIEW OF MOLESG As shown in Figure 2, MoleSG learns features jointly from SMILES and graph by performing masked reconstruction on both modalities with a unified feature extraction backbone network. Concretely, for a given molecule, we first convert its SMILES sequence into tokens and calculate features for nodes and edges in the graph. Then, we randomly mask some node features in the graph and then mask a portion of SMILES tokens corresponding to the remaining unmasked atoms in the graph, so that we can perform non-overlapping masking to facilitate the interaction of information between the two modalities. Figure 2: Overview of MoleSG. The SMILES sequence and the graph of a molecule are first randomly masked using the non-overlapping masking strategy. Then they are individually encoded by independent encoders, and the SMILES embeddings and the graph embeddings are concatenated and inputted into a transformer backbone for joint processing. Finally, processed features belonging to each modality are decoded into token ids and graph nodes for the reconstruction proxy task. During pre-training, we employ a symmetric joint encoder-decoder framework to perform further feature extraction. The framework consists of two independent branches for the two modalities and a shared backbone for feature fusion. The independent encoder branches encode the data of two different modalities into a unified form i.e. embedding, which is suitable for understanding by a transformer backbone (Hu et al., 2023; Huang et al., 2022). The shared transformer backbone can learn the dependencies between atoms within and across the modalities and output features for the subsequent independent decoders. Finally, the SMILES decoder and the graph decoder reconstruct the original SMILES sequence and graph based on the output of the backbone. Different from prior works (Liu et al., 2021; Zhu et al., 2021; Zhang et al., 2023), the core of MoleSG lies in the specially designed masking strategy and the unified network capable of handling data of different modalities. We will introduce the details of our masking strategy in section 3.2, followed by a comprehensive presentation of our network architectures in section 3.3-3.5. Figure 3: Non-overlapping masking strategy. (a) Non-overlapping masking strategy: Masks in the SMILES sequence and the graph for the same molecule do not overlap. (b) Non-overlapping masking strategy pipeline: First, we establish a correspondence between atom index in both modalities. Then, random masking is applied to the graph, followed by mapping the masked atoms from the graph to the SMILES sequence. Finally, random masking on the SMILES sequence is implemented on the remaining unmasked atoms of the graph. 3.2 NON-OVERLAPPING MASKING STRATEGY The non-overlapping masking strategy we propose is illustrated in Figure 3, which can be divided into two steps, first performing atom index alignment between the two modalities, and then performing non-overlapping masking. Step 1: Atom index alignment. Initially, for a given input molecule, we define its molecular graph as $G = (V, E)$, where $V$ and $E$ represent the sets of atoms and edges, respectively. Following the method of CoMPT (Chen et al., 2021), we precompute the node features $V_{feature} = \{v_{f0}, v_{f1}, ..., v_{f(m-1)}\}$, where $m$ is the number of atoms and then represent the SMILES sequence as the set of a series of tokens $S_1 = \{s_0, s_1, ..., s_{n-1}\}$, where $n$ is the total number of tokens. The SMILES tokens can be categorized into three classes: (1) Atoms, including single-character atoms like C and N, as well as multi-character atoms like Ca and Au, and ions like [Cl-] and [Fe+3]; (2) Chemical bonds, represented by symbols like ‘#’ and ‘=’; (3) Other symbols, such as numbers ‘1’ and ‘2’ indicating the positions of atoms in a ring and parentheses ‘(’ and ‘)’ denoting containing side chains. Given that single bonds are often omitted in SMILES, achieving a one-to-one correspondence between two modalities for chemical bonds is not practical. Therefore, in this paper, we focus on aligning the atom index. Therefore, we gather the tokens representing the atoms and assign indexes to them to establish a consistent correspondence between atoms in graph $G_1$ and those in filtered SMILES tokens $S_2$. Step 2: Masking strategy. We randomly mask atomic features on the graph $M_G : G_1 \mapsto G_2$, where $G_2$ is the masked graph, and the set of masked atom indexes on $G_2$ is defined as $I_G$. Following that, we randomly mask atomic tokens on the SMILES sequence $M_S : S_2 \mapsto S_3$, where $S_3$ is the preliminary masked SMILES sequence, and the set of masked atom indexes on $S_3$ is denoted as $I_S$. To encourage better interaction between the two modalities, we set the overlap ratio between masked atoms in both modalities to be 0, forcing one modality to learn the “correct answer” from the other modality. Specifically, based on the one-to-one correspondence of atom index, we localize the positions of masked atoms onto the SMILES sequence. Through operation $P : I_S - I_G \cap I_S, S_3 \mapsto S_4$, where $S_4$ is the final masked SMILES sequence, we avoid masking atoms on the SMILES sequence that are already masked on the graph. 3.3 Encoder To facilitate the interaction of fine-grained features across different modalities, we use two independent encoders to convert the data of two entirely different modalities into embeddings of the same dimensions for being further processed by transformer. For the SMILES sequence, we adopt the method used in Roberta (Liu et al., 2019b). We first convert the masked SMILES sequence into a sequence of token ids following ChemBERTa (Chithramanda et al., 2020), and we expand its vocabulary by conducting a comprehensive analysis of all tokens in our dataset, as detailed in Appendix E. Then, we calculate their corresponding embeddings $F_S \in \mathbb{R}^{N_S \times d}$ by a vanilla transformer, where $N_S$ represents the number of SMILES tokens, and $d$ is the feature dimension. For the graph, we precompute the same node features and edge features as CoMPT (Chen et al., 2021) does. After that, a portion of node features are randomly masked, and then we feed them into the graph encoder. Our graph encoder is the same as that used in CoMPT (Chen et al., 2021), which consists of many message-passing layers. After repeating message-passing in the graph encoder, we finally obtain token embeddings $F_G \in \mathbb{R}^{N_G \times d}$ for nodes, where $N_G$ is the number of atoms, and $d$ is the feature dimension. 3.4 Unified backbone Given that two modalities are treated as embeddings of the same dimension, we can easily use a simple unified network to learn fine-grained features in both modalities. We first add trainable parameters to $F_S \in \mathbb{R}^{N_S \times d}$ and $F_G \in \mathbb{R}^{N_G \times d}$ and then concatenate them. The concatenated embeddings $F_{S,G} \in \mathbb{R}^{(N_S+N_G) \times d}$ are then fed into the backbone. Here, we use the transformer encoder employed in Roberta (Liu et al., 2019b) as the backbone network, and its multi-head self-attention mechanism can facilitate information interaction between token embeddings both within the same modalities and across different modalities. 3.5 Decoder After feature extraction in the backbone, we split the output features $F'_{S,G} \in \mathbb{R}^{(N_S+N_G) \times d}$ into features $F'_S \in \mathbb{R}^{N_S \times d}$ for SMILES and features $F'_G \in \mathbb{R}^{N_G \times d}$ for graph. $F'_S$ and $F'_G$ are features for individual modality-specific mask reconstruction tasks. Specifically, $F'_S$ is fed into LMhead in Roberta (Liu et al., 2019b) to predict the masked token ids, while $F'_C$ is inputted into a lightweight network GIN (Xu et al., 2018) after re-masking (Hou et al., 2022) to reconstruct the masked node features. We calculate the entropy loss $L_{EN}$ (Liu et al., 2019b) in SMILES reconstruction and the SCE loss $L_{SCE}$ (Hou et al., 2022) in graph reconstruction. Finally, the overall loss for the entire task is as follows: $L_{Total} = L_{EN} + L_{SCE}$. ### 3.6 Fine-tuning We conduct fine-tuning on 14 downstream tasks of predicting molecular properties. Since previous works only utilize a single modality in the downstream tasks, we also take a single modality as input to achieve a fair comparison. Moreover, as single modality input has an inconsistent distribution with two modalities, the backbone that takes two modalities as input during pre-training may suffer from performance decrease during fine-tuning. Therefore, we also discard the backbone during fine-tuning and inference. In other words, we only reserve a single special encoder during fine-tuning and inference. Our following experiment in section 4.3.3 also verifies it. ## 4 Experiments ### 4.1 Implementation Details Datasets setup: During the pre-training stage, we sample 250,000 unlabeled molecules from ZINC15 (Sterling & Irwin, 2015), which is a comprehensive collection of chemical compounds for drug discovery and computational chemistry research. During the fine-tuning stage, we utilize 14 benchmark datasets from MoleculeNet (Wu et al., 2018), covering molecular data from various domains, including pharmaceuticals, biology, chemistry, and physics. These downstream datasets include 678 binary classification tasks and 19 regression tasks. For more detailed information about benchmark datasets, please refer to Appendix A. We partition each benchmark dataset into the train, validation, and test sets in an 8:1:1 ratio. For all datasets except QM9, we employ scaffold splitting, reporting the mean and standard deviation of results from three random seeds for each benchmark. Scaffold splitting is a more challenging and realistic data partitioning method (Ramsundar et al., 2019). For the QM9 dataset, we follow the approach used in most prior work (Wang et al., 2022; Fang et al., 2023) for random splitting. Pre-training: We train MoleSG for 90k iterations using the AdamW optimizer with a base learning ratio of 1e-3. We set the masking ratio for graph at 25% and for SMILES at 15%. The details of the mask ratio setting experiments for the two modes are shown in Appendix C. Downstream: We set a maximum of 150 training epochs, with early stopping applied when the validation set’s best value is not improved for more than 20 epochs. We use the AdamW optimizer with a base learning rate of 1e-3 and a warmup factor of 0.1 for the first 30 epochs. Competitors: We compare MoleSG with both supervised (training from scratch) baselines and pre-trained baselines. Supervised methods include MPNN (Gilmer et al., 2017), DMPNN (Yang et al., 2019), CMPNN (Song et al., 2020), and CoMPT (Chen et al., 2021). Pre-training methods include N-gram (Liu et al., 2019a), PretrainGNN (Hu et al., 2019), MGSSL (Zhang et al., 2021), GROVER (Rong et al., 2020), GraphMVP (Liu et al., 2021), MolCLR (Wang et al., 2022), GEM (Fang et al., 2022), DVMP (Zhu et al., 2021), KANO (Fang et al., 2023), and Uni-mol (Zhou et al., 2023). The specific configurations for these competitors can be found in Appendix B. Additionally, for a fair comparison, we implement new MolCLR and DVMP by replacing the original encoders in them with the same networks we use, which are denoted as MolCLR$_{CoMPT}$ and DVMP$_{MoleSG}$. We also utilize our non-overlapping masking strategy in DVMP$_{MoleSG}$. ### 4.2 Results of Molecular Property Prediction Table 1 presents the test results in classification tasks. It can be observed that MoleSG consistently outperforms other methods across all eight datasets, demonstrating its effectiveness. It’s worth noticing that though the Toxcast dataset benchmark with 617 binary classification tasks is challenging, our method still performs better than the current SOTA method KANO. Complementary information Table 1: Performance of different models on eight classification benchmarks in physiology and biophysics. The mean and standard deviation of ROC-AUC (%) from three independent runs are reported. (Higher values indicate better performance.) \| Category \| Physiology \| Biophysics \| \|----------\|------------\|------------\| \| Dataset \| BBBP \| Tox21 \| ToxCast \| SIDER \| ClinTox \| BACE \| MUV \| HIV \| \| Molecules\| 2039 \| 7831 \| 8575 \| 1427 \| 1478 \| 1513 \| 93807 \| 41127 \| \| Tasks \| 1 \| 12 \| 617 \| 27 \| 2 \| 1 \| 17 \| 1 \| \| MPNN \| 91.3±4.1 \| 80.8±2.4 \| 69.1±3.0 \| 59.5±3.0 \| 87.9±5.4 \| 81.5±1.0 \| 75.7±1.3 \| 77.0±1.4 \| \| DMPNN \| 91.9±3.0 \| 75.9±0.7 \| 63.7±0.2 \| 57.0±0.7 \| 90.6±0.6 \| 85.2±0.6 \| 78.6±1.4 \| 77.1±0.5 \| \| CMPNN \| 92.7±1.7 \| 80.1±1.6 \| 70.8±1.3 \| 61.6±0.3 \| 89.8±0.8 \| 86.7±0.2 \| 79.0±2.0 \| 78.2±2.2 \| \| CoMPT \| 96.1±0.4 \| 84.5±0.7 \| 72.2±0.8 \| 66.1±0.9 \| 97.3±2.5 \| 94.1±3.6 \| 82.6±1.6 \| 86.4±1.2 \| \| N-Gram \| 91.2±0.3 \| 76.9±2.7 \| - \| 63.2±0.5 \| 87.5±2.7 \| 79.1±1.3 \| 76.9±0.7 \| 78.7±0.4 \| \| PretrainGNN \| 70.8±1.5 \| 78.7±0.4 \| 65.7±0.6 \| 62.7±0.8 \| 72.6±1.5 \| 84.5±0.7 \| 81.3±2.1 \| 79.9±0.7 \| \| MGSSL \| 70.5±1.1 \| 76.4±0.4 \| 64.1±0.7 \| 61.8±0.8 \| 80.7±2.1 \| 79.7±0.8 \| 78.7±1.5 \| 79.5±1.1 \| \| GEM \| 88.8±0.4 \| 78.1±0.4 \| 68.6±0.2 \| 63.2±1.5 \| 90.3±0.7 \| 87.9±1.1 \| 75.3±1.5 \| 81.3±0.3 \| \| GROVER \| 86.8±2.2 \| 80.3±2.0 \| 56.8±3.4 \| 61.2±2.5 \| 70.3±13.7 \| 82.4±3.6 \| 67.3±1.8 \| 68.2±1.1 \| \| GraphMVP \| 72.4±1.6 \| 75.9±0.5 \| 63.1±0.4 \| 63.9±1.2 \| 79.1±2.8 \| 81.2±0.9 \| 77.7±0.6 \| 77.0±1.2 \| \| Uni-mol \| 72.9±0.6 \| 79.6±0.5 \| 69.6±0.1 \| 65.9±1.3 \| 91.9±1.8 \| 85.7±0.2 \| 82.1±1.3 \| 80.8±0.3 \| \| DVMP \| 77.8±0.3 \| 79.1±0.4 \| - \| 69.8±0.6 \| 95.6±0.7 \| 89.4±0.8 \| - \| 81.4±0.4 \| \| DVMPMoleSG \| 80.9±2.1 \| 84.4±1.2 \| 73.3±0.9 \| 66.9±1.2 \| 98.4±2.0 \| 93.5±2.8 \| 80.9±2.1 \| 87.6±1.8 \| \| MolCLR \| 73.3±1.0 \| 74.1±5.3 \| 65.9±2.1 \| 61.2±3.6 \| 89.8±2.7 \| 82.8±0.7 \| 78.9±2.3 \| 77.4±0.6 \| \| MolCLRCoMPT \| 97.2±0.2 \| 82.4±1.8 \| 72.7±0.5 \| 57.1±8.7 \| 77.0±14.5 \| 85.5±0.9 \| 75.8±15.0 \| 81.8±2.2 \| \| KANO \| 96.0±1.6 \| 83.7±1.3 \| 73.2±1.6 \| 65.2±0.8 \| 94.4±0.3 \| 93.1±2.1 \| 83.7±2.3 \| 85.1±2.2 \| \| MoleSG \| 97.9±0.3 \| 85.0±1.2 \| 74.2±0.5 \| 70.0±0.2 \| 99.1±0.9 \| 95.1±2.1 \| 85.1±0.8 \| 87.7±1.9 \| Table 2: Performance of different models on six regression benchmarks in physical chemistry and quantum mechanics. The mean and standard deviation of root mean square error (RMSE) (for ESOL, FreeSolv, and Lipophilicity) or mean absolute error (MAE) (for QM7, QM8, and QM9) from three independent runs are reported. (Lower values indicate better performance.) \| Category \| Physical chemistry \| Quantum mechanics \| \|----------\|--------------------\|-------------------\| \| Dataset \| ESOL \| FreeSolv \| Lipophilicity \| QM7 \| QM8 \| QM9 \| \| Molecules\| 1128 \| 642 \| 4200 \| 6830 \| 21786 \| 133885 \| \| Tasks \| 1 \| 1 \| 1 \| 1 \| 12 \| 3 \| \| MPNN \| 1.167±0.043 \| 1.621±0.952 \| 0.672±0.051 \| 111.4±0.9 \| 0.0148±0.001 \| 0.00522±0.00003 \| \| DMPNN \| 1.050±0.008 \| 1.673±0.082 \| 0.683±0.016 \| 103.5±8.6 \| 0.0156±0.001 \| 0.00514±0.00001 \| \| CMPNN \| 0.798±0.112 \| 1.570±0.442 \| 0.614±0.029 \| 75.1±3.1 \| 0.0153±0.002 \| 0.00405±0.00002 \| \| CoMPT \| 0.643±0.051 \| 0.970±0.207 \| 0.572±0.058 \| 32.7±7.4 \| 0.0120±0.001 \| 0.00353±0.00067 \| \| N-Gram \| 1.100±0.030 \| 2.510±0.191 \| 0.880±0.121 \| 125.6±1.5 \| 0.0320±0.003 \| 0.00964±0.00031 \| \| PretrainGNN \| 1.100±0.006 \| 2.764±0.002 \| 0.739±0.003 \| 113.2±0.6 \| 0.0215±0.001 \| 0.00992±0.00004 \| \| GEM \| 0.813±0.028 \| 1.748±0.114 \| 0.674±0.022 \| 60.0±2.7 \| 0.0163±0.001 \| 0.00562±0.00007 \| \| GROVER \| 1.423±0.288 \| 2.947±0.615 \| 0.823±0.010 \| 91.3±1.9 \| 0.0182±0.001 \| 0.00719±0.00208 \| \| Uni-mol \| 0.788±0.029 \| 1.480±0.048 \| 0.603±0.010 \| 41.8±0.2 \| 0.0156±0.000 \| - \| \| DVMP \| 0.817±0.024 \| 1.952±0.061 \| 0.653±0.002 \| 74.4±1.2 \| 0.0171±0.004 \| - \| \| DVMPMoleSG \| 0.669±0.114 \| 0.942±0.110 \| 0.594±0.018 \| 30.2±3.0 \| 0.0123±0.001 \| 0.00323±0.00006 \| \| MolCLR \| 1.113±0.023 \| 2.301±0.247 \| 0.789±0.009 \| 90.9±1.7 \| 0.0185±0.013 \| 0.00480±0.00003 \| \| MolCLRCoMPT \| 0.849±0.062 \| 1.135±0.163 \| 0.657±0.012 \| 32.7±2.8 \| 0.0141±0.001 \| 0.00350±0.00000 \| \| KANO \| 0.670±0.019 \| 1.142±0.258 \| 0.566±0.007 \| 56.4±2.8 \| 0.0123±0.000 \| 0.00320±0.00001 \| \| MoleSG \| 0.599±0.067 \| 0.932±0.131 \| 0.545±0.014 \| 29.6±2.9 \| 0.0117±0.001 \| 0.00313±0.00006 \| Table 3: Comparison of our approach with two single-modality pre-training approaches on classification tasks. The mean and standard deviation of ROC-AUC (%) over three independent runs are reported. (Higher values indicate better performance.) \| \| BBBP \| Tox21 \| ToxCast \| SIDER \| Clintox \| BACE \| MUV \| HIV \| \|----------------\|--------\|--------\|---------\|--------\|---------\|--------\|--------\|--------\| \| SMILES scratch \| 63.6±4.3 \| 75.5±0.5 \| 64.2±2.5 \| 54.0±2.4 \| 88.1±6.3 \| 79.2±6.6 \| 63.6±4.3 \| 72.7±3.5 \| \| SMILES pre-train \| 61.5±4.9 \| 77.6±2.5 \| 66.8±0.9 \| 55.0±3.1 \| 93.3±2.8 \| 83.8±0.9 \| 61.5±4.9 \| 75.1±2.5 \| \| Ours SMILES \| 65.3±3.1 \| 77.9±2.5 \| 67.0±0.9 \| 59.6±3.8 \| 94.3±2.0 \| 85.3±1.1 \| 65.3±3.1 \| 77.3±0.7 \| \| \| Graph scratch \| Graph pre-train \| Ours graph \| \|----------------\|---------------\|-----------------\|------------\| \| SMILES scratch \| 96.1±0.4 \| 84.5±0.7 \| 72.2±0.8 \| \| SMILES pre-train \| 96.8±1.8 \| 84.2±0.1 \| 72.6±1.0 \| \| Ours SMILES \| 97.9±0.3 \| 85.0±1.2 \| 74.2±0.5 \| Table 4: Comparison of our approach with two single-modality pre-training approaches on regression tasks. The mean and standard deviation of RMSE or MAE over three independent runs are reported. (Lower values indicate better performance.) \| \| ESOL \| Freesolv \| Lipophilicity \| QM7 \| QM8 \| QM9 \| \|----------------\|------------\|------------\|---------------\|---------\|---------\|---------\| \| SMILES scratch \| 0.946±0.226 \| 2.581±0.286 \| 1.028±0.030 \| 160.2±6.8 \| 0.0146±0.001 \| 0.01017±0.00045 \| \| SMILES pre-train \| 1.030±0.336 \| 1.942±0.450 \| 1.034±0.015 \| 159.3±5.7 \| 0.0141±0.001 \| 0.01080±0.00010 \| \| Ours SMILES \| 0.873±0.172 \| 1.889±0.590 \| 0.964±0.036 \| 155.7±3.9 \| 0.0139±0.001 \| 0.00973±0.00059 \| \| \| Graph scratch \| Graph pre-train \| Ours graph \| \|----------------\|---------------\|-----------------\|------------\| \| SMILES scratch \| 0.643±0.051 \| 0.970±0.207 \| 0.572±0.058 \| \| SMILES pre-train \| 0.635±0.104 \| 0.939±0.225 \| 0.585±0.031 \| \| Ours SMILES \| 0.599±0.067 \| 0.932±0.131 \| 0.545±0.014 \| of the two modalities in MoleSG contributes to outstanding results, surpassing methods injecting additional 3D information. Table 2 shows the test results in regression tasks. We can observe that MoleSG achieves the best scores among both supervised and self-supervised pre-training models, with a relative improvement of 14.4% over KANO across all six regression tasks. MoleSG greatly benefits tasks with limited label information, achieving a 18.4% improvement over KANO on the small dataset FreeSolv, which contains only 642 labeled molecules. Moreover, it is worth noting that our method still outperforms MolCLRCoMPT, which is a version of the typical contrastive learning method MolCLR with the same encoder as ours, verifying the superiority of our method. We also compare with another contrastive learning competitor DVMPMoleSG, which utilizes the same encoders as ours. In addition, both MolCLRCoMPT and DVMPMoleSG outperform their original counterpart MolCLR and DVMP in most tasks, demonstrating the effectiveness of the corresponding strategies proposed in this paper. ### 4.3 Ablation Experiments #### 4.3.1 Single-modality vs. Multi-modality To further reveal the superiority of our method, we compare our multi-modality pre-training with single-modality pre-training. The results are shown in Table 3 and Table 4. Our method successfully achieves the best performance on all downstream tasks. Moreover, it is worth noting that single modality pre-training may cause performance degradation. However, by fully leveraging the complementary information among different modalities, our method can improve performance on all downstream tasks, showing more potential for practical applications. We present visualization results of our method’s feature extraction capability in Appendix D. #### 4.3.2 Overlap vs. Non-overlap To validate whether our non-overlapping masking strategy benefits pre-training, we conduct experiments on different overlap ratios on all downstream tasks. We define overlap ratio as a metric measuring the proportion of jointly masked atoms in both modality inputs. We conduct experiments at overlap ratios at 0%, 25%, 50%, 75%, and 100% across all benchmarks, where our non-overlapping masking strategy is equivalent to setting the overlap ratio to 0. The experimental results shown in Figure 4 indicate that the performance on downstream tasks is the best when the overlap ratio is 0. ### 4.3.3 WITH vs. WITHOUT BACKBONE As analyzed above, fine-tuning both the encoder and backbone may cause suboptimal performance due to the inconsistent distributions. Therefore, we conduct an experiment to validate it. Specifically, section 4.3.1 has shown that the graph encoder has better performance than the SMILES encoder. Therefore, we only consider two combinations in this section. The former is fine-tuning a single graph encoder, and the other is fine-tuning both the graph encoder and the backbone. We perform experiments on all benchmarks, and the results are shown in Table 5 and Table 6. The results show that using only the graph encoder achieves higher performance in all tasks. #### Table 5: Comparison of results on classification tasks with and without the backbone network. The mean and standard deviation of ROC-AUC (%) from three independent runs are reported. \| \| BBBP \| Tox21 \| ToxCast \| SIDER \| ClinTox \| BACE \| MUV \| HIV \| \|------------------\|----------\|----------\|----------\|----------\|----------\|----------\|----------\|----------\| \| Graph encoder+backbone \| 97.23±0.6 \| 84.8±1.8 \| 73.6±0.9 \| 65.6±0.4 \| 98.8±0.6 \| 89.7±5.2 \| 81.9±1.9 \| 85.8±1.4 \| \| Graph encoder \| 97.9±0.3 \| 85.0±1.2 \| 74.2±0.5 \| 70.0±0.2 \| 99.1±0.9 \| 95.1±2.1 \| 85.1±0.8 \| 87.7±1.9 \| #### Table 6: Comparison of results on regression tasks with and without the backbone network. The mean and standard deviation of RMSE (or MAE) from three independent runs are reported. \| \| ESOL \| FreeSolv \| Lipophilicity \| QM7 \| QM8 \| QM9 \| \|------------------\|----------\|----------\|--------------\|----------\|----------\|----------\| \| Graph encoder+backbone \| 0.661±0.011 \| 0.988±0.250 \| 0.560±0.017 \| 31.9±3.8 \| 0.0119±0.001 \| 0.00353±0.00015 \| \| Graph encoder \| 0.599±0.067 \| 0.932±0.131 \| 0.545±0.014 \| 29.6±2.9 \| 0.0117±0.001 \| 0.00313±0.00006 \| ## 5 CONCLUSION In this study, we address the challenges of learning fine-grained information from two complementary modalities: SMILES and graph. To better capture rich molecular features from the interaction between these two modalities, we design a simple and efficient multi-modality pre-training framework called MoleSG, which utilizes a unified feature processing network to fuse both modalities. 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EWTFMkTdkT	In the performance comparison figure 4, is the X axis the time axis? The results surprised me because the prediction from the first 3 models is bad starting from the beginning of the prediction, which means the error mostly comes from decoder reconstruction rather than dynamics prediction. Especially for ODE2VAE, VAE should be able to reconstruct the pendulum image fairly well.	INVARIANCE-BASED LEARNING OF LATENT DYNAMICS Kai Lagemann∗ Statistics and Machine Learning, German Center for Neurodegenerative Diseases (DZNE), Bonn, Germany kai.lagemann@dzne.de Christian Lagemann∗ Department of Mechanical Engineering, University of Washington, Seattle, USA Sach Mukherjee Statistics and Machine Learning, German Center for Neurodegenerative Diseases (DZNE), Bonn, Germany MRC Biostatistics Unit, University of Cambridge, Cambridge, UK sach.mukherjee@dzne.de ABSTRACT We propose a new model class aimed at predicting dynamical trajectories from high-dimensional empirical data. This is done by combining variational autoencoders and (spatio-)temporal transformers within a framework designed to enforce certain scientifically-motivated invariances. The models allow inference of system behavior at any continuous time and generalization well beyond the data distributions seen during training. Furthermore, the models do not require an explicit neural ODE formulation, making them efficient and highly scalable in practice. We study behavior through simple theoretical analyses and extensive empirical experiments. The latter investigate the ability to predict the trajectories of complicated systems based on finite data and show that the proposed approaches can outperform existing neural-dynamical models. We study also more general inductive bias in the context of transfer to data obtained under entirely novel system interventions. Overall, our results provide a new framework for efficiently learning complicated dynamics in a data-driven manner, with potential applications in a wide range of fields including physics, biology, and engineering. 1 INTRODUCTION Dynamical models are central to our ability to understand and predict natural and engineered systems. A key question in studying dynamical systems is predicting future behavior. Real-world systems often show time-varying behavior that is much too complex for straightforward statistical forecasting or extrapolation approaches. This is due to the fact that the temporal behavior, while potentially explained by an underlying dynamical model, can show strong, possibly abrupt changes in the observation/data space, precluding effective modeling via traditional curve-fitting or extrapolation. Furthermore, different instances or realizations of a single scientific/engineering system (e.g. with different initial conditions or constants) can show large differences in terms of data distributions, hence going beyond standard in-distribution assumptions of traditional data-fitting approaches. Against this background, in recent years a wide range of sophisticated dynamical machine learning approaches have been proposed, including in particular neural ordinary differential equations (Chen et al., 2018) and a wider class of related models (see for example Zhi et al., 2022; Finlay et al., 2020; Duong and Atanasov, 2021; Choi et al., 2022; Chen et al., 2021; Kim et al., 2021b). Broadly speaking, these models go beyond simple curve-fitting/extrapolation schemes by leveraging suitable inductive biases to allow learning of latent dynamical models. There has been rapid progress in this area but key challenges remain for complicated real-world systems, due to multiple factors, including data limitations, generalization to unseen settings, irregular time sampling and issues relating to long-horizon trajectories (Iakovlev et al., 2023). ∗These authors contributed equally to this work. Motivated by these challenges, we propose a new framework, called “Latent Dynamics via Invariant Decomposition” or LaDID, for learning latent dynamics from empirical data. LaDID leverages certain scientifically-motivated invariances to permit efficient learning and effective generalization. In numerous real-world dynamical systems, longitudinal trajectories may exhibit significant variability, e.g. due to differences in initial conditions or model constants. Each temporal trajectory, which we refer to as a “realization”, represents a particular manifestation of the system’s dynamics under certain conditions. A key notion underpinning LaDID is that, even when temporal trajectories from a class of scientific systems appear diverse, they can still be effectively explained by an appropriate, in a sense “universal”, model; such a model is therefore realization-invariant. To facilitate broad generalization, LaDID introduces factors specific to each realization as inputs to its universal model. These factors are hence realization-specific and play the role of (implicitly) encoding aspects such as the initial states of the system or specific model constants. A transformer-based architecture is used to learn all model aspects from data, including both realization-specific (RS) and realization-invariant (RI) information. At inference-time LaDID can output predictions for any continuous time. Due to the universal nature of the RI model, LaDID can effectively handle substantial variations in system behavior and data distributions (e.g. due to changes in initial conditions or system constants). We empirically validate LaDID on various spatio-temporal systems with dynamics on regular and irregular time grids governed by ordinary or partial differential equations. The LaDID architecture is fast and easy to train and, as we show, substantially outperforms existing neural-dynamical models over a range of challenging tasks. Thus, our main contributions are: • We present a novel framework, and associated transformer-based network, for the separation of realization-specific information and (realization-invariant) latent dynamical systems. • We systematically study performance on short- and longer-horizon prediction of a wide range of complex temporal and spatio-temporal problems, comparing against a range of state-of-the-art neural-dynamical baselines. • We study the challenging case of transfer to data obtained under entirely novel system interventions via a few-shot learning (FSL) approach. 2 RELATED WORK Flexible neural models have been exploited to learn dynamical models, with connections drawn between deep architectures and numerical solvers for ODEs, PDEs and SDEs (Chen et al., 2018; Weinan, 2017; Lu et al., 2018; Ruthotto and Haber, 2020; Haber and Ruthotto, 2017; Richter-Powell et al., 2022). Algorithms rooted in neural differential equations (NODEs) have been shown to offer benefits relative to standard recurrent neural networks (RNNs) and their variants. However, since NODEs directly relate to the problem formulation of standard ODEs, they inherit some associated limitations. Specifically, the temporal dynamics only depend on the current state but not on the history putting a limit on the complexity of the trajectories that NODEs can model (Holt et al., 2022). Improvements have been proposed that augment the latent state space to broaden the range of dynamical systems that can be learned (Dupont et al., 2019) while Rubanova et al. (2019) suggested a combination with an autoregressive RNN updated at irregularly sampled time points. Complementary work has proposed neural controlled differential equations, a mechanism to adjust the trajectory based on subsequent observation (Kidger et al., 2020; Morrill et al., 2021; Massaroli et al., 2021) transferred the concept of multiple shooting to solve differential equations to the conceptual space of NODEs and Takovlev et al. (2023) extended this concept to sparse Bayesian multiple shooting, with both works evaluating latent NODEs. However, for certain types of dynamics numerical instability poses challenges for NODEs (and their variants) (Li et al., 2020). This is due to the fact that NODEs rely on numerical ODE solvers to predict the latent trajectory (forward pass) and this becomes unstable with longer time horizons (Takovlev et al., 2023). In contrast, by exploiting RS and RI invariances our model eschews explicit neural ODEs altogether, providing an arguably simpler and faster transformer-based scheme that can be trained in a straightforward fashion, as we detail below. The idea of leveraging invariances is a core notion in scientific modeling and is seen throughout the natural sciences at a conceptual and practical level. For instance, in the field of AI, it has been utilized in the context of video frame prediction as demonstrated by various studies (van der Wilk et al., 2018; Franceschi et al., 2020; Kabra et al., 2021). LaDID differs from these approaches because it uses invariances to model a kind of generalized initial condition (motivated by scientific uses of dynamical models; see below) and because it learns continuous latent trajectories (as opposed to an autoregressive model), including in the irregular time-sampling case. See Section A of the Appendix for further details on related work. 3 PROBLEM STATEMENT AND CONCEPTUAL OUTLINE We start with a high-level problem statement and outline the motivating concepts behind LaDID, deferring a detailed description of the architecture itself to subsequent Sections. We focus on settings in which we have (finite) observations of a system of interest at time points $ t \in T $ (potentially irregularly spaced). We do not require any specific prior information on the underlying model; rather our approach is data-driven, informed by certain very general invariances as detailed below. For an instance/realization $ r $ of a dynamical system of interest let $ X_r \in \mathbb{R}^{T \times C \times H \times W} $ denote a high-dimensional trajectory in the observation space; $ T $ denotes the number of time steps (possibly irregular), and $ C, H, $ and $ W $ respectively the number of channels, frame height and frame width (in empirical examples we focus on image-like data inputs) of the observations. Let $ X = \{X_r\}_{r \in R} $ denote the collection of available training trajectories; the notation emphasizes the possibility that available data spans multiple instances/realizations. Given these initial data, LaDID seeks to predict future observations $ x_t^r $ for any continuous time $ t $ and for any realization $ r $. From invariances to a simple learning framework. We start by studying a basic, conceptual set-up that sheds light on how our assumptions lead to a simple, but very general, learning framework. Intuitively, we demonstrate that while we cannot guarantee recovery of the true underlying model parameters, under mild invariance assumptions, there exists a function capable of reconstructing the true observations, even when dealing with potentially highly heterogeneous parameters and data. Importantly, we do not make any prior assumptions about the nature of these potentially complex and nonlinear functions. Instead, our learning framework simultaneously uncovers and refines these functions in a data-driven, end-to-end manner, as elaborated in Section 4. Consider an entirely general system $ f $ in which some aspects are realization-specific (RS) while others are realization-invariant (RI); the latter model aspects are general to all instances/realizations of the model, while the former are not. We assume also that the RS aspects are the same for all times; i.e. these are time-invariant. Our model is not specific to physical ODE-like models, but rather generalizes invariances to permit flexible learning in a wide range of settings. To fix ideas, it is nevertheless useful to consider the specific example of a physical model class described by ODEs. For this latter setting, the model itself would be RI, while the initial conditions or system constants could be thought of as RS. As a result, the same model class can describe a wide range of actual systems which share an underlying scientific basis while differing (perhaps strongly) in details. Let $ x^r_t = f(t; \Theta_r) $ denote the fully general model. Here, $ \Theta_r $ is the complete parameter set needed to specify the time-evolution, including both RS and RI parts. To make the separation clear, we write the two parts separately as $ x^r_t = f(t; \theta_r, \theta) $, where $ \theta_r, \theta $ are respectively the RS and RI parameters and $ r $ indexes the realization. Suppose $ \hat{\theta}_r $ is a candidate encoding of the RS information. We now assume that the encoding, while possibly incorrect (i.e. such that $ \hat{\theta}_r \neq \theta_r $) satisfies the property $ \exists m, \exists \theta_m : \theta_r = m(\theta_r; \theta, \theta_m) $, where $ m $ is a function that “corrects” $ \hat{\theta}_r $ to give the correct RS parameter. This essentially demands that while the encoding $ \hat{\theta}_r $ might be very different from the true value (and may even diverge from it in a complicated way that depends on unknown system parameters), there exists an RI transformation that recovers the true RS parameter from it, and in this sense the encoding contains all RS information. We call this the sufficient encoding assumption (SEA). In the context of dynamics involving latent variables $ z \in \mathbb{R}^q $ again consider a model with RS and RI parts but at the level of the latents, i.e. $ z^r_t = f(t; \theta_r, \theta) $. Assume the observables are given by (an unknown) function $ g $ of the hidden state $ z $. Then, as shown in the Appendix B assumptions similar in spirit to SEA above imply existence of a mapping function that allows arbitrary queries to in principle be predicted via a straightforward learning scheme. In brief, these assumptions require that we have access to an approximation $ \hat{g} $ of the observation process that while possibly very far away from the true function, can nonetheless be “corrected” in a specific way (see Appendix B for details). We emphasize that as for SEA, it is not required that we have a good approximation in the usual sense of being close to the true function $ g $, only that there exists a correction of a certain form. Importantly, at no point do we actually require prior knowledge of the underlying dynamical system, its true latent variables nor any system constants or initial conditions; rather, the assumptions imply existence of mapping functions that can be learned from data, even when the underlying model is entirely unknown at the outset. Analogy to traditional ODE formulations. To facilitate a more intuitive understanding of our framework, we discuss further analogies to standard ODE problems. Typical ODE solvers comprise a formulated ODE function and some initial values (IVs) which are evolved over time using well-known integration methods, e.g., Euler, Runge-Kutta, or DOPRI schemes. From a high-level perspective, the IVs relate to our RS representation while the ODE functions and its corresponding integration approach is our RI part. However, please note that this analogy only holds on a superficial level due to two fundamental differences: First, LaDID relies only on observations of a specific system, for which underlying state variables required to apply standard ODE solvers are not known (and in fact not observed). Hence, our RS representation benefits from access to a collection of high-dimensional observations. Second, our RI function is continuous in time and therefore unites temporal integration and dynamics function on an abstract level. To do so, we condition our RI model on specific RS representations and only query time points from it to obtain a discretized latent trajectory. 4 METHODS Based on these initial arguments, we now put forward a specific architecture to allow learning in practice. At a high level, the architecture implements the general mapping approach outlined above (further details in the Appendix), learning, in a data-driven manner, both RS and RI model components and putting these together to allow prediction at any continuous time in a query realization. Implementation details and a full ELBO derivation can be found in Section C–F of the Appendix. 4.1 MODEL, INFERENCE AND PREDICTION Model. The LaDID architecture is composed of three main components: the encoder $ f_{\phi_{enc}} $, the invariant dynamical system $ f_{\phi_{dyn}} $, and the decoder $ f_{\phi_{dec}} $, respectively governed by parameters $ \phi_{enc}, \phi_{dyn}, $ and $ \phi_{dec} $. The encoder is a collection of three NNs: a CNN processing spatial information in... the observation space, a transformer utilizing temporal attention and a learnable mapping function. Since we want to predict future observations based on a few observations, we only use the first $K$ datapoints in time and process these in a shared convolutional encoder (green trapezoid in Figure 1(ii)). We employ a shallow CNN that compresses the input to $1/16$ of the initial input size using four ReLU activated and batch-normalized convolutional layers. The resulting tensors are then flattened and mapped linearly to a single vector. Next, we use a transformer on the $K$ output vectors of the convolutional encoder, applying temporal attention to reweigh vectors. We tested two approaches (Bulat et al., 2021; Iakovlev et al., 2023) with comparable performance which are discussed in the Appendix in more detail. For each of the $k \in K$ time aware representations $\rho^{TA}_k$, we sample a latent embedding using the reparameterization trick, i.e. $l_{emb}^k \sim N(f_\mu(\rho^{TA}_k), f_\sigma(\rho^{TA}_k))$. The final trajectory representation $\psi^r$ is the output of an aggregation over all $K$ tokens. In our implementation, we choose a simple yet effective mean-aggregation which can be changed based on the task at hand. The second important part of our proposed framework is the dynamical model $f_{\phi_{dyn}}$. We utilized a three layer MLP which can also be interchanged by other functions. To obtain a latent trajectory, we condition the latent dynamical model on our end-to-end learned trajectory representation $\psi^r$ and roll-out the latent trajectory $z$ based on the queried time points $t_q$ represented through a time encoding which we choose as a set of different sine and cosine waves with different wave length. Finally, we map all data points of our latent trajectory back to the original observation space. Our decoder module $f_{\phi_{dec}}$ is kept very simple consisting of four deconvolutional layers. The key novelty of our approach lies in the unique structure of the latent space mimicking the interplay of realization-specific information and a realization-invariant dynamical model similar to the frame of differential equations. However, we can significantly reduce computational costs as we are not forced to solve explicitly any differential equation since we rely on an effective end-to-end learning scheme. Generative model, inference and optimization. We now turn the descriptive technical context of our method to a probabilistic model. Our graphical model (see Figure A.1 in the Appendix) consists of trainable parameters $\Phi = \phi_{enc} \cup \phi_{dyn} \cup \phi_{dec}$, a random variable $\psi^r$ which additionally acts as global random variable at the level of latent states $z_{t_q}$ and observations $x_{t_q}$. The index $t_q$ refers to a specific queried time point within a trajectory. The joint distribution is given by $$p(x, z, \psi^r) = p(x \| z, \psi^r)p(z \| \psi^r)p(\psi^r) = p(x \| z)p(z \| \psi^r)p(\psi^r).$$ (1) Our graphical model assumes these independencies: (i) The dataset contains i.i.d. trajectories of varying length. (ii) The observation of trajectory $x^r_{t_q}$ at time $t_q$ is conditionally independent of $x^r_{t_{q-1}}$ at time $t_{q-1}$, given latent states $z^r_{t_q}$ and trajectory representation $\psi^r$: $p(x_{t_q} \| z_{t_q}, \psi^r) \perp \psi^r \| x_{t_{q-1}}, \psi^r)$. Analyzing data with this graphical model involves computing posterior distributions of hidden variables given observations $$p(z, \psi^r \| x) = \frac{p(x, z, \psi^r)}{\int p(x \| z)p(z \| \psi^r)p(\psi^r)dzd\psi^r}. $$ (2) To effectively process long-horizon time series data, we apply a variant of multiple shooting. However, since our model does not rely on an explicit ODE formulation, we are not concerned with turning an initial value problem into a boundary value problem (Massaroli et al., 2021). Instead, we incorporate a Bayesian continuity prior (Hegde et al., 2022; Iakovlev et al., 2023) to extend the multiple-shooting framework from deterministic neural ODEs to a probabilistic context. Our approach dissects each realization $x^r_{t:T}$ into a series of $N$ overlapping subtrajectories and independently condenses each patch into a latent representation. Within this Bayesian multiple shooting framework, the smoothness prior connects the patches via $$p(z \| \psi^r) = \prod_{n=1}^{N} p(z_n \| \psi^r_n)p(z_n \| z_{n-1}, \psi^r_{n-1})$$ (3) to form a cohesive global trajectory. We leverage the independence of trajectory representations in subpatches i.e. $p(z_i \| \psi^r_i) \perp p(z_j \| \psi^r_j)$. For the continuity prior, we follow Hegde et al. (2022) and place a Gaussian prior on the error between consecutive subtrajectories, i.e. $\Delta \sim N(0, \sigma_\Delta)$ entailing exact overlapping if $\Delta \rightarrow 0$. This yields our continuity prior $$p(z_n \| z_{n-1}, \psi^r_{n-1}) = N((z^t_n \| z^t_{n-1}, \psi^r_{n-1}), \sigma_\Delta),$$ (4) where the time index $-t$ refers to the last time point of a subpatch. The prior trajectory representation is set to a Gaussian, i.e. $p(\psi^r) \sim \mathcal{N}(0, 1)$. With the priors introduced above, we get the following generative model (we drop the subpatch index $n$ for improved readability): \begin{align} p(l_{emb}^K \| x) &= \mathcal{N}(f_{\phi_{enc}, \mu}(x_K), f_{\phi_{enc}, \sigma}(x_K)) \\ p(\psi^r \| x) &= f_{agg}(l_{emb}^K) \\ p(z \| \psi^r, x) &= f_{\phi_{dyn}}(\psi^r, t_q) \\ p(x \| z) &= \mathcal{N}(f_{\phi_{dec}}(z), \sigma_{dec}) \end{align} For inference, we use Gaussian approximations and set $\sigma_{dec} = 10^{-2}$. We then seek to minimize the KL divergence $\text{KL}[q(z, \psi^r)\|\|p(z, \psi^r\|x)]$, essentially equivalent to maximizing the ELBO in eq. 9. A full derivation of this ELBO can be found in Section C of the Appendix. \begin{equation} \max \mathbb{E}_{q(z, \psi^r)} \sum_{n=1}^{N} \ln p_n(\hat{x}_n) - \sum_{n=1}^{N} \text{KL}(q(\psi^r_n)\|\|p(\psi^r_n\|x_n)) - \sum_{n=2}^{N} \mathbb{E}_{q(z, \psi^r)} \text{KL}(q(z_n)\|\|p(z_n\|z_{n-1}, \psi^r_n)) \end{equation} 5 EXPERIMENTAL SET-UP Experiments are structured into four different series that shed light on the performance of LaDiD. We provide a short overview of the experimental set-up in the following. Further details can be found in Section D in the Appendix. Datasets. We consider a wide range of physical systems ranging from relatively simple ODE-based datasets to complex turbulence-driven fluid flows. Specifically, we consider high-dimensional observations ($p=16,384$) from: a nonlinear swinging pendulum; a swinging double pendulum; realistic simulations of the two-dimensional wave equation; a lambda-omega reaction-diffusion system; the two-dimensional incompressible Navier-Stokes equations; and the fluid flow around a blunt body solved via the latticed Boltzmann equations. This extensive range sheds light on performance on complex datasets relevant to real-world use-cases, including models frequently used in the literature on dynamical modeling. Regular and irregular time grids are included. We study also the challenging problem of making predictions in a completely novel setting obtained by intervention on the system. This is similar in spirit to experiments seen in causal AI and in this case involves generating small datasets on intervened dynamical systems (either via modifying the underlying systems, for example by changing the gravitational constant or the mass of a pendulum, or via augmenting the realization-specific observation, e.g. by changing the length of a pendulum or the location of a simulated cylinder) and fine-tuning a pre-trained model on a fraction of the data in the target setting. We direct the interested reader to Appendix I for further details. Training. Training is carried out in a multi-phase schedule w.r.t. the multiple shooting loss in eq. 9. In the different phases, we split the input trajectory into overlapping patches and start learning by predicting one step ahead. We double the number of prediction steps per patch every 3000 epochs meaning that learning is done on longer patches with decreased number of patches per trajectory (where trajectory length is not divisible by number of steps, we omit the last patch and scale the loss accordingly). In the final phase, training is carried out on the entire trajectory. All network architectures are implemented in the open source framework PyTorch (Paszke et al., 2019). Further training details and hyperparameters can be found in Appendix E. Testing. We test the trained models on entirely unseen trajectories. During testing, the first $k=10$ trajectory points are provided to the trained model. Based on these samples, an RS representation $\psi^r$ is computed and used to roll out the trajectory to the time points of interest. Finally, predictions and ground truth observations are compared using the evaluation metrics below. Evaluation metrics. We consider mean squared error (MSE) over trajectories: inference runs over $2T$ steps with MSE computed over the last $T$ timesteps, allowing assessment for relatively distant times (relative to the reconstruction MSE). We set $T = 60$ for all experiments, with MSE normalized w.r.t. average (true) intensity, as recommended in Zhong et al. (2021), Botev et al. (2021). Additionally, we provide time history diagrams plotting root mean square error (RMSE) against normalized time (mapping the interval $[T, 2T]$ to the unit interval). Metrics are averaged across all test trajectories and five runs, with mean and 75% inter-quantile ranges (IQR) reported. Subsampled predictions and pixelwise $L_2$ error of one (randomly chosen) trajectory is shown for visual inspection; we acknowledge that these cannot always be representative and should be considered alongside formal metrics. See Figure 2 for intuition on the train/test procedure and metrics. Comparisons. We compare our approach to recent models from the literature, including ODE-RNN (Rubanova et al., 2019), NDP (Norcliffe et al., 2021), ODE2VAE (Yildiz et al., 2019), and MSVI (Iakovlev et al., 2023). In common with LaDID, these models feature encode-simulate-decode structures and seek to learn low-dimensional latent dynamics. ODE2VAE simulates latent trajectories in a straightforward fashion using a BNN to model the underlying dynamics. In contrast, ODE-RNN, NDP and MSVI leverage neural ODE solvers to integrate latent states forward in time. Further details regarding these baselines can be found in Section G of the Appendix. ## 6 RESULTS First, we examined performance on synthetic data for which the training and test data come from the same dynamical system. This body of experiments test whether the model can learn to map from a finite, empirical dataset to an effective latent dynamical model. Second, we examine few-shot generalization to data obtained from systems subject to nontrivial intervention (and in that sense strongly out-of-distribution). In particular, we train our model on a set of trajectories under interventions, i.e. interventions upon the mass or length of the pendulum, changes to the Reynolds number, or variations to the camera view on the observed system, and apply the learned inductive bias to new and unseen interventional regimes in a few-shot learning setting. This tests the hypothesis that the inductive bias of our learned latent dynamical models can be a useful proxy for dynamical systems exposed to a number of interventions. ### 6.1 Benchmark comparisons to state-of-the-art models for ODE and PDE problems We begin by investigating whether LaDID can learn latent dynamical models in the conventional case in which the training and test data come from the same system. We evaluate the performance of ODE-RNN, ODE2VAE, NODEP, MSVI and LaDID on the data described in Section 5 and Section H of the Appendix with increasing order of difficulty, starting with the non-linear mechanical swing systems with underlying ODEs, before moving to non-linear cases based on PDEs (reaction-diffusion system, 2D wave equation, von Kármán vortex street at the transition from laminar to turbulent flows, and Naiver-Stokes equations). Due to limited space, we only present results for a subset of performed experiments but refer the interested reader to Appendix K for a detailed presentation of all results. Applications to ODE-based systems. For visual inspection and intuition, Figure 4 provides predicted observations $\hat{x}_t^r$ of a few time points of one test trajectory of the single pendulum dataset for all tested algorithms, followed by the ground truth trajectory and the pixelwise $L_2$-error. In addition, Figure 3 presents the normalized MSE over entire trajectories averaged across the entire test dataset and the evolution of the RMSE over time for the second half of the predicted observations averaged over all test trajectories (see Section 5) is provided in the Appendix. Across all ODE-based datasets LaDID achieves the lowest normalized MSE. The time history diagram (see Figure K.1 in the Appendix) reveals gains using LaDID for long-horizon predictions relative to all other algorithms tested. This can also be seen by visual inspection in Figure 4, as for other approaches the predicted states at later time points deviate from the ground truth trajectory substantially while LaDID’s predictions essentially follow the ground truth. Considering only the baselines, one can observe that MSVI (a sophisticated, recently proposed approach), predicts accurately within a short-term horizon but nonetheless fails on long-horizon predictions. The results for the challenging double pendulum test case can be found in the Appendix. ![Figure 3: Test errors - Normalized MSE](image) Figure 4: Left: predicted test trajectory at various timesteps $ t $, right: corresponding pixelwise error. Applications to PDE-based processes. We additionally evaluated all baselines and our proposed method on PDE-based processes. Due to space restrictions, we focus our analysis on the flow evolution characterized by the Navier-Stokes equation in the two-dimensional case, which is of great importance in many engineering tasks, e.g., the analysis of internal airflow in a combustion engine (Lagemann et al., 2022), drag reduction concepts in the transportation and energy sector (Gowree et al., 2018; Lagemann et al., 2023a; Mateling et al., 2023), and many more. Results in Figure 5 show that LaDID clearly outperforms all considered comparators. The normalized MSE is the lowest and the averaged RMSE is also the lowest at any time. This is echoed in the other experiments whose results are presented in detail in Section K in the Appendix. Overall, these results support the notion that LaDID achieves good performance for challenging ODE and PDE-based systems. We direct the interested reader to Section K for the complete collection of experimental results supporting this statement. In this context, Table H.2 of the Appendix highlights the massively reduced computational resources required during training and inference since LaDID eschews an explicit neural ODE formulation (including the costly ODE solvers), making it efficient and highly scalable in practice. Performance on regular and irregular time grids. Here, we study the performance of LaDID on regular and irregular time grids and compare it to other neural-dynamical models (which are able to deal with irregular time series data). As shown in Figure L.1 and Figure L.2 in the Appendix, the proposed LaDID performs very similarly on both types of the time grids relative to both ODE-based benchmark examples and challenging PDE-based real-world systems, outperforming existing methods demonstrating strong and robust performance on irregularly sampled data. Effects of relevant network modules. LaDID leverages three key features: a reconstruction embedding, a spatio-temporal attention module and a specifically designed loss heuristic to learn temporal dynamics from empirical data. We investigated the importance of these modules (results appear in the Section N of the Appendix). First, we compared LaDID with ablated counterparts, e.g. a pure reconstruction loss and loss combinations either using the representation or smoothness loss. Overall, the proposed loss heuristic appears to stabilize training and yields the lowest MSE and IQR values. Second, we compared LaDID to counterparts trained on ablated attention modules. Empirical results underline the utility of the applied spatio-temporal attention. Finally, Table N.3 further shows the usefulness of the representation-specific encoding. This representation encoding can be thought of a learning-enhanced initial value stabilizing the temporal evolution of latent trajectory dynamics. Moreover we study the effect of restricted training trajectories on the performance of LaDID in Section K.7 of the Appendix to better understand efficiency under limited data. 6.2 Generalizing to novel systems via few-shot learning Here, we assess LaDID’s ability to generalize to a novel system obtained by nontrivial intervention on the system coefficients themselves (e.g., mass, length, Reynolds number). Such changes can induce large changes to data distributions and can be viewed through a causal lens (see also Appendix O). In particular, we train a dynamical model on a set of interventions and fine-tune it to new intervention regimes with only a few samples, finally evaluating performance on an entirely unseen dataset. We compare the performance of our prior-based few-shot learning model with a model trained solely on the fine-tuning dataset (“scratch-trained” model). In our first experiment, we use the single pendulum dataset and test the transferability hypothesis on fine-tuning datasets of varying sizes. The results show that the prior-based model outperforms the scratch-trained model at all fine-tuning dataset sizes, and achieves comparable performance to the model trained on the full dataset with a fine-tuning dataset size of 32%. At a fine-tuning dataset size of 8%, LaDID produces partially erroneous but still usable predictions, which are only slightly worse than the predictions of an advanced NODE based model, MSVI, trained on the full dataset. Further results including robustness to input noise and color shifts in the observation space appear in Section M in the Appendix. Second, we investigate the effect of interventions on the observation process by testing the transferability to new observation settings on the von Kármán vortex street dataset. We re-simulate different cylinder locations (shifted the cylinder to the left, right, up, and down) and evaluate the performance under different fine-tuning dataset sizes. The results show that the prior-based model consistently outperforms the scratch-trained model and produces accurate and usable predictions under new observation conditions with a fine-tuning dataset size of as little as 8%. These findings support our hypothesis that LaDID is capable of extracting general dynamical models from training data. Additional transfer learning experiments are detailed in the Section P of the Appendix, studying the model’s performance when jointly trained on a dataset encompassing Reynolds numbers of $Re = [100, 250, 500]$ and subsequently applied for zero-shot predictions on unseen $Re$ numbers. ![Figure 6: Test errors for a set of transfer learning experiments.](image) 7 Conclusions In this paper, we presented a novel approach called LaDID aimed at end-to-end learning of latent dynamical models. LaDID uses a novel transformer-based architecture that leverages certain scientifically-motivated invariances to allow separation of a universal dynamics module and encoded realization-specific information. We demonstrated strong performance on several new and challenging test cases and well-known benchmarks. Additionally, we showed that LaDID can generalize to systems under nontrivial intervention (when trained on the un-intervened system) using few-shot learning. Currently, while LaDID accommodates irregular time sampling, data acquired from irregular spatial grids will need further work. A future research direction is to explore graph-based methodologies to address this specific challenge. ACKNOWLEDGEMENTS This work was partly supported by the German Federal Ministry of Education and Research (BMBF) project “LODE”, the UK Medical Research Council (MC-UU-00002/17) and the National Institute for Health Research (Cambridge Biomedical Research Centre at the Cambridge University Hospitals NHS Foundation Trust). The work of CL was funded by the Deutsche Forschungsgemeinschaft within the Walter Benjamin fellowship LA 5508/1-1. We gratefully acknowledge the Gauss Centre for Supercomputing e.V. for supporting this project via computing time on the GCS Supercomputers. REFERENCES Shaojie Bai, J Zico Kolter, and Vladlen Koltun. Deep equilibrium models. Advances in Neural Information Processing Systems, 32, 2019. 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vyGp9Mty2t	How does the method deal with registration problems in CT imaging? The validated datasets in the paper seem to be already registered. If we consider the real CT scanning in practice, for different patients, the patient’s positions will always be different. How can this method deal with the position shift when learning the prior from different patients?	Implicit Neural Representations for Joint Sparse-View CT Reconstruction Anonymous authors Paper under double-blind review Abstract Computed Tomography (CT) plays a crucial role in both medical diagnostics and industrial quality control. Sparse-view CT, in particular, has advantages over standard CT for its reduced ionizing radiation but poses challenges due to its inherently ill-posed nature arising from undersampled measurement data. Implicit Neural Representations (INRs) have emerged as a promising solution, demonstrating effectiveness in sparse-view CT reconstruction. Given that modern CT often scans similar subjects, we propose to improve reconstruction quality via joint reconstruction of multiple objects using INRs. This approach can potentially leverage both the strengths of INRs and the statistical regularities across multiple objects. While existing techniques of INR joint reconstruction focus on enhancing convergence rates through meta-initialization, they do not optimize for final reconstruction quality. To fill this gap, we introduce a novel INR-based Bayesian framework that incorporates latent variables to capture inter-object relationships. These latent variables act as a continuously updated reference during the optimization process, thereby enhancing the quality of individual reconstructions. We conduct extensive experiments to evaluate various aspects such as reconstruction quality, susceptibility to overfitting, and generalizability. Our results demonstrate significant improvements over baselines in common numerical metrics, suggesting a step forward in CT reconstruction techniques. Our code will be released. 1 Introduction Computed Tomography (CT) serves as a crucial non-invasive imaging tool in both medical diagnosis and industrial quality control. In CT, a series of X-ray projection images are captured from various angles to reconstruct an object’s internal structure, solving an inverse problem. In specific situations, limiting the number of CT measurements can offer benefits such as reduced radiation exposure and cost management, which may lead to the use of sparse data. This sparsity complicates the reconstruction process, making it an ill-posed inverse problem. Such challenges arise not only in CT reconstruction but also across diverse computational tasks. Hence, while our study centers on sparse-view CT reconstruction, the core ideas are transferable to numerous inverse problems. Various strategies tackle this challenge by incorporating auxiliary information. While many approaches learn the mapping from sparse-view to dense-view images using supervised learning (Zhang et al., 2018; Han & Ye, 2018; Zhu et al., 2018; Wu et al., 2021) or learn the image distribution solely from dense-view images (Song et al., 2022), they often necessitate extensive, domain-specific datasets which are difficult to obtain in practice. There are also works that adopt heuristic image priors, e.g. Total Variation (TV) (Sidky & Pan, 2008; Liu et al., 2013; Zang et al., 2018), or dense view images as priors (Chen et al., 2008; Shen et al., 2022) to assist in the reconstruction. They often lack domain-specific enhancements or require information from dense-view images. On a different tangent, many works explore the potential of implicit neural representations (INRs). Thanks to the continuous representation nature of INRs, these methods have consistently delivered promising results with limited data (Zang et al., 2021; Zha et al., 2022; Rückert et al., 2022; Wu et al., 2023). Given INRs’ proven capabilities in CT reconstruction and the known advantages of leveraging auxiliary information, we try to merge these two paradigms. Modern CT machines routinely scan similar subjects, such as patients in hospitals or analogous industrial products. This observation motivates us to investigate a novel question in this work: In our exploration of this research avenue, we found that several existing methods can be adapted for our purpose (Zhang et al., 2013; Ye et al., 2019; Tancik et al., 2021; Martin-Brualla et al., 2021; Kundu et al., 2022). Some previous works have exploited the statistical regularities among different objects borne in the INR networks’ weights, but target different problems such as convergence rate (Tancik et al., 2021; Lee et al., 2021). A common practice in these methods is to find a network initialization that outperforms random initialization. However, these approaches may not fully capitalize on the available statistical regularities, as such information could be lost during the adaptation phase of the individual reconstructions. To address our research question, we introduce a novel INR-based Bayesian framework designed to adaptively integrate prior information related to network weights throughout the training process. Specifically, we employ latent variables that capture common trend among different objects’ neural representations, and subsequently apply this prior information to improve the accuracy of individual reconstructions. Both of these objectives are achieved by minimizing the Kullback-Leibler (KL) divergence between the prior and the approximated posterior distributions associated with the neural representation networks. Importantly, our framework can automatically adjust the regularization effect of the prior information based on the similarity among neural representation networks, allowing for a broader range of applications in reconstructing diverse images. Overall, our framework provides a robust solution to the challenges posed by sparse data and varied reconstructions in CT imaging. An illustration of our proposed method is provided in Figure 1. Our Contributions: i) We explore a novel problem of INR-based joint reconstruction in the context of CT imaging, supported by a comprehensive review of existing methods that could be adapted to address this challenge. ii) We propose a principled Bayesian framework that adaptively integrates prior information throughout the training process for enhanced reconstruction quality. iii) Through extensive experiments, we evaluate various facets of reconstruction performance using common numerical metrics. Our results establish that our method either outperforms or is competitive with existing INR-based baselines, suggesting notable advancements in the field of CT reconstruction. ### 2 RELATED WORKS We briefly outline key studies related to our focal areas, with a comprehensive understanding of NeRF and INR available in the survey (Tewari et al., 2022). Neural Radiance Fields. Coordinates-based Multi-Layer Perceptrons (MLPs) have transitioned from traditional discrete representations to implicit neural representations (INRs) by addressing high-frequency function detailing issues (Jacot et al., 2018; Tancik et al., 2020). Neural Radiance Fields (NeRF), a state-of-the-art INR approach, models continuous scenes using spatial coordinates and viewing angles, incorporating transmittance effects during ray-tracing (Mildenhall et al., 2021; Barron et al., 2021, 2022). Specifically, NeRF-wild (Martin-Brualla et al., 2021) differentiated between static and transient scene aspects, an approach echoed in video representations (Li et al., 2021; Mai & Liu, 2022). INR for CT Reconstruction. INRs’ potential in CT reconstruction has been exploited in various ways. While Sun et al. (2021) focused on representing sparse measurements, Zang et al. (2021) combined INRs with total variation and non-local priors for CT reconstruction. Notable advancements include cone-beam CT optimization (Zha et al., 2022), and adaptive hierarchical octree representation (Rückert et al., 2022). Wu et al. (2023) also improved reconstruction precision using reprojectons on inferred density fields. Building on the groundwork laid by INR-based approaches, several techniques have emerged to leverage prior information in joint CT reconstruction. Meta-learning’s application in CT reconstruction was first introduced by Tancik et al. (2021), using techniques like MAML (Nichol et al., 2018). Later, Lee et al. (2021) introduced sparsity to the initialization, while Chen & Wang (2022) proposed using transformers. For scene representation, Kundu et al. (2022) applied federated learning to obtain the prior information. Lastly, while other INR-based CT reconstructions like Shen et al. (2022) use priors from pre-reconstructed images, and Reed et al. (2021) rely on finding a template image from 4DCT, their practical limitations led to their exclusion from our comparative analysis. 3 PROBLEM STATEMENT AND PRELIMINARIES Mathematically, the CT acquisition process can be formulated as a linear equation: $ y = Ax + \epsilon $, where $ x \in \mathbb{R}^m $ represents the unknown object of interest and $ y \in \mathbb{R}^n $ symbolizes the noisy measurements. These measurements arise from the interaction between the measurement matrix $ A \in \mathbb{R}^{n \times m} $ and the object, with $ \epsilon \in \mathbb{R}^n $ accounting for the associated measurement noise. The task in CT is to infer the unknown object $ x $ from the acquired CT measurements $ y $. The inherent challenge lies in the common sparsity of these measurements, resulting in $ m > n $. This makes the reconstruction problem ill-posed. The INR designed for CT reconstruction is a function $ f_w : \mathbb{R}^3 \rightarrow \mathbb{R}^1 $ parameterized by $ w $. It maps the spatial coordinates of the object to its intensity in a continuous three-dimensional space. INR consists of two components, formulated as $ f_w = M \circ \gamma $. Here, $ \gamma : \mathbb{R}^3 \rightarrow \mathbb{R}^d $ serves as the position encoding (Tancik et al., 2020; Barron et al., 2022; Martel et al., 2021; Müller et al., 2022), while $ M : \mathbb{R}^d \rightarrow \mathbb{R}^1 $ acts as the neural representation. Typically, $ M $ is a multi-layer perceptron (MLP). The function $ f_w(\cdot) $ takes a coordinate $ c_i \in \mathbb{R}^3 $ and maps it to the intensity value $ v \in \mathbb{R}^1 $. For a full set of coordinate $ C := \{c_1, c_2, \ldots, c_N\} $, the INR outputs the representation of the entire object as $ F_w(C) := \{f_w(c_1), f_w(c_2), \ldots, f_w(c_N)\} $. The optimization procedure for INR-based reconstruction involves minimizing the loss function: $ \ell(w) := \\|AF_w(C) - y\\|_2^2 $. Joint Reconstruction Problem. We aim to simultaneously recover $ J $ objects $ x_{1:J} $ using their corresponding measurements $ y_{1:J} $ and measurement matrices $ A_{1:J} $. The joint reconstruction problem can be mathematically formulated as: \[ w_{1:J}^* = \arg \min_{w_{1:J}} \sum_{j=1}^{J} \ell_j(w_j), \quad \ell_j(w_j) := \\|A_jF_{w_j}(C) - y_j\\|_2^2. \] We believe that by introducing a dynamic prior that not only links all the models $ w_{1:J} $ during training but also updates in response to their optimization, a Bayesian framework can provide a principled way to exploit the shared statistical regularities among different objects, thereby enhancing the quality of joint reconstruction. 3.1 EXISTING METHODS AVAILABLE FOR JOINT RECONSTRUCTION Although several existing methods are originally designed for different problems and do not employ a Bayesian framework, they also align well with the objective highlighted in Equation (1). In the following sections, we delve into these methods in greater detail. Empirical evaluations suggest that some of these techniques can outperform the individual reconstruction approach, as discussed in Section 5. Thus, we also benchmark these methods against our proposed Bayesian framework. Composite of Static and Transient Representations. Martin-Brualla et al. (2021) introduce a composite representation approach, known as NeRFWild, designed to manage variable illumination and transient occluders in a collective of observations. While CT does not involve variable illumination, their concept of combining “static” and “transient” components can be adapted for our context, which we term INRWild. Let $ G_\phi $ represent the neural representation for the static component and $ H_w $ signify the transient component. For a given set of $ J $ objects, each object-associated reconstruction node has its distinct transient network $ w_j $ and corresponding transient feature $ b_j $. In contrast, the static network $ \phi $ is shared across all nodes. The objective for this framework is formulated as: \[ \phi^, w_{1:J}^, b_{1:J}^* = \arg \min_{\phi, w_{1:J}, b_{1:J}} \sum_{j=1}^{J} \\| A_j \left( H_{w_j} (b_j, G_\phi^\tau(C)) + G_\phi^\tau(C) \right) - y_j \\|_2^2. \] Here, $ G_\phi^\tau(C) $ represents the static intensity, and $ G_\phi^\tau(C) $ serves as intermediate features for the transient network. For a more detailed explanation and a schematic depiction of this framework, readers can refer to the Appendix C.1. At its core, INRwild emphasizes training the static network $ \phi $, which embodies most learnable parameters, using aggregated losses. Concurrently, the individual parameters, characterized by $ w_{1:J} $ and $ b_{1:J} $, are refined based on $ \phi $'s characteristics. Model-agnostic Meta-learning (MAML): Meta-learning aims to train a network in a way that it can quickly adapt to new tasks [Nichol et al., 2018; Fallah et al., 2020]. Several INR-based works have employed MAML to obtain a meta-learned initialization, thereby accelerating the convergence or enabling model compression [Tancik et al., 2021; Lee et al., 2021]. In the MAML framework, computational cycles are organized into “inner loops” and “outer loops”, indexed by $ k = 1 \ldots K $ and $ t = 1, \ldots, T $ respectively. For each node $ j = 1, \ldots, J $, the networks $ w_{1:J} $ are initialized according to the meta neural representation $ w_{1:J}^{(0)} = \theta $. These networks then undergo $ K $ steps of inner-loop learning: $ w_{j}^{(k)} = w_{j}^{(k-1)} - \eta \nabla_{w_j} \ell_j(w_{j}^{(k-1)}) $, where $ \eta $ is the inner-loop learning rate. After these $ K $ steps, the meta network $ \theta $ updates as follows: \[ \theta^t = \theta^{t-1} - \alpha \frac{1}{J} \sum_{j=1}^{J} \nabla_{\theta} \ell_j(w_{j}^{(K)}), \] where $ \alpha $ is the outer-loop learning rate. After $ T $ steps of outer-loop optimization, the meta-learned neural representation $ \theta^T $ serves as an effective initialization for individual reconstructions. Federated Averaging (FedAvg): [Kundu et al., 2022] suggested to employ FedAvg [McMahan et al., 2017] as the optimization framework of the meta-learned initialization. Like MAML, FedAvg also consists of inner and outer loops. The inner loop is identical to MAML. Whereas, the outer loop simplifies the meta network optimization by averaging all individual networks, represented as $ \theta = \frac{1}{J} \sum_{j} w_{j}^{(K)} $. Essentially, the meta network acts as the centroid of all networks.$^1$ ### 4 A NOVEL BAYESIAN FRAMEWORK FOR JOINT RECONSTRUCTION In this section, we introduce INR-Bayes, our Bayesian framework for INR joint reconstruction. Motivation. A method that uses a composition of static and transient components operates under the assumption that all the representations substantially overlap. This may be true in 3D scene reconstruction, where observations are taken from different viewpoints of the same object. Our empirical findings on INRwild indicate such methods do not work efficiently in CT reconstruction and other image-level reconstruction tasks. Meta-learned initialization methods train a meta-model to capture a conceptual common representation, which can then be flexibly adapted to individual objects. However, such methods subsequently adapt the models purely based on local measurements, making them prone to the notorious overfitting issue in iterative methods of CT reconstruction [Herman & Odhner, 1991; Effland et al., 2020], as demonstrated in Section 5. By considering the meta-model, which we denote by $ \omega $ in the sequel, a latent variable that updates based on individual networks and uses it as a reference for individual training, a Bayesian framework provides a principled way to conduct this process. Definition and Notation. We introduce distribution to the networks $ w_{1:J} $ for $ J $ objects, and define latent variables $ \{\omega, \sigma\} $ that parameterize an axis-aligned multivariate Gaussian prior $ N(\omega, \sigma) $. $^1$It is noteworthy that FedAvg can also be regarded as using a specific first-order algorithm of MAML called Reptile [Nichol et al., 2018] and setting the outer-loop learning rate to 1. from which the weights are generated. These latent variables collectively serve to capture the shared trends within the network, effectively quantifying the mutual information across different objects. To simplify the model, we assume the conditional independence among all objects: \[ p(w_{1:J} \| \omega, \sigma) = \prod_{j=1}^{J} p(w_j \| \omega, \sigma). \] This assumption of conditional independence allows us to decompose the variational inference into a separable optimization problem, thereby facilitating more efficient parallel computing. Given that the measurements of the objects $ y_1, \ldots, y_J $ are mutually independent and that each network focuses on a specific object, the posterior distribution of network weights and latent variables can be derived using the Bayes’ rule as \[ p(w_{1:J}, \omega, \sigma \| y_{1:J}) \propto p(\omega, \sigma) \prod_{j=1}^{J} p(y_j \| w_j) p(w_j \| \omega, \sigma). \] While this posterior enables various forms of deductive reasoning, inferring the true posterior is often computationally challenging or intractable. Moreover, the selection of an appropriate prior $ p(\omega, \sigma) $ poses its own difficulties (Wenzel et al., 2020; Fortuin et al., 2022). To tackle these issues, we present an algorithm that aims at maximizing the marginal likelihood $ p(y_{1:J} \| \omega, \sigma) $ in the sequel. The details of derivations are provided in Appendix B. ### 4.1 Optimization Method To optimize the marginal likelihood $ p(y_{1:J} \| \omega, \sigma) $, we approximate the posterior distribution of the network weights $ w_{1:J} $ using variational inference techniques (Kingma & Welling, 2013; Blei et al., 2017). Specifically, we introduce the factorized variational approximation $ q(w_{1:J}) = \prod_{j=1}^{J} q(w_j) $, employing an axis-aligned multivariate Gaussian for the variational family, i.e. $ q(w_j) = \mathcal{N}(\mu_j, \rho_j) $. Variational Expectation Maximization. To maximize the marginal likelihood, we use the evidence lower bound (ELBO): \[ ELBO(q(w_{1:J}), \omega, \sigma) = \mathbb{E}_{q(w_{1:J})} \log \frac{p(y_{1:J}, w_{1:J} \| \omega, \sigma)}{q(w_{1:J})}. \] (4) The ELBO is optimized using Expectation Maximization (EM) (Dempster et al., 1977), a two-stage iterative algorithm involving an E-step and an M-step. Generally, each EM cycle improves the marginal likelihood $ p(y_{1:J} \| \omega, \sigma) $ unless it reaches a local maximum. E-step. At this stage, the latent variables $ \{\omega, \sigma\} $ are held fixed. The aim is to maximize ELBO by optimizing the variational approximations $ q(w_{1:J}) $. By assumption, the objective can be separately optimized for each network. Specifically, each network minimizes: \[ L(q(w_j)) = -\mathbb{E}_{q(w_j)} \log p(y_j \| w_j) + D_{KL}(q(w_j) \|\| p(w_j \| \omega, \sigma)). \] (5) The minimization of the negative log-likelihood term is achieved through the minimization of the squared error loss of reconstruction (see Equation (1)). The KL divergence serves as a regularization constraint on the network weights, pushing $ w_j $ to be closely aligned with a conditional prior determined by $ \{\omega, \sigma\} $. These parameters represent the collective mean and variance of all the networks in the ensemble. The KL divergence thus serves to couple the neural representations across networks, allowing them to inform each other. M-step. After obtaining the optimized variational approximations $ q(w_{1:J}) $, we proceed to maximize the ELBO with respect to the latent variables $ \{\omega, \sigma\} $: \[ ELBO(\omega, \sigma) \propto \sum_{j=1}^{J} \mathbb{E}_{q(w_j)} \log p(w_j \| \omega, \sigma). \] (6) Equation (6) allows for a closed-form solution of $ \{\omega, \sigma\} $, derived by setting the derivative of the ELBO to zero: \[ \omega^* = \frac{1}{J} \sum_{j=1}^{J} \mu_j, \quad \sigma^* = \frac{1}{J} \sum_{j=1}^{J} \rho_j + (\mu_j - \omega^)^2. \] (7) In our framework, $ \omega $ serves as a collective mean of individual network weights, while $ \sigma $ provides an adaptive measure of dispersion, factoring in both individual variances and deviations from the collective mean. We note the KL divergence term, introduced in the preceding E-step objective (see Equation (5)), operates element-wise. During the training process, weight elements with larger values of $ \sigma $ are less regularized, thereby offering a flexible, self-adjusting regularization scheme that pushes all weights toward the latent mean $ \omega $.* Algorithm 1 INR-Bayes: Joint reconstruction of INR using Bayesian framework Input: $\mu^{(0,0)}_{1:J}$, $\pi^{(0,0)}_{1:J}$, $\omega^0$, $\sigma^0$, $\eta$, $\beta$, $T$, $R$ Output: $\mu^{(R,T)}_{1:J}$, $\pi^{(R,T)}_{1:J}$, $\omega^R$, $\sigma^R$ 1: for $r = 1$ to $R$ do 2: for $j = 1, \ldots, J$ in parallel do 3: NodeUpdate($\omega^{r-1}$, $\sigma^{r-1}$) 4: After the E-step of each network, collecting $\mu^{(r,T)}_{1:J}$, $\pi^{(r,T)}_{1:J}$. 5: ▷ Compute the optimal latent variable $\omega$, $\sigma$. 6: $\omega^r = \frac{1}{J} \sum_{j=1}^{J} \mu^{(r,T)}_j$ 7: $\sigma^r = \frac{1}{J} \sum_{j=1}^{J} \log \left(1 + \exp(\pi^{(r,T)}_j)\right) + (\mu^{(r,T)}_j - \omega^r)^2$ 8: NodeUpdate($\omega^r$, $\sigma^r$): 9: for $t = 1, \ldots, T$ in parallel do 10: ▷ Sample $\hat{w}_j$. 11: $\hat{w}_j^{(r,t)} \sim \mu^{(r,t)}_j + \log \left(1 + \exp(\pi^{(r,t)}_j)\right) N(0, I)$ 12: ▷ Compute the loss function. 13: $\mathcal{L}(\mu_j, \rho_j) = \\|A_j(F_{\hat{w}_j}(C)) - y_j\\|_2^2 + \beta D_{KL}(q(w_j)\|\|p(w_j\|\omega^r, \sigma^r))$ 14: ▷ SGD on the variational approximation $\mu_j$, $\rho_j$ with learning rate $\eta$. 15: $\mu^{(r,t+1)}_j = \mu^{(r,t)}_j - \eta \frac{\partial \mathcal{L}}{\partial \mu_j}, \pi^{(r,t+1)}_j = \pi^{(r,t)}_j - \eta \frac{\partial \mathcal{L}}{\partial \pi_j}$ 4.2 IMPLEMENTATION We delve into the intricacies of implementation, addressing in particular the computational challenges associated with Equation (5). A summary of our method can be found in Algorithm 1. Variational Approximation. Given that the expected likelihood in Equation (5) is generally intractable, we resort to Monte Carlo (MC) sampling to provide an effective estimation. Moreover, we introduce an additional hyperparameter $\beta$ for the KL divergence to balance the trade-off between model complexity and overfitting. Linking the likelihood with the square error loss, for any node $j$, the effective loss function can be expressed as: $$\mathcal{L}(q(w_j)) \approx \\|A_j(F_{\hat{w}_j}(C)) - y_j\\|_2^2 + \beta D_{KL}(q(w_j)\|\|p(w_j\|\omega, \sigma)), \quad (8)$$ where $\hat{w}_j$ denotes a sample from $q(w_j)$. We only do MC sampling once at each iteration, which works efficiently in practice. Reparameterization Tricks. To facilitate the gradient-based optimization schemes, we utilize the reparameterization trick (Kingma & Welling, 2013): $$q(w_j) = \mu_j + \log \left(1 + \exp(\pi_j)\right) N(0, I). \quad (9)$$ Here, we additionally deploy the softplus function in parameterizing the variance $\sigma_j$ with the variable $\pi_j$ to ensure the non-negativity of the variance of the variational approximation. The EM algorithm operates through alternating E and M steps. In the E-step, we perform $T$ iterations to achieve the locally optimal variational approximations. Following this, the M-step utilizes the closed-form solution (see Equation (7)) to achieve an efficient parameter update. The entire cycle is executed for $R$ rounds to ensure convergence. Finally, the parameters $\mu_{1:J}$ serve as the weights for individual neural representation, while $\omega$ is used as the weights for meta neural representation. 5 EXPERIMENTS Dataset. Our study utilizes three CT datasets: 4DCT on the lung area (Castillo et al., 2009), LungCT from the Medical Segmentation Decathlon (Antonelli et al., 2022), and BrainCT from the Brain CT Hemorrhage Challenge (Flanders et al., 2020). Additionally, we include a natural image dataset CelebA (Liu et al., 2015) to evaluate the generalizability to broader applications. \| Experiment \| Metrics \| FBP \| SIRT \| SingleINR \| INRWild \| FedAvg \| MAML \| INR-Bayes \| \|------------\|---------\|-----\|------\|-----------\|---------\|-------\|------\|-----------\| \| Intra-patient \| PSNR \| 26.50 ±0.06 \| 28.81 ±0.06 \| 32.80 ±0.11 \| 28.46 ±0.07 \| 32.42 ±0.08 \| 33.26 ±0.10 \| 33.90 ±0.10 \| \| SSIM \| 0.568 ±0.002 \| 0.719 ±0.002 \| 0.815 ±0.002 \| 0.674 ±0.003 \| 0.808 ±0.002 \| 0.825 ±0.002 \| 0.840 ±0.002 \| \| Inter-patient \| PSNR \| 24.97 ±0.14 \| 28.32 ±0.13 \| 32.64 ±0.27 \| 25.05 ±0.18 \| 31.68 ±0.19 \| 33.13 ±0.22 \| 33.75 ±0.20 \| \| SSIM \| 0.503 ±0.004 \| 0.678 ±0.005 \| 0.821 ±0.007 \| 0.560 ±0.008 \| 0.807 ±0.006 \| 0.833 ±0.006 \| 0.847 ±0.006 \| \| Lung \| PSNR \| 17.68 ±0.34 \| 20.74 ±0.49 \| 25.56 ±1.14 \| 21.81 ±0.37 \| 24.46 ±0.96 \| 25.81 ±1.14 \| 25.84 ±1.15 \| \| SSIM \| 0.398 ±0.004 \| 0.498 ±0.004 \| 0.801 ±0.017 \| 0.638 ±0.009 \| 0.761 ±0.011 \| 0.823 ±0.013 \| 0.823 ±0.014 \| \| Inter-patient \| PSNR \| 25.19 ±0.03 \| 28.61 ±0.03 \| 34.07 ±0.04 \| 33.76 ±0.04 \| 34.67 ±0.04 \| 34.35 ±0.05 \| 34.79 ±0.04 \| \| SSIM \| 0.529 ±0.001 \| 0.746 ±0.001 \| 0.877 ±0.001 \| 0.863 ±0.001 \| 0.885 ±0.001 \| 0.881 ±0.001 \| 0.894 ±0.001 \| Table 1: Results from intra-patient, inter-patient joint reconstruction, and joint reconstruction across temporal phases in 4DCT. The highest average PSNR/SSIM values that are statistically significant are bolded. Our method, INR-Bayes, consistently achieves the best performance across datasets. Comparison Methods. We compare our approach with the following methods: i) Classical techniques: Filtered Back Projection (FBP) and Simultaneous Iterative Reconstruction Technique (SIRT); ii) Naive INR-based single reconstruction method, denoted as SingleINR; iii) FedAvg, a federated averaging approach proposed by Kundu et al. (2022); iv) MAML, A meta-learning technique as discussed by Tancik et al. (2021); v) INRWild, a method adapted from NeRFWild (Martin-Brualla et al., 2021). FBP and SIRT are classical methods that do not use neural networks, while all other methods employ an identical INR network as described in the next paragraph. INR Network Configuration. The same backbone and associated configurations are applied to ensure a fair comparison. All INR-based methods employ the SIREN architecture (Sitzmann et al., 2020) coupled with the same positional embedding (Tancik et al., 2020). In alignment with the INRWild design, we utilize an 8-layer SIREN network for the static segment and a 4-layer SIREN for each transient component. Additional details are available in the Appendices C.2 and C.3. CT Configuration. We simulate CT projections using the Tomosipo package (Hendriksen et al., 2021) with a parallel beam. Experiments on 4DCT and BrainCT use projections from 40 angles across 180°, while others use 60 angles. Practical applicability is further tested under a 3D cone-beam CT setting, detailed in Appendix E.1. Experiment Configurations. i) Intra-patient: 10 equidistant slices from a patient’s lung center. ii) Inter-patient: 10 slices from different patients, each from a similar upper-body/head position. iii) 4DCT: 10 temporal phases from one 4DCT slice. Metrics. We primarily evaluate using Peak Signal to Noise Ratio (PSNR) and Structural Similarity Index Measure (SSIM), with metrics referenced against ground-truth images. We calculate mean and standard error over all reconstructed images in each experiment. 5.1 RESULTS Reconstruction Performance. Table 1 presents the average metrics across various datasets. Our method consistently achieves the top average PSNR/SSIM values, underscoring its proficiency in exploiting inherent trends across slices. The superiority of our approach is more pronounced when the images demonstrate an inherent transition pattern, as observed in both inter-patient and intra-patient experiments. Meta-learning mostly ranks as the second-best joint reconstruction method, with an exception in the 4DCT dataset where pronounced image similarities exist. This highlights the advantages of utilizing averaging as a prior under such circumstances. The visual comparisons in Figure 2 and Figure 3 further substantiate our findings. The reconstruction of SingleINR shows noticeable artifacts. Although FedAvg and MAML achieve higher PSNR and SSIM due to smoother reconstructions, they also sacrifice some image details. In contrast, our INR-Bayes method consistently delivers superior visual quality, balancing smoothness and detail. Comparison with Different Numbers of Nodes and Angles. Figure 4a demonstrates that all methods see an improvement in average PSNR values as the number of scanning angles increases. Methods that leverage prior information, such as FedAvg, MAML, and ours INR-Bayes outperform singleINR when the number of angles is limited. With only 20 angles, FedAvg’s performance is Figure 2: Visual comparison for intra-patient joint reconstruction. Enlarged areas are highlighted in red insets. PSNR values are on the top left, with SSIM values on the bottom left. Figure 3: Visual comparison for joint reconstruction across 4DCT temporal phases. Enlarged areas are highlighted in red insets. PSNR values are on the top left, with SSIM values on the bottom left. (a) Performance across different numbers of scanning angles. (b) Different numbers of nodes. Individual reconstruction methods are presented as reference. Figure 4: Results of the impact of varying scanning angles and nodes on intra-patient LungCT. on par with our method, indicating that simple averaging can be effective in extremely data-scarce scenarios. However, as the number of angles grows, both ours INR-Bayes and MAML surpass FedAvg. Remarkably, our INR-Bayes method generally yields the best results. It is also worth noting that the performance gap between singleINR and ours INR-Bayes narrows as more data becomes available, suggesting that while the prior information is useful in sparse data situations, its advantage diminishes in the data-rich environment. In Figure 4b, our method consistently delivers superior performance compared to other methods across a range of node counts. MAML shows strong results when the node count is between 5 and 25, but experiences a decline in performance, eventually matching that of FedAvg when the node count reaches 40. This drop indicates that MAML might struggle to capture the shared features when many nodes are participating in the joint reconstruction. Overfitting. Iterative reconstruction methods tend to overfit when applied to limited data (Herman & Odhner [1991]; Effland et al. [2020]). In contrast, Bayesian frameworks have demonstrated robustness against overfitting (MacKay [1992]; Neal [2012]; Blundell et al. [2015]). To validate this, we extend the training iterations from 30K to 60K, designating the latter half as a pure adaptation phase. As shown in Figure 5, on inter-patient LungCT, the learning curves of baselines deteriorate in the long run, indicating overfitting on the measurement noise. Conversely, our approach maintains a consistent level of reconstruction quality once the optimal performance is achieved, underscoring the robustness of our framework. We note that determining an exact stopping criterion is challenging without reference ground truth, making such robustness highly valuable in practice. Applying to Unseen Data using Learned Prior. We apply the acquired prior from the inter-patient experiment to guide the reconstruction of test subjects in the LungCT dataset. Specifically, we select 5 consecutive slices from new patients, choosing slices from the same anatomical location the prior has been trained. The prior information is solely utilized to guide the reconstruction and is not updated during the process. Table 2 shows that FedAvg fails to improve the reconstruction quality compared with singleINR, suggesting its learned meta neural representation struggles to generalize to unseen data. In contrast, both MAML and ours INR-Bayes effectively leverage their trained priors for improved reconstruction, with our method showing notably better metrics. Figure 6 presents performance curves of different methods. All joint reconstruction methods converge faster than individual reconstruction. Initially, FedAvg converges the fastest, but as training progresses, both MAML and INR-Bayes surpass it. Additionally, the results reconfirm the robustness of ours INR-Bayes against overfitting, a problem that other methods cannot avoid. Broader Application. We also conduct experiments on the CelebA dataset to evaluate the generalizability of different methods to natural images. Results are relegated to Appendix E.2. ## 6 Discussion and Conclusion We introduced a novel INR-based Bayesian framework tailored for joint CT reconstruction in this study. Through extensive experiments, our method has effectively showcased its ability to leverage the statistical regularities inherent in the sparse measurement of multiple objects to improve individual reconstructions. This capability allows our approach to outperform competing methods in terms of reconstruction quality, robustness to overfitting as well as generalizability. While the primary focus of our method has been on joint CT reconstruction, its underlying principles hold potential applicability across a variety of inverse problems plagued by the challenges of sparse measurements. Limitation. We recognize that INR-based methods outperform conventional ones but require more computation, making their efficiency a crucial focus for future research. Additionally, the metrics employed in our study may not always correlate with clinical evaluations (Renieblas et al., 2017; Verdun et al., 2015). 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B1VWS7ZRm6	Also, the LGB, RTDL, and Resnet methods are listed alongside knowledge transfer baselines and I'm not sure how to interpret them. I assume they refer to scores of models without cross-modality knowledge transfer, but I also assume LGB and RTDL are supposed to target tabular data while Resnet image data. Since the target scenario is to evaluate the model on image data only, I'm not sure how LGB and RTDL are useful here.	ON TRANSFERRING EXPERT KNOWLEDGE FROM TABULAR DATA TO IMAGES Anonymous authors Paper under double-blind review ABSTRACT Transferring knowledge across modalities has gained considerable attention in machine learning. Expert knowledge in fields like medicine is often represented in tabular form, and transferring this information can enhance the accuracy of image-based learning. Unlike general knowledge reuse scenarios, tabular data is divided into numerical and categorical variables, with each column having a unique semantic meaning. In addition, not all columns in tabular can be accurately represented in images, making it challenging to determine “how to reuse” and “which subset to reuse”. To address this, we propose a novel method called CHannel tAbulaR alignment with optiMal tranSport (CHARMS) that automatically and effectively transfers relevant tabular knowledge. Specifically, by maximizing the mutual information between a group of channels and tabular features, our method modifies the visual embedding and captures the semantics of tabular knowledge. The alignment between channels and attributes helps select the subset of tabular data which contains knowledge to images. Experimental results demonstrate that CHARMS effectively reuses tabular knowledge to improve the performance and interpretability of visual classifiers. 1 INTRODUCTION Data takes on various forms, such as images, text, video, and audio, providing rich and diverse sources of information for a given task. In contrast to using a single modality, multimodal learning aims to fuse information from different data modalities to create more comprehensive and accurate models (Baltrušaitis et al., 2018; Ngiam et al., 2011; Ramachandram & Taylor, 2017; Yang et al., 2020). This approach has demonstrated exceptional performance across many domains, including recommender systems (Salah et al., 2020; Huang et al., 2019; Baltescu et al., 2022), healthcare (Zhang et al., 2022; Han et al., 2022), and visual question answering (Li et al., 2019; Zheng et al., 2020; Jing et al., 2020). In practical applications, obtaining data from multiple modalities can be challenging (Zhou, 2018), as expert knowledge or specialized equipment may be required, such as medical images. The high acquisition cost of such data makes the traditional multimodal fusion approach impractical. To address this, one solution is to employ multiple modalities during training, enabling expert knowledge to transfer from one modality to another and improving the performance of a single modality during testing. The current research on crossmodal transfer primarily focuses on images and text (Karpathy & Fei-Fei, 2015; Wang et al., 2016; Radford et al., 2021), but limited exploration has been done with tabular data (Häger et al., 2023). Tabular data is a common type of structured data, usually organized in a table format, where each column represents an attribute or feature and each row represents a sample of data (McKinney et al., 2010). Tabular data often involves some expert knowledge, for example, in the medical field, an attribute of tabular data may represent position information in an MRI image that needs to be focused on, which requires expert annotation. Therefore, transferring expert knowledge from tables to images will improve detection efficiency and reduce the burden on doctors. However, tabular data’s structured format distinguishes it from existing unstructured data such as text, making existing crossmodal transfer methods unsuitable for tabular data (Kimball & Ross, 2011; Shwartz-Ziv & Armon, 2022). Specifically, we face two challenges in transferring tabular knowledge for images. Firstly, we must address “how to reuse” the tabular data. As each column in tabular data has a unique semantic meaning, relying on standard RNN (Hopfield [1982], Zarembo et al. [2014]) or Transformer (Vaswani et al. [2017]) methods to construct a coarse feature space would result in a loss of interpretability of certain attributes. Moreover, categorical and numerical variables in tabular data require different processing methods. Secondly, we must identify “what subset to reuse” from the vast amount of information contained in tabular data since not all of it is relevant to the corresponding image. For example, in a pet adoption scenario, the tabular data contains not only the type of the pet but also information such as whether the pet is vaccinated or not. Therefore it is crucial to identify the useful information that can be transferred to instruct the learning of images. We expect that by transferring tabular knowledge to an image model, the model can learn corresponding semantics more effectively and achieve better performance on correlation tasks. To overcome the aforementioned challenges, we propose a novel method named CHannel tAbulaR alignment with optiMal tranSport (CHARMS) that aligns tabular data attributes with image channels which automatically transfers relevant expert knowledge in tabular data to images. Specifically, we modify the visual embedding with the instruction of tabular data as auxiliary information and learning tabular features with a group of channels, maximizing the mutual information between them. Additionally, we utilize the optimal transport algorithm (Bonneel et al. [2011], Caffarelli & McCann [2010]) to match the representation of each channel with the representation of each attribute, where a distinction is made between categorical and numerical variables. We strengthen the corresponding channels to ensure a focused learning of the tabular knowledge. In this way, our approach can automatically and effectively utilize expert knowledge from tabular data in the learning process, outperforming previous methods. To summarize, our contribution is three-fold: - We emphasize the importance of knowledge transfer from tabular data to image data, as this can lead to improved performance when tabular data is missing due to high costs. - We propose CHARMS method to automatically transfers relevant tabular knowledge to images. It aligns attributes and channels by leveraging optimal transport and utilizes tabular data as auxiliary information during transfer. - Experimental results demonstrate that CHARMS effectively reuses tabular knowledge to improve the performance of visual classifiers. Moreover, our approach offers insightful explanations of the learned visual embedding space with tabular instruction. This paper is organized as follows: the related work is introduced in Section 2. Section 3 and Section 4 provide the setting formalization, discovery experiment and our method. In Section 5, we present experiment results and discuss our findings. Finally, Section 6 concludes our study results. 2 RELATED WORK Multimodal Learning. Data of different modalities, such as image, video, audio, and text, usually overlap in some content, while some information is complementary. Multimodal learning aims to leverage the information in different modalities to learn a better representation and improve the performance of the task for different scenarios. An important task in multimodal learning is the fusion of modalities. Some previous work used BERT (Li et al. [2020a], Su et al. [2019]) or co-attention (Li et al. [2019], Tan & Bansal [2019]) to fuse different modal information simply. Subsequently, some large models (Li et al. [2021], Jia et al. [2021], Li et al. [2022]) were created to align the information of different modalities in terms of their semantic relationships using contrastive learning approach (Tsai et al. [2018]). Different pre-training approaches have also been extensively studied (Bao et al. [2022], Huang et al. [2021], Yao et al. [2021], Liang et al. [2020]). Crossmodal Transfer. The modality fusion approach directly depends on the integrity of the data from different modalities. However, the reality is often that we do not have access to the data of all modalities. Therefore, another direction of multimodal learning is to construct robust models to cope with missing modalities or crossmodal transfer. For example, knowledge in missing modalities can be complemented using autoencoders or generative adversarial approaches (Cai et al. [2018], Pan et al. [2021], Li et al. [2020b]). Ma et al. (Ma et al. [2021]) improves the robustness of Transformer models by automatically searching for an optimal fusion strategy regarding input data. Wang et al. (Wang et al. [2020]) proposed a framework based on knowledge distillation, utilizing the supplementary information from all modalities, and avoiding imputation and noise associated with it. Hager et al. (Hager et al., 2023) proposes the first self-supervised contrastive learning framework that takes advantage of images and tabular data to train unimodal encoders. But most of these approaches consider Vision-Language scenarios, audio or video, which have been well investigated and are not suitable for tabular data due to their structured character and the difference between numerical and categorical variables. Our approach fills the gap of multimodal learning on tabular modality by taking it into account. Learning with Tabular Data. The learning of tabular data has become an important research direction in the field of machine learning and data science for a long time. Traditional machine learning methods have been widely used on some tabular data, such as decision trees (Quinlan, 1986), support vector machines (Vapnik, 1999), and random forests (Breiman, 2001). These methods usually rely on pre-processing steps such as manual feature engineering and data cleaning, followed by model training and prediction using supervised learning. With the development of deep learning, tabular modeling approach using deep learning (Wang & Sun, 2022; Huang et al., 2020; Gorishniy et al., 2021) is very appealing because this allows tabular data to be used as input to a single modality and trained end-to-end by gradient optimization, which is competitive with GDBT methods (Chen & Guestrin, 2016; Ke et al., 2017; Prokhorenkova et al., 2018). In recent years, more and more approaches for tabular data have been proposed (Arik & Pfister, 2021; Hollmann et al., 2022; Yan et al., 2023; Jeffares et al., 2023). However, tabular data usually contains expert knowledge, such as medical diagnosis information of doctors and seismic waveform information, making it costly to acquire. So we consider such a scenario. Expert knowledge from the tabular data is used to guide the learning of the image data during training, with the expectation that good performance can be efficiently obtained even when the tabular data is missing during testing. ### 3 Preliminaries In this section, we first introduce the crossmodal transfer task, followed by some existing methods and some analysis. #### 3.1 Transfer Knowledge from Table to Images Formally, we define the crossmodal transfer training dataset $D_{train} = \{x^I_i, x^T_i, y_i\}_{i=1}^{N}$, where $x^I \in \mathbb{R}^{H_0 \times W_0 \times C_0}$ represent image data, $x^T \in \mathbb{R}^D$ represent tabular data and $y \in Y$ is the label space of the task. The image data is represented as a three-dimensional tensor with height $H_0$, width $W_0$, and RGB channels $C_0 = 3$, while the tabular data is a vector of dimension $D$, where each dimension corresponds to an attribute. We define the test dataset $D_{test} = \{x^I_j\}_{j=1}^{M}$, where tabular modality is missing due to high collection cost and the need for expert annotation. During training, we aim to minimize the empirical risk of model $f(x)$ over the training set: $$\sum_{(x^I_i, x^T_i, y_i) \in D_{train}} L(f(x^I_i), y_i \| x^T_i),$$ where $L$ is the loss function that measures the discrepancy between prediction and ground-truth label such as cross-entropy loss and $\|.$ indicates conditioning on the tabular data. The model can be decomposed into embedding and linear classifier: $f(x) = W^\top \phi(x)$, where $\phi(\cdot) : \mathbb{R}^D \rightarrow \mathbb{R}^d$ is the feature extractor to extract the embedding of the images and $W \in \mathbb{R}^{d \times \|Y\|}$. Our objective is to transfer relevant tabular information into the image model $f$. In situations where expert knowledge is not available, we expect the model to provide better predictions when only given the image data $x^I$ on the test set. #### 3.2 Methods for Crossmodal Transfer One of the main challenges in this task is how to transfer the tabular knowledge to the image model. It is feasible to align the two modality and then select the appropriate part for knowledge transfer. So we explore methods with alignment from different perspectives, including output-based transfer, parameter-based transfer, and embedding-based transfer. Output-based Transfer. To transfer knowledge from tabular data to image models, we aim to ensure that the predictions of image model $f$ and tabular model $g$ are aligned. To achieve this, we first train a classifier $g$ on the tabular data such as LightGBM (Ke et al., 2017). We then fit the prediction results of the image model $ f $ to $ g $ during the training. Knowledge Distillation (KD) (Hinton et al., 2015) is an output-based method: \[ L(x^I, x^T, y) = (1 - \lambda)L(x^I, y) + \lambda L_{KD}(f(x^I), g(x^T)). \] $ L_{KD} $ measures the similarity between the prediction of two models with Kullback-Leibler (KL) divergence $ g $ is called teacher network and $ f $ is student network. Aligning the output of the tabular model and the current model helps to reuse the knowledge in tabular data. So as Modality Focus Hypothesis (MFH) (Xue et al., 2022), the modality general decisive information is set according to the feature importance (Breiman, 2001; Wojtas & Chen, 2020) in tabular data as the teacher network, selecting subset of the tabular data. Then only use $ L_{KD} $ for distillation to fully observe the tabular’s influence on image. Parameter-based Transfer. The parameters of the model may contain part of the knowledge in the data, so the knowledge can be transferred from the perspective of the parameters of the model as well. For example, Fixed Model Reuse (FMR) (Yang et al., 2017) utilizes the learning power of deep models to implicitly grab the useful discriminative information from fixed models/features. In our setting, the fixed features referred to here are the tabular data: \[ L = y \log h(f(x^I) + g(x^T)) + \frac{1}{2} \\|x^T - \phi(x^I)U\\|_F^2. \] $ h $ is a soft-max operator and $ U $ is the linear connections between the tabular features and embedding of images. To transfer the influence of the fixed features $ x^T $ to images during the training procedure, FMR removes those connected parts corresponding to features $ x^T $ gradually and finally vanish all related components with the knockdown method. Embedding-based Transfer. The method expects to find a subspace in which the embedding of similar images and tabular data is as close as possible, while the embedding of dissimilar images is as far as possible. For example, Multimodal Contrastive Learning (MMCL) (Hager et al., 2023) proposes the self-supervised contrastive learning framework that takes advantage of images and tabular data to train unimodal encoders: \[ L = \lambda \ell_{I,T} + (1 - \lambda)\ell_{T,I}, \quad z_{jI} = f_{\phi_I}(\phi(x^I)), \\ \ell_{I,T} = - \sum_{j \in N} \log \frac{\exp (\cos(z_{jI}, z_{jT}) / \tau)}{\sum_{k \in N, k \neq j} \exp (\cos(z_{jI}, z_{kT}) / \tau)}, \] where embeddings are propagated through separate projection heads $ f_{\phi_I} $ and $ f_{\phi_T} $ and brought into a shared latent space as projections $ x_{jI}, z_{jT} $. $ \ell_{I,T} $ is calculated analogously. $ N $ denotes all subjects in a batch. Then MMCL uses linear probing of frozen networks to evaluate the quality of the learned representations. By mapping tabular and image data to the same space and utilizing contrastive learning methods, the knowledge in tabular data can be transferred into an image feature extractor. While the output-based, parameter-based, and embedding-based methods offer perspectives on transferring knowledge between modalities, each method has its own limitations. The output-based approach offers a simple and straightforward alignment based on the output of the model, but it may not capture detailed information for a certain attribute. The MFH method considers important features, but it completely discards other information during knowledge distillation. Parameter-based methods such as FMR cannot address the significant differences between tabular and image models, and the information contained in the parameters may be limited. The embedding-based approach attempts to find a common subspace for alignment but may lose some attribute information in the tabular data when changing the space, potentially ignoring valuable expert knowledge during transfer. By exploring these different transfer methods and their respective limitations, we can gain a deeper understanding of the challenges and opportunities in multimodal learning and develop more effective approaches for transferring knowledge from table to images. ### 4 Transferring Knowledge after Alignment Motivated by the unique characteristics of tabular data, we leverage it as auxiliary information in our approach to transfer knowledge to the image modality. Specifically, we minimize the mutual information between the image and each attribute of the table data, effectively transferring the relevant table knowledge to the image modality. Additionally, we use Optimal Transport to match the expert knowledge that can be expressed in the image data, allowing us to select a subset of the image features and strengthen the learning of the corresponding channels. Our approach highlights the importance of leveraging the specific characteristics of each modality to develop effective transfer. The flowchart is shown in Figure 1. 4.1 Preliminary Experiments We evaluate the quality of crossmodal transfer with MINE method, which uses mutual information, a measure of information in information theory that quantifies the amount of information contained in one variable about another (Belghazi et al., 2018). In our setting, a good image model based on tabular knowledge transfer should contain more tabular knowledge, resulting in higher mutual information both with the image and tabular data. To evaluate our approach, we conduct experiments on MFEAT dataset (van Breukelen et al., 1998), using two types of tabular data: 76 Fourier coefficients of character shapes and 6 morphological features. The image modality is reconstructed from 240 pixel averages of images from $2 \times 3$ windows. The result is shown in Figure 2. The Tab-Only and Img-Only methods are the result of models trained on a single modality. Our experiments indicate that existing methods for transferring tabular knowledge to image models yield low mutual information between the representations and tabular data. This suggests that these methods are not effective at transferring all types of tabular knowledge to the image modality and that feature selection is crucial. To validate this hypothesis, we perform knowledge distillation of the image model using two models trained on different parts of the tabular data. We find that morphological features in the tabular data can effectively promote image information, while other non-morphological features can make the tabular information more comprehensive. These results highlight the importance of the careful selection of different tabular attributes and their relationship with the image modality. Similarly, different channels exist for the images, and the choice of different channels can... also impact the final performance of the model. Since these methods do not transfer table information well, it is important to know how to use tabular knowledge. Based on these findings, we propose our method for transferring knowledge between modalities, which takes into account the specific characteristics of each modality and transfers expert knowledge to guide the image model. 4.2 Channel Tabular Alignment To extract the relevant information from the tabular data that is beneficial to the image model, we also use alignment-based methods for feature selection. This task consists of two main parts: first, obtaining the intermediate embedding of the image and tabular data; and second, performing alignment-based feature selection. To extract representations of the different channels, we use convolutional neural networks (CNNs). CNNs leverage convolutional filters to scan over the input data and extract local features. By stacking multiple convolutional layers, CNNs can learn increasingly complex and abstract features, allowing us to obtain different channels that capture different aspects of the image. Specifically, the channels of image data $ x^I $ are defined as $ \phi^{-1}(x^I) \in \mathbb{R}^{H \times W \times C} $, where $ C $ is the number of channels, and each channel corresponds to a high-level feature such as edges, whose shape is $ H \times W $. Similarly, we use a neural network to obtain the representation of each attribute of the tabular data. This involves transforming all features, including both categorical and numerical variables, into embeddings. The resulting attributes are defined as $ \psi(x^T) \in \mathbb{R}^{D \times E} $, where $ D $ is the number of attributes and $ E $ is the embedding dimension. We assume that the first $ p $ attributes are numerical variables $ x^T_{\text{num}} $, and the remaining $ q $ attributes are categorical variables $ x^T_{\text{cat}} $. Secondly, we use the optimal transport to align the channels of the image with the attributes of the tabular data [Benamou et al., 2015]. OT is a mathematical framework for measuring the similarity between probability distributions and finding the optimal way to transport mass from one distribution to another. The basic idea behind OT is to find a mapping between the elements of two distributions that minimizes the cost of moving one distribution to the other. The cost is typically defined as a distance metric between the elements. However, not all tabular attributes can be displayed on the image, and in some cases, there may be missing or irrelevant attributes that cannot be aligned with the image data. For example, on the PetFinder-adoption dataset, the photo of the pet can reflect the pet’s hair, body size, and other attributes, but not the health condition or vaccination status. To address this issue, we use the partial optimal transport (POT) algorithm [Chapel et al., 2020]. Specifically, To address the issue that different channels of an image may have repeated semantics with some redundancy, we use K-Means [Lloyd, 1982; MacQueen, 1967] clustering to group similar channels together. This allows us to obtain fewer distinct channels, each capturing a distinct aspect of the image data. Then we compute the cosine similarity of the dataset on each channel, resulting in a matrix $ S_I \in \mathbb{R}^{C' \times N \times N} $, where $ C' $ is the number of clustered channels and $ N $ is the length of the dataset. In parallel, we process the attributes of the tabular data similarly to obtain the attribute-wise similarity matrix $ S_T \in \mathbb{R}^{D \times N \times N} $. Then the cost matrix is constructed from the channel-wise similarity between attribute-wise similarity. Then the OT transfer matrix is calculated: \[ C_{ij} = \\| S_{T_i} - S_{I_j} \\|_2^2, \quad T = \arg \min_T \langle C, T \rangle_F, \] where $ \langle \cdot \rangle_F $ denotes the Frobenius norm. After aligning the distributions of the image and tabular data using optimal transport, we obtain the transfer matrix $ T \in \mathbb{R}^{D \times C'} $. Based on the clustering results, we can restore the corresponding relationship between the tabular attributes and the original channels of the image as $ A \in \mathbb{R}^{D \times C} $. Then the channels and attributes are aligned and relevant features are selected. 4.3 Learning with Auxiliary Information To leverage the knowledge of each attribute of the tabular data, we construct auxiliary tasks to learn this information. Specifically, we use the matrix $ A $ to weigh the image channels, allowing us to focus the attention of the relevant tabular attributes on the corresponding image channels. We use the feature extractor of an existing image network $ \phi(\cdot) $ to learn a classifier that maps from the image with a certain mask to the corresponding attributes of the tabular data. By doing so, we enhance the image network’s knowledge of the attributes of the tabular data and transfer this knowledge into the image modality. This allows the learned model to handle missing tabular modalities and improve its overall performance on complex tasks. In summary, the loss can be written in the following form \[ L = L(f(x^I), y) + L(g(x^T), y) + L_{i2t}, \] \[ L_{i2t} = \sum_p L_{MSE}(A_p \cdot \phi(x^I), x^T_{num,p}) + \sum_q L_{CE}(A_q \cdot \phi(x^I), x^T_{cat,q}). \] (6) Here, $L$ is the label prediction loss function such as cross entropy loss for classification tasks or mean square error loss for regression tasks. Since there may be numerical and categorical attributes for tabular data, we model them separately when constructing the loss to guide the image model to learn more information, expecting that the processing of different types is reasonable. The tabular model is updated in order to get a more accurate representation of each tabular attribute. $L_{CE}$ is cross entropy loss for categorical attributes and $L_{MSE}$ is mean square error loss for numerical attributes. This style of updating ensures that the model learns increasingly accurate channel-attribute correspondences, allowing the tabular data to guide the image data with increasing precision. By leveraging this approach, we can effectively transfer expert knowledge to images to develop more accurate and comprehensive image models for complex tasks. To sum up, our method leverages OT to align the distributions of different modalities and select relevant tabular attributes that are closely related to the image data. We then use the alignment to enhance the image learning of the relevant attributes, thus transferring expert knowledge from the tabular data to the image model. 5 EXPERIMENTS In this section, we compare Charms with crossmodal transfer methods on several datasets. The analysis experiment and ablations verify the effectiveness of our method. Moreover, we visualized the result of the alignment of attributes and channels. 5.1 EXPERIMENTS AND RESULTS Dataset. Totally six datasets are used in the experiment: Data Visual Marketing (DVM) (Huang et al., 2022) is created from 335,562 used car advertisements. The tabular data includes some car parameters such as the number of doors and some advertising data such as the year. Different from (Hager et al., 2023), only the new version DVM dataset is available. Car models with less than 700 samples were removed, resulting in 129 target classes, a classification task. SUNAttribute (Paterson et al., 2014). We use the table modality in this experiment to help images more accurately predict whether a scene is an open space, which is a binary classification task. CelebA (Liu et al., 2015) is the abbreviation of CelebFaces Attribute, meaning celebrity face attribute dataset. It’s a large-scale dataset with more than 200K celebrity images, each with 40 attribute annotations. We use Attractive as the label, which is a binary classification task. PetFinder-adoption dataset comes from a kaggle competition where the task is to predict the speed at which a pet is adopted, which is a five-class classification task. Tabular data contains information about the pet such as the type and vaccination status. PetFinder-pawpularity dataset also comes from a kaggle competition where the task was to predict the popularity of a pet based on that pet’s profile and photo. Avito is a challenge to predict demand for an online advertisement based on its full description, its context and historical demand for similar ads in similar contexts. The target deal_probability can be any float from zero to one. It’s also a regression task. Evaluation metrics. For classification tasks, we compute accuracy to measure the performance. For the regression task, we use root mean square error (RMSE) for performance evaluation. Implementation Details. In the course of the experiment, we implement CHRAMS with PyTorch and conduct experiments with a single GPU. Moreover, we utilize the grid search to find the hyper-parameters and we choose the best models from the validation set by using early stopping. Specifically, the batch size $k$ is searched in \{32, 64, 128\} and the learning rate is searched in \{1e-5, 5e-5, 1e-4, 5e-4, 1e-3, 5e-3\}. More details can be seen in Appendix A. Table 1: Comparisons with baseline methods on DVM, SUN, CelebA, Adoption, Pawpularity, and Avito datasets. The first four are classification tasks while the last two are regression tasks. RTDL means the FT-transformer (Gorishniy et al., 2021) model trained on the tabular modality. \| \| DVM ↑ \| SUN ↑ \| CelebA ↑ \| Adoption ↑ \| Pawpularity ↓ \| Avito ↓ \| \|----------\|-------\|-------\|----------\|------------\|---------------\|--------\| \| LGB \| 0.9748\| 0.8501\| 0.7963 \| 0.4101 \| 20.0720 \| 0.2290 \| \| RTDL \| 0.9682\| 0.8563\| 0.7936 \| 0.4107 \| 20.0844 \| 0.2317 \| \| Resnet \| 0.8743\| 0.8361\| 0.8146 \| 0.3477 \| 18.6150 \| 0.2512 \| \| KD \| 0.8390\| 0.8382\| 0.8118 \| 0.3532 \| 19.0683 \| 0.2499 \| \| MFH \| – \| 0.8312\| 0.7507 \| 0.3041 \| 43.1455 \| 0.2873 \| \| FMR \| 0.8427\| 0.8347\| 0.8003 \| 0.3526 \| 19.3517 \| 0.2937 \| \| MMCL \| 0.8203\| 0.8431\| 0.8041 \| 0.2981 \| – \| – \| \| CHARMS \| 0.9175 \| 0.8661 \| 0.8220 \| 0.3603 \| 18.4314 \| 0.2495 \| Table 2: Visualization by GradCAM. We conducted experiments on CelebA dataset and PetFinder-adoption. The results show that the OT algorithm can indeed align the tabular attributes with the image channels automatically. \| Tabular Attribute \| 5_o_Clock_Shadow \| Arched_Eyebrows \| Big_Nose \| Blond_Hair \| \|-------------------\|------------------\|-----------------\|----------\|------------\| \| Aligned Channel \| 65, 87, 119, 236…\| 33, 76, 78, 115,…\| 50, 224, 258, …\| 684 \| Visualization \| Tabular Attribute \| Type \| Color \| \|-------------------\|------\|-------\| \| Aligned Channel \| 399, 413, 414, 521…\| 400, 412, 425, 448…\| Visualization Results. To demonstrate the superiority of CHARMS, we compare it with other popular methods on six datasets as shown in Table 1. The result in the form of mean plus standard deviation are shown in Appendix Table 4. Our results show that CHARMS consistently achieves the best performance on all datasets. In contrast, the baseline methods we compared with do not significantly improve the performance compared to direct training with images. In fact, some of them even decrease the results. This is likely because these methods only use the tabular data to guide the image model at a coarse level, without considering the complex relationships and interactions between the modalities. As a result, the guidance provided by these methods is not sufficient for the image model to learn useful information, which can lead to confusion and poor results. The MFH approach only learns the KL divergence between the teacher and student networks, which may not be sufficient for handling complex tasks, as evidenced by its poor performance on the DVM 129 classification task. The experiment on the regression task is one of MMCL’s limitations according to (Hager et al., 2023). What is particularly surprising about our approach is that it can outperform the tabular modality on the SUNAttribute dataset. Similarly, on the CelebA and Pawpularity datasets, our approach can improve the performance of the image modality, even though the tabular data is weaker than images. It is possible that our approach can outperform the tabular modality even if it is a strong modality. These findings suggest that we indeed transfer tabular knowledge to images. Visualization. To verify the effectiveness of OT in matching tabular attributes and image channels, we used GradCAM (Selvaraju et al., 2017) to visualize the results of OT, as shown in Table 3. On the CelebA dataset, our model can accurately capture various table attributes for the same image. On the PetFinder-adoption dataset, we demonstrate our model’s ability to recognize the same attribute across different images. Our results demonstrate that OT is able to accurately match image channels with the relevant tabular attributes, highlighting the validity of our approach in integrating tabular knowledge into the image model. This supports the rationale behind our approach and highlights the importance of carefully aligning the distributions of different modalities to effectively transfer knowledge between them. 5.2 Experiments Analysis Comparison for CHArMS and other methods. During the training process, we visualize the mutual information to understand how the mutual information changes during the training process. Specifically, we take ten models from the beginning of training to convergence and calculated the mutual information. The results are shown in Figure 3. Our results show that the mutual information in CHArMS increases steadily during training, demonstrating the effectiveness in transferring knowledge between modalities and improving the accuracy and interpretability of the model. Comparing our approach with the MFH and FMR methods, we found that the MFH method initially selects important features using feature importance, leading to higher mutual information with the table, but as the model focuses more on the image information, the mutual information with the table decreases. The FMR method obtains a good initialization using the tabular data, but as the table modality is down-weighted, the mutual information with both the table and image decreases. Overall, visualizing mutual information provides important insights into the learning process of knowledge transfer models and can enhance the interpretability and effectiveness of these models, highlighting the importance of aligning the distributions of modalities and transferring knowledge between them. More discussions with attention method and CLIP (Radford et al., 2021) method are provided in Appendix B. The ablation study of components in CHArMS. To demonstrate the applicability and robustness of our proposed method, CHArMS, we conducted experiments using different network structures, including Densenet-121, Inception-v1, and MobileNet-v2, in addition to ResNet50. Our results, shown in Figure 4, demonstrate that the performance improvements achieved by our method are consistent across different network structures, highlighting the robustness of our approach. More visualisation and interpretative experiments are provided in Appendix C. 6 Conclusion In this work, we propose the CHArMS, a novel method that automatically transfers relevant tabular knowledge to images. Our method leverages tabular data as auxiliary information during transfer, enabling the transfer of expert knowledge in tabular data to images. Since not all attributes contained in tabular data are relevant to the corresponding image, we utilize optimal transport to align the attributes with channels, strengthening the correlated channels during transfer. Experimental results demonstrate that CHArMS outperforms previous methods in crossmodal transfer and our method enables insightful explanations of the learned visual embedding space with tabular instruction. We hope this work motivates future research on the challenges of multimodal encountered in real-world problems, with a particular focus on tabular data and knowledge transfer. REFERENCES Sercan Ö Arik and Tomas Pfister. Tabnet: Attentive interpretable tabular learning. In Proceedings of the AAAI conference on artificial intelligence, volume 35, pp. 6679–6687, 2021. Paul Baltescu, Haoyu Chen, Nikil Pancha, Andrew Zhai, Jure Leskovec, and Charles Rosenberg. Itemsage: Learning product embeddings for shopping recommendations at pinterest. In Proceedings of the 28th ACM SIGKDD Conference on Knowledge Discovery and Data Mining, pp. 2703–2711, 2022. Tadas Baltrušaitis, Chaitanya Ahuja, and Louis-Philippe Morency. Multimodal machine learning: A survey and taxonomy. 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Zh047FhXqI	As mentioned in the paper, PCM adopt more advanced network architecture to enhance performance so that even the PAM achieves better performance. Does this mean that the advantage of PCM may not come from the algorithmic novelty but from the better network architecture instead?	EFFECTIVE OFFLINE ENVIRONMENT RECONSTRUCTION WHEN THE DATASET IS COLLECTED FROM DIVERSIFIED BEHAVIOR POLICIES Anonymous authors Paper under double-blind review ABSTRACT In reinforcement learning, it is crucial to have an accurate environment dynamics model to evaluate different policies’ value in tasks like offline policy optimization and policy evaluation. However, the learned model is known to have large value gaps when evaluating target policies different from data-collection policies. This issue has hindered the wide adoption of models as various policies are needed for evaluation in these downstream tasks. In this paper, we focus on one of the typical offline environment model learning scenarios where the offline dataset is collected from diversified policies. We utilize an implicit multi-source nature in this scenario and propose an easy-to-implement yet effective algorithm, policy-conditioned model (PCM) learning, for accurate model learning. PCM is a meta-dynamics model that is trained to be aware of the evaluation policies and on-the-fly adjust the model to match the evaluation policies’ state-action distribution to improve the prediction accuracy. We give a theoretical analysis and experimental evidence to demonstrate the feasibility of reducing value gaps by adapting the dynamics model under different policies. Experiment results show that PCM outperforms the existing SOTA off-policy evaluation methods in the DOPE benchmark with a large margin, and derives significantly better policies in offline policy selection and model predictive control compared with the standard model learning method. 1 INTRODUCTION Environment model learning, which learns a model to approximate state transitions and reward functions of the environment, has extensive applications in Offline Policy Evaluation (OPE) (Thomas et al., 2015; Doroudi et al., 2017) and offline Reinforcement Learning (offline RL) (Lange et al., 2012; Levine et al., 2020). In OPE, policy value is estimated by calculating the return of simulated trajectories from the learned model. In offline RL, approaches utilize the model for planning or optimizing policy to maximize the return. Model accuracy significantly affects the efficacy of these methodologies. However, a learned model is known to have a large value gap when used to evaluate a target policy different from the data-collection policies (Xu et al., 2020; Clavera et al., 2018; Janner et al., 2019; Yu et al., 2020). This issue has hindered the adoption of models in many scenarios. In this article, we focus on learning an accurate dynamics model for offline policy optimization and policy evaluation. Most current model learning methods fit the whole dataset with a unified model and then utilize it for evaluation no matter what target policy is faced with. We refer to it as Policy-Agnostic dynamics Model (PAM). In many realistic situations such as robotic manipulation (Mandlekar et al., 2018; 2021), autonomous driving (Yu et al., 2018; Sun et al., 2020) and sequential recommendation (Saito et al., 2020; Gao et al., 2022), the offline data is collected by a wide range of different policies (including parameterized policies, rule-based policies and also human policies), and this inherent multi-source nature of the dataset remains rarely explored in recent works on dynamics model learning. In these learning tasks where the diverse data-collection behavior policies in fact correspond to different sources of state-action distributions (Ho & Ermon, 2016), the distribution of the offline dataset will be broad. Due to the state-action visitation frequency shift among different policies, state-action pairs visited by one policy may be infrequently observed by another policy, resulting in the offline data not always being beneficial for accurately evaluating the current policy. As shown later, attempting to learn samples from all policies may even impair the accuracy of predictions for the current policy. From this perspective, model learning from offline datasets collected by numerous behavior policies implies a feature of data fitting from a mixture of multi-source distributions, which is ignored in current learning paradigms. For accurate model learning in this scenario, we utilize the implicit multi-source nature caused by numerous data-collection policies and propose an easy-to-implement yet effective algorithm, policy-conditioned model (PCM) learning. PCM is a meta-dynamics model that is trained to be aware of the evaluation policies and make predictions by adapting to the evaluation policies’ state-action distribution to improve the prediction accuracy. In practice, we implement PCM via policy representation techniques (Duan et al., 2016; Chen et al., 2021; Nagabandi et al., 2019), which adopt an extra policy-aware module to on-the-fly encode policies’ representation and input the policy representations as well as a state-action pair into the meta-dynamics model. PCM produces different dynamics models given different policy representations. We theoretically show that PCM can achieve a smaller value gap for a target policy compared with PAM. Experiments are conducted based on MuJoCo (Todorov et al., 2012). We first conducted a proof-of-concept experiment, utilizing our custom-made dataset, which verified the effectiveness of the policy-aware mechanism for improving the model prediction accuracy. Then apply PCM in several downstream tasks. Results show that PCM improves the performance of off-policy evaluation in the DOPE benchmark with a large margin, and derives significantly better policies in offline policy selection and model predictive control compared with the standard model learning method. 2 PRELIMINARIES 2.1 Markov Decision Process and Reinforcement Learning We consider a Markov decision process (MDP) (Sutton & Barto, 2018) specified by the tuple $ M = (S, A, r, T, \gamma, \rho_0) $, where $ S $ is the state space, $ A $ is the action space, $ r(s, a) $ is the reward function, $ T(s'\|s, a) $ is the transition function, $ \gamma \in (0, 1) $ is the discount factor, and $ \rho_0(s) $ is the initial state distribution. In reinforcement learning (RL), we are typically concerned with optimizing or estimating the value of a policy $ \pi $ in a policy space $ \Pi $. Specifically, value is defined as: \[ V^\pi = \mathbb{E}_{s_0 \sim \rho_0, s_1 \sim \pi, a_0 \sim \pi} \left[ \sum_{t=0}^{\infty} \gamma^t r(s_t, a_t) \right]. \] (1) For a fixed policy $ \pi $, the MDP becomes a Markow chain, and we define the occupancy measure $ \rho^\pi(s, a) = (1 - \gamma) \sum_{t=0}^{\infty} \gamma^t \pi_t(s_t = s, a_t = a) $ then the policy value can be rewritten as $ V^\pi = \mathbb{E}_{s, a \sim \rho^\pi} [r(s, a)] $. When different dynamics are involved, we use an additional subscript to indicate the transition, e.g. $ V^\pi_T $ and $ V^\pi_T $. 2.2 Off-policy Evaluation Off-policy evaluation (OPE) (Le et al., 2019; Precup et al., 2000; Jiang & Li, 2016; Kostrikov & Nachum, 2020; Yang et al., 2020; Wen et al., 2020) aims at estimating the value $ V^\pi $ of a target policy $ \pi $, based on a fixed dataset of transitions $ D $ collected from some behavior policies $ \{\mu_i\}_{i=1}^n $ (or named data-collection policies). This problem is of great practical significance for several reasons, including providing high-confidence guarantees prior to deployment, performing policy improvement, and model selection. A major challenge in OPE is the distribution shift between the behavior policy and the target policy, which induces a large value gap between the estimated value and the true value. 3 RELATED WORKS Off-policy Evaluation (OPE): OPE research is relevant to many practical domains such as recommendation systems (Li et al., 2011), health (Liao et al., 2019), and education (Mandel et al., 2014). There exists a large body of work on OPE, including methods based on fitted q-evaluation (Le et al., 2019; Hao et al., 2021) and importance sampling (Kostrikov & Nachum, 2020). Another class of OPE is the model-based approach (also referred to as the direct method), which is focused on in this paper. While model-based OPE has been considered by many previous works (Thomas & Brunskill, 2016; Hanna et al., 2017), they are confined to simple tasks and produce biased predictions owing to the restricted range of state and action space in offline trajectories (Fu et al., 2021b). By contrast, our approach is applied to more intricate tasks and proves that model-based OPE can also do well in challenging continuous tasks. Model should be aware of policies: There are some previous works in other fields also proposing the idea that the dynamics model should be aware of or focus on certain policies rather than all the policies. PAML (Abachi et al., 2020) proposes that model learning should incorporate the way the planner is going to use the model. PDML (Wang et al., 2022) dynamically adjusts the historical policy mixture distribution to ensure the learned model can continually adapt to the state-action visitation distribution of the evolving policy. However, in contrast to us, both of them are concerned with the online RL setting and achieving the policy-aware mechanism by adjusting the sampling distribution from the replay buffer. Our method considers the offline RL setting and explicitly incorporates policy representation as an extra input for model learning. Model-based Offline RL: Model-based Offline RL (MBORL) algorithms also involve dynamics models for some downstream tasks. From the perspective of model usage, MBORL can generally be categorized into two groups: model predictive control (MPC) (Camacho & Alba, 2013) and Policy learning (PL). In MPC, Argenson & Dulac-Arnold (2021) directly performs planning in a learned dynamics model. In PL, a policy can be trained either in an in-support region by utilizing a conservative surrogate MDP (Yu et al., 2020; Kidambi et al., 2020; Yu et al., 2021), or in out-of-policy regions by learning an adaptive policy (Chen et al., 2021). Some works also utilize dynamics models with off-the-shelf model-free algorithms for better policy learning (Lyu et al., 2022; Wang et al., 2021). Recent studies (Rigter et al., 2022; Yang et al., 2022) also adopt an adversarial framework that alternates between dynamic-model training and policy learning. However, these works pay more attention to optimizing policy under a restricted dynamics model instead of directly learning a faithful model when using it, where the latter is what our work focuses on. ### 4 POLICY-CONDITIONED DYNAMICS MODEL LEARNING In this section, we first give the metric to evaluate the gap between true dynamics and a learned model in Sec. 4.1 and the intuition for policy-conditioned model (Sec. 4.2). Then we formally introduce the policy adaptation mechanism of PCM from an error reduction perspective (Sec. 4.3) and show this mechanism also leads to a better generalization to out-of-distribution data (Sec. 4.4). #### 4.1 VALUE GAPS BETWEEN TRUE DYNAMICS AND A LEARNED MODEL Offline dataset $ \mathcal{D} = \{\tau_m\}_{m=1}^{M} $ consists of previously collected trajectories $ \tau_m = (s_0, a_0, r_0, s_1, \ldots) $, each of which is generated by the interaction between one of the behavior policies $ \Omega = \{\mu_i \| i \in I\} $ and the environment. Here we consider the multiple diversified behavior policy case, which coincides with many realistic situations. It should be noted that this is not a setting raising new challenges but a refined description of the existing problem, which provides more information that could be utilized for dynamics modeling compared to simply ignoring the multi-source property of the dataset. We follow the basic idea in OPE to define the performance metric of a dynamics model: in an MDP, a good dynamics model means for any target policy $ \pi $, the gap between the value under true transition $ T^* $ and the value estimation under $ \hat{T} $ is small, i.e., $ \|V_{T^}^\pi - V_{\hat{T}}^\pi\| $ is small. Inspired by a previous study (Janner et al., 2019), the value gaps between true dynamics and a learned model is bounded by \[ \|V_{T^}^\pi - V_{\hat{T}}^\pi\| \leq \frac{2R_{\text{max}}}{(1-\gamma)^2} l(\pi, T^, \hat{T}), \] where $ l(\pi, T^, \hat{T}) = \mathbb{E}_{s,a \sim \rho^\pi} D_{\text{TV}}(T^(\cdot\|s,a), \hat{T}(\cdot\|s,a)) $ is total variation divergence between true and learned transitions under the state-action occupancy of the target policy $ \pi $ to measure the model error. Eq. (2) implies that as long as we reduce the model error $ l(\pi, T^, \hat{T}) $ under the target policy’s distribution $ \rho^\pi $, we can guarantee the reduction of the corresponding upper bound of the value gap. The bound is an extension of previous bounds in Janner et al. (2019); Xu et al. (2020; 2021), where we further consider the generalization ability of the learned models. The full derivation is in App. A.1. #### 4.2 THE INTUITION FOR POLICY-CONDITIONED MODEL LEARNING In Fig. 1, we use an example to illustrate why policy-conditioned model (PCM) learning is superior to policy-agnostic model (PAM) learning. Suppose we wish to learn an environment model where a biped robot is asked to move forward from an offline dataset including different locomotion patterns, such as walking, running, jumping, etc. Currently, the standard dynamics model, i.e., the policy-agnostic model (PAM), learns to predict all of the transitions coming from different locomotion patterns in one unified model. However, we notice that different locomotion patterns usually correspond to quite different transition patterns though these patterns can be regarded as a single task. For instance, jumping requires both legs to be folded and unfolded at the same time while running involves alternate flexion and extension of the legs. If we can utilize this nature, the learning complexity will be reduced. Based on the above motivation, instead of learning a single model for the whole dataset, we propose to “divide” the dataset according to the data-collection policy and learn a model for each subset. Figure 1: An illustration of the difference between the policy-agnostic model (left) and the policy-conditioned model (right). Suppose we wish to learn an environment where a biped robot is asked to move forward from an offline dataset including different locomotion patterns, such as jumping, walking, running, etc. Different locomotion patterns usually correspond to quite different transition patterns even though they can be regarded as a single task. We regard each locomotion pattern as a subtask and respectively learn a model for each subtask. In this way, we can reduce the learning difficulty of each model, which is expected to obtain a more accurate model for each data-collection policy. The rationale behind this is that each data-collection policy only focuses on a relatively small subregion of the support set of the whole mixed state-action distribution, thus training the model under the state-action occupancy of each policy should be an easier task than the global model training and tends to obtain more accurate models. Moreover, if the target policy to be evaluated is unseen before in the dataset, e.g., jogging, which is a locomotion pattern between walking and running, it is hoped to yield a new model to adapt to the jogging policy by combining the walking model and the running model. 4.3 THE POLICY ADAPTATION MECHANISM FOR MODEL LEARNING With a dataset $D$ collected by a set of diversified behavior policies $\Omega = \{\mu_i \| i \in I\}$, the training data distribution is a mixture of occupancy measures $\rho^{\text{mix}}(s, a) = \sum_{i \in I} w_i \rho^{\mu_i}(s, a)$, where $w_i$ is data proportion of policy $\mu_i$. Conventional model learning fits a universal transition model directly under the whole mixed data distribution and rolls out whatever target policy in this policy-agnostic model: $$\hat{\psi} = \arg\min_{\psi \in \Psi} \sum_{\mu_i \in \Omega} w_i l(\mu_i, T^, T_\psi),$$ where model $T$ is parameterized by $\psi \in \Psi$. This is sufficient for simple environments where the model capacity is rich enough to completely recover true transitions. However, in realistic large-scale tasks, the model’s capacity is limited in comparison to true transition, resulting in a non-zero error, which will be further compounded during long-horizon rollout (Janner et al., 2019; Xu et al., 2020). With an adequate model capacity, it is possible to accurately fit true transition dynamics, which is the unique optimal model for any target policy. Nevertheless, the usually limited model capacity prevents perfect transition modeling and requires a proper allocation of the finite accuracy budget to facilitate the target policy rollout as much as possible. Since different policies perform distinct behaviors and access varied subregions of the state-action space, their optimal models within the model space are different, resulting in an optimal model inconsistency, i.e., there does not exist a unique model within the model space that is optimal for general target policies. A consequent idea is to select dynamics models adaptively for different policies, where each model is optimized specially for the occupancy measure of its corresponding policy. We name it policy-conditioned model (PCM). This "model selection" procedure can be expressed through a mapping $F : \Pi \rightarrow \Psi$, where each policy $\pi$ is associated with a model $T_{F(\pi)}$. Learning a PCM is therefore translated into finding an optimal $F$ to minimize model error on the data distribution of each policy: $$\hat{F} = \arg\min_{F \in \mathcal{F}} \sum_{\mu_i \in \Omega} w_i l(\mu_i, T^, T_{F(\mu_i)}),$$ where $\mathcal{F}$ is function space of $F$. For behavior policies $\mu_i$, model error $l(\pi, T^, T_{F(\mu_i)})$ can be reduced to achieve smaller value gaps compared to PAM as shown empirically in the experiments in Sec. 5.1 and 5.2. This is intuitive since PAM attempts to fit global transition dynamics, which is more difficult than local transition modeling that PCM specializes in. For those new target policies $\pi$, the learned models have to extrapolate to the data distribution $\rho^\pi$, resulting in an extra generalization error. We show a generalization benefit brought by the adaptation mechanism in the next section. Remark 1 (Varied dynamics models for different policies): The formulation of PCM is similar to a meta-learning objective with $F$ representing the meta module (Rakelly et al., 2019). At first glance, it is counterintuitive to build the problem as a meta-optimization problem since all policies are deployed in the same environment $T^$, meaning that the ground-truth parameter of the dynamics model $T_\psi$ among different policies $\pi$ should be the same, while the PCM gives varied models adapted to different policies. In fact, this adaptation method under limited model capacity resembles human behaviors under limited attention. For example, when a man drives a car, his attention focuses on the road, and therefore the predictions of vehicle movement are relatively clear in his mind while the flight trajectories of the birds in the sky are blurred. On the contrary, when the man stops the car and starts to observe the flying birds, his focus will be shifted. Even in the same environment, the models in his brain differ in the accuracy assignment when performing different tasks. This adaptability reflects an attempt to maximize efficiency in the use of limited attention (or capacity), and our policy-conditioned model actually shares a common idea. Implementation: In real-world applications, the corresponding white-box policies are typically unknown. It is impractical to learn a mapping function $F(\pi)$ which directly takes policy $\pi$ as the input. Inspired by many previous works (Duan et al., 2016; Chen et al., 2021; Nagabandi et al., 2019) which have successfully utilized RNN as an extra representation extractor module to map the interaction trajectories to some task-specific meta-parameters, we use similar RNN structure to learn and infer policy representations from given interaction trajectories, and a policy-representation-conditioned dynamics model is learned to adapt its predictions based on the input policy representation. Formally, let $\tau_{0:t} = (s_0, a_0, s_1, a_1, ..., s_t, a_t)$ be a trajectory generated by a data-collection policy up to timestamp $0 \leq t \leq H - 1$ ($H$ is the horizon of the MDP) and the offline dataset is a set of $N$ trajectories $\mathcal{D} = \{\tau^{(j)}\}_{j=1}^N$. For any timestamp $t$, trajectories $\tau_{0:t-1}$ will be fed into a recurrent neural network $q_\phi(\tau_{0:t-1})$ to obtain an embedding $z_t$. After that, an adaptive dynamics model $T_\psi(s_{t+1}\|s_t, a_t, z_t)$ is learned to adapt its predictions of $s_{t+1}$ based on $z_t$. Recall that we expect to get a representation of a policy, the embedding $z_t$ should encode salient information about the policy. To this end, we simply incorporate a policy decoder $p_\theta(a_t\|s_t, z_t)$ and the encoder and decoder are jointly optimized to reconstruct the specified policy. In summary, the overall learning objective of PCM is: $$\min_{\phi, \theta, \psi} \mathbb{E}_{t \sim [0,H-2], \tau_{0:t+1} \sim \mathcal{D}}[-\log T_\psi(s_{t+1}\|s_t, a_t, q_\phi(\tau_{0:t-1})) - \lambda \log p_\theta(a_t\|s_t, q_\phi(\tau_{0:t-1}))],$$ where $\lambda$ is a hyperparameter. Note that the gradients would be backpropagated from $T_\psi$ and $p_\theta$ to $z$ if optimal models’ or policies’ parameters in different trajectories are inconsistent but have the same representation of $z$, then the parameters of $\phi$ will be updated automatically to distinguish them. Our pseudo-code of the overall PCM learning is shown in Alg. 1. Remark 2 (Model complexity): The module $F$ in PCM (i.e. the RNN module in our implementation) introduces additional model complexity compared to the model space of PAM. One may suspect that it is the increased model capacity helps the dynamics model learning, instead of the policy-conditioned mechanism. In fact, we find that simply increasing the model capacity by using a larger network without mechanism changes cannot bring significant improvement (as shown in Sec. 5.3.1). The additional module works mainly because it allows an adaptive and therefore effective utilization of the limited capacity for different target policies, which reduces the in-distribution model error and also brings generalization benefits for new target policies as we show in the next subsection. 4.4 Adaptation Effect Improves the Generalization In this section, we show that the adaptation effect from PCM can provide additional generalization benefits when the learned model extrapolates to the data distribution of new target policies absent from the training dataset. We introduce an assumption on the smoothness of well-trained models: Assumption 4.1. For the learned model $T$, the point-wise model error $D_{TV}(T^(\cdot\|s,a), T(\cdot\|s,a))$ is $L$-Lipschitz with respect to the state-action pairs, i.e., $$\|D_{TV}(T^, T)(s_1, a_1) - D_{TV}(T^, T)(s_2, a_2)\| \leq L \cdot D((s_1, a_1), (s_2, a_2)),$$ where $D(\cdot, \cdot)$ is some kind of distance defined on the state-action space $S \times A$. Assump. 4.1 measures the local generalization ability of a learned model. Generally speaking, if we say the learned model $T_\psi$ generalizes well w.r.t. the state-action inputs, we mean that for some unseen $(s_2, a_2)$ deviating from a training data $(s_1, a_1)$, the point-wise model error will not increase much, reflected by a bounded $L$. Based on this assumption, we find that the expected model error of PCM under the target policy data distribution can be controlled: Figure 2: Illustration of model error, value gap of policy learned in with or w/o policy embedding model, and the heatmap about the performance of evaluating policy with different policy embedding. Proposition 4.2.* Under Assump. 4.1, for any policy $ \pi \in \Pi $, model error of PCM $ T_{F(\pi)} $ is bounded: \[ l(\pi, T^, T_{F(\pi)}) \leq \min_{\mu_i \in \Omega} \left\{ l(\mu_i, T^, T_{F(\mu_i)}) + L \cdot W_1(\rho^\pi, \rho^{\mu_i}) - C(\pi, \mu_i) \right\}, \] where adaptation gain $ C(\pi, \mu_i) := l(\pi, T^, T_{F(\mu_i)}) - l(\pi, T^, T_{F(\pi)}) $, $ W_1 $ is Wasserstein-1 metric. The adaptation gain $ C(\pi, \mu_i) $ summarizes the benefit of the policy adaptation effect based on the insight that when testing on a new policy $ \pi $ within some effective region, the model $ T_{F(\pi)} $ customized for $ \pi $ should have a smaller model error under the target distribution $ \rho^\pi $ than any $ T_{F(\mu_i)} $. PAM does not include the policy-conditioned mechanism and the adaptation gain is always zero. Simply fine-tuning the PAM parameters for a new policy is not practical because it requires the interaction of the new policy with the environment to collect the target domain experiences, which is prohibitive in general. In contrast, the policy representation serves as an additional covariate in PCM, which enables an extra adaptation ability to target policies and hence a non-zero adaptation gain term with no need for the real experiences in the target domain and also the model parameter fine-tuning. Therefore, Prop. 4.2 shows that the model error of PCM for a new target policy $ \pi $ is reduced by the adaptation gain $ C_i $, if $ C > 0 $, compared with PAM. However, it is hard in general to rigorously analyze the adaptation gain $ C(\pi, \mu_i) $ because of the complexity of neural networks and the optimization process. Empirically, as the target policy $ \pi $ gradually diverges from $ \Omega $, the adaptation gain will increase from zero and partially reduce the extrapolation error within an effective adaptation region. When $ \pi $ leaves far enough from $ \Omega $, $ C $ will reach the maximum and then start to decrease. This trend exhibits the efficacy of policy adaptation to a reasonable degree. We provide experimental evidence in Sec. 5.2, which aligns with the intuition. We also discuss two extreme cases of zero adaptation effect and complete cancellation of extrapolation error in App. A.3, and the realistic case lives between the two extremes. ## 5 EXPERIMENT In this section, we first justify the efficacy of the policy adaptation mechanism for model learning via a proof-of-concept experiment (Sec. 5.1). In Sec 5.2, we conduct experimental studies to verify PCM enjoys smaller value gaps as analyzed in Sec. 4.4. Then we evaluate PCM on specific downstream tasks including off-policy evaluation (OPE), offline policy selection (OPS) and model predictive control (MPC), in contrast to PAM (Sec 5.3). Finally, we analyze the learned policy embedding by PCM to verify whether it learns reasonable policy representation (Sec 5.4). ### 5.1 Proof-of-Concept Verification on the Policy Adaptation Mechanism We consider a simplified setting that does not involve generalization to unseen policies to justify the idea of the policy adaptation mechanism for model learning. We collect a dataset sampled by 10 different policies in HalfCheetah and solely choose one of the 10 policies for evaluation. Since there is no need for generalization, we can use a simple policy representation scheme called vector policy embedding, $ F(\mu_i) $. Specifically, we employ a $ n \times m $ matrix to represent the policies, where $ n $ is the number of policies in the dataset and $ m $ is the dimension of the policy representation. The matrix can be updated by backpropagation. We compared the performance of the model with and without embedding. Fig. 2(a) and 2(b) show even with such a simple policy representation scheme, PCM can significantly outperform PAM on the model error as well as the value gap. Furthermore, we show that the vector policy embedding indeed helps the model adapt to a specific policy. We first train and obtain an embedding for each policy, After training, we have 10 different --- 1 code: the code will be published after the acceptance. vector policy embeddings for these 10 policies, respectively. Then we evaluate each policy under models given different vector embeddings and record the value gap under each case. The results are shown in the mismatch heatmap below. Fig. 2(c) shows that the model performs better under policy with better-matched embedding (for any two policies, the closer their numbers are, the more similar they are), indicating that the vector policy embedding helps the model adapt to a specific policy. 5.2 Empirical Evidence of PCM Having Smaller Value Gaps Prop. 4.2 indicates that the value gap of PCM for an unseen policy $\pi$ can be reduced by 1) a smaller model error on the training dataset; 2) a positive adaptation gain $C$. We now present empirical evidence to support our analysis and demonstrate that PCM indeed has smaller value gaps. All experiments in this section are conducted in the HalfCheetah environment. We first compare the model error of PAM and PCM on the training dataset. As shown in Fig. 3(a), PCM enjoys a smaller model error than PAM. We then analyze the adaptation gain quantitatively by fixing a data-collection policy $\mu_i$ and computing $C(\pi, \mu_i)$ for different policies $\pi$. We refer to Appx. E.2 for more details. As illustrated in Fig. 3(b), the gain gradually increases with the policy divergence, reaches a maximum, and decreases as the policy divergence continues increasing. This confirms the analyzed cases in Sec. 4.4. Finally, we directly compare the value gaps of PAM and PCM and also investigate the influence of different levels of dataset diversity on them. To do so, we construct datasets with varying levels of diversity (0%, 20%, 50%, 80%, 100%), where the percentage indicates that the dataset is created from the replay buffer of SAC (Haarnoja et al., 2018) until the policy reaches the specific level of performance. Appx. E.1 presents details of the data collection process. We train PAM and PCM on each dataset and test them on other 11 policies provided by the DOPE benchmark (Fu et al., 2021a), which were unseen before in the datasets. Fig. 3(c) depicts the value gap of each model trained on each dataset, demonstrating that PCM can achieve smaller value gaps. Moreover, the results show that as the diversity of the dataset increases, both PAM and PCM achieve smaller value gaps, with PCM exhibiting a more substantial advantage. ![Figure 3](image) Figure 3: Left illustrates model errors (computed by mean squared error) of PAM and PCM on the training dataset. Medium illustrates the adaptation gain of PCM for different unseen policies $\pi$, relative to a data-collection policy $\mu_i$. The right illustrates normalized value gaps of PAM and PCM trained on datasets with different levels of diversity when testing on 11 target unseen policies. 5.3 Evaluation on Downstream Tasks 5.3.1 Off-policy Evaluation We compare PCM with several OPE methods, including: Fitted Q-Evaluation (FQE) (Le et al., 2019), that estimates the policy value via iteratively performing Bellman update, Doubly Robust (DR) (Jiang & Li, 2016), that combines the importance sampling technique with a value estimator for variance reduction, Importance Sampling (IS) (Kostrikov & Nachum, 2020), that performs importance sampling with a learned behavior policy, DICE (Yang et al., 2020), that uses a saddle-point objective to estimate marginalized importance weights $d^\pi(s, a)/d^{\pi_B}(s, a)$, Variational Power Method (VPM) (Wen et al., 2020), that runs a variational power iteration algorithm to estimate the importance weights without the knowledge of the behavior policy, Policy-Agnostic Model (PAM), that removes the policy representation module in PCM and serves as the ablation method. We evaluate these approaches on a variety of tasks from DOPE-D4RL and DOPE-RL-Unplugged benchmarks (Fu et al., 2021a). The data in these tasks is collected by diverse policies, which aligns with the multi-source assumption in our theoretical analysis. Fig. 4 shows the performance of PCM and other methods in three metrics (details of the metrics and results separated by tasks are in App. C). We find that PCM outperforms other methods by a large margin. Specifically, the results of the absolute error provide direct evidence that PCM can reduce the value gap effectively. Besides, PCM obtains a higher rank correlation and lower regret, indicating that PCM can not only perform accurate evaluation but also select the competitive policies among the policies to be evaluated. Note that PAM also shows competitive performance among these algorithms, which contradicts results from most previous works (Fu et al., 2021a; Voloshin et al., 2019). This is because we incorporate some components of modern neural networks both into PAM and PCM. To be more specific, we find that classical MLP (denoted as PAM(old arch)) is not well-suited for autoregressive predictions when evaluating a policy and is susceptible to compounding errors, as shown in Fig. 5. After introducing components of modern neural networks, including residual connection, layer normalization, and dropout, into our baseline (denoted as PAM(new arch)), we observe a significant reduction in the compounding error as well as a remarkable improvement in overall performance, as illustrated in Fig. 5 and Tab. 1. Table 1: OPE performance of different networks, in terms of absolute error (value gap), rank correlation, and regret. We bold the best scores for each metric. \| Method \| Absolute error \| Rank correlation \| Regret \| \|----------------------\|----------------\|------------------\|----------\| \| PAM (old arch) \| 0.51±0.04 \| 0.47±0.10 \| 0.22±0.07\| \| PAM (new arch) \| 0.29±0.06 \| 0.62±0.09 \| 0.13±0.10\| \| PAM (new arch larger)\| 0.26±0.05 \| 0.61±0.07 \| 0.12±0.08\| \| PCM \| 0.19±0.03 \| 0.77±0.05 \| 0.07±0.03\| Furthermore, to keep a balanced capacity, we increase the size of the network for PAM (from 200 hidden size & 4 layers to 400 hidden size & 4 layers (denoted as PAM(larger))) and the result is shown in Tab. 1. It shows that even with increasing the size of PAM, PAM still falls behind PCM. 5.3.2 Offline Policy Selection In this section, we explore the efficacy of using PCM on offline policy selection (OPS) for a practical offline RL algorithm. Specifically, we train MOPO (Yu et al., 2020) for 1000 epochs and record policy snapshots at the latest 20 epochs for OPS. We compare our method against PAM and FQE as well as directly selecting the last-epoch policy. Tab. 2 shows the performance gains by different methods. The performance gain is computed by $\frac{V_{\text{selected}} - V}{V_{\max} - V} \times 100\%$, where $V_{\text{selected}}$ is the value of the selected policy and $V, V_{\max}$ are the average and max values of the evaluated policies, respectively. It is noteworthy that the gains of FQE and PAM are even lower than directly selecting the last-epoch policy, also indicated in another work (Qin et al., 2022). In contrast, our approach shows a brilliant performance, implying that it reliably chooses a better policy for an offline RL algorithm to deploy. Table 2: Performance gain of offline policy selection for MOPO (Yu et al., 2020) by different methods. \| Task Name \| Last Epoch \| FQE \| IS \| DICE \| PAM \| PCM (Ours) \| \|--------------------------\|------------\|-----\|-----\|------\|-----\|------------\| \| halfcheetah-medium-replay\| 39.3% \| 23.0%\| 87.8%\| 1.6% \| 1.6%\| 98.4% \| \| hopper-medium-replay \| 56.0% \| 34.1%\| 56.0%\| 19.8%\| 47.3%\| 64.8% \| \| walker2d-medium-replay \| -4.6% \| 4.6% \| 34.3%\| 13.0%\|-30.6%\| 51.9% \| \| Average \| 30.2% \| 20.6%\| 59.4%\| 39.3%\| 11.5%\| 71.7% \| 5.3.3 Model Predictive Control An accurate model can also be expected to perform effective model predictive control (MPC). We therefore compare our proposed PCM against PAM and the true dynamics (using MuJoCo simulator itself as true dynamics). Following Chua et al. (2018), we use the cross-entropy method (CEM) as the optimization technique in MPC, which samples actions from a distribution closer to previous action samples yielding high rewards. More details on MPC and CEM are discussed in App. F. Fig. 6(a) shows the cumulative rewards of the three methods during an episode, from which we can see that PCM performs similarly to the true dynamics and significantly outperforms PAM. To further explore why our approach works better, we calculate regret values of the evaluation of action sequences for PCM and PAM respectively. We track several planning processes and compute regret \[ \sum_{i=t}^{t+T} \mathbb{E}_{T^}[r(s_i, a_i^)] - \sum_{i=t}^{t+T} \mathbb{E}_{T^}[r(s_i, \hat{a}_i)] \] for both PAM and PCM, where $ \hat{a}_{t:t+T} $ and $ a^_{t:t+T} $ are the optimal action sequences selected by the learned model and true dynamics respectively. Regret is the difference between the real value of the action sequence selected by the model and the value of the optimal action sequence. Results in Fig. 6(b) shows that PCM has lower regret than PAM, meaning that our approach tends to pick out actions that are closer to the optimal policy. 5.4 Analysis of Learned Policy Representation In this section, we conduct a study to verify whether the PCM can learn reasonable policy representations. We select several policies with different performance and feed the trajectories generated by these policies into the policy encoder module of PCM. We visualize the outputted policy representations via the t-SNE (van der Maaten & Hinton, 2008) technique in Fig. 7. We find that the policies with similar performance have similar policy representations since there is a degree of resemblance between their performed actions, while the representations of policies with widely different performance are far apart due to their quite different behavior. This result demonstrates that PCM can effectively identify similar policies and distinguish different policies. We provide results on more tasks in Appx. D. 6 Discussion and Future Work This paper handles the challenge that a learned dynamics model tends to have a large value gap when used to evaluate a target policy different from the data-collection policies when the offline dataset is collected by diverse behavior policies. We propose training a Policy-Conditioned Model (PCM) that generates distinct dynamics models based on different target policies. We demonstrate that PCM can achieve smaller value gaps by reducing training errors and better generalization to out-of-distribution data. Empirical results across domains and algorithms validate the superiority of our approach. It should be noted that several possible ways exist to implement the policy-conditioned mechanism, and the RNN-based policy encoding employed in this work is just one of them. Another limitation is that we analyze the generalization of PAM and PCM based on the infinite sample assumption for each behavior policy. However, for realistic situations where only finite samples are available, the data from each policy are limited, and additional estimation errors occur in the model learning, which requires further analysis to compare PAM with PCM. In the future, we aim to find a more efficient policy representation scheme to enhance the model’s generalization ability. REFERENCES Romina Abachi, Mohammad Ghavamzadeh, and Amir-massoud Farahmand. Policy-aware model learning for policy gradient methods. CoRR, abs/2003.00030, 2020. Arthur Argenson and Gabriel Dulac-Arnold. Model-based offline planning. In 9th International Conference on Learning Representations (ICLR’21), virtual event, 2021. E.F. Camacho and C.B. Alba. Model Predictive Control. Advanced Textbooks in Control and Signal Processing. Springer London, 2013. ISBN 9780857293985. 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i92ssjkZCz	Table 8 (a): the detection performance wrt. masking ratio didn't change much when the masking ratio ranged from 0.1 to 0.7. A small masking ratio leads to little information loss, but this pre-training strategy still works well compared to other methods. Hence it is uncertain whether this performance improvement really comes from the masking-and-completion paradigm or other tricks.	UNiPAD: A UNIVERSAL PRE-TRAINING PARADIGM FOR AUTONOMOUS DRIVING Anonymous authors Paper under double-blind review ABSTRACT In the context of autonomous driving, the significance of effective feature learning is widely acknowledged. While conventional 3D self-supervised pre-training methods have shown widespread success, most methods follow the ideas originally designed for 2D images. In this paper, we present UniPAD, a novel self-supervised learning paradigm applying 3D volumetric differentiable rendering. UniPAD implicitly encodes 3D space, facilitating the reconstruction of continuous 3D shape structures and the intricate appearance characteristics of their 2D projections. The flexibility of our method enables seamless integration into both 2D and 3D frameworks, enabling a more holistic comprehension of the scenes. We manifest the feasibility and effectiveness of UniPAD by conducting extensive experiments on various downstream 3D tasks. Our method significantly improves lidar-, camera-, and lidar-camera-based baseline by 9.1, 7.7, and 6.9 NDS, respectively. Notably, our pre-training pipeline achieves 73.2 NDS for 3D object detection and 79.4 mIoU for 3D semantic segmentation on the nuScenes validation set, achieving state-of-the-art results in comparison with previous methods. 1 INTRODUCTION Self-supervised learning for 3D point cloud data is of great significance as it is able to use vast amounts of unlabeled data efficiently, enhancing their utility for various downstream tasks like 3D object detection and semantic segmentation. Although significant advances have been made in self-supervised learning for 2D images (He et al., 2022; 2020; Chen & He, 2021; Chen et al., 2020a), extending these approaches to 3D point clouds have presented considerably more significant challenges. This is partly caused by the inherent sparsity of the data, and the variability in point distribution due to sensor placement and occlusions by other scene elements. Previous pre-training paradigms for 3D scene understanding adapted the idea from the 2D image domain and can be roughly categorized into two groups: contrastive-based and MAE-based. Contrastive-based methods (Zhang et al., 2021; Chen et al., 2022c) explore pulling similar 3D points closer together while pushing dissimilar points apart in feature space through a contrastive loss function. For example, PointContrast (Xie et al., 2020) directly operates on each point and has demonstrated its effectiveness on various downstream tasks. Nonetheless, the sensitivity to positive/negative sample selection and the associated increased latency often impose constraints on the practical applications of these approaches. Masked AutoEncoding (MAE) (He et al., 2022), which encourages the model to learn a holistic understanding of the input beyond low-level statistics, has been widely applied in the autonomous driving field. Yet, such a pretext task has its challenges in 3D point clouds due to the inherent irregularity and sparsity of the data. VoxelMAE (Hess et al., 2022) proposed to divide irregular points into discrete voxels and predict the masked 3D structure using voxel-wise supervision. The coarse supervision may lead to insufficient representation capability. In this paper, we come up with a novel pre-training paradigm tailored for effective 3D representation learning, which not only avoids complex positive/negative sample assignments but also implicitly provides continuous supervision signals to learn 3D shape structures. The whole framework, as illustrated in Figure 2, takes the masked point cloud as input and aims to reconstruct the missing geometry on the projected 2D depth image via 3D differentiable neural rendering. Specifically, when provided with a masked LiDAR point cloud, our approach employs a 3D encoder to extract hierarchical features. Then, the 3D features are transformed into the voxel space via voxelization. We further apply a differentiable volumetric rendering method to reconstruct the complete geometric representation. The flexibility of our approach facilitates its seamless integration for pre-training 2D backbones. Multi-view image features construct the 3D volume via lift-split-shoot (LSS) (Philton & Fidler, 2020). To maintain efficiency during the training phase, we propose a memory-efficient ray sampling strategy designed specifically for autonomous driving applications, which can greatly reduce training costs and memory consumption. The novel sampling strategy boosts the accuracy significantly. Extensive experiments conducted on the competitive nuScenes (Caesar et al., 2020) dataset demonstrate the superiority and generalization of the proposed method. For pre-training on the 3D backbone, our method yields significant improvements over the baseline, as shown in Figure 1, achieving enhancements of 9.1 NDS for 3D object detection and 6.1 mIoU for 3D semantic segmentation, surpassing the performance of both contrastive- and MAE-based methods. Notably, our method achieves the state-of-the-art mIoU of 79.4 for segmentation on nuScenes dataset. Furthermore, our pre-training framework can be seamlessly applied to 2D image backbones, resulting in a remarkable improvement of 7.7 NDS for multi-view camera-based 3D detectors. We directly utilize the pre-trained 2D and 3D backbones to a multi-modal framework. Our method achieves 73.2 NDS for detection, achieving new SoTA results compared with previous methods. Our contributions are summarized as follows: • To the best of our knowledge, we are the first to explore a novel 3D differentiable rendering approach for self-supervised learning in the context of autonomous driving. • The flexibility of the method makes it easy to be extended to pre-train a 2D backbone. With a novel sampling strategy, our approach exhibits superiority in both effectiveness and efficiency. • We conduct comprehensive experiments on the nuScenes dataset, wherein our method surpasses the performance of six pre-training strategies. Experimentation involving seven backbones and two perception tasks provides convincing evidence for the effectiveness of our approach. 2 RELATED WORK Self-supervised learning in point clouds has gained remarkable progress in recent years (Chen et al., 2022c; Li & Heizmann, 2022; Liang et al., 2021; Liu et al., 2022a; Pang et al., 2022; Tian et al., 2023b; Xu et al., 2023c; Yin et al., 2022; Zhang et al., 2021; Huang et al., 2023). PointContrast (Xie et al., 2020) contrasts point-level features from two transformed views to learn discriminative 3D representations. Point-BERT (Yu et al., 2022) introduces a BERT-style pre-training strategy with standard transformer networks. MSC (Wu et al., 2023a) incorporates a mask point modeling strategy into a contrastive learning framework. PointM2AE (Zhang et al., 2022) utilizes a multiscale strategy to capture both high-level semantic and fine-grained details. STRL (Huang et al., 2021b) explores the rich spatial-temporal cues to learn invariant representation in point clouds. GD-MAE (Yang et al., 2023a) applies a generative decoder for hierarchical MAE-style pre-training. ALSO (Boulch et al., 2023) regards the surface reconstruction as the pretext task for representation learning. Unlike previous works primarily designed for point clouds, our pre-training framework is applicable to both image-based and point-based models. Representation learning in image has been well-developed (He et al., 2022; Tian et al., 2023a; Bachmann et al., 2022; Bao et al., 2022; He et al., 2020; Chen et al., 2020b), and has shown its capabilities in all kinds of downstream tasks as the backbone initialization (Liang et al., 2022; Li et al., 2022a; Yan et al., 2023). Contrastive-based methods, such as MoCo (He et al., 2020) and MoCov2 (Chen et al., 2020b), learn images’ representations by discriminating the similarities between different augmented samples. MAE-based methods, including MCMAE (Gao et al., 2022) and SparK (Tian et al., 2023a), obtain the promising generalization ability by recovering the masked patches. In autonomous driving, models pre-trained on ImageNet (Deng et al., 2009) are widely utilized in image-related tasks (Liu et al., 2022b; Li et al., 2022a). For example, to compensate for the insufficiency of 3D priors in tasks like 3D object detection, depth estimation (Park et al., 2021) and monocular 3D detection (Wang et al., 2021b) are used as extra pre-training techniques. Neural rendering for autonomous driving utilizes neural networks to differentially render images from 3D scene representation (Chen et al., 2022a; Mildenhall et al., 2020; Oechsle et al., 2021; Xu et al., 2023a; Yang et al., 2023c). Those methods can be roughly divided into two categories: perception and simulation. Being capable of capturing semantic and accurate geometry, NeRFs are gradually utilized to do different perception tasks including panoptic segmentation (Fu et al., 2022), object detection (Xu et al., 2023a,b), segmentation (Kundu et al., 2022), and instance segmentation (Zhi et al., 2021). For simulation, MARS (Wu et al., 2023b) models the foreground objects and background environments separately based on NeRF, making it flexible for scene controlling in autonomous driving simulation. Considering the limited labeled LiDAR point clouds data, NeRF-LiDAR (Zhang et al., 2023) proposes to generate realistic point clouds along with semantic labels for the LiDAR simulation. Besides, READ (Li et al., 2023b) explores multiple sampling strategies to make it possible to synthesize large-scale driving scenarios. Inspired by them, we make novel use of NeRF, with the purpose of universal pre-training, rather than of novel view synthesis. 3 METHODOLOGY The UniPAD framework is a universal pre-training paradigm that can be easily adapted to different modalities, e.g., 3D LiDAR point and multi-view images. Our framework is shown in Figure 2, which contains two parts, i.e., a modality-specific encoder and a volumetric rendering decoder. For processing point cloud data, we employ a 3D backbone for feature extraction. In the case of multi-view image data, we leverage a 2D backbone to extract image features, which are then mapped into 3D space to form the voxel representation. Similar to MAE (He et al., 2022), a masking strategy is applied for the input data to learn effective representation. For decoders, we propose to leverage off-the-shelf neural rendering with a well-designed memory-efficient ray sampling. By minimizing the discrepancy between rendered 2D projections and the input, our approach encourages the model to learn a continuous representation of the geometric or appearance characteristics of the input data. 3.1 MODAL-SPECIFIC ENCODER UniPAD takes LiDAR point clouds $ P $ or multi-view images $ I $ as input. The input is first masked out by the mask generator (detailed in the following) and the visible parts are then fed into the modal-specific encoder. For the point cloud $P$, a point encoder, e.g., VoxelNet (Yan et al., 2018), is adopted to extract hierarchical features $F_p$, as shown in Figure 2(a). For images, features $F_c$ are extracted from $I$ with a classic convolutional network, as illustrated in Figure 2(b). To capture both high-level information and fine-grained details in data, we employ additional modality-specific FPN (Lin et al., 2017) to efficiently aggregate multi-scale features in practice. Mask Generator Prior self-supervised approaches, as exemplified by He et al. (He et al., 2022), have demonstrated that strategically increasing training difficulty can enhance model representation and generalization. Motivated by this, we introduce a mask generator as a means of data augmentation, selectively removing portions of the input. Given points $P$ or images $I$, we adopt block-wise masking (Yang et al., 2023a) to obscure certain regions. Specifically, we first generate the mask according to the size of the output feature map, which is subsequently upsampled to the original input resolution. For points, the visible areas are obtained by removing the information within the masked regions. For images, we replace the traditional convolution with the sparse convolution as in (Tian et al., 2023a), which only computes at visible places. After the encoder, masked regions are padded with zeros and combined with visible features to form regular dense feature maps. ### 3.2 Unified 3D Volumetric Representation To make the pre-training method suitable for various modalities, it is crucial to find a unified representation. Transposing 3D points into the image plane would result in a loss of depth information, whereas merging them into the bird’s eye view would lead to the omission of height-related details. In this paper, we propose to convert both modalities into the 3D volumetric space, as shown in Figure 2(c), preserving as much of the original information from their corresponding views as possible. For multi-view images, the 2D features are transformed into the 3D ego-car coordinate system to obtain the volume features. Specifically, we first define the 3D voxel coordinates $X_p \in \mathbb{R}^{X \times Y \times Z \times 3}$, where $X \times Y \times Z$ is the voxel resolution, and then project $X_p$ on multi-view images to index the corresponding 2D features. The process can be calculated by: $$V = G(T_{c2i}T_{l2c}X_p, F_c),$$ where $V$ is the constructed volumetric feature, $T_{l2c}$ and $T_{c2i}$ denote the transformation matrices from the LiDAR coordinate system to the camera frame and from the camera frame to image coordinates, respectively, $F_c$ is the image features, and $G$ represents the bilinear interpolation. For the 3D point modality, we follow Li et al. (2022a) to directly retain the height dimension in the point encoder. Finally, we leverage a projection layer involving $L$ conv-layers to enhance the voxel representation. ### 3.3 Neural Rendering Decoder Differentiable Rendering We represent a novel use of neural rendering to flexibly incorporate geometry or textural clues into learned voxel features with a unified pre-training architecture, as shown in Figure 2(c). Specifically, when provided the volumetric features, we sample some rays... from multi-view images or point clouds and use differentiable volume rendering to render the color or depth for each ray. The flexibility further facilitates the incorporation of 3D priors into the acquired image features, achieved via supplementary depth rendering supervision. This capability ensures effortless integration into both 2D and 3D frameworks. Figure 3 shows the rendered RGB images and depth images based on our rendering decoder. Inspired by Wang et al. (2021a), we represent a scene as an implicit signed distance function (SDF) field to be capable of representing high-quality geometry details. The SDF symbolizes the 3D distance between a query point and the nearest surface, thereby implicitly portraying the 3D geometry. For ray $ r_i $ with camera origin $ o $ and viewing direction $ d_i $, we sample $ D $ ray points $ \{ p_j = o + t_j d_i \| j = 1, ..., D, t_j < t_{j+1} \} $, where $ p_j $ is the 3D coordinates of sampled points, and $ t_j $ is the corresponding depth along the ray. For each ray point $ p_j $, the feature embedding $ f_j $ can be extracted from the volumetric representation by trilinear interpolation. Then, the SDF value $ s_j $ is predicted by $ \phi_{\text{SDF}}(p_j, f_j) $, where $ \phi_{\text{SDF}} $ represents a shallow MLP. For the color value, we follow Oechsle et al. (2021) to condition the color field on the surface normal $ n_j $ (i.e., the gradient of the SDF value at ray point $ p_j $) and a geometry feature vector $ h_j $ from $ \phi_{\text{SDF}} $. Thus, the color representation is denoted as $ c_j = \phi_{\text{RGB}}(p_j, f_j, d_i, n_j, h_j) $, where $ \phi_{\text{RGB}} $ is parameterized by a MLP. Finally, we render RGB value $ \hat{Y}_i^{\text{RGB}} $ and depth $ \hat{Y}_i^{\text{depth}} $ by integrating predicted colors and sampled depth along rays: \[ \hat{Y}_i^{\text{RGB}} = \sum_{j=1}^{D} w_j c_j, \quad \hat{Y}_i^{\text{depth}} = \sum_{j=1}^{D} w_j t_j, \] where $ w_j $ is unbiased and occlusion-aware weight (Wang et al., 2021a) given by $ w_j = T_j \alpha_j $. $ T_j = \prod_{k=1}^{j-1}(1 - \alpha_k) $ is the accumulated transmittance, and $ \alpha_j $ is the opacity value computed by: \[ \alpha_j = \max \left( \frac{\sigma_s(s_j) - \sigma_s(s_{j+1})}{\sigma_s(s_j)}, 0 \right), \] where $ \sigma_s(x) = (1 + e^{-sx})^{-1} $ is a Sigmoid function modulated by a learnable parameter $ s $. Memory-friendly Ray Sampling Previous novel view synthesis methods prioritize dense supervision to enhance image quality. However, rendering a complete set of $ S \times H \times W $ rays — where $ S $ represents the number of camera views and $ H \times W $ is the image resolution — presents substantial computational challenges, especially in the context of autonomous driving scenes. To alleviate computational challenges, we devise three memory-friendly ray sampling strategies to render a reduced subset of rays: Dilation Sampling, Random Sampling, and Depth-aware Sampling, illustrated in Figure 4. 1) Dilation Sampling traverses the image at intervals of $ I $, thereby reducing the ray count to $ \frac{S \times H \times W}{I^2} $. 2) In contrast, Random Sampling selects $ K $ rays indiscriminately from all available pixels. 3) Although both dilation and random sampling are straightforward and significantly cut computation, they overlook the subtle prior information that is inherent to the 3D environment. For example, instances on the road generally contain more relevant information over distant backgrounds like sky and buildings. Therefore, we introduce depth-aware sampling to selectively sample rays informed by available LiDAR information, bypassing the need for a full pixel set. To implement this, we project the 3D LiDAR point cloud onto the multi-view images and acquire the set of projection pixels with a depth less than the $ \tau $ threshold. Subsequently, rays are selectively sampled from this refined pixel set as opposed to the entire array of image pixels. In doing so, our approach not only alleviates computational burden but also optimizes the precision of neural rendering by concentrating on the most relevant segments within the scene. Pre-training Loss The overall pre-training loss consists of the color loss and depth loss: \[ L = \frac{\lambda_{\text{RGB}}}{K} \sum_{i=1}^{K} \|\hat{Y}_i^{\text{RGB}} - Y_i^{\text{RGB}}\| + \frac{\lambda_{\text{depth}}}{K^+} \sum_{i=1}^{K^+} \|\hat{Y}_i^{\text{depth}} - Y_i^{\text{depth}}\|, \] where $ Y_i^{\text{RGB}} $ and $ Y_i^{\text{depth}} $ are the ground-truth color and depth for each ray, respectively. $ \hat{Y}_i^{\text{RGB}} $ and $ \hat{Y}_i^{\text{depth}} $ are the corresponding rendered ones in Eq. 2. $ K^+ $ is the count of rays with available depth. Table 1: Comparisons of different methods with a single model on the nuScenes val set. We compare with classic methods on different modalities without test-time augmentation. †: denotes our reproduced results based on MMDetection3D (Contributors, 2020). L, C, CS, and M indicate the LiDAR, Camera, Camera Sweep, and Multi-modality input, respectively. \| Methods \| Present at \| Modality \| CS \| CBGS \| NDS† \| mAP† \| \|--------------------------\|------------\|----------\|----\|------\|------\|------\| \| PVT-SSD (Yang et al., 2023b) \| CVPR’23 \| L \| ✓ \| \| 65.0 \| 53.6 \| \| CenterPoint (Yin et al., 2021a) \| CVPR’21 \| L \| ✓ \| \| 66.8 \| 59.6 \| \| FSDv1 (Fan et al., 2022) \| NeurIPS’22 \| L \| ✓ \| \| 68.7 \| 62.5 \| \| VoxelNeXt (Chen et al., 2023b) \| CVPR’23 \| L \| ✓ \| \| 68.7 \| 63.5 \| \| LargeKernel3D (Chen et al., 2023a) \| CVPR’23 \| L \| ✓ \| \| 69.1 \| 63.3 \| \| TransFusion-L (Bai et al., 2022) \| CVPR’22 \| L \| ✓ \| \| 70.1 \| 65.1 \| \| CMT-L (Yan et al., 2023) \| ICCV’23 \| L \| ✓ \| \| 68.6 \| 62.1 \| \| UVTR-L (Li et al., 2022a) \| NeurIPS’22 \| L \| ✓ \| \| 67.7 \| 60.9 \| \| UVTR-L+UniPAD (Ours) \| - \| L \| ✓ \| \| 70.6 \| 65.0 \| \| BEVFormer-S (Li et al., 2022b) \| ECCV’22 \| C \| ✓ \| \| 44.8 \| 37.5 \| \| SpatialDETR (Doll et al., 2022) \| ECCV’22 \| C \| ✓ \| \| 42.5 \| 35.1 \| \| PETR (Liu et al., 2022b) \| ECCV’22 \| C \| ✓ \| \| 44.2 \| 37.0 \| \| Ego3RT (Lu et al., 2022) \| ECCV’22 \| C \| ✓ \| \| 45.0 \| 37.5 \| \| 3DPPE (Shu et al., 2023) \| ICCV’23 \| C \| ✓ \| \| 45.8 \| 39.1 \| \| CMT-C (Yan et al., 2023) \| ICCV’23 \| C \| ✓ \| \| 46.0 \| 40.6 \| \| FCOS3D (Wang et al., 2021b) \| ICCVW’21 \| C \| ✓ \| \| 38.4 \| 31.1 \| \| FCOS3D+UniPAD (Ours) \| - \| C \| ✓ \| \| 40.1 \| 33.2 \| \| UVTR-C (Li et al., 2022a) \| NeurIPS’22 \| C \| ✓ \| \| 45.0 \| 37.2 \| \| UVTR-C+UniPAD (Ours) \| - \| C \| ✓ \| \| 47.4 \| 41.5 \| \| UVTR-CS (Li et al., 2022a) \| NeurIPS’22 \| C \| ✓ \| \| 48.8 \| 39.2 \| \| UVTR-CS+UniPAD (Ours) \| - \| C \| ✓ \| \| 50.2 \| 42.8 \| \| FUTR3D (Chen et al., 2022b) \| arXiv’22 \| C+L \| ✓ \| \| 68.3 \| 64.5 \| \| PointPainting (Vora et al., 2020) \| CVPR’20 \| C+L \| ✓ \| \| 69.6 \| 65.8 \| \| MVP (Yin et al., 2021b) \| NeurIPS’21 \| C+L \| ✓ \| \| 70.8 \| 67.1 \| \| Transfusion (Bai et al., 2022) \| CVPR’22 \| C+L \| ✓ \| \| 71.3 \| 67.5 \| \| AutoAlign V2 (Chen et al., 2022d) \| ECCV’22 \| C+L \| ✓ \| \| 71.2 \| 67.1 \| \| BEVFusion (Liang et al., 2022) \| NeurIPS’22 \| C+L \| ✓ \| \| 71.0 \| 67.9 \| \| BEVFusion (Liu et al., 2023) \| ICR’23 \| C+L \| ✓ \| \| 71.4 \| 68.5 \| \| DeepInteraction (Yang et al., 2022) \| NeurIPS’22 \| C+L \| ✓ \| \| 72.6 \| 69.9 \| \| CMT-M (Yan et al., 2023) \| ICCV’23 \| C+L \| ✓ \| \| 72.9 \| 70.3 \| \| UVTR-M (Li et al., 2022a) \| NeurIPS’22 \| C+L \| ✓ \| \| 70.2 \| 65.4 \| \| UVTR-M+UniPAD (Ours) \| - \| C+L \| ✓ \| \| 73.2 \| 69.9 \| Table 2: Comparisons of different methods with a single model on the nuScenes segmentation dataset. \| Split \| SPVNAS (Tang et al., 2020) \| Cylinder3D (Zhu et al., 2021) \| SphereFormer (Lai et al., 2023) \| SpUNet (Choy et al., 2019) \| SpUNet+UniPAD (Ours) \| \|-------\|----------------------------\|-------------------------------\|---------------------------------\|---------------------------\|----------------------\| \| val \| - \| 76.1 \| 78.4 \| 73.3 \| 79.4 \| \| test \| 77.4 \| 77.2 \| 81.9 \| - \| 81.1 \| 4 EXPERIMENTS 4.1 DATASETS AND EVALUATION METRICS We conduct the experiments on the NuScenes (Caesar et al., 2020) dataset, which is a challenging dataset for autonomous driving. It consists of 700 scenes for training, 150 scenes for validation, and 150 scenes for testing. Each scene is captured through six different cameras, providing images with surrounding views, and is accompanied by a point cloud from LiDAR. The dataset comes with diverse annotations, supporting tasks like 3D object detection and 3D semantic segmentation. For detection evaluation, we employ nuScenes detection score (NDS) and mean average precision (mAP), and for segmentation assessment, we use mean intersection-over-union (mIoU). 4.2 IMPLEMENTATION DETAILS We base our code on the MMDetection3D (Contributors, 2020) toolkit and train all models on 4 NVIDIA A100 GPUs. The input image is configured to $1600 \times 900$ pixels, while the voxel dimensions for point cloud voxelization are [0.075, 0.075, 0.2]. During the pre-training phase, we implemented several data augmentation strategies, such as random scaling and rotation. Additionally, we partially mask the inputs, focusing only on visible regions for feature extraction. The masking size and ratio for images are configured to 32 and 0.3, and for points to 8 and 0.8, respectively. ConvNeXt-small (Liu et al., 2022c) and VoxelNet (Yan et al., 2018) are adopted as the default image and point encoders, respectively. A uniform voxel representation with the shape of $180 \times 180 \times 5$ is constructed across modalities. The feature projection layer reduces the voxel feature dimensions to 32 via a 3-kernel size convolution. For the decoders, we utilize a 6-layer MLP for SDF and a 4-layer MLP for RGB. In the rendering phase, 512 rays per image view and 96 points per ray are randomly selected. We maintain the loss scale factors for $\lambda_{RGB}$ and $\lambda_{depth}$ at 10. The model undergoes training for 12 epochs using the AdamW optimizer with initial learning rates of $2e^{-5}$ and $1e^{-4}$ for point and image modalities, respectively. In the ablation studies, unless explicitly stated, fine-tuning is conducted for 12 epochs on 50% of the image data and for 20 epochs on 20% of the point data, without the implementation of the CBGS (Zhu et al., 2019) strategy. ### 4.3 Comparison with State-of-the-Art Methods #### 3D Object Detection. In Table 1, we compare UniPAD with previous detection approaches on the nuScenes validation set. We adopt UVTR (Li et al., 2022a) as our baselines for point-modality (UVTR-L), camera-modality (UVTR-C), Camera-Sweep-modality(UVTR-CS) and fusion-modality (UVTR-M). Benefits from the effective pre-training, UniPAD consistently improves the baselines, namely, UVTR-L, UVTR-C, and UVTR-M, by 2.9, 2.4, and 3.0 NDS, respectively. When taking multi-frame cameras as inputs, UniPAD-CS brings 1.4 NDS and 3.6 mAP gains over UVTR-CS. Our pre-training technique also achieves 1.7 NDS and 2.1 mAP improvements over the monocular-based baseline FCOS3D (Wang et al., 2021b). Without any test time augmentation or model ensemble, our single-modal and multi-modal methods, UniPAD-L, UniPAD-C, and UniPAD-M, achieve impressive NDS of 70.6, 47.4, and 73.2, respectively, surpassing existing state-of-the-art methods. #### 3D Semantic Segmentation. In Table 2, we compare UniPAD with previous point clouds semantic segmentation approaches on the nuScenes Lidar-Seg dataset. We adopt SpUNet (Choy et al., 2019) as our baseline. Benefits from the effective pre-training, UniPAD improves the baselines by 6.1 mIoU, achieving state-of-the-art performance on the validation set. Meanwhile, UniPAD achieves an impressive mIoU of 81.1 on the test set, which is comparable with existing state-of-the-art methods. ### 4.4 Comparisons with Pre-training Methods. #### Camera-based Pre-training. In Table 3, we conduct comparisons between UniPAD and several other camera-based pre-training approaches: 1) Depth Estimator: we follow Park et al. (2021) to inject 3D priors into 2D learned features via depth estimation; 2) Detector: the image encoder is initialized using pre-trained weights from MaskRCNN (He et al., 2017) on the nuImages dataset (Caesar et al., 2020); 3) 3D Detector: we use the weights from the widely used monocular 3D detector (Wang et al., 2021b) for model initialization, which relies on 3D labels for supervision. UniPAD demonstrates superior knowledge transfer capabilities compared to previous unsupervised or supervised pre-training methods, showcasing the efficacy of our rendering-based pretext task. #### Point-based Pre-training. For point modality, we also present comparisons with recently proposed self-supervised methods in Table 4. 1) Occupancy-based: we implement ALSO (Boulch et al., 2023) in our framework to train the point encoder; 2) MAE-based: the leading-performing method (Yang et al., 2023a) is adopted, which reconstructs masked point clouds using the chamfer distance. 3) Contrast-based: (Liu et al., 2021) is used for comparisons, which employs pixel-to-point contrastive learning to integrate 2D knowledge into 3D points. Among these methods, UniPAD achieves the best NDS performance. While UniPAD has a slightly lower mAP compared to the contrast-based method, it avoids the need for complex positive-negative sample assignments in contrastive learning. ### 4.5 Effectiveness on Various Backbones. #### Different View Transformations. In Table 5, we investigate different view transformation strategies for converting 2D features into 3D space, including BEVDet (Huang et al., 2021a), BEVDepth (Li et al., 2023a), and BEVformer (Li et al., 2022b). Consistent improvements ranging from 5.2 to 6.3 Table 3: Comparison with different camera-based pre-training methods. \| Methods \| Label 2D \| NDS \| mAP \| \|--------------------------\|----------\|-------\|-------\| \| UVTR-C (Baseline) \| \| 25.2 \| 23.0 \| \| +Depth Estimator \| \| 26.9 \| 21.7 \| \| +Detector \| ✓ \| 29.4 \| 27.7 \| \| +3D Detector \| ✓ \| 31.7 \| 29.0 \| \| +UniPAD \| \| 32.9 \| 32.6 \| Table 4: Comparison with different point-based pre-training methods. \| Methods \| Support 2D \| NDS \| mAP \| \|--------------------------\|------------\|-------\|-------\| \| UVTR-L (Baseline) \| \| 46.7 \| 39.0 \| \| +Occupancy-based \| ✓ \| 48.2 \| 41.2 \| \| +MAE-based \| ✓ \| 48.8 \| 42.6 \| \| +Contrast-based \| ✓ \| 49.2 \| 48.8 \| \| +UniPAD \| ✓ \| 55.8 \| 48.1 \| Table 5: Pre-training effectiveness on different view transform strategies. \| Methods \| View Transform \| NDS \| mAP \| \|---------------\|----------------\|-------\|-------\| \| BEVDet \| Pooling \| 27.1 \| 24.6 \| \| +UniPAD \| Pooling \| 32.7 \| 32.8 \| \| BEVDepth \| Pooling & Depth\| 28.9 \| 28.1 \| \| +UniPAD \| Pooling & Depth\| 34.1 \| 33.9 \| \| BEVformer \| Transformer \| 26.8 \| 24.5 \| \| +UniPAD \| Transformer \| 33.1 \| 31.9 \| Table 6: Pre-training effectiveness on different input modalities. \| Methods \| Modality \| NDS \| mAP \| \|---------------\|--------------\|-------\|-------\| \| UVTR-L \| LiDAR \| 46.7 \| 39.0 \| \| +UniPAD \| LiDAR \| 55.8 \| 48.1 \| \| UVTR-C \| Camera \| 25.2 \| 23.0 \| \| +UniPAD \| Camera \| 32.9 \| 32.6 \| \| UVTR-M \| LiDAR-Camera\| 49.9 \| 52.7 \| \| +UniPAD \| LiDAR-Camera\| 56.8 \| 57.0 \| NDS can be observed across different transformation techniques, which demonstrates the strong generalization ability of the proposed approach. Different Modalities. Unlike most previous pre-training methods, our framework can be seamlessly applied to various modalities. To verify the effectiveness of our approach, we set UVTR as our baseline, which contains detectors with point, camera, and fusion modalities. Table 6 shows the impact of UniPAD on different modalities. UniPAD consistently improves the UVTR-L, UVTR-C, and UVTR-M by 9.1, 7.7, and 6.9 NDS, respectively. Scaling up Backbones. To test UniPAD across different backbone scales, we adopt an off-the-shelf model, ConvNeXt, and its variants with different numbers of learnable parameters. As shown in Table 7, one can observe that with our UniPAD pre-training, all baselines are improved by large margins of +6.0∼7.7 NDS and +8.2∼10.3 mAP. The steady gains suggest that UniPAD has the potential to boost various state-of-the-art networks. ### 4.6 Ablation Studies Masking Ratio. Table 8a shows the influence of the masking ratio for the camera modality. We discover that a masking ratio of 0.3, which is lower than the ratios used in previous MAE-based methods, is optimal for our method. This discrepancy could be attributed to the challenge of rendering the original image from the volume representation, which is more complex compared to image-to-image reconstruction. For the point modality, we adopt a mask ratio of 0.8, as suggested in Yang et al. (2023a), considering the spatial redundancy inherent in point clouds. Rendering Design. Our examinations in Tables 8b, 8c, and 8d illustrate the flexible design of our differentiable rendering. In Table 8b, we vary the depth ($D_{SDF}, D_{RGB}$) of the SDF and RGB decoders, revealing the importance of sufficient decoder depth for succeeding in downstream detection tasks. This is because deeper ones may have the ability to adequately integrate geometry or appearance cues during pre-training. Conversely, as reflected in Table 8c, the width of the decoder has a relatively minimal impact on performance. Thus, the default dimension is set to 32 for efficiency. Additionally, we explore the effect of various rendering techniques in Table 8d, which employ different ways for... Table 8: Ablation studies of the volume-based neural rendering. (a) Mask ratio. A masking ratio of 0.3 is more accurate. \| ratio \| NDS \| mAP \| \|-------\|-----\|-----\| \| 0.1 \| 31.9\| 32.4\| \| 0.3 \| 32.9\| 32.6\| \| 0.5 \| 32.3\| 32.6\| \| 0.7 \| 32.1\| 32.4\| (b) Decoder depth. A deep decoder can improve accuracy. \| layers \| NDS \| mAP \| \|--------\|-----\|-----\| \| (2, 2) \| 31.3\| 31.3\| \| (4, 3) \| 31.9\| 31.6\| \| (5, 4) \| 32.1\| 32.7\| \| (6, 4) \| 32.9\| 32.6\| (c) Decoder width. The decoder width has minor impact. \| dim \| NDS \| mAP \| \|-------\|-----\|-----\| \| 32 \| 32.9\| 32.6\| \| 64 \| 32.5\| 32.8\| \| 128 \| 32.9\| 32.6\| \| 256 \| 32.4\| 32.9\| (d) Rendering technique. Representation benefits from well-designed rendering methods. \| Methods \| NDS \| mAP \| \|------------------\|-----\|-----\| \| UniSurf (Oechsle et al., 2021) \| 32.5\| 32.1\| \| VolSDF (Yariv et al., 2021) \| 32.8\| 32.4\| \| NeuS (Wang et al., 2021a) \| 32.9\| 32.6\| (e) Sampling strategy. Depth-aware sampling outperforms other sampling strategies. \| Methods \| NDS \| mAP \| \|-----------------\|-----\|-----\| \| Dilation Sampling \| 31.9\| 32.4\| \| Random Sampling \| 32.5\| 32.1\| \| Depth-aware Sampling \| 32.9\| 32.6\| (f) Feature projection. Feature projection is crucial for enhancing voxel representation. \| Methods \| NDS \| mAP \| \|--------------------\|-----\|-----\| \| Baseline \| 32.9\| 32.6\| \| w/o Projection<sub>FT</sub> \| 30.2<sup>±2.7</sup>\| 29.7<sup>±2.9</sup>\| \| w/o Projection<sub>PT</sub> \| 31.1<sup>±1.8</sup>\| 30.5<sup>±2.1</sup>\| \| Shared Projection \| 32.1<sup>±0.8</sup>\| 32.0<sup>±0.6</sup>\| (g) Pre-trained components. Each of the pre-trained components is essential for fine-tuning. \| Methods \| NDS \| mAP \| \|--------------------\|-----\|-----\| \| Baseline \| 25.2\| 23.0\| \| +Encoder \| 32.0<sup>±6.8</sup>\| 31.8<sup>±8.8</sup>\| \| +Encoder & FPN \| 32.2<sup>±0.2</sup>\| 32.2<sup>±0.4</sup>\| \| +Encoder & FPN & VT\| 32.9<sup>±0.7</sup>\| 32.6<sup>±0.4</sup>\| Ray point sampling and accumulation. Using NeuS (Wang et al., 2021a) for rendering records a 0.4 and 0.1 NDS improvement compared to UniSurf (Oechsle et al., 2021) and VolSDF (Yariv et al., 2021), respectively, showcasing the learned representation can be improved by utilizing well-designed rendering methods and benefiting from the advancements in neural rendering. Memory-friendly Ray Sampling. Instead of rendering the entire set of multi-view images, we sample only a subset of rays to provide supervision signals. Table 8e outlines the different strategies explored to minimize memory usage and computational costs during pre-training. Our observations indicate that depth-aware sampling holds a substantial advantage, improving scores by 0.4 and 1.0 NDS compared to random and dilation sampling, respectively. The sampling excludes regions without well-defined depth, like the sky, from contributing to the loss. This allows the representation learning to focus more on the objects in the scene, which is beneficial for downstream tasks. Feature Projection. The significance of feature projection is shown in Table 8f. Removing projection from pre-training and fine-tuning leads to drops of 1.8 and 2.7 NDS, respectively, underscoring the essential role it plays in enhancing voxel representation. Concurrently, utilizing shared parameters for the projection during pre-training and fine-tuning induces reductions of 0.8 NDS and 0.6 mAP. This phenomenon is likely due to the disparity between the rendering and recognition tasks, with the final layers being more tailored for extracting features specific to each task. Pre-trained Components. In Table 8g, the influence of pre-trained parameters on each component is investigated. Replacing the pre-trained weights of the FPN and view transformation (VT) with those from a random initialization induces declines of 0.2 and 0.7 NDS, respectively, thereby highlighting the crucial roles of these components. 5 CONCLUSION We introduce an innovative self-supervised learning method, named UniPAD, which demonstrates exceptional performance in a range of 3D downstream tasks. UniPAD stands out for its ingenious adaptation of NeRF as a unified rendering decoder, enabling seamless integration into both 2D and 3D frameworks. Furthermore, we put forward the depth-aware sampling strategy that not only reduces computational demands but also enhances overall performance. The adaptability inherent in our approach opens the door to future investigations into cross-modal interactions utilizing paired image-point data in the domain of autonomous driving. 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bYwEpQ96ng	Equation 1 seems to only describe how to obtain the prediction but does not involve how to reduce the difference between the prediction and the ground truth. This is not entirely consistent with what ERM describes. Therefore, another issue arises. It seems that no loss function was mentioned. It should be cross-entropy. Use CE loss to make the mixed concat features and mixed labels close, thereby updating $\mathbf{W}^{cls}$?	Hierarchical Long-tailed Classification with Visual Language Models Anonymous authors Paper under double-blind review Abstract Vision Language Models (VLMs) have shown promising capabilities in handling open vocabulary tasks but struggle with imbalanced data tuning, particularly when dealing with highly skewed label distributions. To address the challenges, we propose a hierarchical long-tailed classification framework, named HLC, which prioritizes candidate categories before conducting fine-grained classification using detailed textual descriptions. Specifically, we fine-tune a linear classifier based on the CLIP encoder, incorporating visual prompt tokens and leveraging shared feature space mixup for multimodal feature interactions. Based on candidates given by the coarse classifier, we query large language models to generate corresponding fine-grained descriptions to refine the final predictions. Importantly, we introduce a reweighting mechanism to filter out invalid descriptions generated by language models. Extensive evaluations demonstrate that our approach achieves state-of-the-art performance by fine-tuning only a few parameters on the PlacesLT, ImageNet-LT, and iNaturalist 2018 datasets. 1 Introduction Real-world visual data typically exhibits an instance-imbalanced long-tailed distribution. Models trained on such skewed datasets over-focus on the majority (head) classes while neglecting the minority (tail) ones, resulting in the bias for the head and poor generalization on the tail (Liu et al., 2019; Cui et al., 2019; Xu et al., 2021; Cao et al., 2019). Researchers struggle to alleviate the Long-Tailed Recognition (LTR) problem by leveraging visual datasets alone to train classifiers with elaborately designed strategies, briefly taxonomized into four categories: 1) rebalancing by reweighting (Cui et al., 2019; Xu et al., 2023b; Ma et al., 2023), or resampling (Cao et al., 2019; Kang et al., 2020), 2) enhancing the tail with head categories (Chou et al., 2020; Park et al., 2022), 3) decoupling feature learning and downstream tasks with two-stage framework (Kang et al., 2020; Zhou et al., 2023), and 4) integrating multi-experts to focus on different aspects (Li et al., 2022a; Jin et al., 2023; Xu et al., 2023a). In this paper, we observe that while previous methods can achieve satisfactory top-5 classification accuracy, the true challenge lies in fine-grained predictions from the candidates (Figure 1a). However, naively enumerating possible fine-grained classifiers and training each candidate yield exponential computational overheads. The recent success of Open-Vocabulary Classification (OVC) can fine-tune a few parameters of the pre-trained VLMs, such that the classifier can be ready for arbitrary category numbers with satisfactory Few-Shot Learning (FSL) capabilities. One may intuitively resort to the versatile large-scale vision language models (VLMs) (Radford et al., 2021; Alayrac et al., 2022; Jia et al., 2021) as auxiliary to support image classification. Is it possible to embrace OVC to compensate the inherently imbalanced datasets without extreme computational overheads? Unfortunately, while OVC excels at handling fine-grained recognition with label prompts (Yao et al., 2021), it is incapable of catching up with the state-of-the-art approaches (Table 1) in the challenging LTR, because the OVC fails to 1) take advantage of existing annotation information; 2) overcome the long-tail bias problem; and 3) fully encapsulate the visual information with existing language description. In this paper, we make an affirmative answer that VLMs are strong enough to promote state-of-the-art approaches in LTR, by proposing tailored solutions to address the above issues. We follow previous methods (Tian et al., 2022; Zhou et al., 2022b; Pratt et al., 2022) to collect class-wise corpora from the Web, such that the LTR datasets (Liu et al., 2019) are equipped with reasonable language information to conduct mix-modality tuning. We then prompt the Large Language Models (LLMs) to generate fine-grained descriptions for each class and propose the novel Hierarchical Long-tailed Classification (HLC) framework, which combines the traditional coarse classifiers and OVC in a hierarchical manner. After obtaining top-k candidate categories from the trained coarse classifier in the first stage, instead of training numerous candidate classifiers, we alternatively employ OVC for further fine-grained classification without parameter tuning (Figure 1b). Our HLC is composed of several crucial components. First, we integrate the advantages of the abovementioned class-wise corpora to fine-tune additional visual prompt tokens (Jia et al., 2022) and the coarse classifier. In this way, the model trained with mixed-modal data absorbs the reasonable features by Vision-Language (V-L) shared space compared to naive image space. Second, given the empirical observation that the classifier performs better on text than on visual data, we propose Shared Feature Space Mixup (SFM) to enhance the correspondence of multi-modality data and specialize HLC for LTR. Further, we adopt the post-hoc logit adjustment (Menon et al., 2021) to eliminate the preference for the head and improve the robustness of the tail. Third, considering the inevitable mismatched descriptions given by LLMs, we propose adaptive weights tuning for each description and jointly optimize weights with coarse classifier and visual prompt tokens, such that crucial descriptions are emphasized while negligible parts are weakened. Once trained, we calculate the expectations of all weighted image-descriptions similarity to integrate the final prediction for explainable reasoning (Figure 1c). We present extensive experiments to demonstrate the advances of the proposed method, with detailed ablation studies to manifest the effectiveness of our proposals. In summary, our contributions are three-fold. First, we supplement class-wise corpora for the missing text information in large-scale LTR benchmarks and leverage the versatile LLMs to construct informative and detailed descriptions of each category. Second, we propose novel HLC for OVC to perform fine-grained recognition in LTR, with two tailored solutions, shared feature space mixup and adaptive weights tuning mechanism, to prevent head-prejudice and side-effect of class-irrelevant descriptions. Finally, with the organic integration of these crucial insights and techniques, we demonstrate the state-of-the-art performance of OVC with HLC on challenging imbalanced benchmarks w.r.t. Places-LT, ImageNet-LT, and iNaturalist 2018. Our class-wise corpora, descriptions from LLMs, and models will be publicly available for research purposes. ## 2 Preliminaries Task Definition. We focus on the LTR task, where the training data lies in long-tailed distribution w.r.t. the class labels. Given a visual dataset with $C$ classes, $\mathcal{D}_V := \{(x_i^V, y_i)\}_{i=1}^N$, where $x_i^V \in \mathbb{R}^{H \times W \times 3}$ and $y_i \in \mathbb{R}^C$, we denote the instance number of the $i$-th class as $n_i$ and $n_1 \geq n_2 \geq \ldots \geq n_C$, where $n_1 \gg n_C$ typically in LTR. Vision Language Model. Our proposals are based on pre-trained vision-language models, e.g., CLIP, with the ViT-B as the visual encoder $E^V$. The language encoder $E^L$ maps the textual corpora into the visual language (V-L) shared feature space $v^V \rightarrow L \in \mathbb{R}^d$. For the visual branch, given a query image $I \in \mathbb{R}^{H \times W \times 3}$, the visual encoder $E^V$ splits the image $I$ and embedding it into $M$ patches tokens $E_0$. Combined with a CLS token $c_0$, we get the input visual token sequence $[E_0, c_0]$, which will be processed by $L_t$ transformer layers. The output CLS token $c_{L_t}$ of the final transformer layer will be projected to V-L shared latent embedding space. Figure 2: The finetuning pipeline. 1) We conduct mixed training with collected category-specific corpora. 2) We adopt visual prompt tuning to better fit the training data. 3) We propose V-L shared feature space mixup to increase the interaction between the two modalities’ features. 4) We construct a fine-grained text description feature cache and optimize the weight of each description ($M = 10$) to ignore the impact of the training image-unrelated descriptions. 3 METHODOLOGY 3.1 CROSS-MODAL TRAINING. Although the open-vocabulary classification is promising, the performance is still far from the SOTA on the LTR benchmarks, e.g., zero-shot CLIP 37.9% v.s. VL-LTR 50.1% on the PlacesLT in Table[1]. Hence, we attempt to amalgamate prior research with OVC to leverage the VLMs’ multimodal capabilities. Initially, we aim to train a high-performance classifier that can yield reliable candidates. We consider a cosine classifier $W_{cls}$ and the training framework is shown in Figure 2. We utilize the shared feature space of CLIP (Radford et al., 2021) to assist the classifier training because the mixed-modal data can benefit downstream uni-modality tasks (Lin et al., 2023). Some tail images are challenging to collect, while the relevant textual descriptions can be obtained from the Internet easily. Therefore, we construct corpora to supplement the few-shot visual features and train the classifier with mixed modality data (See section 4.1 for details). Given mini-batch input images $x^V$ and texts $x^L$, the empirical risk minimization will be formulated as follows: $$\hat{y} = \arg\max \frac{W_{cls} \cdot v^{V-L}}{\\|W_{cls}\\|_2 \cdot \\|v^{V-L}\\|_2},$$ where $v^{V-L} := \text{concat}(E^V(x^V), E^L(x^L))$ is the feature batch given by the CLIP encoders. Experimentally, the classifier $W_{cls}$ performs much better on textual modality than on images, with slight improvement in the tail classes. Hence, we propose the Shared Feature space Mixup (SFM) to enhance the feature interaction between the two modalities (see Figure 2). Give the corresponding ground-truth labels $y^V, y^L$, the embedding-level mixup will be: $$v^{V-L}_{mixed} = \lambda \cdot v^{V-L} + (1 - \lambda) \cdot \phi(v^{V-L})$$ $$y_{mixed} = \lambda \cdot y + (1 - \lambda) \cdot \phi(y),$$ where $y := \text{concat}(y^V, y^L)$, $\lambda$ is sampled from Beta distribution and the $\phi(\cdot)$ is the batch shuffle operation. Note that the mixing operation takes place in the shared space rather than at the input level in vanilla mixup (Zhang et al., 2017), thereby avoiding the contradiction arising from disparate modalities. While MixGen (Hao et al., 2023) attempts to directly concatenate text to achieve input-level mixup, our experimental results suggest that embedding-level mixup is more effective in training high-performance classifiers when the ground-truth labels are available. Considering the similarity between the few-shot learning and tail learning of LTR, we employ deep Visual Prompt Tuning (VPT) (Jia et al., 2022) in the few-shot learning area to adapt the downstream data distribution. Concretely, we introduce a group of learnable visual prompts tokens $\tilde{p} := [p_0, p_1, \cdots, p_{N-1}]$ to build the input sequence of the visual branch and more learnable tokens $\{\tilde{p}_i\}_{i=0}^{L_p}$ will be introduced in deeper transformer layers up to depth $L_p$. Hence, we derive the token sequence and the flow in the vision branch will be: $$[\_, E_i, c_i] = L_i([\tilde{p}_{i-1}, E_{i-1}, c_{i-1}]) \quad i = 1, 2, \cdots, L_p.$$ $$[E_j, c_j] = L_j([E_{j-1}, c_{j-1}]) \quad j = L_p + 1, \cdots, L_t.$$ $$v^{V-L} = \text{ImageProj}(c_{L_t}) \quad v^{V-T} \in \mathbb{R}^d.$$ Figure 3: The inference pipeline. 1) We perform a coarse-grained inference to obtain features and debiased logits. Post-hoc adjustment is used for long-tail debiasing. 2) We retrieve the descriptions features and corresponding weights of the top k candidates based on the debiased logits. 3) We calculate the weighted average similarity between the image and M text descriptions feature of candidate classes. The results have good interpretability with matching scores of different descriptions. 3.2 Recognition with Weighted Descriptors. To perform fine-grained classification, we further construct descriptor sets that characterize M detailed features of the category. Following VCD (Menon & Vondrick, 2023), we prompt large language models, e.g., GPT-3.5-turbo, to generate fine-grained descriptions to verify class labels. To match the fine-tuning data, we let LLMs generate sentence-level descriptions instead of phrases in VCD. Let’s take the query prompt of PlacesLT (Liu et al., 2019) as an illustrative example: Q: List 10 useful features for distinguishing {CLS} in a photo. In contrast to VCD, we observe that LLMs tend to provide descriptions that are unrelated to the images in the datasets (see examples in Figure 4). To mitigate their impact on the overall performance, we introduce a set of learnable weights $W_{lm}$ for each descriptor and update parameters during classifier training (see Figure 2). The effectiveness of our descriptions and the adaptive weights mechanism has been demonstrated to be superior to VCD by ablation experiments (Table 7). In contrast to previous methods that employ the text encoder in intricate loss optimization processes (Zhou et al., 2022a; Khattak et al., 2023), our approach relies solely on a single forward pass of the text encoder for all descriptions and caches the corresponding features. This allows us to perform open-vocabulary classification without incurring any additional computational overhead. 3.3 Pipeline. Based on the aforementioned proposals, we present our hierarchical LTR framework. First, we train a base classifier with SFM and VPT. To mitigate bias incurred by long-tail data, we further adopt the post-hoc logit adjustment (Menon et al., 2021; Ren et al., 2020; Xu et al., 2021) for brevity (Equation 4). The $\pi_i$ is the statistical proportion of training set labels and we set $\tau = 1$ by default. $$\hat{z}_i = z_i - \tau \cdot \log(\pi_i), \quad \pi_i = n_i / \sum_{j=1}^{C} n_j$$ Second, we depict the inference process in Figure 3 and outlined in Algorithm 1. Given the top k candidates, we retrieve the descriptor features $v^L$ from the cache and perform OVC with reweighted average similarities between the image and descriptor features. $$z_i = \frac{1}{M} \sum_{m=1}^{M} W_{lm}^{i,m} \cdot \frac{v^V_i \cdot v^L_{i,m}}{\\|v^V_i\\|_2 \cdot \\|v^L_{i,m}\\|_2}$$ The results show satisfactory interpretability (Figure 1c), as we can identify the model’s decision-making basis by examining the similarity ranking of M descriptor matching. We can also determine which fine-grained features the model misidentified that led to the final incorrect prediction. Algorithm 1 Inference pseudo code of HLC in a PyTorch-like style. ``` # Input: Visual_Encoder, p_vpt, W_cls(d x C), W_llm(C x M), D_llm(C x M x d), K, tau, pi # Output: prediction y for x in loader: # load images x from test set p_img = PatchEmbedding(x) # Project vanilla image to tokens p = torch.cat([p_vpt, p_img, p_cls], dim=1) # Add visual prompt tokens and CLS token v = Visual_Encoder(p) # Get image features from CLS token v_norm = v.norm(dim=-1, keepdim=True) # Get normalized feature for similarity z = W_cls @ v.T - tau * log(pi) # Calculate debiasing logits via Eq. 4 idxs = torch.topk(z, K).indices # Get top K candidates classes # Calculate weighted similarity between v_norm and M descriptor features of each class score = [torch.mean(W_llm[i].softmax(dim=-1) * (v_norm @ D_llm[i].T)) for i in idxs] y = idxs[torch.argmax(torch.tensor(score))] # Get final decision from the K candidates ``` 4 EXPERIMENT 4.1 DATASET Long-tailed visual datasets. We conduct comprehensive experiments on 3 LTR benchmarks, namely Places-LT (Lu et al., 2019), ImageNet-LT (Liu et al., 2019), and iNaturalist 2018 (Horn et al., 2018). Places-LT is a long-tailed version of the large-scale scene recognition Places dataset (Zhou et al., 2017). It contains 62.5K images from 365 categories, with the instance number ranging from 5 to 4,980. ImageNet-LT is created by subsampling from ImageNet-2012 (Deng et al., 2009), consisting of 1,000 classes. The training set comprises 115.8K images, with the class image number ranging from 1,280 to 5. The validation and test sets are balanced, containing 20K and 50K images, respectively. iNaturalist 2018 (Horn et al., 2018) is a naturally long-tailed real-world dataset, comprising 8,142 fine-grained species and 437.5K images. We employ the official validation set for fair comparisons. Class-level text datasets. We build the class-wise text corpus to fine-tune the classifier, visual prompt tokens, and descriptor cache weights for hierarchical classification. For the Text Corpus, we utilize the corpus given by VL-LTR (Tian et al., 2022), which leverages class names as the query to retrieve entities from Wikipedia. After irrelevant section cleaning, we split the wiki sentences to construct the original text candidate set for each category. We incorporate handcraft-prompted (from CoOp (Zhou et al., 2022b)) and generated sentences (from CuPL (Pratt et al., 2022)) to achieve class balance. For Descriptor Cache, we query the GPT-3.5-turbo for $M = 10$ fine-grained label descriptions. We request the LLMs to generate longer sentences instead of phrases to match the Text Corpus style. We utilize the CLIP text encoder to extract description features and cache them after normalisation to avoid loading the text encoder during inference. 4.2 COMPARISON WITH SOTA We conduct comprehensive experiments on PlacesLT, ImageNet-LT and iNaturalist 2018 benchmarks. Our HLC outperforms state-of-the-art (full-finetuning methods) remarkably with minor parameters (visual prompt tokens and classifier) to tune. We provide detailed comparisons in terms of model size, tuning parameters, and inference required parameters. By default, we finetune the CLIP ViT-B-16 for 30 epochs with the initial learning rate $5e^{-4}$ and cosine decay schedule. We set the number of visual prompt tokens as 20 and the VPT layer depth $L_p = 12$. The sub-groups are split by instance number according to the SADE (Zhang et al., 2022). Comparisons on Places-LT. Table 1 shows the experimental comparisons with previous SOTA on PlacesLT. The zero-shot CLIP does not require any fine-tuning but its performance is far from satisfactory. BALLAD (Ma et al., 2021) and VL-LTR (Tian et al., 2022) both fully fine-tuned CLIP using additional textual corpus and propose unique techniques to address the long-tail problem. However, our HLC outperforms them significantly without further complex LTR designs (only post-hoc logit adjustment Equation 4). Compared to VL-LTR, the HLC demonstrates significant advantages in terms of training epochs (30 v.s. 400) and tuning parameters (0.42M v.s. 149.62M). The LPT (Dong et al., 2023) has a similar number of tuning parameters to ours (1.01M v.s. 0.42M), while the longer input token sequence slows down its inference speed (Table 8). In contrast, our approach utilizes a few numbers of visual prompt tokens and conducts inference end to end. Comparisons on ImageNet-LT. Table 2 presents the quantitative results on ImageNet-LT. Compared to previous works, multimodal methods (VL-LTR (Tian et al., 2022), IVLM (Wang et al., 2023)) show significant advantages for the excellent performance of CLIP on ImageNet. Both full fine-tuning and VPT methods ameliorate zero-shot CLIP, while our approach achieves state-of-the-art results. Note that our HLC is deployed without a text encoder, which ensures the model size is on par with multi-expert methods based on ResNet-50, such as NCL (Li et al., 2022a). Comparisons on iNaturalist 2018. Table 3 presents the evaluation on the iNaturalist 2018 dataset. Due to the domain gap between iNaturalist 2018 and CLIP training data, direct zero-shot or linear probe yield unsatisfactory performance (Dong et al., 2023; Wang et al., 2023). Hence, we employ the full finetuning strategy (visual branch and classifier) to facilitate comparisons with other methods based on ImageNet21k. We fail to reproduce the LPT and VL-LTR@384 (we leave '-' in the table) because the training is overly complex for the $4 \times 2080$Ti. Given the default resolution (@224), the HLC outperforms the previous SOTA remarkably. Note that the performance of HLC@336 is on par with VL-LTR@384 resolution. ### Table 1: Performance on the PlacesLT dataset. All methods are grouped by model type. VLMs refer to the dual encoder of vision (ViT-B/16) and text architecture. Our HLC achieves state-of-the-art results on all shots while requiring significantly fewer fine-tuning parameters. \| Method \| Model \| Tuning Params. \| Model Params. \| Many \| Med. \| Few \| Acc. \| \|--------\|-------\|----------------\|---------------\|------\|------\|-----\|------\| \| OLTR \| Liu et al., 2019 \| \| \| 44.7 \| 37.0 \| 25.3 \| 35.9 \| \| SADE \| Zhang et al., 2022 \| \| \| 42.8 \| 39.0 \| 31.2 \| 38.8 \| \| MisLAS \| Zhong et al., 2021 \| ResNet152 \| 60.34M \| 60.34M \| 39.6 \| 43.3 \| 36.1 \| 40.4 \| \| ALA \| Zhao et al., 2022 \| \| \| 43.9 \| 40.1 \| 32.9 \| 40.1 \| \| PaCo \| Cui et al., 2021 \| \| \| 36.1 \| 47.9 \| 35.3 \| 41.2 \| \| MAE \| He et al., 2022 \| 111.66M \| \| 48.9 \| 24.6 \| 8.7 \| 30.3 \| \| DeiT III \| Touvron et al., 2022 \| 86.66M \| 86.66M \| 51.6 \| 31.0 \| 9.4 \| 34.2 \| \| LiVT \| Xu et al., 2020 \| ViT-B/16 \| 111.66M \| \| 48.1 \| 40.6 \| 27.5 \| 40.8 \| \| VPT \| Jia et al., 2022 \| 0.09M \| 86.75M \| 50.4 \| 33.8 \| 23.3 \| 37.5 \| \| LPT \| Dong et al., 2023 \| 1.01M \| 87.58M \| 49.3 \| 52.3 \| 46.9 \| 50.1 \| \| RAC \| Long et al., 2022 \| 86.57M \| 236.19M \| 48.7 \| 48.3 \| 41.8 \| 47.2 \| \| CLIP \| Radford et al., 2021 \| VLMs \| 0M \| \| 35.0 \| 37.3 \| 44.2 \| 37.9 \| \| BALLAD \| Ma et al., 2021 \| 149.62M \| 149.62M \| 49.3 \| 50.2 \| 48.4 \| 49.5 \| \| VL-LTR \| Tian et al., 2022 \| 149.62M \| \| 54.2 \| 48.5 \| 42.0 \| 50.1 \| \| HLC (ours) \| \| ViT-B/16 \| 0.42M \| 86.99M \| 53.1 \| 52.1 \| 48.6 \| 51.5 \| ### Table 2: Performance on the ImageNet-LT. Our HLC achieves state-of-the-art without backbone parameter tuning. \| Method \| Many \| Med. \| Few \| Acc. \| \|--------\|------\|------\|-----\|------\| \| CE \| 64.0 \| 33.8 \| 5.8 \| 41.6 \| \| c-RT \| 61.8 \| 46.2 \| 27.3 \| 49.6 \| \| RIDE \| 68.3 \| 53.5 \| 35.9 \| 56.8 \| \| PaCo \| 68.0 \| 56.4 \| 37.2 \| 58.2 \| \| GCL \| 63.0 \| 52.7 \| 37.1 \| 54.5 \| \| BCL \| 67.6 \| 54.6 \| 36.6 \| 57.2 \| \| NCL \| 67.3 \| 55.4 \| 39.0 \| 57.7 \| \| SADE \| 66.5 \| 57.0 \| 43.5 \| 58.8 \| \| DLSA \| 67.8 \| 54.5 \| 38.8 \| 57.5 \| \| DeiT III \| 70.4 \| 40.9 \| 12.8 \| 48.4 \| \| LiVT \| 73.6 \| 56.4 \| 41.0 \| 60.9 \| \| CLIP \| 65.4 \| 63.5 \| 63.2 \| 64.2 \| \| VL-LTR \| 84.5 \| 74.6 \| 59.3 \| 77.2 \| \| LPT \| 76.6 \| 73.3 \| 67.6 \| 73.7 \| \| MARC+IVLM \| 83.9 \| 78.3 \| 70.0 \| 79.3 \| \| HLC \| 84.1 \| 79.1 \| 71.1 \| 79.9 \| ### Table 3: Performance on the iNaturalist 2018. We report higher resolution results (@224 by default) for fair comparisons. \| Method \| Many \| Med. \| Few \| Acc. \| \|--------\|------\|------\|-----\|------\| \| CE \| 72.2 \| 63.0 \| 57.2 \| 61.7 \| \| OLTR \| 59.0 \| 64.1 \| 64.9 \| 63.9 \| \| RIDE \| 70.9 \| 72.5 \| 73.1 \| 72.6 \| \| TADE \| 74.4 \| 72.5 \| 73.1 \| 72.9 \| \| PaCo \| 75.0 \| 75.5 \| 74.7 \| 75.2 \| \| GCL \| 67.5 \| 71.3 \| 71.5 \| 71.0 \| \| BCL \| 66.7 \| 71.0 \| 70.7 \| 70.4 \| \| NCL \| 72.0 \| 74.9 \| 73.8 \| 74.2 \| \| DOC \| 72.8 \| 71.7 \| 70.0 \| 71.0 \| \| CLIP \| 9.9 \| 5.3 \| 4.6 \| 5.5 \| \| LiVT \| 78.9 \| 76.5 \| 74.8 \| 76.1 \| \| LPT \| - \| - \| 79.3 \| 76.1 \| \| VL-LTR \| 81.6 \| 78.0 \| 74.4 \| 76.8 \| \| VL-LTR@384 \| - \| - \| - \| 81.0 \| \| HLC \| 78.3 \| 81.8 \| 77.5 \| 79.8 \| \| HLC@336 \| 79.1 \| 81.8 \| 80.6 \| 81.1 \| Table 4: Ablation study on the PlacesLT based on CLIP (ViT/B-16). LP: linear probe. VPT: visual prompt tuning. LA: logit adjustment (Equation 4). Corpus: training with textual data. SFM: shared feature space mixup. Descriptor / Reweight: inference with (reweighted) feature descriptors. \| ID \| Method \| Many \| Med. \| Few \| Acc. \| \|----\|--------\|------\|------\|-----\|------\| \| a) \| Zero-shot CLIP \| 35.0 \| 37.3 \| 44.2 \| 37.9 \| \| b) \| CLIP + Linear Probe \| 55.7 \| 34.5 \| 14.4 \| 38.4 \| \| c) \| CLIP + Full Finetune \| 54.3 \| 34.6 \| 20.3 \| 39.1 \| CLIP + Linear Probe \| ID \| VPT \| LA \| Corpus \| SFM \| Descriptor \| Reweight \| Many \| Med. \| Few \| Acc. \| \|----\|-----\|----\|--------\|-----\|------------\|----------\|------\|------\|-----\|------\| \| d) \| ✓ \| \| \| \| \| \| 55.1 \| 37.5 \| 22.7 \| 41.2 \| \| e) \| ✓ \| ✓ \| \| \| \| \| 50.2 \| 47.2 \| 40.7 \| 47.1 \| \| f) \| ✓ \| ✓ \| ✓ \| \| \| \| 51.2 \| 48.9 \| 43.9 \| 48.8 \| \| g) \| ✓ \| ✓ \| ✓ \| ✓ \| \| \| 53.1 \| 50.2 \| 46.4 \| 50.5 \| \| h) \| ✓ \| ✓ \| ✓ \| ✓ \| ✓ \| \| 49.9 \| 48.2 \| 40.3 \| 47.3 \| \| i) \| ✓ \| ✓ \| ✓ \| ✓ \| ✓ \| ✓ \| 53.1 \| 52.1 \| 48.6 \| 51.5 \| Table 5: Performance comparisons with few-shot learning methods on the LTR and FSL datasets. \| Dataset \| PlacesLT \| ImageNet-16 shot \| \|---------\|----------\|------------------\| \| Method \| Many \| Med. \| Few \| Acc. \| Acc. \| Δ \| \| CLIP \| 35.0 \| 37.3 \| 44.2 \| 37.9 \| 72.43 \| - \| \| CoOp \| 52.8 \| 34.7 \| 29.3 \| 40.1 \| 76.47 \| 4.04 \| \| CoCoOp \| 50.0 \| 33.4 \| 33.0 \| 39.3 \| 75.98 \| 3.55 \| \| MaPLe \| 53.8 \| 36.2 \| 29.5 \| 41.2 \| 76.66 \| 4.23 \| \| Ours \| 52.7 \| 41.1 \| 36.8 \| 44.5 \| 77.06 \| 4.63 \| Table 6: Ablation study of candidate number $K$ on the PlacesLT. \| Top K \| Many \| Med. \| Few \| Acc. \| \|-------\|------\|------\|-----\|------\| \| N/A \| 52.7 \| 41.1 \| 36.8 \| 44.5 \| \| 1 \| 53.1 \| 50.2 \| 46.4 \| 50.5 \| \| 3 \| 53.1 \| 50.6 \| 46.9 \| 50.8 \| \| 5 \| 53.3 \| 51.4 \| 48.3 \| 51.5 \| \| 7 \| 52.6 \| 49.7 \| 46.2 \| 50.1 \| \| 9 \| 52.5 \| 49.1 \| 45.1 \| 49.5 \| \| 11 \| 52.3 \| 47.8 \| 43.2 \| 48.5 \| 4.3 Ablation Study In this paper, we construct text corpora and feature description cache to assist VLMs. We propose shared feature space mixup (SFM) and weighted feature descriptors to align image features with language. Besides, we employ VPT (Liu et al., 2022) and LA (Menon et al., 2021) on top of the baseline to enhance fine-tuning effectiveness. Hence, we conduct experiments on PlacesLT to verify the efficacy of all proposals and the results are presented in Table 4. Combined with Table 1, we observe that the zero-shot CLIP has surpassed the previous baseline performance (OLTR (Liu et al., 2019) 35.9% vs. CLIP 37.9%), but fails to catch up with SOTA (e.g., LPT 50.1%). Considering the types a-c, the linear probe and full fine-tuning exhibit similar performance (38.4% vs. 39.1%), albeit with a significant difference in optimization parameters. Consequently, we adopt the CLIP + linear probe in the following experiments. The effectiveness of VPT is corroborated by type d, which indicates that VPT inspires better adaptation of the downstream distribution without compromising the CLIP feature extraction capability (Dong et al., 2023). Type e demonstrates that LA can effectively calibrate the model’s prior biases (see sub-group performance), which serves as a concise post-processing approach. Type f and g demonstrate that the multimodal data consistently improves overall performance, particularly on the few-shot subgroup. Mixed with high-quality textual features, the tail image features have better anchor (centre) representation, and our SFM further facilitates the convergence of tail image features. Directly utilizing the average similarity of descriptors results in performance degradation (type g and h), which is incurred by the erroneous or non-informative descriptions that do not contribute to the visual recognition (Figure 6). We optimize each description weight to mitigate the aforementioned adverse effects (type i). 4.4 Further Discussion Comparisons with FSL methods. Our HLC is inspired by few-shot learning (FSL). Therefore, we reproduce the FSL methods, e.g., CoOp (Zhou et al., 2022b), CoCoOp (Zhou et al., 2022a) and MaPLe (Khattak et al., 2023) based on the MaPLe code repository. We conduct sufficient experiments on both LTR and few-shot benchmarks and present the results in Table 5. We retain only VPT and reweighted descriptors of the HLC to ensure comparable complexity with FSL methods. The descriptors’ weights are trained on the LTR benchmarks. From Table 5, the FSL methods have consistently improved the zero-shot CLIP performance on both datasets. Our HLC significantly outperforms FSL methods on the LTR dataset (+3.3% compared to MaPLe), thereby providing com- Figure 4: Ablation study of VPT tokens ($N$) and layer depth ($L_p$) on the PlacesLT. We adopt ViT-B/16 as the backbone for all settings. Figure 5: Ablation study of the visual encoder backbone on the PlacesLT. @336px means the input image size is $336 \times 336$. Table 7: Zero-shot OVC performance on the PlacesLT. †: GPT-3. ‡: GPT-3.5-turbo. We further reweight the descriptors given by LLMs. \| Method \| CLIP \| VCD† \| VCD‡ \| Ours‡ \| \|--------\|------\|------\|------\|-------\| \| Acc. \| 37.91\| 40.34\| 40.51\| 43.12 \| \| ∆ \| - \| +2.43\| +2.6 \| +5.21 \| Table 8: Inference performance on the iNaturalist 2018 with batch size 64. We outperform SOTA in computation and reasoning time. \| Method \| VL-LTR \| LPT \| HLC \| \|--------\|--------\|-----\|-----\| \| FLOPs (T) \| 1.301 \| 2.406 \| 1.189 \| \| Inf. time (ms) \| 354.17 \| 513.62 \| 269.32 \| Pelling evidence for the crucial role of fine-grained descriptors and corresponding weights. Our approach also demonstrates its effectiveness on the ImageNet 16-shot base classes. Our text branch has no learnable tokens and relies on the basic prompt "a photo of {CLS}". Nevertheless, our proposal achieves superior performance compared to MaPLe (76.66% v.s. 77.06%). Number of Candidates $K$. Table 6 shows the effect of the candidate number $K$ given by the LTR classifier. N/A means that we directly calculate the reweighted average similarity for all classes (no LTR classifier). $K = 1$ means the LTR classifier performance. Note that larger $K$ can not always lead to better performance. More candidate categories introduce more noise information. The $K$ mainly affects med and few groups. Hence, we set $K = 5$ by default for our experiments. Number of Prompt Tokens $N$ and Depth $L_p$. We conduct experiments on the PlacesLT to investigate the impact of visual prompt tokens $N$ and layer depth $L_p$. As shown in Figure 4, the impact of $L_p$ on the performance is significant, while the impact of token number $N$ is minor. Note that the maximum $N$ is 30, as longer token sequences will remarkably slow down the inference speed. Therefore, we set $N = 20$ and $L_p = 12$ by default for other experiments. Effect of Visual Encoder. Figure 5 demonstrates the effect of different visual encoders. The backbone shows minimal effect on both zero-shot CLIP and proposed HLC. In contrast, proper text features will serve as effective anchors to guide model classification. This conclusion aligns with prior works such as CoOp and VCD. Comparisons with VCD. VCD (Menon & Vondrick, 2023) is the first to utilize the LLMs (GPT-3) to generate fine-grained descriptions for assisting VLMs in open vocabulary classification. Our HLC is different in 2 aspects: 1) we employ more advanced LLMs (GPT-3.5-turbo) to generate more apt descriptors; 2) we reweight each description to filter out the irrelevant ones. The comparisons are shown in Table 7. Both descriptions provided by GPT-3 and GPT-3.5-turbo enhance the performance of zero-shot CLIP. However, the reweighting operation significantly improves the OVC performance, demonstrating its success in mitigating the influence of irrelevant descriptions on visual recognition. Inference Analysis. Table 8 shows the model FLOPs and inference time for an epoch. We evaluate the validation dataset of iNaturalist 2018 with batch size 64 on 2080Ti. Based on ViT-B-16, our HLC is much smaller on both FLOPs and average inference time than the previous SOTA. Failure Case Analysis. Why is it necessary to reweight descriptors? We observe that some descriptors are not beneficial for visual recognition and thus result in lower average V-L similarity. These cases encompass descriptions that do not contribute to visual recognition (Fig. 6a, description of the sound.) and that are inconsistent with the images in the benchmark datasets (Fig. 6b, no signs are Figure 6: Examples of 4 type failure descriptors given by large language models. Our reweighting operation effectively mitigates the negative influence of types (a) and (b). present in the category Airfield.). By reweighting, we can decrease the similarity of such descriptors and emphasize the ones that are truly useful for visual recognition. There are also failure cases due to label word ambiguity (Fig. 6c), mistakenly interpreting the commercial brand “Vespa” as a type of wasp) and factual inaccuracies resulting from hallucinations (Fig. 6d). However, we believe these issues can be alleviated along with LLMs’ rapid development. 5 RELATED WORK Long-tailed Visual Recognition. The most straightforward approach is to address the issue through rebalancing techniques, which encompass resampling via balanced or inverse sampler (Cao et al., 2019; Kang et al., 2020; Zhang et al., 2021; Li et al., 2022b; Dong et al., 2023) and reweighting via loss weight (Cui et al., 2019; Tang et al., 2020; Tan et al., 2020) or margin (Menon et al., 2021; Xu et al., 2023b; Li et al., 2022b) methods. The LTR data augmentation enhances the tail samples via feature mixing (Chou et al., 2020; Chu et al., 2020) or generation (Li et al., 2021; Park et al., 2022). The Mixture of Experts (MoE) method proposes to learn different parts of LTR data (Wang et al., 2021; Li et al., 2022a; Jin et al., 2023; Xu et al., 2023a). Visual Language Models have been proposed to facilitate downstream tasks by introducing extra language data (Radford et al., 2021; Alayrac et al., 2022; Jia et al., 2021). CoOp (Zhou et al., 2022b) learns soft text prompts to improve the zero-shot CLIP performance. CoCoOp (Zhou et al., 2022a) formulates the text prompts with instance-level conditions. VPT (Jia et al., 2022) introduces visual prompts to effectively fine-tune the visual branch to fit the downstream data distribution. MaPLE (Khattak et al., 2023) jointly optimizes the visual and text prompts and employs mapping to establish a correspondence between the two types of prompts. The CuPL (Pratt et al., 2022), VCD (Menon & Vondrick, 2023) and CHiLS (Novack et al., 2023) have further leveraged large language models to generate fine-grained descriptions of class labels to enhance the text branch of VLMs. These FSL methods hold implications for learning from tail classes. Learning LTR data with Visual Language Models. The impressive zero-shot capabilities exhibited by VLMs have inspired a series of excellent works on long-tail recognition. The VL-LTR (Tian et al., 2022) conducts full fine-tuning of CLIP (Radford et al., 2021), followed by a language-guided recognition head to adapt the long-tail data. Similarly, BALLAD (Ma et al., 2021) leverages a linear adapter to mitigate the impact of long-tail bias. LMPT (Xia et al., 2023) introduces an embedding loss with class-aware soft margin and re-weighting to learn class-specific contexts. IVLM (Wang et al., 2023) incorporates a lightweight decoder to accommodate previous work on LTR and provides a comprehensive assessment of their performance based on VLMs. 6 CONCLUSION In this paper, we propose a hierarchical long-tailed classification (HLC) framework to address the long-tailed recognition problem. We employ visual prompt tuning and propose the shard space mixup to train an effective coarse classifier. Then, we utilize large language models to generate fine-grained descriptions for each class and train corresponding weights to filter out irrelevant ones. Given top k candidate classes from the coarse classifier, we perform fine-grained open vocabulary classification based on descriptions. Our approach achieves state-of-the-art performance with minimal parameters to tune and enhances the interpretability of the prediction results. 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Nxn6vGgpI9	Compared to the vision-based action recognition datasets, the scale of WEAR is relatively small, with only 18 people operating 18 workout activities, and the total duration of each activity is 90 seconds, which does not show the minimum times the activity appears in this period.	WEAR: AN OUTDOOR SPORTS DATASET FOR WEARABLE AND EGOCENTRIC ACTIVITY RECOGNITION Anonymous authors Paper under double-blind review ABSTRACT Though research has shown the complementarity of camera- and inertial-based data, datasets which offer both egocentric video and inertial-based sensor data remain scarce. In this paper, we introduce WEAR, an outdoor sports dataset for both vision- and inertial-based human activity recognition (HAR). The dataset comprises data from 18 participants performing a total of 18 different workout activities with untrimmed inertial (acceleration) and camera (egocentric video) data recorded at 10 different outside locations. Unlike previous egocentric datasets, WEAR provides a challenging prediction scenario marked by purposely introduced activity variations as well as an overall small information overlap across modalities. Benchmark results obtained using each modality separately show that each modality interestingly offers complementary strengths and weaknesses in their prediction performance. Further, in light of the recent success of temporal action localization models following the architecture design of the ActionFormer, we demonstrate their versatility by applying them in a plain fashion using vision, inertial and combined (vision + inertial) features as input. Results demonstrate both the applicability of vision-based temporal action localization models for inertial data and fusing both modalities by means of simple concatenation, with the combined approach (vision + inertial features) being able to produce the highest mean average precision and close-to-best F1-score. The dataset and code to reproduce experiments is publicly available via: https://www.anonymous.edu/anon 1 INTRODUCTION The physical activities that we perform in our daily lives have been identified as valuable information for a number of research fields and applications, such as work processes support, preventive healthcare, cognitive science or workout monitoring (e.g., Bao & Intille (2004); Patterson et al. (2005); Ward et al. (2006)). Research efforts have till now shown that physical activities can be detected using either wearable inertial sensors or camera-based approaches. The inertial sensors can continuously observe motion and gestures at particular body locations, whereas camera-based systems can typically observe the user’s entire body, but can be hindered by (self-)occlusions. Inertial data manifests itself as multidimensional timeseries, while image data can be interpreted more easily afterwards. Even though research has shown (e.g. Spriggs et al. (2009); Song et al. (2016a); Diete et al. (2019); Nakamura et al. (2017)) that both modalities are complementary to each other, available benchmark datasets that provide both egocentric video and inertial-based sensor data remain scarce. We therefore introduce WEAR, an outdoor sports human activity recognition (HAR) dataset featuring workout activities performed by 18 participants while wearing inertial sensors on both wrists and ankles as well as a head-mounted camera capturing egocentric vision using a wide field-of-view - see Figure 1. In light with one of the key challenges in HAR, namely the NULL-class problem (Bulling et al. (2014), WEAR provides continuous data streams of each workout session including all breaks and interruptions. Our dataset features a challenging prediction scenario marked by purposely introduced activity variations, activities consisting of within-activity sequences (i.e. a sequence of multiple base activities) and an overall small information overlap across modalities. Unlike previous egocentric datasets, included activities are not defined by human-object-interaction Figure 1: Setup and example data from the two types of wearable sensors in our dataset. Participants were equipped with four open-source smartwatches (one per limb) and a head-mounted camera. (e.g., DelPreto et al., 2022; de la Torre et al., 2009)) nor originate from inherently distinct activity categories (e.g., Possas et al., 2018; Xu et al., 2023). WEAR was collected at 10 different outdoor recording locations, with each location introducing different visual and surface conditions, yet not providing cues about the activity being performed. With these dataset traits in place we deem the WEAR dataset being an exemplary dataset to assess methods on how to combine both inertial- and vision-based features in the context of HAR. Our contributions in this paper are three-fold: 1. We introduce a new inertial- and vision-based HAR dataset called WEAR. The dataset features data of 18 participants, each performing 18 different sports activities. 2. We provide benchmark scores using both wearable- (Bock et al., 2021; Abedin et al., 2021) and vision-based (Zhang et al., 2022; Shi et al., 2023) state-of-the-art models. 3. We demonstrate that state-of-the-art temporal action localization models from computer vision are excellently suited to not only process raw inertial data, but even successfully fuse multi-modal information significantly outperforming the best single-modality approach as well as beating the best possible (oracle) late fusion approach in terms of mAP. 2 RELATED WORK Inertial-based HAR Compared to video-based modalities body-worn sensor systems bear a great potential in analyzing our daily activities with minimal intrusion, yielding various applications from the provision of medical support to supporting complex work processes (Bulling et al., 2014). Within the last decade deep learning based-methods have established themselves as the de facto standard in inertial-based HAR as they have shown to outperform classical machine learning algorithms (Ordóñez & Roggen, 2016; Hammerla et al., 2016; Guan & Plötz, 2017). One of the most well-known deep learning approaches for inertial-based HAR is the DeepConvLSTM which is a hybrid model combining both convolutional and recurrent layers (Ordóñez & Roggen, 2016). By combining both types of layers the network is able to automatically extract discriminative features and model temporal dependencies. Following the success of the original DeepConvLSTM, researchers worked on extending the architecture (Murahari & Plötz, 2018; Xi et al., 2018) or build up on the idea of combining convolutional and recurrent layers by proposing their own architectures (Xu et al., 2019; Abedin et al., 2021; Yuki et al., 2018; Zhou et al., 2022). Within this publication we are reporting benchmark scores using the WEAR dataset inertial sensor-streams as input for two popular HAR models (Bock et al., 2021; Abedin et al., 2021). Contrary to the belief that one needs to employ multiple recurrent layers when dealing with sequential data (Karpathy et al., 2015), Bock et al. (2021) proposed an altered shallow DeepConvLSTM architecture which proved to outperform the original architecture by a significant margin. Differently, Abedin et al. (2021) chose to build up on the idea of the DeepConvLSTM and introduced the Attend-and-Discriminate architecture which exploits interactions among different sensor modalities by introducing self-attention through a cross-channel interaction encoder and adding attention to the recurrent parts of the network. Vision-based HAR Predicting activities performed by humans based on visual-cues can broadly be categorized into three main application scenarios: action recognition, localization and anticipation. Action recognition systems (Liu et al., 2021b; Wang et al., 2021; Li et al., 2022) aim to assign a set of trimmed action segments an activity label. Contrarily, temporal action localization systems (Zhang et al., 2022; Yang et al., 2022; Liu et al., 2022b) are tasked to identify start and end times of all activities in a untrimmed video by predicting a set of activity triplets (start, end, activity label). Lastly, action anticipation systems (Girdhar & Grauman, 2021; Roy & Fernando, 2022) aim to predict the label of a future activity having observed a segment preceding its occurrence. Though sensor-based HAR systems are employed using a sliding window approach and thus assign activity labels to a set of trimmed inertial-sequences, their ultimate goal is to identify a set of activities within a continuous timeline. We therefore deem vision-based temporal action localization to be most comparable to inertial-based HAR and will focus on it in our benchmark analysis. Existing temporal action localization methods can be divided into two categories: two- and single-stage approaches. Two-stage approaches (Lin et al., 2019; 2020; Xu et al., 2020; Bai et al., 2020; Zhao et al., 2020; Zeng et al., 2019; Gong et al., 2020; Liu et al., 2021a; Qing et al., 2021; Sridhar et al., 2021; Zhu et al., 2021; Zhao et al., 2021; Tan et al., 2021) divide the process of temporal action localization into two subtasks. First, during the action segment proposal generation, candidate video segments are generated which are then, classified with an activity label as well as refined regarding their temporal boundaries. Contrarily, single-stage approaches (Yang et al., 2022; Shi et al., 2022; Nag et al., 2022; Liu et al., 2022b; Liu & Wang, 2020; Long et al., 2019; Lin et al., 2021; Chen et al., 2022; Zhang et al., 2022; Shi et al., 2023) aim to localize actions in a single shot without using action proposals. In light with the success of transformer architectures in natural language processing (see e.g. Vaswani et al., 2017; Devlin et al., 2019) and computer vision (see e.g. Kolesnikov et al., 2021; Yuan et al., 2021; Liu et al., 2021b), researchers have demonstrated their applicability for temporal action localization (Cheng & Bertasius, 2022; Liu et al., 2022a,b; Shi et al., 2022; Tan et al., 2021; Zhang et al., 2022) breaking previously held benchmark scores of numerous popular datasets (Heilbron et al., 2015; Damen et al., 2022; Jiang et al., 2014) without any additional training data by a significant margin. One of such architectures is the ActionFormer proposed by Zhang et al., 2022, which is an end-to-end trainable transformer-based architecture, which unlike other single-stage approaches, does not rely on pre-defined anchor windows. The architecture combines multiscale feature representations with local self-attention and is trained through a classification and regression loss calculated by a light-weighted decoder. Building up on the works of Zhang et al., 2022, Shi et al., 2023 proposed the TriDet model which suggest to replace the transformer layers of the ActionFormer with fully-convolutional, so-called SGP layers, as well as use a trident regression head which claims to improve imprecise boundary predictions via an estimated relative probability distribution around the boundary. Given the rapid rise in popularity of single-stage temporal action localization such as the ActionFormer, we decided said models to be a suited option to deliver a first benchmark for the WEAR dataset. Multimodal (Inertial and RGB Video) HAR In Table 1 we show a curated list of datasets which provide both egocentric vision- (e.g. RGB, depth) and IMU-based (e.g. accelerometer, gyroscope, magnetometer) modalities in the context of HAR. We compare datasets regarding their recency, number of participants, number and type of activities performed, recording environment, camera and IMU position and whether the dataset is provided on a clip-basis or a continuous stream. As evident by the rise in popularity of commercial head-mounted cameras and wrist-worn smartwatches for tracking sports, we decided to position the camera and IMU sensors used during collection of the WEAR dataset in line with the recent trends in real-world application scenarios. With the head and limbs being positions which do not limit participants in their freedom of movement, we deem said positions to further be most suited in capturing how participants interact with their environment and/ or objects. This makes the works of de la Torre et al., 2009; Song et al., 2015; Diete et al., 2019 and DelPretto et al., 2022 to be most comparable to the WEAR dataset. DelPretto et al., 2022 and de la Torre et al., 2009 both provide datasets of participants cooking food recipes. Different from the WEAR dataset, recording takes places indoors in an artificial kitchen environment, which by nature Table 1: List of available egocentric vision datasets, which provide inertial data, compared with the WEAR dataset. We differentiate between recency (year), number and type of activity classes (S = Sports, G = Gestures, L = Locomotion, D = Daily Living, C = Cooking, O = Other), number of subjects, recording environment (laboratory, outside or inside), location of the Camera and IMU sensor (Multi = multiple locations on body) and recording type (trimmed or untrimmed video sequences). \| Dataset \| Year \| Sbjs \| Cls \| Type \| Where \| Camera \| IMU \| Recording \| \|------------------\|------\|------\|-----\|------\|-------\|--------\|-------\|----------\| \| CMU-MMAC [de la Torre et al., 2009] \| 2009 \| 16 \| 29 \| C \| Lab \| Head \| Multi \| Untrimmed \| \| MEAD [Song et al., 2015] \| 2015 \| 2 \| 20 \| A \| In/Out\| Head \| Head \| Trimmed \| \| Stanford-ECM [Nakamura et al., 2017] \| 2017 \| 10 \| 24 \| S \| In/Out\| Chest \| Chest \| Trimmed \| \| Daily Intention [Worr et al., 2017] \| 2017 \| 12 \| 34 \| D \| In \| Wrist \| Arm \| Trimmed \| \| DataGen [Zhang et al., 2018] \| 2018 \| 84 \| 10 \| D \| In/Out\| Head \| Head \| Trimmed \| \| ADL Dataset [Diete et al., 2019] \| 2019 \| 2 \| 6 \| D \| In \| Head \| Wrists\| Untrimmed \| \| Ego4D [Grauman et al., 2021] \| 2021 \| 931 \| 110 \| D \| In/Out\| Head \| Head \| Untrimmed \| \| ActionSense [DelPrete et al., 2022] \| 2022 \| 10 \| 20 \| C \| Lab \| Head \| Multi \| Untrimmed \| \| EPIC-Kitchen [Damen et al., 2022] \| 2022 \| 37 \| ≈149\| C \| In \| Head \| Head \| Untrimmed \| \| UESTC-MMEA-CL [Xu et al., 2023] \| 2023 \| 10 \| 32 \| D \| In/Out\| head \| Head \| Trimmed \| WEAR \| Year \| Sbjs \| Cls \| Type \| Where \| Camera \| IMU \| Recording \| \|------\|------\|-----\|------\|-------\|--------\|-------\|----------\| \| 2023 \| 18 \| 18 \| S \| Out \| Head \| Limbs \| Untrimmed \| limits the amount of variety captured in the visual data as lighting conditions and surroundings remain the same throughout all participants. Further, as cooking usually involves object-centric activities, we deem said datasets be more biased towards vision-based prediction scenario, with most of the action taking place in the POV of the user. Compared to [Song et al., 2015] and [Diete et al., 2019], WEAR provides a larger participant count and, unlike [Song et al., 2015] continuous instead of clip-based data streams. Especially the latter ensures that algorithms are assessed in their ability to differentiate unrelated actions (like breaks) from relevant activities, being a necessary trait of HAR prediction algorithm in order to be applied in-the-wild [Bulling et al., 2014]. With early works such that of [Spriggs et al., 2009] having shown the complementarity of inertial- and camera-based features, research has followed up by exploring different ways of combining the two modalities. One can categorize such methods broadly by the point in time at which the fusion of both modalities is performed. Late fusion approaches usually follow a two-stream architecture training both vision- and inertial-based modalities separately before merging together outputs of each stream through such as produced softmax probabilities e.g. via a weighted combination [Wei & Kehitarnavaz, 2020], pooling operations [Song et al., 2016a; Imran & Raman, 2020b], majority voting [Diete et al., 2018] or a concurrent classifier [Wu et al., 2017; Diete & Stuckenschmidt, 2019; Ijaz et al., 2022]. Early fusion approaches aim at jointly learning from both modalities by using feature embeddings calculated on one (or both) modalities to e.g. use the concatenation of both to train a concurrent network [Imran & Raman, 2020a; Xu et al., 2023; Nakamura et al., 2017; Lu & Velipasalar, 2018; Hu et al., 2023; Ehatisham-Ul-Haq et al., 2019; Diete & Stuckenschmidt, 2019; Diete et al., 2019; Song et al., 2016b; Yu et al., 2019; Chen et al., 2016; Islam & Iqbal, 2022, 2021], enhance softmax probabilities used during late fusion [Diete & Stuckenschmidt, 2019; Diete et al., 2019] or adding intermediate cross-view connections amongst the two modality streams [Ijaz et al., 2022]. With experiments showing that single-stage temporal action localization models are able to produce competitive results on raw inertial data, this paper also tests the applicability of two state-of-the-art models, namely the ActionFormer and TriDet model, to fuse and combine cues of both modalities in an early-fusion style. Unlike other early fusion techniques, our approach is the first to directly use the raw inertial data by means of simple concatenation together with a vision-based feature embedding. 3 METHODOLOGY Study Design & Scalable Pipeline Participants were recorded during separate recording sessions. Prior to their first session, participants were handed a recording plan which outlined the study protocol as well informed about any risks of harm, data collection, usage, anonymisation and publication, as well as how to revoke their data usage rights at any point in the future. The study design involving human participants was reviewed and approved by [Anonymized]. All participants were briefed and provided their written informed consent. Each participant was asked to perform 18 workout activities. The location and the time of day at which the sessions were performed, were not fixed and thus vary across subjects. Participants were suggested to follow a two-session setup, i.e. 9 activities per session. Nevertheless, it was allowed to differ from this setup and split the 18 activities across as many (or as few) sessions as participants liked. This caused the amount of recording sessions to vary across subjects, but also increased the amount of captured variability in weather conditions and recording locations. In order to avoid misunderstandings in the execution of the activities, the authors discussed all activities prior to each session and encouraged participants to ask questions during the session if something remained unclear. Participants were tasked to perform each activity for roughly 90 seconds. As activities varied in their intensity, it was not required to perform activities for 90 seconds straight and participants could include breaks as needed. Furthermore, to ensure that each participant was able to perform all workout activities properly, the recording plan detailed how activities could be altered in their execution, for instance so that they required less physical strength. The recording plan provided with our dataset (see Section E in the supplementary material) includes all necessary materials and is written in such a way that all activities and sessions can easily be reproduced by persons other than the authors. Besides the used sensors for video and acceleration recording, the exercises only require a yoga mat and a chair (or similar items). Sessions can be recorded at any location outside as long as the privacy of the participants as well as pedestrians is ensured. We argue that this facilitates reproducibility, and with a minimal setup ensures that it is possible for others to extend our dataset at a later date. Participant Information We recorded data for 18 participants (10 male, 8 female) at 10 different locations and under varying weather conditions over a stretch of 5 months (October till February), totalling more than 15 hours, with each participant on average contributing roughly 50 minutes of data. The participants were at the time of recording on average 28 years old (± 5), 175.4 cm tall (± 10.8) and weighed 69.26 kg (± 12.43). In order to assess their sports level, participants filled in a post-session questionnaire. The questionnaire contained questions related to vital information (such as body height, weight and age), weekly workout frequency (min. 15 minutes duration) and experience in particular workout activities. On average, participants which took part in the study tend to work out 3.6 times per week (± 2.1), already knew 15.06 (± 3.75) out of the 18 activities in advance, and regularly conduct 5.5 (± 3.74) of the recorded activities as part of their private workouts. Participants reported for their personal workout schedules a wide-range of cardio- (running, hiking, cycling, dancing), strength- (weight lifting, freeletics, rowing), team- (volleyball, basketball, table-tennis) and flexibility-focused (yoga, ballet) exercise types. Dataset Collection & Structure The WEAR dataset provides subject-wise raw and processed acceleration and egocentric-video data (see Figure 1). We focus on 3D accelerometers especially as they cover a substantial amount of commercial fitness devices worn at the wrists and ankles. They furthermore are used in a large set of existing research and datasets focusing on wearable data for activity recognition, and they do not suffer from noise, drift, and other device-specific characteristics. 3D accelerometer data was collected at 50 Hz with a sensitivity of ± 8g using four open-source Bangle.js smartwatches running a custom, open-source firmware (Van Laerhoven et al., 2022). The watches were placed by the researchers in a fixed orientation on the left and right wrists and ankles of each participant. Egocentric video data was captured using a GoPro Hero 8 action camera, which was mounted using a head strap on each participant’s head. The resulting `.mp4`-videos were recorded at 1080p resolution with 60 frames per second and the camera being tilted downwards in a 45 degree angle. A second tripod-mounted camera was placed within the proximity of each participant to facilitate annotation recording the environment in which the workout was performed from a third-person-perspective. For privacy reasons, the second camera’s video and all audio captured are not part of the WEAR dataset. During postprocessing, the delta-compressed inertial data, extracted from the watch’s memory, was decompressed to `.csv`-format. Inspired by the works of Scholl et al. (2019) and Morshed et al. (2022), we made use of the similarities between inertial sensor and audio data and converted the 3D accelerometer data to `.wav`-files, which allowed to import both modalities into a standard video editing software. By having participants perform synchronization jumps, i.e. jumping 3 times while raising the arms during the jump, at the start and end of each session, peaks in the inertial data were able to be mapped to timestamps in the video stream. Lastly, activity labels, which were added as video subtitles, were exported along with the synchronized video and inertial data streams and appended as an additional column to the inertial data as well as provided as `.json`-format files, following the THUMOS-14 (Jiang et al., 2014) formatting-style. 4 BENCHMARKS AND BASELINE RESULTS Though the WEAR dataset provides the possibility for a multitude of HAR use cases, this paper focuses on introducing one sample application scenario per data modality, namely: (1) inertial-based wearable activity recognition, (2) vision-based temporal action localization, as well as, (3) a combined approach using both data modalities as input simultaneously. We chose to use said application scenarios because of their similarities with each other as they both aim to detect a set of activities in an untrimmed sequence of data. Nevertheless, other HAR-specific (e.g. action anticipation and classification) and non-HAR application scenarios (e.g. hand detection, pose estimation or simultaneous localization and mapping (SLAM)) are applicable. During each experiment we employ a three-fold validation split each time using 12 subjects for training while reserving 6 subjects for validation. The validation is applied in such a way that each subject becomes part of the validation set exactly once with the final evaluation metrics being the average across the three splits. In order to minimize the risk of performance differences between experiments being the result of statistical variance, evaluation metrics are averaged across three runs each time employing a different random seed. With the standard error of evaluation metrics amongst runs being at maximum 2.5% and the majority of runs being below 1%, we only report average evaluation metrics in this paper. All mentioned experiments were conducted on a single NVIDIA Tesla V100 GPU and lasted no longer than 24 hours. Though sharing inherent similarities, vision-based action localization algorithms predict a collection of activity segments defined by a start and end time, while, contrarily, inertial-based HAR systems provide labels based on the pre-defined windowed segmentation. Given their difference in prediction output, different evaluation metrics are applied, with mean average precision (mAP) being most prominent metric in vision-based temporal action localization and accuracy/F1-score being the most prominent metric in inertial-based activity recognition. Therefore, to guarantee comparability amongst application scenarios and architectures, predictions of each algorithm are converted such that both vision- and inertial-based evaluation metrics can be calculated. More specifically, our reported benchmark evaluation metrics are (1) a record-based calculated recall, precision and F1-score, and (2) segment-based mean average precision (mAP) at different temporal intersection over union (tIoU) thresholds, commonly used to evaluate temporal action localization datasets. Vision-based Temporal Action Localization Same as Zhang et al. (2022) and Shi et al. (2023), we chose to train the vision-based benchmark models using two-stream 13D feature embeddings pretrained on Kinetics-400 applying three different clip lengths (0.5, 1 and 2 seconds) with a 50% overlap between clips. Besides increasing the number of epochs to 300, we chose to use the same training strategy which produced best performing results on the EPIC-Kitchens dataset (Damen et al., 2022) as reported by both architectures. Different from inertial-based approaches, temporal action localization models are not trained and able to predict an explicitly modelled NULL-class. With both models being set to predict up to 2000 action segments per video, each timestamp ended up being classified by an action segment causing prediction performance of the NULL-class being (close to) 0% accuracy. We therefore eliminated low-scoring segments by increasing the scoring threshold of both models to be 0.2, which significantly increased the accuracy of the NULL-class, while only marginally affecting prediction performance of all other activity classes (see Section C.3 of the supplementary material for an ablation study). Looking at results presented in Table 2, one can see that for the vision-based models, a clip length of 1 second delivered the best predictive performance. Analysing per-class results, one can see that the vision-based approaches struggle differentiating between different running styles, activities which do not take place within the field of view of the participant (e.g. triceps stretches) as well as normal and complex sit-ups. Inertial-based Wearable Activity Recognition As our inertial-based benchmark algorithms of choice we use the shallow DeepConvLSTM proposed by Bock et al. (2021) and the Attend-and-Discriminate model proposed by Abedin et al. (2021). During all experiments we employed the same training strategy as suggested by Bock et al. (2021), which showed to produce reliable results on a multitude of inertial-based HAR datasets, only increasing the number of epochs to be the same as during the vision-based experiments (see Section C.2 of the supplementary material). To compensate with longer training times, we applied a step-wise learning rate schedule. Further, incorporating architecture changes suggested by Bock et al. (2021), we altered the Attend-and-Discriminate model to use a one-layered instead of a two-layered recurrent module and scaled the convolutional kernel size according to the sliding window and sampling rate of the WEAR dataset. Table 2: Results of human activity recognition approaches based on body-worn IMU (Inertial), vision (Camera) and combined (Inertial + Camera) features for different clip lengths (CL) on our WEAR dataset evaluated in terms of precision (P), recall (R), F1-score and mean average precision (mAP) for different temporal intersection over union (IoU) thresholds. The results underline the complementarity of the inertial and camera modalities. Best results per modality are in bold. \| Model \| CL \| P \| R \| F1 \| mAP \| \|----------------\|----\|-----\|-----\|-----\|-----\| \| \| \| 0.3 \| 0.4 \| 0.5 \| 0.6 \| 0.7 \| Avg \| \| Shallow D. \| 0.5\| 86.77\| 75.42\| 79.18\| 54.36\| 51.67\| 49.42\| 47.40\| 44.70\| 49.51\| \| A-and-D \| 0.5\| 87.54\| 75.98\| 79.59\| 53.57\| 51.08\| 48.51\| 45.82\| 42.87\| 48.37\| \| ActionFormer \| 0.5\| 78.73\| 70.50\| 72.51\| 63.71\| 61.28\| 53.90\| 39.81\| 26.40\| 49.02\| \| TriDet \| 0.5\| 86.06\| 70.95\| 75.38\| 60.91\| 63.71\| 57.49\| 50.95\| 47.44\| 55.56\| \| Shallow D. \| 1 \| 88.74\| 77.03\| 80.86\| 57.09\| 55.35\| 53.61\| 50.59\| 47.85\| 52.89\| \| A-and-D \| 1 \| 87.87\| 79.02\| 82.01\| 56.38\| 54.47\| 52.28\| 50.07\| 46.92\| 52.03\| \| ActionFormer \| 1 \| 81.69\| 75.37\| 76.86\| 72.90\| 71.30\| 68.28\| 64.14\| 56.65\| 66.65\| \| TriDet \| 1 \| 83.85\| 73.76\| 77.12\| 73.27\| 71.66\| 69.83\| 66.79\| 62.25\| 68.76\| \| Shallow D. \| 2 \| 87.92\| 78.16\| 81.60\| 59.89\| 57.00\| 54.69\| 51.77\| 48.99\| 54.47\| \| A-and-D \| 2 \| 88.24\| 80.55\| 83.08\| 58.32\| 56.68\| 54.44\| 51.58\| 48.34\| 53.87\| \| ActionFormer \| 2 \| 78.18\| 69.15\| 71.15\| 66.43\| 63.30\| 60.47\| 56.66\| 50.26\| 59.43\| \| TriDet \| 2 \| 81.72\| 69.37\| 72.53\| 65.57\| 63.65\| 61.86\| 59.07\| 54.82\| 60.99\| \| ActionFormer \| 0.5\| 68.06\| 57.68\| 58.47\| 51.27\| 49.45\| 45.74\| 36.10\| 23.38\| 41.19\| \| TriDet \| 0.5\| 73.21\| 57.73\| 60.99\| 53.41\| 51.19\| 47.24\| 40.80\| 35.08\| 45.54\| \| ActionFormer \| 1 \| 72.63\| 68.87\| 67.26\| 63.99\| 62.32\| 60.62\| 57.88\| 52.79\| 59.52\| \| TriDet \| 1 \| 75.32\| 68.07\| 67.95\| 64.36\| 63.30\| 61.38\| 59.13\| 54.64\| 60.56\| \| ActionFormer \| 2 \| 69.67\| 65.79\| 64.15\| 61.32\| 59.92\| 57.96\| 55.91\| 50.39\| 57.10\| \| TriDet \| 2 \| 73.85\| 64.09\| 64.25\| 60.95\| 60.03\| 57.75\| 55.55\| 52.19\| 57.30\| \| ActionFormer \| 0.5\| 82.49\| 70.96\| 73.76\| 64.95\| 63.89\| 58.49\| 44.67\| 31.77\| 52.75\| \| TriDet \| 0.5\| 87.85\| 70.34\| 75.90\| 67.65\| 66.05\| 62.22\| 58.55\| 46.12\| 59.52\| \| ActionFormer \| 1 \| 82.38\| 80.30\| 80.15\| 77.63\| 75.97\| 73.28\| 70.31\| 63.04\| 72.05\| \| TriDet \| 1 \| 84.99\| 79.55\| 81.08\| 78.64\| 77.45\| 75.74\| 73.40\| 68.79\| 74.81\| \| ActionFormer \| 2s \| 79.19\| 73.88\| 74.52\| 71.10\| 68.79\| 66.38\| 63.00\| 57.54\| 65.36\| \| TriDet \| 2s \| 83.10\| 74.55\| 76.72\| 71.20\| 69.69\| 67.88\| 65.49\| 61.77\| 67.20\| \| O-LF(I, C) \| 0.5s\| 96.19\| 89.32\| 92.13\| 75.96\| 74.06\| 71.90\| 69.54\| 68.32\| 71.96\| \| O-LF(I, C) \| 1s \| 95.52\| 91.52\| 93.08\| 74.86\| 74.09\| 72.78\| 71.68\| 70.23\| 72.73\| \| O-LF(I, C) \| 2s \| 94.99\| 91.03\| 92.46\| 73.71\| 72.99\| 71.88\| 70.26\| 68.95\| 71.56\| \| O-LF(I, C, I + C) \| 0.5s\| 97.64\| 91.75\| 94.27\| 82.74\| 81.38\| 79.89\| 78.08\| 77.33\| 79.88\| \| O-LF(I, C, I + C) \| 1s \| 97.08\| 94.52\| 95.59\| 83.56\| 83.16\| 82.38\| 80.96\| 79.83\| 81.98\| \| O-LF(I, C, I + C) \| 2s \| 97.20\| 93.40\| 94.96\| 83.18\| 82.61\| 81.87\| 80.71\| 80.71\| 79.62\| Figure 2: Confusion matrices of the TriDet model (Shi et al., 2023) being applied using inertial, vision (camera) and both combined (inertial + camera) with a one second sliding window and 50% overlap. Compared to inertial-based architectures (Bock et al., 2021; Abedin et al., 2021) overall confusion (except for the NULL-class) is decreased. After combination strengths of each architecture are leveraged with e.g. jogging activities not getting confused anymore and overall confusion with the NULL-class decreases. Note that confusions which are 0 are omitted. (see Section C.1 of the supplementary material for further details). As the inertial-based architectures are providing predictions on a per-window basis, intermediate, short-lasting activity switches occur quite frequently along the time axis causing said architectures to produce only small coherent segments and ultimately lower mAP scores compared to the vision-based models presented in this paper. In order to remove these intermediate switches, predictions made by the inertial-based architectures were smoothed using a majority-vote-filter of 15 seconds (see Section C.3 of the supplementary material for an ablation study on the performed postprocessing). With the confusion of vision-based models being mostly among the activity categories (jogging, stretching and strength), inertial-based models show a larger degree of overall confusion among all workout classes. Caused by per-window predictions and resulting intermediate activity switches, calculated mAP scores of the inertial-architectures are significantly lower than that of the camera-based approaches. Nevertheless, one can see that inertial-based models are on average able to predict all workout activities more consistently and produce the highest classification metrics across all experiments. Multimodal (Inertial and Egocentric Video) HAR Within our last set of experiments, we assess the applicability of single-stage temporal action localization models for inertial-based as well as modality-combined (inertial + camera) HAR. In order to early fuse the two-stream I3D feature embeddings with the inertial data, we flattened the windowed inertial data such that the captured acceleration along each axis of each sensor is appended to become a vector of size $[window\ length \times no.\ sensor\ axis]$. Using the same hyperparameters as used during vision-based experiments, a plain ActionFormer and TriDet network are not only able to produce competitive classification results based on inertial input data, but, unlike the inertial-based architectures, show less confusion amongst the activity classes. Furthermore, with both temporal action localization models predicting segments instead per-window activity labels, mAP scores significantly increase. By means of simple concatenation of both modalities, both architecture achieve the highest average mAP and close-to-best F1-scores across all experiments (see Table 2). Comparing confusion matrices of all three approaches (see Figure 2) reveals that both vision models, applied in a plain fashion, are able to successfully combine inertial and vision data and leveraging the strengths of each modality. To assess how our earl-fusion approach compares to voting-based late-fusion approaches such as proposed by [Ijaz et al. (2022)](IJCAI), we implemented an Oracle-based late fusion, which creates perfectly late fused predictions of different models. The predictions are merged by comparing each of them with the ground truth data and only keeping, if predicted by one of the networks, the correct prediction. Interestingly, the first Oracle-late-fusion $O-LF(I, C)$, which late fuses predictions of the best inertial and best vision model, produces lower mAP scores than that of the best temporal action localization model being trained on both modalities simultaneously. Furthermore, late-fusing the best inertial, vision and early-fusion approach $O(I, C, I + C)$, increases mAP scores of $O(I, C)$ by as much as 10%, suggesting the early-fusion-based approach is capable of learning to differentiate activities both single-modality models failed to classify correctly. Nevertheless, classification results of the Oracle-based late fusion significantly outperform both single- and combined-modality approaches, indicating that the data set is far from being saturated. ## 5 LIMITATIONS Our dataset contributes a benchmark for human activity recognition classifiers, for the two leading wearable modalities of egocentric video and inertial data, using in particular a high variety of fitness exercises and outdoor scenes. With the current selection of participants, the WEAR dataset is biased towards young, healthy people. Given the ease of reproducibility, future extensions of the WEAR dataset could focus on featuring participants (1) of an older and/or younger age, (2) with known physical impairments and (3) sessions recorded at new locations (outside of [Anonymised]) and at different times of the year (e.g., summer). As supplementary experiments already indicate (see Section C.7 in the supplementary material), recording the same participants a second time would allow to analyse how a certain degree of familiarity with the recording setup can be seen in altered movements (e.g., via a smoother execution of activities) as well as give an intuition about robustness of learned approaches. Besides extending the amount of data recorded, further recordings could also involve other sensors such as higher-end commercial smartwatches to enable the study of increased sampling rates, the variability of the capturing devices, and the inclusion of additional modalities such as 3D gyroscopes, 3D magnetometers, or photoplethysmography (PPG) to obtain fitness-relevant information such as heart rate), as well as additional wearables, such as earables. 6 CONCLUSION In this paper, we introduced a benchmark dataset for both inertial- and vision-based Human Activity Recognition (HAR), to explore the learning of HAR across these modalities. The dataset comprises data from 18 participants performing each 18 different sports activities with the two common types of wearable sensors delivering inertial (3D acceleration) and camera (egocentric video) data. Our WEAR dataset provides a challenging prediction scenario across both modalities marked by purposely introduced activity variations along with a small information overlap between the inertial and vision data, putting forward the necessity of exploring techniques to combine both modalities. Benchmark results obtained using each modality separately show that each modality interestingly offers complementary strengths and weaknesses in their prediction performance. In light of the recent successes of temporal action localization following the architecture design as proposed by Zhang et al. (2022), we demonstrate their versatility by applying them in a plain fashion using only inertial data as input. Results show that the vision-based models are not only able to produce competitive results using inertial data, but also can function as an architecture to fuse both modalities by means of simple concatenation with vision data. In experiments that combined raw inertial with extracted vision-based feature embeddings, the plain, vision-based temporal action localization models were able to produce the highest average mAP and close-to-best F1-scores. Lastly, to give an intuition about a possible upper bound for future fusion-approaches, we evaluated an oracle-merged late fusion of the best inertial- and vision-based model predictions. Vision-based temporal action localization such as the ActionFormer (Zhang et al., 2022) have thus far neither been explored in inertial nor in the combination of inertial- and vision-based human activity recognition. With WEAR, we provide both communities (inertial- and vision-based HAR) a common, challenging benchmark dataset to assess the applicability of combined approaches. 7 Ethics Statement Before participating in the study, participants were notified that by nature the data they provide can only be pseudonymised. This means that, though requiring a substantial amount of effort, the identity of a person can be reconstructed. Although participants agreed to include their egocentric videos in a public dataset, it is essential to refrain from actively identifying the individuals featured in the WEAR dataset. If other researchers decide to contribute to the WEAR dataset by recording additional participants, societal and ethical implications should be considered. As with the participants part of the original release of the WEAR dataset, all participants must be briefed before their first recording, making them aware of all necessary information and implications that come with providing to the WEAR dataset. Recording locations should only be chosen if video recordings are allowed at said location and participants are given enough space to perform each activity safely. If the recording location involves pedestrians walking within close proximity, pedestrians should be notified that they are being recorded and, if applicable, captured faces should be blurred during postprocessing. The WEAR dataset and associated code are made public for research purposes. With the accurate detection of physical activities that we perform in our daily lives having been identified as valuable information, the WEAR dataset focuses on one of the most popular application scenarios of wearable smartwatches and action cameras, i.e. self-tracking of workout activities. With the ease of reproducability we hope to make WEAR a collaborative, expanding dataset which researchers from different locations and backgrounds can contribute to. For example, as the current selection of participants is biased towards healthy, young people, we hope to overcome said limitation by including people from more diverse backgrounds and age groups in future iterations of the dataset. Lastly, the authors took great care of avoiding any infringement of rights during the data collection process. Yet, in case of conflicts, they are of course committed to taking appropriate actions, such as promptly removing data associated with such concerns. 8 Reproducibility Statement The source code that was used to conduct all experiments is available via [Anonymized](https://www.anonymous.edu/anon). A snapshot of the code is provided as part of the supplementary material download. The repository is written in such a way that other architectures (both inertial- and vision-based) can be added in the future. The repository provides Readme files which give details on the overall structure of the repository, how collect additional data and how to set up an Anaconda environment with the needed packages to run experiments. Experiments are defined via '.json'-format configuration files which allow for easy sharing of used hyperparameter settings. WEAR and all associated files are offered under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. The dataset is hosted via a cloud-storage platform. It is a non-commercial cloud storage service for research, studying and teaching and is provided to participating institutions exclusively. With locations exclusively in [anonymized], [anonymized] is subject to the strict directives on data protection and data security. The dataset download is structured into the (1) '.json'-formatted annotations, (2) raw, synchronized inertial and vision data and (3) precomputed feature embeddings as mentioned in the main paper. Third party data-hosting services will be explored once the dataset paper is published and in a non-changing state. We will involve the ethics council of [anonymized] during our decision process to ensure a each selected hosting platform is inline with our data privacy standards. Note that to ensure anonymization of affiliated authors, the dataset cannot be shared as part of the review phase, making it not possible to rerun experiments. References Alireza Abedin, Mahsa Ehsanpour, Qinfeng Shi, Hamid Rezatofighi, and Damith C. Ranasinghe. Attend and discriminate: Beyond the state-of-the-art for human activity recognition using wearable sensors. ACM on Interactive, Mobile, Wearable and Ubiquitous Technologies, 5(1):1–22, 2021. URL [https://doi.org/10.1145/3448083](https://doi.org/10.1145/3448083). Yueran Bai, Yingying Wang, Yunhai Tong, Yang Yang, Qiyue Liu, and Junhui Liu. Boundary content graph neural network for temporal action proposal generation. In Andrea Vedaldi, Horst Bischof,
0IaTFNJner	The idea of multi-facet embedding or polysemy embedding has been studied quite extensively in the past. From network embedding (Liu et al. Is a single vector enough? exploring node polysemy for network embedding) to recommender systems (Weston et al. Nonlinear latent factorization by embedding multiple user interests). However, non of the related work on multi-embedding has been discussed in the paper.	On the Embedding Collapse When Scaling up Recommendation Models Anonymous authors Paper under double-blind review Abstract Recent advances in deep foundation models have led to a promising trend of developing large recommendation models to leverage vast amounts of available data. However, we experiment to scale up existing recommendation models and observe that the enlarged models do not improve satisfactorily. In this context, we investigate the embedding layers of enlarged models and identify a phenomenon of embedding collapse, which ultimately hinders scalability, wherein the embedding matrix tends to reside in a low-dimensional subspace. Through empirical and theoretical analysis, we demonstrate that the feature interaction module specific to recommendation models has a two-sided effect. On the one hand, the interaction restricts embedding learning when interacting with collapsed embeddings, exacerbating the collapse issue. On the other hand, feature interaction is crucial in mitigating the fitting of spurious features, thereby improving scalability. Based on this analysis, we propose a simple yet effective multi-embedding design incorporating embedding-set-specific interaction modules to capture diverse patterns and reduce collapse. Extensive experiments demonstrate that this proposed design provides consistent scalability for various recommendation models. 1 Introduction Recommender systems are significant machine learning scenarios that predict users’ actions on items based on multi-field categorical data (Zhang et al., 2016), which play an indispensable role in our daily lives to help people discover information about their interests and have been adopted in a wide range of online applications, such as E-commerce, social media, news feeds, and music streaming. Recently, researchers have developed deep-learning-based recommendation models to dig feature representations flexibly. These models have been successfully deployed across a multitude of application scenarios, thereby demonstrating their widespread adoption and effectiveness. In recommender systems, there is a tremendous amount of Internet data, while mainstream models typically tuned with embedding size 10 (Zhu et al., 2022) do not adequately capture the magnitude of the available data. Motivated by the advancement of large foundation models (Kirillov et al., 2023; OpenAI, 2023; Radford et al., 2021; Rombach et al., 2022), which benefit from increasing parameters, it would be a promising trend to scale up the recommendation model size. However, when scaling up the embedding size, the bottleneck of mainstream recommendation models (Qu et al., 2016; Lian et al., 2018; Wang et al., 2021), we find an unsatisfactory improvement or even performance drop as shown in Figure 1a. This suggests a deficiency in the scalability of existing architecture designs, constraining the maximum potential for recommender systems. We take a spectral analysis on the learned embedding matrices based on singular value decomposition and exhibit the normalized singular values in Figure 1b. Surprisingly, most singular values are significantly small, i.e., the learned embedding matrices are nearly low-rank, which we refer to as the embedding collapse phenomenon. With the enlarged model size, the model does not learn to capture a larger dimension of information, implying a learning process with ineffective parameter utilization, which restricts the scalability. In this work, we study the mechanism behind the embedding collapse phenomenon through empirical and theoretical analysis. We shed light on the two-sided effect of the feature interaction module, the characteristic of recommendation models to model higher-order correlations, on model scalability. On the one hand, interaction with collapsed embeddings will constrain the embedding learning... Figure 1: Unsatisfactory scalability of existing recommendation models. (a): Increasing the embedding size does not improve remarkably or even hurts the model performance. (b): Most embedding matrices do not learn large singular values and tend to be low-rank. and, thus, in turn, aggravate the collapse issue. On the other hand, the feature interaction also plays a vital role in reducing overfitting when scaling up models. Based on our analysis, we conclude the principle to mitigate collapse without suppressing feature interaction about how to design scalable models. We propose multi-embedding as a simple yet efficient design for model scaling. Multi-embedding scales the number of independent embedding sets and incorporates embedding-set-specific interaction modules to jointly capture diverse patterns. Our experimental results demonstrate that multi-embedding provides scalability for extensive mainstream models, pointing to a methodology of breaking through the size limit of recommender systems. Our contributions can be summarized as: • To the best of our knowledge, we are the first to point out the non-scalability issue for recommendation models and discover the embedding collapse phenomenon, which is an urgent problem to address for model scalability. • We shed light on the two-sided effect of the feature interaction process on scalability based on the collapse phenomenon using empirical and theoretical analysis. Specifically, feature interaction leads to collapse while providing essential overfitting reduction. • Following our concluded principle to mitigate collapse without suppressing feature interaction, we propose multi-embedding as a simple unified design, which consistently improves scalability for extensive state-of-the-art recommendation models. 2 PRELIMINARIES Recommendation models aim to predict an action based on features from various fields. Throughout this paper, we consider the fundamental scenario of recommender systems, in which categorial features and binary outputs are involved. Formally, suppose there are $N$ fields, with the $i$-th field denoted as $\mathcal{X}_i = \{1, 2, ..., D_i\}$ where $D_i$ denotes the field cardinality. The value of $D_i$ may vary in a wide range, adding difficulty to recommender systems. Let $$\mathcal{X} = \mathcal{X}_1 \times \mathcal{X}_2 \times ... \times \mathcal{X}_N$$ and $\mathcal{Y} = \{0, 1\}$, then recommendation models aim to learn a mapping from $\mathcal{X}$ to $\mathcal{Y}$. In addition to considering individual features from diverse fields, there have been numerous studies (Koren et al., 2009; Rendle, 2010; Juan et al., 2016; Guo et al., 2017; Lian et al., 2018; Pan et al., 2018; Sun et al., 2021; Wang et al., 2021) within the area of recommender systems to model combined features using feature interaction modules. In this work, we investigate the following widely adopted architecture for mainstream models. A model comprises: (1) embedding layers $E_i \in \mathbb{R}^{D_i \times K}$ for each field, with embedding size $K$; (2) an interaction module $I$ responsible for integrating all embeddings into a combined feature scalar or vector; and (3) a subsequent postprocessing module $F$ used for prediction purposes, such as MLP and MoE. The forward pass of such a model is formalized as $$e_i = E_i^\top 1_{x_i}, \forall i \in \{1, 2, ..., N\},$$ $$h = I(e_1, e_2, ..., e_n),$$ $$\hat{y} = F(h),$$ where $1_{x_i}$ indicates the one-hot encoding of $x_i \in X_i$, in other words, $e_i$ refers to (transposed) $x_i$-th row of the embedding table $E_i$. 3 Embedding Collapse Singular value decomposition has been widely used to measure the collapse phenomenon [Jing et al., 2021]. In Figure 1b, we have shown that the learned embedding matrices of recommendation models are approximately low-rank with some extremely small singular values. To determine the degree of collapse for such matrices with low-rank tendencies, we propose information abundance as a generalized quantification. Definition 1 (Information Abundance) Consider a matrix $E \in \mathbb{R}^{D \times K}$ and its singular value decomposition $E = U\Sigma V = \sum_{k=1}^{K} \sigma_k u_k v_k^T$, then the information abundance of $E$ is defined as \[ IA(E) = \frac{\\|\sigma\\|_1}{\\|\sigma\\|_\infty}, \] i.e., the sum of all singular values normalized by the maximum singular value. Intuitively, a matrix with high information abundance demonstrates a balanced distribution in vector space since it has similar singular values. In contrast, a matrix with low information abundance suggests that the components corresponding to smaller singular values can be compressed without significantly impacting the result. Compared with matrix rank, information abundance can be regarded as a simple extension by noticing that $\text{rank}(E) = \\|\sigma\\|_0$, yet it is applicable for non-strictly low-rank matrices, especially for fields with $D_i \gg K$ which is possibly of rank $K$. We calculate the information abundance of embedding matrices for the enlarged DCNv2 [Wang et al., 2021] and compare it with that of randomly initialized matrices, shown in Figure 2. It is observed that the information abundance of learned embedding matrices is extremely low, indicating the embedding collapse phenomenon. 4 Feature Interaction Revisited In this section, we delve deeper into the embedding collapse phenomenon for recommendation models. Our investigation revolves around two questions: (1) How is embedding collapse caused? (2) How to properly mitigate embedding collapse for scalability? Through empirical and theoretical studies, we shed light on the two-sided effect of the commonly employed feature interaction module on model scalability. 4.1 Interaction-Collapse Theory To determine how feature interaction leads to embedding collapse, it is inadequate to directly analyze the raw embedding matrices since the learned embedding matrix results from interactions with all other fields, making it difficult to isolate the impact of field-pair-level interaction on embedding learning. Under this obstacle, we propose empirical evidences on models with sub-embeddings and theoretical analysis on general models, and conclude that feature interaction causes embedding collapse, named the interaction-collapse theory. Evidence I: Experiments on FFM. Field-aware factorization machines (FFM) [Juan et al., 2016] split an embedding matrix of field $i$ into multiple sub-embeddings with \[ E_i = \begin{bmatrix} E_i^{i+1}, E_i^{i+2}, \ldots, E_i^{(i-1)}, E_i^{(i+1)}, \ldots, E_i^{N} \end{bmatrix}, \] where sub-embedding $E_i^{j} \in \mathbb{R}^{D_i \times K/(N-1)}$ is only used when interacting field $i$ with field $j$ for $j \neq i$. To determine the collapse of sub-embedding matrices, we calculate $IA(E_i^{j})$ for all $i, j$ and show them in Figure 3a. For convenience, we pre-sort the field indices by the ascending order. Figure 3: Visualization of information abundance of sub-embedding matrices for FFM (left) and DCNv2 (right), with field indices sorted by information abundance of corresponding raw embedding matrices. Higher or warmer indicates larger. It is observed that IA($E_i^{j}$) are co-influenced by both IA($E_i$) and IA($E_j$). of information abundance, i.e., $i$ is ordered according to IA($E_i$), similar to $j$. We can observe that IA($E_i^{j}$) is approximately increasing along $i$, which is trivial since $E_i^{j}$ is simply a split of $E_i$. Interestingly, another correlation can be observed that the information abundance of sub-embeddings is co-influenced by the fields it interacts with, reflected by the increasing trend along $j$, especially with larger $i$. This is amazing in the sense that even using independent embeddings to represent the same field features, these embeddings get different information abundance after learning. For instance, we calculate the summation of IA($E_i^{j}$) over $j$ or $i$ to study the effect of the other single variable, shown in Figure 3b and Figure 3c. Both of them show an increasing trend, confirming the co-influence of $i$ and $j$. Evidence II: Experiments on DCNv2. An improved deep & cross network (DCNv2) (Wang et al., 2021) incorporates a crossing network which is parameterized with transformation matrices $W_{i \rightarrow j}$ (Sun et al., 2021) over each field pair to project an embedding vector from field $i$ before interaction with field $j$. By collecting all projected embedding vectors, DCNv2 can be regarded to implicitly generate field-aware sub-embeddings $E_i^{1}, E_i^{2}, ..., E_i^{N}$ to interact with all fields from embedding matrix $E_i$, with $$E_i^{j} = E_i W_{i \rightarrow j}^T.$$ DCNv2 consists of multiple stacked cross layers, and for simplification, we only discuss the first layer throughout this paper. Similar to Evidence I, we calculate IA($E_i^{j}$) together with the axis-wise summations and show them in the right part of Figure 3. Consistent with previous observation as FFM, the information abundance of sub-embedding matrices approximately increases along $j$ with the same $i$, even though they are projected from the same embedding matrix $E_i$. Theoretical analysis: Collapse on non-sub-embedding-based models. We now present how collapse is caused by feature interaction in non-sub-embedding-based recommendation models from a theoretical view. For simplicity, we consider an FM-style (Rendle, 2010) feature interaction. Formally, the interaction process is defined by $$h = \sum_{i=1}^{N} \sum_{j=1}^{i-1} e_i^\top e_j = \sum_{i=1}^{N} \sum_{j=1}^{i-1} 1_x^\top E_i E_j^\top 1_x,$$ where $h$ is the combined feature as mentioned before. Without loss of generality, we discuss one specific row $e_1$ of $E_1$ and keep other embedding matrices fixed. Consider a minibatch with batch size $B$. Denote $\sigma_{i,k}$ as the $k$-th singular value of $E_k$, similar for $u_{i,k}$, $v_{i,k}$. We have \[ \frac{\partial \mathcal{L}}{\partial e_1} = \frac{1}{B} \sum_{b=1}^{B} \frac{\partial \ell(b)}{\partial h(b)} \cdot \frac{\partial h(b)}{\partial e_1} = \frac{1}{B} \sum_{b=1}^{B} \frac{\partial \ell(b)}{\partial h(b)} \cdot \sum_{i=2}^{N} E_i^\top x_i^{(b)} \] \[= \frac{1}{B} \sum_{b=1}^{B} \frac{\partial \ell(b)}{\partial h(b)} \cdot \sum_{i=2}^{N} \sum_{k=1}^{K} \sigma_{i,k} v_{i,k} u_{i,k}^\top 1_{x_i^{(b)}}\] \[= \sum_{i=2}^{N} \sum_{k=1}^{K} \left( \frac{1}{B} \sum_{b=1}^{B} \frac{\partial \ell(b)}{\partial h(b)} u_{i,k}^\top 1_{x_i^{(b)}} \right) \sigma_{i,k} v_{i,k}\] \[= \sum_{i=2}^{N} \sum_{k=1}^{K} \alpha_{i,k} \sigma_{i,k} v_{i,k} = \sum_{i=2}^{N} \theta_i\] The equation means that the gradient can be decomposed into field-specific terms. We analyze the component $\theta_i$ for a certain field $i$, which is further decomposed into spectral for the corresponding embedding matrix $E_i$. From the form of $\theta_i$, it is observed that $\{\alpha_{i,k}\}$ are $\sigma_i$-agnostic scalars determined by the training data and objective function. Thus, the variety of $\sigma_i$ significantly influences the composition of $\theta_i$. For those larger $\sigma_{i,k}$, the gradient component $\theta_i$ will be weighted more heavily along the corresponding spectral $v_{i,k}$. When $E_i$ is low-information-abundance, the components of $\theta_i$ weigh imbalancedly, resulting in the degeneration of $e_1$. Since different $e_1$ affects only $\alpha_{i,k}$ instead of $\sigma_{i,k}$ and $v_{i,k}$, all rows of $E_1$ degenerates in similar manners and finally form a collapsed matrix. To further illustrate, we conduct a toy experiment over synthetic data. Suppose there are $N = 3$ fields, and we set $D_3$ to different values with $D_3 < K$ and $D_3 \gg K$ to simulate low-information-abundance and high-information-abundance cases, which matches the diverse range of the field cardinality in real-world scenarios. We train $E_1$ while keeping $E_2, E_3$ fixed. Details of experiment setups are discussed in Appendix A. We show the information abundance of $E_1$ along the training process for the two cases in Figure 4. It is observed that interacting with a low-information-abundance matrix will result in a collapsed embedding matrix. Summary: How is collapse caused in recommendation models? Evidence I&II highlight that interacting with a field with a low-information-abundance embedding matrix will result in a more collapsed sub-embedding. By further considering the fact that sub-embeddings reflect the effect when fields interact since it originates from raw embeddings, we recognize the inherent mechanism of feature interaction to cause collapse, which is further confirmed by our theoretical analysis. We conclude the interaction-collapse theory: Finding 1 (Interaction-Collapse Theory). In feature interaction of recommendation models, fields with low-information-abundance embeddings constrain the information abundance of other fields, resulting in collapsed embedding matrices. The interaction-collapse theory generally suggests that feature interaction is the primary catalyst for collapse, thereby imposing constraints on the ideal scalability. ### 4.2 IS IT SUFFICIENT TO AVOID COLLAPSE FOR SCALABILITY? Following our discussion above, we have shown that the feature interaction process of recommendation models leads to collapse and thus limits the model scalability. We now discuss its negative proposition, i.e., whether suppressing the feature interaction to mitigate collapse leads to model scalability. To answer this question, we design the following two experiments to compare standard models and models with feature interaction suppressed. Evidence III: Regularization on DCNv2 to mitigate collapse. Evidence II shows that a projection $W_{i \rightarrow j}$ is learned to adjust information abundance for sub-embeddings and thus lead to collapse.\footnote{Further explanation is referred to Appendix F} We now investigate how suppressing such effect would result in scalability by introducing the following regularization with learnable parameter $\lambda_{ij}$ $$\ell_{reg} = \sum_{i=1}^{N} \sum_{j=1}^{N} \\| W_{i \rightarrow j}^T W_{i \rightarrow j} - \lambda_{ij} I \\|_F^2$$ to regularize the projection matrix to be a multiplication of an unitary matrix. This way, $W_{i \rightarrow j}$ will preserve all normalized singular values and maintain the information abundance after projection. We experiment with various embedding sizes and compare the changes in performance, the information abundances, and the optimization dynamics for standard and regularized models. Results are shown in Figure 5. As anticipated, regularization in DCNv2 helps learn embeddings with higher information abundance. Nevertheless, from the performance perspective, the model presents unexpected results whereby the scalability does not improve or worsen as the collapse is alleviated. We further find that such a model overfits during the learning process, with the training loss consistently decreasing and the validation AUC dropping. ![Figure 5](image) (a) IA w/ 10x model size. (b) Test AUC w.r.t. model size. (c) Training vs. validation. Figure 5: Experimental results of Evidence III. Restricting DCNv2 leads to higher information abundance, yet the model suffers from over-fitting, thus resulting in non-scalability. Evidence IV: Scaling up DCNv2 and DNN. We now discuss DNN, which consists of a plain interaction module by concatenating all feature vectors from different fields and processing with an MLP, formulized by $$h = G([e_1, e_2, ..., e_N]).$$ Since DNN does not conduct explicit 2-order feature interaction (Rendle et al., 2020), following our previous interaction-collapse theory, it would suffer less from collapse. We compare the learned embeddings of DCNv2 and DNN and their performance with the growth of embedding size. Considering that different architectures or objectives may differ in modeling, we mainly discuss the performance trend as a fair comparison. Results are shown in Figure 6. DNN learns less-collapsed embedding matrices, reflected by higher information abundance than DCNv2. Yet, perversely, the AUC of DNN drops when increasing the embedding size, while DCNv2 sustains the performance. Such observations show that DNN falls into the issue of overfitting and lacks scalability, even though it suffers less from collapse. ![Figure 6](image) (a) IA w/ 10x model size. (b) Test AUC w.r.t. model size. Figure 6: Experimental results of Evidence IV. Despite higher information abundance, the performance of DNN drops w.r.t. model size. Summary: Does suppressing collapse definitely improve scalability? Regularized DCNv2 and DNN are both models with feature interaction suppressed, and they learn less-collapsed embedding matrices than DCNv2, as expected. Yet observations in evidence III&IV demonstrate that regularized DCNv2 and DNN are both non-scalable with the growth of model size and suffer from serious overfitting. We conclude the following finding: Finding 2. A less-collapsed model with feature interaction suppressed is insufficient for scalability due to overfitting concern. Such a finding is plausible, considering that feature interaction brings domain knowledge of higher-order correlations in recommender systems and helps form generalizable representations. When feature interaction is suppressed, models tend to fit noise as the embedding size increases, resulting in reduced generalization. 5 MULTI-EMBEDDING DESIGN In this section, we present a simple design of multi-embedding, which serves as an effective scaling design applicable to a wide range of model architecture designs. We introduce the overall architecture, present experimental results, and analyze how multi-embedding works. 5.1 MULTI-EMBEDDING FOR BETTER SCALABILITY The two-sided effect of feature interaction for scalability implies a principle for model design. That is, a scalable model should be capable of less-collapsed embeddings within the existing feature interaction framework instead of removing interaction. Based on this principle, we propose multi-embedding or ME as a simple yet efficient design to improve scalability. Specifically, we scale up the number of independent and complete embedding sets instead of the embedding size, and incorporate embedding-set-specific feature interaction layers. Similar to previous works such as group convolution (Krizhevsky et al., 2012), multi-head attention (Vaswani et al., 2017), and other decoupling-based works in recommender systems (Liu et al., 2022; 2019; Weston et al., 2013), such design allows the model to learn different interaction patterns jointly, while a single-embedding model would be limited to the only interaction pattern that causes severe collapse. This way, the model is capable of learning diverse embedding vectors to mitigate collapse while keeping the original interaction modules. Formally, a model with $M$ sets of embeddings is defined as $$e_i^{(m)} = \left(E_i^{(m)}\right)^T 1_{x_i}, \forall i \in \{1, 2, ..., N\},$$ $$h^{(m)} = I^{(m)}(e_1^{(m)}, e_2^{(m)}, ..., e_N^{(m)}),$$ $$h = \frac{1}{M} \sum_{m=1}^{M} h^{(m)}, \quad \hat{y} = F(h),$$ where $m$ stands for the index of embedding set. One requirement of multi-embedding is that there should be non-linearities such as ReLU in interaction $I$; otherwise, the model is equivalent to single-embedding and hence does not capture different patterns (see Appendix B). As a solution, we add a non-linear projection after interaction for the model with linear interaction layers and reduce one MLP layer for $F$ to achieve a fair comparison. An overall architecture comparison of single-embedding and mult-embedding models with $N = 2$ and $M = 2$ is shown in Figure 7. Figure 7: Architectures of single-embedding (left) and multi-embedding (right) models with $N = 2$ and $M = 2$. Figure 8: Scalability of multi-embedding on Criteo dataset. 5.2 Experiments Setup. We conduct our experiments on two datasets for recommender systems: Criteo (Jean-Baptiste Tien, 2014) and Avazu (Steve Wang, 2014), which are large and hard benchmark datasets widely used in recommender systems. We experiment on baseline models including DNN, IPNN (Qu et al., 2016), NFwFM (Pan et al., 2018), xDeepFM (Lian et al., 2018), DCNv2 (Wang et al., 2021), FinalMLP (Mao et al., 2023) and their corresponding multi-embedding variants with 2x, 3x, 4x and 10x model size. Here NFwFM is a variant of NFM (He & Chua, 2017) by replacing FM with FwFM. All experiments are performed with 8/1/1 training/validation/test splits, and we apply early stopping based on validation AUC. More details are shown in Appendix C.2. Results. We repeat each experiment 3 times and report the average test AUC with different scaling factors of the model size. Results are shown in Table 1. For the experiments with single-embedding, we observe that all the models demonstrate poor scalability. Only DCNv2 and NFwFM show slight improvements with increasing embedding sizes, with gains of 0.00036 on Criteo and 0.00090 on Avazu, respectively. For DNN, xDeepFM, and FinalMLP, which rely highly on non-explicit interaction, the performance even drops (0.00136 on Criteo and 0.00118 on Avazu) when scaled up to 10x, as discussed in Section 4.2. In contrast to single-embedding, our multi-embedding shows consistent and remarkable improvement with the growth of the embedding size, and the highest performance is always achieved with the largest 10x size. For DCNv2 and NFwFM, multi-embedding gains 0.00099 on Critio and 0.00202 on Avazu by scaling up to 10x, which is never obtained by single-embedding. Overall models and datasets, compared with baselines, the largest models averagely achieve 0.00106 improvement on the test AUC. Multi-embedding provides a methodology to break through the non-scalability limit of existing models. We visualize the scalability of multi-embedding on Criteo dataset in Figure 8. The standard deviation and detailed scalability comparison are shown in Appendix C.3. Table 1: Test AUC for different models. Higher indicates better. Underlined and bolded values refer to the best performance with single-embedding (SE) and multi-embedding (ME), respectively. \| Model \| Criteo \| Avazu \| \|-----------\|-----------------\|----------------\| \| \| base 2x 3x 4x 10x \| base 2x 3x 4x 10x \| \| DNN \| SE 0.81228 0.81207 0.81213 0.81142 \| ME 0.78744 0.78759 0.78752 0.78728 0.78648 \| \| \| SE 0.81261 0.81288 0.81289 0.81287 \| ME 0.78805 0.78826 0.78862 0.78844 \| \| IPNN \| SE 0.81272 0.81272 0.81271 0.81262 \| ME 0.78732 0.78741 0.78738 0.78750 0.78745 \| \| \| SE 0.81268 0.81270 0.81273 0.81311 \| ME 0.78806 0.78868 0.78902 0.78894 \| \| NFwFM \| SE 0.81059 0.81087 0.81100 0.81112 \| ME 0.78684 0.78757 0.78783 0.78794 – \| \| \| SE 0.81128 0.81153 0.81171 0.81210 \| ME 0.78868 0.78901 0.78932 – \| \| xDeepFM \| SE 0.81217 0.81180 0.81167 0.81116 \| ME 0.78743 0.78750 0.78714 0.78735 0.78693 \| \| \| SE 0.81236 0.81239 0.81255 0.81299 \| ME 0.78848 0.78886 0.78894 0.78927 \| \| DCNv2 \| SE 0.81341 0.81345 0.81346 0.81357 \| ME 0.78786 0.78835 0.78854 0.78852 0.78856 \| \| \| SE 0.81348 0.81361 0.81382 0.81385 \| ME 0.78862 0.78882 0.78907 0.78942 \| \| FinalMLP \| SE 0.81259 0.81248 0.81240 0.81175 \| ME 0.78751 0.78797 0.78795 0.78742 0.78662 \| \| \| SE 0.81290 0.81302 0.81303 0.81303 \| ME 0.78821 0.78831 0.78836 0.78830 \| 5.3 Analysis Information abundance. Multi-embedding models achieve remarkable scalability compared with single-embedding. We verify that such scalability originates from the mitigation of collapse. We compare the information abundance of single-embedding and multi-embedding DCNv2 with the 10x embedding size. As shown in Figure 9a, multi-embedding offers higher information abundance and indicates less collapsed embedding matrices. Variations of embeddings. Multi-embedding utilizes embedding-set-specific interactions to enrich embedding learning. We analyze the information abundance for each embedding set as shown --- 2The embedding of NFwFM with 10x size on Avazu costs nearly 37.6GB memory, which exceeds our GPU memory limit. Therefore, we do not conduct 10x NFwFM on Avazu. On the other hand, the existing experiment with 4x is already sufficient for NFwFM on Avazu. 3A slightly higher AUC at 0.001-level is regarded significant (Cheng et al., 2016; Guo et al., 2017; Song et al., 2019; Tian et al., 2023). in Figure 9b. It is observed that the embedding matrices of different sets vary in information abundance. Different interaction patterns. To justify that the scalability of multi-embedding originates from different interaction patterns, we visualize $\\|W_{i \rightarrow j}^{(m)}\\|_F$ as the interaction pattern (Wang et al., 2021) for a multi-embedding DCNv2 model in Figure 9c. It is shown that the interaction layers learn various patterns. To further illustrate, we conduct an ablation study by restricting the divergence of $\\|W_{i \rightarrow j}^{(m)}\\|_F$ across all embedding sets. From results in Figure 9d, it is observed that the divergence-restricted multi-embedding model does not show similar scalability as standard multi-embedding models, indicating multi-embedding works from the diversity of interaction layers. Ablation study on sharing one interaction layer across all embedding sets are provided in Appendix H. ![Figure 9](image) (a) IA($E_i$). (b) IA($E_i^{(m)}$). (c) $\\|W_{i \rightarrow j}^{(m)}\\|_F$. (d) Restricting diversity. Figure 9: Analysis of multi-embedding. (a): Multi-embedding learns higher information abundance. (b): Each embedding set learns diverse embeddings, reflected by varying information abundance. (c): Embedding-set-specific feature interaction layers capture different interaction patterns. (d): Restricting diversity of $\\|W_{i \rightarrow j}^{(m)}\\|_F$ across all embedding sets leads to non-scalability. 6 RELATED WORKS Modules in recommender systems. Plenty of existing works investigate the module design for recommender systems. A line of studies focuses on feature interaction process (Koren et al., 2009; Rendle, 2010; Juan et al., 2016; Qu et al., 2016; He & Chua, 2017; Guo et al., 2017; Pan et al., 2018; Lian et al., 2018; Song et al., 2019; Cheng et al., 2020; Sun et al., 2021; Wang et al., 2021; Mao et al., 2023; Tian et al., 2023), which is specific for recommender systems. These works are built up to fuse domain-specific knowledge of recommender systems. In contrast to proposing new modules, our work starts from a view of machine learning and analyzes the existing models for scalability. Collapse phenomenon. Neural collapse or representation collapse describes the degeneration of representation vectors with restricted variation. This phenomenon is widely studied in supervised learning (Papyan et al., 2020; Zhu et al., 2021; Tirer & Bruna, 2022), unsupervised contrastive learning (Hua et al., 2021; Jing et al., 2021; Gupta et al., 2022), transfer learning (Aghajanyan et al., 2020; Kumar et al., 2022), and generative models (Mao et al., 2017; Miyato et al., 2018). Chi et al. (2022) discuss the representation collapse in sparse MoEs. Inspired by these works, we realize the embedding collapse of recommendation models when regarding embedding vectors as representations by their definition, yet we are facing the setting of field-level interaction, which has not previously been well studied. Intrinsic dimensions and compression theories. To describe the complexity of data, existing works include intrinsic-dimension-based quantification (Levina & Bicke, 2004; Ansuini et al., 2019; Pope et al., 2020) and pruning-based analysis (Wen et al., 2017; Alvarez & Salzmann, 2017; Sun et al., 2021). Our SVD-based concept of information abundance is related to these works. 7 CONCLUSION In this paper, we highlight the non-scalability issue of existing recommendation models and identify the embedding collapse phenomenon that hinders scalability. From empirical and theoretical analysis around embedding collapse, we conclude the two-sided effect of feature interaction on scalability, i.e., feature interaction causes collapse while reducing overfitting. We propose a unified design of multi-embedding to mitigate collapse without suppressing feature interaction. Experiments on benchmark datasets demonstrate that multi-embedding consistently improves model scalability. REPRODUCIBILITY STATEMENT For toy experiments, we show the detailed settings in Appendix A. For experiments on benchmark datasets, we follow the default data pre-processing according to the repository of pytorch-fm[^1]. We present the general model architecture in Section 5.1, and demonstrate the specific design and all hyperparameters in Appendix C.2. We show the confidence of results with empirical standard deviations in Appendix C.3. We will release our code in case our paper is accepted. 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D1w3huGGpu	For the `HARD HOLDOUTS` set-up, will the performance further improve if there are more easy combinations available for training? Are there any other possible solutions to address the easy-to-hard transfer problem?	COMPOSITIONAL INTERFACES FOR COMPOSITIONAL GENERALIZATION Anonymous authors Paper under double-blind review ABSTRACT With recent work such as GATO (Reed et al., 2022), we see the development of agents that can accomplish a variety of tasks, and are able to perceive the world and act in multiple observation and action spaces. We would want such agents to exhibit compositional generalization to unseen combinations of observation and action spaces, and adapt quickly to novel observation spaces by transferring knowledge. In this work, we demonstrate how these abilities can be achieved through the use of end-to-end modular architectures: the encoding of observations and the prediction of actions are handled by differentiable modules specialized to that space, with a single shared controller between them. To study the properties of such modular architectures in a controlled manner, we construct an environment with compositional structure, where each instance of the environment is created by combining an observation, action, and instruction space from a large set of options. We demonstrate that through the use of modularity, agents can generalize to unseen combinations of observation, action and instruction spaces; even when the unseen combinations are more challenging. Moreover, we demonstrate that modularity enables quick integration of novel observation modalities, requiring only adaptation of the modules encoding the new observation. 1 INTRODUCTION In recent years, there has been remarkable successes with scaling model and data sizes. Across a wide variety of domains, the state-of-the-art is dominated by large models (pre-)trained on billions of samples (Brown et al., 2020; Goyal et al., 2021; Bommasani et al., 2021). This also holds true in the setting of multi-domain “generalist” agents, that can integrate perceptual information across multiple modalities, and can accomplish a variety of tasks (Reed et al., 2022; Shridhar et al., 2023). In this multi-domain setting, transferring common knowledge between domains while respecting the particularities of each domain is still not a solved problem. For example, in the setting of agents that are either virtually or physically embodied, one of the most important special cases is simulation-to-real transfer. Practitioners would like to use gradient-based end-to-end learned controllers, but it is difficult to collect large amounts of training data on a physical robot in the real world. While there has been great progress in some tasks (Wijmans et al., 2019), driven in large part by ever higher-fidelity simulations (Savva et al., 2019; Shen et al., 2021), there are still no out-of-the-box solutions for generic sim-to-real transfer.\footnote{see sim for lively debates on this issue.} More generally, one would like to be able to train agents as much as possible in domains where training is cheap, and deploy after minimal training in domains where training is expensive. Even more generally, between domain transfer may allow sample efficiency via composition and abstraction. For example, a wheeled robot, a legged robot, and a digital assistant “embodied” in AR glasses all might be able to share some knowledge about indoor navigation despite differences in their locomotion. However, it is not possible to entirely abstract away all these differences- the wheeled robot cannot traverse the same terrain as the human with AR glasses. Based on the above cited results in large-scale sequence modeling, one might wonder if researchers need to worry explicitly about transfer; maybe it is better to just scale token-based monolithic models like (Reed et al., 2022), or scale language models and some task/domain-specific models, with Figure 1: Illustration of the dataset formulation and the COIN (Compositional Interfaces) architecture for compositional generalization. (a) Each environment instance is defined by a tuple $(O_m, A_n, I_k)$: a combination of observation $O_m$, action $A_n$ and instruction $I_k$ spaces. The agent is trained on data from a subset of all possible combinations, with the expectation of generalizing to combinations not included in the training dataset. (b) The agent architecture consists of: perception modules (one for each observation space), action modules (one for each action space) and a controller (shared across all environment instances). The controller takes the observation embedding, instruction and action space identifier as input, while outputting action embedding. When acting in an environment consisting of observation space $O_m$ and action space $A_n$, $m$-th perception module is used to predict the create the observation embedding and $n$-th action module is used to predict action from the action embedding. inter-model text interfaces as in (Ahn et al., 2022; Zeng et al., 2022). We are sympathetic to these viewpoints, but as much as the bitter lesson has been that scale can be more important than good inductive bias, it has also been that optimizing end-to-end leads to the best results. Properly designed modular architectures can be both scalable, and allow end-to-end training (Pfeiffer et al., 2023). Furthermore, one of the benefits of modern attention-based architectures is that they are conducive to modular inductive biases without radical changes (Alayrac et al., 2022; Jaegle et al., 2021; Shridhar et al., 2023), and can differentiably interface various domains and still directly take advantage of pre-trained language models. Thus, we might hope to both encourage transfer through abstraction and composition, and allow end-to-end fine-tuning to handle the necessary details that cannot be abstracted; without giving up any of the benefits of large-scale pre-training. In this work, we study the effectiveness of such a modular architecture for compositional generalization and transfer learning in the embodied agent setting. We develop an environment that allows us to independently vary perceptual modalities and action and task specifications, and use it to carefully analyze the agent’s performance in these compositions. We show that we can compose the agent’s perceptual suite, its task specifications, and its action spaces. Our experiments demonstrate zero-shot performance on held-out combinations of perception/instruction/action-space, and demonstration of fast adaptation (requiring less samples) to new perceptual or action spaces (with or without freezing the controller), and without excessive negative transfer. 2 SETTING Our goal is to solve tasks defined by an environment instance $(O_m, A_n, I_k)$, which is constructed by combining $m$-th observation $(O)$, $n$-th action $(A)$ and $k$-th instruction $(I)$ space. Given an observation $o^{(m)}$ from space $O_m$, action space id $n$ for $A_n$, and instruction $i^{(k)}$ from space $I_k$, the goal is to find a policy that will predict an optimal action $\pi(o^{(m)}, n, i^{(k)}) \rightarrow a \in A_n$. The agent is trained using imitation learning (Schaal 1999) on a dataset of expert trajectories $\{D_{m,n,k}\}$ collected on $(O_m, A_n, I_k)$. We make sure that during training, the agent will be trained on samples from environment instances containing at least one of each of the individual spaces $O_m$, $A_n$, and $I_k$, but not all possible combinations $(O_m, A_n, I_k)$. This allows us to test compositional generalization by deploying the agent in environments containing unseen combinations, as demonstrated in Figure 1(a). Alternatively, we can measure how quickly it adapts to newly added space. Note that in this setting, generalization can mean two different things: (1) in-domain generalization where the agent is trained on trajectories from $(O_m, A_n, I_k)$, but a particular test sample $(o^{(m')}, n, i^{(k')})$ is never seen due to random procedural generation of environments. And (2), compositional generalization where test samples are from environment combinations that never seen during training. For example, learning to predict the right action in action space $A_{n'}$ given observation $o^{(m')}$ and instruction $i^{(k')}$ when the training dataset does not contain samples from environment $(O_{m'}, A_{n'}, I_{k'})$. In this work, we are particularly interested in the compositional generalization to unseen combinations of spaces. ### 2.1 Environment with Composable Observation, Action and Instruction Spaces To study compositional generalization to unseen combinations of spaces, we construct a grid-world environment that supports multiple interfaces for observation and actions. The state is a 7x7 grid containing up to four different objects in addition to the agent itself. The objects can be picked up by the agent given that they are next to each other. The agent’s inventory show object that are picked up, which later can be dropped down. Each object has a shape (box, ball, snake, key) and a color (red, green, yellow, blue). Each environment combination is constructed by selection on observation, action, and instruction space from one of the available options. Observation spaces ($O_m$): There are six possible observation spaces, in which positions, shapes, and colors of the objects and agent in the grid are represented by: Text, Symbols, List, Grid, Top View, or Side View. These spaces are detailed in Table 1. Text space describes everything in human understandable language. Symbol space is similar, but uses compact symbols instead of words. To build an observation in List space, we represent everything with one-hot representation first. Then, for each object, we concatenate all its properties into a single vector. Finally, we stack all such vectors from all object and the agent together to give complete description of the state. Grid space also builds a vector for each object first, but then arranges them by their location instead, producing a 3D tensor. The remaining Top and Side view spaces are simply image rendering of the environment. These image spaces and Grid space assumes spatial location, which does not apply to inventory objects. As a workaround, we use List representation of inventory for those spaces. We made sure each observation contains sufficient information for completing the instruction, hence the tasks are fully observable. Instruction spaces ($I_k$): In a given environment instance, the agent is tasked with completing an instruction from one of the eight possible instruction spaces. The simplest instruction, “Go to (x,y)”, requires the agent to reach the specified location, while more complex instructions like “Pick up in order: red box, yellow snake, and green box” involves multiple steps and require the agent to distinguish shapes and colors. For the full list of instructions, please refer to the Appendix A.1. All instruction spaces involve manipulating objects and positions of the agent, with individual instructions being randomly sampled from the instruction space, while satisfying the constraint of instruction completion being possible given the initial state. \| Observation Space \| Description \| \|-------------------\|-------------\| \| Text \| A natural language description, e.g., “The agent is at (3, 5), facing east. There is a yellow snake at (2, 0). The agent has following items in the inventory: a blue box.” \| \| Symbol \| A sequence of symbols, e.g., “A @ E x3y5. y S @ x2y0. I: b B.” \| \| List \| A 2D tensor $o$, where $o[i] = \text{concat}([\text{object}_i\text{position}, \text{one\_hot}(\text{object}_i\text{shape}), \text{one\_hot}(\text{object}_i\text{color})])$ \| \| Grid \| A 3D tensor $o$ where the object at position $(x, y)$ is indicated by $o[x, y] = \text{concat}([\text{one\_hot}(\text{object\_shape}) \text{one\_hot}(\text{object\_color})])$ \| \| Top View \| An image made by projecting 3D space from the top (see Figure 2 right). \| \| Side View \| An image made by projecting 3D space from the side (see Figure 2 left). \| Table 1: Observation Spaces \| Action Space \| Description \| \|--------------------\|-----------------------------------------------------------------------------\| \| Cardinals \| Move one step in one of the 4 cardinal directions (north, east, south, west) \| \| Move NW \| Move one step north, move one step west, or teleport to the south-east corner of the grid \| \| Rotations \| Rotate left, rotate right, or move one step forward in the direction of facing \| \| Teleport Direction \| Rotate left, rotate right, or teleport to a certain distance from the wall currently facing (0-6 steps from the wall) \| \| Knight Rotations \| Rotate left, rotate right, knight move left, or knight move right (i.e. two steps forward in the direction of facing + one step left or right). \| Table 2: Action Spaces Action spaces ($A_n$): Completing each of the instructions can be accomplished by using one of the five possible action spaces. The type of movement available varies between spaces, as described in Table 2. Additionally, each action space has three shared actions for picking and dropping objects, and indicating the episode is done (Pick, Drop, and Done actions respectively). Successful completion requires the agent to complete the instruction and then output Done action. ### 3 ARCHITECTURE WITH COMPOSITIONAL INTERFACES In our work, we use COIN (Compositional Interfaces) architecture. It is a modular architecture consisting of three main components: the perception modules, the controller, and the action modules, as demonstrated in Figure 1(b), and detailed in the Appendix A.2. There is a different perception module for each observation space and a different action module for each action space. The controller is shared between all spaces and has a transformer architecture (Vaswani et al., 2017) (although any architecture that can handle variable-length inputs and tokens can be used). Since instructions and action descriptions (we use textual descriptions, such as e.g. "The action space is cardinals.") are expressed in text, we can directly feed to the controller via simple word embedding layers. The perception modules take in an observation and output a fixed-size embedding. The architecture for each perception module is chosen to best fit the modality of the corresponding observation space (the inventory, when represented as a list, is embedded with a 2D convolutional network). However, the number of vectors output by those specialized architectures vary from space to space (or even sample to sample for List space). In order to unify these outputs as input to the controller, we use adapter networks for each observation space. An adapter takes input of embedding of variable length and outputs embedding of fixed length, which is then input to the controller. That is achieved by using the network as cross-attention layers in enc-dec transformers. The observation embedding is then concatenated with instruction and action space description embeddings. We also concatenate a fixed number of special padding tokens before feeding it into the controller. Among output vectors from the controller, we select the ones that correspond to the padding tokens, which gives us a fixed number of embeddings vectors to work with. Those embeddings are then fed into the action space specific action module, whose output corresponds to the dimensions of the corresponding action space. The fixed size of action embedding enables faster adaptation to new action spaces. 4 EXPERIMENTS In the following set of experiments, we examine the compositional generalization properties of COIN agent. First, we examine the ability of COIN to generalize to unseen environment instances $(O_m, A_n, I_k)$, where combinations seen during training are selected uniformly at random. Next, we examine the case where the samples held out from training are a selected group of particularly challenging instruction and observation spaces. Lastly, we test the ability of COIN to adapt to new, completely unseen observation spaces $O_{new}$ through finetuning. All the experiments use the compositional environments described in Section 2.1. For training, we use a dataset of 2,048 episodes with near-optimal trajectories $\{\tau^{(t)}_{(m,n,k)}\}_{t=1}^{2048}$ for each of the 240 = 6 × 5 × 8 possible combinations of observation, action and instruction spaces (6, 5 and 8 options respectively); some of which will be held out from training. The near-optimal trajectories are generated using the A* algorithm or hand-engineered optimal policy. We evaluate the performance of the trained agent by measuring the rate of successful completion on both unseen and seen environment combinations. The task is considered completed if the agent reaches the goal defined by the instruction within the first 100 steps. When evaluating the trained agent on seen environment combinations, we measure the performance on that environment instance generated using a different random seed, i.e. the exact initial state and instruction are likely to differ from train time. As an architecture for the perception modules, we use a pre-trained ResNet-18 network [He et al., 2015] for image spaces; for Grid and List spaces we use a 2D and 1D convolutional networks; for Text and Symbol spaces, we use 1D convolutions. The controller is a pre-trained Distilled-GPT-2 [Sanh et al., 2020], while each action module is a simple feed-forward network with the output corresponding to the dimensionality of the action space. The dimensions of observation embedding are 10 × 768, and the dimensions of action embedding are 4 × 768, where 768 is the dimension of GPT-2 token embedding. Each network is trained for 80 epochs. More details about the architecture and the training procedure can be found in the Appendix A.2. Figure 5: Comparison of performance between individual observation, action and instruction spaces. For each space, we report the performance averaged over all environment combinations containing that space (the error bars represent standard deviation). For trained on samples from 75% environment combinations, we report the completion rate on environment instances included (green) and not included (blue) in the training data. The performance of an agent trained on only one environment instance is shown in red. As a baseline, we use an agent trained on individual environment combinations using the same architecture, i.e. we train a separate agent on each of the 240 environment combinations. Note that in these cases, there is no weight sharing and each of such networks will contain only one perceptual and action module. 4.1 RANDOM HOLDOUTS We start by examining the case where the environment instances $(O_m, A_n, I_k)$ included in the dataset have been chosen randomly by chance. To ensure relatively uniform coverage of all spaces, the procedure for selecting environment instances guarantees that each space individually has been included in at least four combinations. We vary the percentage of environment instances held out from the training: either 25, 50 or 75% of all possible combinations. We report separately the performance on the environments included in the training data (seen) or held out for training (unseen). In Figure 5, we take a look at the performance difference for each of the spaces individually, i.e. we fix one of the spaces (observation, action or instruction) and average over the rest. We consider the case where 25% combinations are held out (results with 50 and 75% held out combinations can be found in the Appendix A.3.1). Here we can see that COIN outperforms or matches the individual agents in all but one space (instruction space Sort by Property). We can also see that performance on unseen combinations matches the performance on seen combinations, which implies that the agent achieved near-perfect generalization to unseen compositions (with the remaining generalization gap being a consequence of either optimization difficulties or poor generalization to unseen observations and instructions). The greatest performance gains are seen on token observation spaces (Text, Symbol), which are particularly challenging for optimization and may particularly benefit from additional supervision provided by co-training on multiple observation spaces: the learning may be bootstrapped by learning a good controller on other, easier spaces. The completion rates averaged over all the environment instances can be seen in Figure 3. From there, we can see that the COIN agent generalizes to unseen environment combinations extremely well, even outperforming the agents trained on individual environment combinations when the holdout rate is over 50%. The performance of COIN agent drops as we decrease the number of combinations included in the training data as expected. The error bars represent standard deviation over 240 environment instances. To evaluate the relative importance of using more data for each of the training environment combinations versus using more environment combinations in the training dataset, we run an experiment where the total number of episodes seen during training is kept constant while varying the holdout rate and number of episodes used in training. The total number of episodes used in training is always 192k, with the percentage of environment combinations used in training and number of episodes being: $(80\%, 2^{10})$, $(40\%, 2^{11})$ and $(20\%, 2^{12})$. The results can be seen in Figure 4. We find that for compositional generalization, it is more advantageous to use more environment combinations. 4.2 Hard Holdouts Next, we consider the hard case where the holdout set is composed of particularly challenging combinations, either in terms of data collection or training time. We are particularly interested in this case, as in practice, there may be cases where data collection is much more challenging for some combinations (e.g. when some observation spaces correspond to data collected on real robots instead of data collected in simulation, where it can be hard to evaluate completion of some instructions outside of simulation). In these cases, it might be advantageous to collect the data on easier combinations for training and obtain good zero-shot performance on hard combinations without requiring data collection or training. To construct hard combinations, we selected image observation spaces ($O_{\text{hard}} = \{\text{Top View}, \text{Side View}\}$) and two of the hardest instruction spaces ($I_{\text{hard}} = \{\text{Bring Object}, \text{Pickup In Order}\}$) as the performance on these spaces is generally the lowest, training trajectories are the longest and training on images takes more time. The hold-out set $E_{\text{hard}}$ then consists of all combinations $(O_m, A_n, I_k)$ where $O_m \in O_{\text{hard}}$ and $I_k \in I_{\text{hard}}$. In our case, this will be a total of 20 environment combinations. We train the COIN agent on the remaining 200 combinations, or a randomly sampled set of 75 and 50% of the remaining environments (in total, this corresponds to 8, 32 and 55% of all possible combinations being held out respectively). For hard holdouts, we report results on 5 different random seeds. As shown in Table 6, we find that while not matching the performance of agents trained individually on those combinations or when the combinations are held out randomly, we still observe good transfer from easier to hard combinations, despite never seeing the particularly hard combination of observation and instruction in the training dataset. \| Method \| Completion Rate \| \|-------------------------\|-----------------\| \| Individual Envs \| 0.35 ± 0.18 \| \| Random Holdouts (25%) \| 0.34 ± 0.07 \| \| Hard Holdouts (8%) \| 0.26 ± 0.08 \| \| Hard Holdouts (32%) \| 0.21 ± 0.12 \| \| Hard Holdouts (55%) \| 0.20 ± 0.10 \| Figure 6: Agent performance on a set of 20 particularly challenging environment instances $E_{\text{hard}}$. For COIN with both random and hard holdouts, we report zero-shot performance. In hard holdouts, the entire $E_{\text{hard}}$ was held out from the training data; whereas in random holdouts, a random selection of 25% of combinations was held out. For hard holdouts, we report results where a total of 8, 32 and 55% of environment instances (including $E_{\text{hard}}$), were held out. We also report the performance of agents trained on individual environments from $E_{\text{hard}}$. 4.3 New Perception Spaces Lastly, we examine if COIN agent can effectively and efficiently incorporate new observation spaces. This is particularly relevant in a continual learning setting, where over a lifetime, new perceptual spaces may need to be added, without hurting the performance on spaces the agent has been already trained on or requiring training again from scratch on the entire dataset including the new observation spaces. Modular architectures have the potential to integrate new observational spaces without affecting the performance on other spaces by training only the new perceptual module, while freezing the controller and action modules. Moreover, training only the perception modules may require less data and converge more quickly. To test this, for each observation space $O_{\text{new}}$, we first take out all the samples with that observation space from the training data (in total 40 different environment combination) and train on the data from randomly selected 75% of the environment combinations. Next, we take the trained COIN network and add the freshly initialized perception module for $O_{\text{new}}$, which is trained on the data from all the environment combinations containing $O_{\text{new}}$. The weights of the controller and action modules are kept constant. To compare the data requirements of adding the new perceptual spaces to already trained controller and action modules, to training the entire network from scratch, we train the new module using 2048, 1024, or 516 episodes from the dataset (full, half, or one-fourth of the dataset respectively). For comparison, we also try fine-tuning the entire network on the combinations with the new observation space (i.e. without freezing weights of the controller and action modules). We also compare the results to the modular network trained on the same 40 environments from scratch (i.e. without transfer from other observation spaces). Figure 7: Performance of COIN agent on the 40 environment combinations $\mathcal{E}^O$ containing a newly added observation space $O$, for each of the six available observation spaces. The controller and action modules are trained on 75% of all randomly selected combinations not including $\mathcal{E}^O$. In the top figure, we only train the newly added perceptual module (i.e., without affecting the performance on other tasks), whereas in the bottom figure, we fine-tune the entire network. We report the results using 2048, 1024, or 516 episodes from each environment in $\mathcal{E}^O$ for training. We contrast these results to an agent trained from scratch on $\mathcal{E}^O$ and agents trained individually on each task in $\mathcal{E}^O$. The results are reported over 3 random seeds, with the error bar representing standard deviation over all environment instances in $\mathcal{E}^O$. Results on the new observation spaces with freezing of the controller and action modules can be found in Figure 7 Top and without freezing in Figure 7 Bottom. We find that, when averaged over the observation spaces, by training only the new perception module, we can match the performance obtained by training from scratch and outperform training on individual environments. Moreover, we can match the training from scratch with one-fourth of the data. This is likely due to transfer from other tasks, where the new observation just needs to be mapped to a representation already understandable by the controller. We are finding that finetuning the entire network is not necessary for achieving good performance, hence a new observation space can be incorporated without affecting performance on other environment instances. 5 RELATED WORK Single Modality: Compositional generalization is often studied at the level of a single domain. In vision domain, models are tested if they can recognize an image that contains an unseen combination of different visual properties (e.g., shape, color), with emphasis on disentangled representation (Xu et al., 2022). Instruction and task is another domain where compositional generalization is well studied (Zhang et al., 2018; Zhou et al., 2022). At test time, the agent is given an unseen instruction or task that usually can be accomplished by chaining together already learned skills (Lake & Baroni, 2017). This idea of learning a set of skills that can be composed together can be traced back to options framework (Sutton et al., 1999) and other hierarchical RL methods (Sukhbaatar et al., 2018). Given the recent success of large language models, language is increasingly being used: Jiang et al. (2019) leverages natural language for hierarchical RL; Huang et al. (2022a) uses large language models to decompose tasks into smaller subtasks; Mezghani et al. (2023) has a single model for both policy and language reasoner. Unlike these, the focus our paper is composition generalization across of different modalities. Modular Architectures for Multi-task Learning: Modular architectures can be viewed as composition of internal modules of the agent, and often studied together with multi-task generalization. PathNet (Fernando et al., 2017) uses genetic algorithm to select a subset of a neural network to be used for a specific task, showing positive transfer from one task to another. Rusu et al. (2016) grows a neural network by adding new modules with each new task, but allowing them to connect all previous modules. Continual learning is enabled by freezing previous modules, but still positive transfer is observed between different Atari games. Similarly, Gesmundo & Dean (2022) both grows and selects subsets of the network. However those methods require training on the target environment, while our method enables zero-shot generalization to an unseen environment while utilizing a simple end-to-end training. LegoNN (Dalmia et al., 2022) is an encoder-decoder model with decoder modules that can be reused across machine translation and speech recognition tasks. Our approach for connecting perceptual modules to the controller recalls Alayrac et al. (2022), where the authors use cross-attention to connect a vision model to a text Transformer, and Ieagle et al. (2021) where this idea is discussed more generally. Multi-Embodiment Continuous Control: Devin et al. (2017); Huang et al. (2020) used Graph Neural Networks (GNN) to build modular architecture that can control many different physical bodies. Furthermore Huang et al. (2020) shows such architectures are capable of zero-shot generalization to a new physical body. Our work, as Kurin et al. (2020), uses a Transformer in place of the GNN. The “action spaces” in this work are analogous to the body morphologies in those. However, here, we study composable generalization not just to different action spaces, but to perceptual and task spaces as well. Language Model as Controller and Planner via Text Interfaces: Several works have shown how a language model used as can be a nexus between modalities, and controller or planner for embodied agents. The general theme is to use text as glue, and the language model as a central processor. For example Socratic agents (Zeng et al., 2022) combines multiple pre-trained models from different domains to create a system that can solve unseen task involving a novel combination of domains. Similarly Huang et al. (2022b) deploy a pretrained LM as a robotic controller by augmenting it with additional models that can interpret visual scenes in language. In Ahn et al. (2022), the language model is used to score affordances based on a task description, and as a planner, following Huang et al. (2022a). In this work, rather than connecting modules via text, we use self-attention, allowing end-to-end learning. Transformers in Behaviorally Cloned Generalist Agents: Our work is closely related to Reed et al. (2022) and Shridhar et al. (2023), where the authors showed that end-to-end Transformers can be effective controllers for embodied agents with multi-modal perception and/or actions. As in those works, we train via behavioral cloning. While Reed et al. (2022) tokenizes all inputs and treats the Transformer controller as a monolith, we allow passing gradients to perceptual or task-specific submodules. In this, we are similar to Shridhar et al. (2023), but rather than consider a fixed perceptual and action space as in that work, we show that our setup allows compositional generalization between perceptual, action and task spaces, and fast adaptation to new spaces. 6 CONCLUSION In this paper, we proposed a modular architecture with differentiable interfaces to various modalities of perception and action. These interfaces are connected to a shared controller, enabling passing gradients and end-to-end backpropagation, while supporting knowledge sharing. We developed a new environment in which perceptual modalities, sets of actions and types of instructions can be independently varied. This environment allowed us to systematically study compositional generalization across different modalities. 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GOt2kP383R	x_i in Eq. (2) denotes the feature (activation). However, I wonder if it will lead to the homogenization of features since they are expected to have a low standard deviation. Did the authors try to minimize the difference in the mean of each channel?	ABSTRACT Quantization is a promising approach to reduce the high computational complexity of image super-resolution (SR) networks. However, compared to high-level tasks like image classification, low-bit quantization leads to severe accuracy loss in SR networks. This is because feature distributions of SR networks are significantly divergent for each channel or input image, and is thus difficult to determine a quantization range. Existing SR quantization works approach this distribution mismatch problem by dynamically adapting quantization ranges to the variant distributions during test time. However, such dynamic adaptation incurs additional computational costs that limit the benefits of quantization. Instead, we propose a new quantization-aware training framework that effectively Overcomes the Distribution Mismatch problem in SR networks without the need for dynamic adaptation. Intuitively, the mismatch can be reduced by directly regularizing the variance in features during training. However, we observe that variance regularization can collide with the reconstruction loss during training and adversely impact SR accuracy. Thus, we avoid the conflict between two losses by regularizing the variance only when the gradients of variance regularization are cooperative with that of reconstruction. Additionally, to further reduce the distribution mismatch, we introduce selective distribution offsets to layers with a significant mismatch, which selectively scales or shifts channel-wise features. Our algorithm effectively reduces the mismatch in distributions with minimal computational overhead. 1 INTRODUCTION Image super-resolution (SR) is a core low-level vision task that aims to reconstruct the high-resolution (HR) images from their corresponding low-resolution (LR) counterparts. Recent advances in deep learning (Dong et al., 2015; Kim et al., 2016; Lim et al., 2017; Zhang et al., 2018a;b) have led to astonishing achievements in producing high-fidelity images. However, the remarkable performance relies on heavy network architectures with significant computational costs, which limits the practical viability, such as mobile deployment. To mitigate the computational complexity of neural networks, quantization has emerged as a promising avenue. Network quantization has proven effective in reducing computation costs without much loss in accuracy, particularly in high-level vision tasks, such as image classification (Choi et al., 2018; Hou & Kwok, 2018; Zhou et al., 2016). Nonetheless, when it comes to quantizing SR networks to lower bit-widths, a substantial performance degradation (Ignatov et al., 2021) occurs, posing a persistent and challenging problem to be addressed. Such degradation can be attributed to the significant variance present in the activation (feature) distributions of SR networks. The feature distribution of a layer exhibits substantial discrepancies across different channels and images, which makes it difficult to determine a single quantization range for a layer. Early approach on SR quantization (Li et al., 2020a) adopts quantization-aware training to learn better quantization ranges. However, as observed in Figure 1, despite careful selection, the quantization ranges fail to align with the diverse values within the channel and image dimension, which we refer to as distribution mismatch. Recent approaches aim to address this challenge by incorporating dynamic adaptation methods to accommodate the varying distributions. For instance, Hong et al. (2022b) leverage distribution mean and variance to dynamically adjust quantization ranges for each channel and Zhong et al. (2022) employ input-adaptive dynamic modules to determine quantization ranges on a per-image basis. However, such dynamic adaptation modules introduce significant computational overhead. While adapting the quantization function to each image during inference might handle the variable distributions, the overhead compromises the computational benefits of quantization. In this study, we propose a novel quantization-aware training framework that addresses the distribution mismatch problem, by introducing a new loss term that regulates the variance in distributions. While direct regularization of distribution variance demonstrates potential in reducing quantization errors in each quantized feature, its relationship with the reconstruction loss is questionable. We observe that concurrently optimizing the network with variance regularization and reconstruction loss can disrupt the image reconstruction process, as shown in Figure 2. Therefore, we introduce a cooperative variance regularization strategy, where the variance is regulated only when it collaborates harmoniously with the reconstruction loss. To determine the cooperative behavior, we assess whether the sign values of the gradients from each loss are the same. Consequently, we can effectively update the SR network to optimize both quantization-friendliness and reconstruction accuracy. To further reduce the distribution mismatch in SR networks, we introduce the concept of selective distribution offsets for features that exhibit severe mismatch. We first observe that the distribution mismatch problem is more critical in the channel dimension compared to the image dimension (Figure 1). Moreover, we find that the degree of channel-wise mismatch varies across different convolutional layers. As shown in Figure 3, certain layers exhibit a large mismatch between the distribution means, while others show a large mismatch between the distribution deviations. Intuitively, the mismatch in distribution mean can be reduced by applying channel-wise shifting of the distributions and that of the deviation be reduced by scaling. On this basis, we leverage additional offset parameters that selectively shift or scale the channel-wise distributions based on the specific mismatch aptitude of the layer. While these selectively-applied offsets effectively mitigate the distribution mismatch, they only incur negligible overhead, around $\times 30$ smaller storage size overhead or $\times 100$ fewer BitOPs compared to existing works with dynamic modules. The contributions of our work include: - We introduce the first quantization framework to address the distribution mismatch problem in SR networks without dynamic modules. Our framework updates the SR network to be quantization-friendly and accurate at the same time. - We identify the distinct distribution mismatch among different layers and further reduce the distribution mismatch by shifting or scaling largely mismatching features. - Compared to existing approaches on SR quantization, ours achieves state-of-the-art performance with similar or less computations. 2 RELATED WORKS Image super-resolution. Convolutional neural network (CNN) based approaches (Ledig et al., 2017; Lim et al., 2017) have exhibited remarkable advancements in image super-resolution (SR) task, but at the cost of substantial computational resources. The massive computations of SR networks have led to a growing interest in developing lightweight SR architectures (Dong et al., 2014; Hui et al., 2019; 2018; Zhang et al., 2018a; Jo & Kim, 2021). Furthermore, various lightweight networks are investigated through neural architecture search (Chu et al., 2021; Kim et al., 2019; Li et al., 2020b; Song et al., 2020; Li et al., 2021), knowledge distillation (Hui et al., 2018; 2019; Zhang et al., 2021), and pruning (Oh et al., 2022). While these methods mostly focus on reducing the network depth or the number of channels, our focus in this work is to lower the precision of floating-point operations with network quantization. Network quantization. By mapping 32-bit floating point values of input features and weights of convolutional layers to lower-bit values, network quantization provides a dramatic reduction in computational resources (Cai et al., 2017; Choi et al., 2018; Esser et al., 2020; Jung et al., 2019; Zhou et al., 2016; Zhuang et al., 2018). Recent works successfully quantize various networks with low bit-widths without much compromise in network accuracy (Cai & Vasconcelos, 2020; Dong et al., 2019; Habi et al., 2020; Jin et al., 2020; Lou et al., 2020; Wang et al., 2019; Yang & Jin, Figure 1: Distribution mismatch in SR networks. SR networks exhibit a large mismatch inside the feature distributions, which results in a large quantization error. The mismatch is observed in both channel-dimension and image-dimension, but channel-wise mismatch is larger in magnitude and also more critical. Channels and images of a layer are randomly selected for visualization. However, these works primarily focus on high-level vision tasks, while networks for low-level vision tasks remain vulnerable to low-bit quantization. Quantized super-resolution networks. In contrast to high-level vision tasks, super-resolution poses different challenges due to inherently high accuracy sensitivity to quantization (Ignatov et al., 2021; Ma et al., 2019; Xin et al., 2020; Wang et al., 2021). A few works have attempted to recover the accuracy by modifying the network architecture (Ayazoglu, 2021; Jiang et al., 2021; Xin et al., 2020) or by adopting different bits for each image (Hong et al., 2022a; Tian et al., 2023) or network stage (Liu et al., 2021). However, the key challenge of quantizing SR networks is in the vastly distinct feature distributions of SR networks. To deal with such issue, Li et al. (2020a) adopt a learnable quantization range for different layers. More recently, Hong et al. (2022b) recognize that the distributions are not only distinct per layer, but per channel and per input image and adopts dynamic quantization function for each channel. Moreover, Zhong et al. (2022) employ an input-adaptive dynamic module to adapt the quantization ranges differently for each input image. However, these dynamic adaptations of quantization functions during test-time cost non-negligible computational overheads. In contrast, instead of designing input-adaptive quantization modules, we focus on mitigating the feature variance itself. Our framework reduces the inherent distribution mismatch in SR networks with minimal overhead, accurately quantizing networks without dynamic modules. 3 PROPOSED METHOD 3.1 PRELIMINARIES To reduce the heavy computations of convolutional layers in neural networks, the input feature (activation) and weight of each convolutional layer are quantized to low-bit values (Cai et al., 2017; Choi et al., 2018; Jung et al., 2019; Gholami et al., 2021). The input feature of the $i$-th convolutional layer $X_i \in \mathbb{R}^{B \times C \times H \times W}$, where $B$, $C$, $H$, and $W$ denote the dimension of input batch, channel, height, and width, a quantization operator $Q(\cdot)$ quantizes the feature $X_i$ with bit-width $b$: $$Q(X_i) = \text{Int}\left(\frac{\text{clip}(X_i, \alpha_l, \alpha_u) - \alpha_l}{s}\right) \cdot s + \alpha_l,$$ where $\text{clip}(\cdot, \alpha_l, \alpha_u)$ truncates the input into the range of $[\alpha_l, \alpha_u]$ and $s = \frac{\alpha_u - \alpha_l}{2^b - 1}$. After truncation, the truncated feature is scaled to $[0, 2^b - 1]$, then rounded to integer values with $\text{Int}(\cdot)$, and it is rescaled to range $[\alpha_l, \alpha_u]$. To obtain better quantization ranges for SR networks, range parameters $\alpha_l, \alpha_u$ for each layer are generally learned through quantization-aware training (Li et al., 2020a; Zhong et al., 2022). Since the rounding function is not differentiable, a straight-through estimator (STE) (Bengio et al., 2013) is used to train the range parameters in an end-to-end manner. Following Zhong et al. (2022), we initialize $\alpha_u$ and $\alpha_l$ as the $j$-th and $100-j$-th percentile value of feature averaged among the training data. $j$ is set as 1 in our experiments to avoid outliers from corrupting Figure 2: Conflict between variance regularization and reconstruction loss. Variance regularization updates a number of parameters in the opposite direction of reconstruction loss, which we refer to as gradient conflict. We plot the ratio of conflicted gradients during training when the two losses are jointly used. Nearly half of the parameters undergo gradient conflict which indicates that simply leveraging variance regularization and the reconstruction loss together can limit SR accuracy. the quantization range. Similarly, to quantize the weight of the $i$-th convolutional layer $W_i$, quantization operator $Q(\cdot)$ is used. However, instead of setting range parameters as learnable parameters, $\alpha_l, \alpha_u$ for weights are fixed as the $j$-th and $100-j$-th percentile of weights. ### 3.2 Distribution mismatch in SR networks Quantization unfriendliness of SR networks is from the diverse feature (activation) distributions, as reported in previous studies (Li et al., 2020a; Hong et al., 2022b; Zhong et al., 2022), mainly due to the absence of batch normalization layers in SR networks. Existing SR quantization methods address this issue by employing one (Li et al., 2020a) or two (Zhong et al., 2022) learnable quantization range parameters for each convolutional layer feature. However, despite that the quantization-aware training process aims to find the optimal range for each feature, it fails to account for the channel-wise and input-wise variance in distributions. As illustrated in Figure 1, where notable discrepancies exist between layer-wise and channel-wise distributions, quantization grids are needlessly allocated to regions with minimal feature density. This mismatch in inter-channel distributions leads to performance degradation when quantizing SR networks. In the following sections, we introduce a new quantization-aware training scheme to address the distribution mismatch problem. ### 3.3 Cooperative variance regularization Instead of focusing on finding a better quantization range parameter capable of accommodating the diverse feature distributions, our approach aims to regularize the distribution diversity beforehand. Obtaining an appropriate quantization range for a feature with low variance is an easier task compared to that of high variance. In this work, we define the overall mismatch of a feature distribution with the standard deviation, $$M(X_i) = \sigma(X_i),$$ where $\sigma(\cdot)$ calculates the standard deviation of the feature. Thus, variance regularization can be directly applied to the feature to be quantized ($X_i$), which is formulated as follows: $$L_V(X_i) = \lambda_V \cdot M(X_i),$$ where $\lambda_V$ is the hyperparameter that denotes the weight of regularization. The overall $L_V = \sum_{\text{layers}} L_V(X_i)$ is obtained by summing over all quantized convolutional layers. The variance regularization loss can be used in line with the reconstruction loss, which is originally used in the general quantization-aware training process. The optimization of parameter $\theta^t$ is formulated as: $$\theta^{t+1} = \theta^t - \alpha^t (\nabla_\theta L_R(\theta^t) + \nabla_\theta L_V(\theta^t)),$$ where $\nabla_\theta L_R(\theta^t)$ denotes the gradient from the original reconstruction loss and $\nabla_\theta L_V(\theta^t)$ denotes the gradient from variance regularization loss and $\alpha^t$ denotes the learning rate. Updating the network to minimize the variance regularization loss will reduce the quantization error of each feature. Figure 3: The distribution mismatch of different layers in EDSR. While layer (a) shows an overall small mismatch between channels, layer (b) shows a large mismatch on deviation, and layer (c) exhibits a large mismatch on average. This motivates us to selectively scale features with large deviation mismatch ($m^\sigma$) and shift features with a large average mismatch ($m^\mu$). However, then a question arises, does reducing the quantization error of each feature lead to improved reconstruction accuracy? The answer is, according to our observation in Figure 2, not necessarily. During the training process, the variance regularization loss can collide with the original reconstruction loss. That is, for some parameters, the sign of the gradient from reconstruction loss and that of variance regularization are opposing, referred to as gradient conflict (Du et al., 2018). As in Figure 2b, the ratio of parameters that undergo gradient conflict is not minor and such ratio persists throughout training, which means that regularization loss can hinder the reconstruction loss. However, we want to avoid the conflict between two losses, in other words, minimize the variance as long as it does not hinder the reconstruction loss. Thus, we determine whether the two losses are cooperative or not by examining the sign of the gradients of each loss. If the signs of the gradients are equal, then the parameter is updated in the same direction by two losses. By contrast, if the sign values are inverse, the two losses restrain each other, thus we only employ the reconstruction loss. In summary, we leverage variance regularization for parameters that the gradients have the same sign value as that from the reconstruction loss: $$\theta^{t+1} = \begin{cases} \theta^t - \alpha^t (\nabla_\theta L_R(\theta^t) + \nabla_\theta L_V(\theta^t)), & \nabla_\theta L_R(\theta^t) \cdot \nabla_\theta L_V(\theta^t) \geq 0, \\ \theta^t - \alpha^t (\nabla_\theta L_R(\theta^t)), & \nabla_\theta L_R(\theta^t) \cdot \nabla_\theta L_V(\theta^t) < 0. \end{cases}$$ This allows the network to reduce the quantization error cooperatively with the reconstruction error. ### 3.4 Selective Distribution Offsets The variance of the distribution can be reduced to a certain extent via variance regularization in Section 3.3. However, since regularization is applied only when it is cooperative with the SR reconstruction, the gap between distributions remains. In this section, we explore the remaining gap between distributions. First, as visualized in Figure 1, we observe that the distribution gap is larger (and more critical) in the channel dimension compared to the image dimension. Also, we find that this extent of the channel-wise gap in the distribution is different for each layer of the SR network, as shown in Figure 3. Some layers (Figure 3b) exhibit a larger mismatch in the distribution deviation, while others (Figure 3c) show a larger mismatch in the distribution average. The quantization errors of the layer with a large mismatch in distribution mean can be decreased by shifting the channel-wise feature. Similarly, the mismatch in layers with large divergence in distribution deviation can be reduced by scaling each channel-wise distribution. Since channel-wise shifting and scaling incur computational overhead and not all layers are in need of additional shifting and scaling (Figure 3a), we selectively apply offset scaling/shifting to layers that can maximally benefit from it. The standards for our selection are derived by feeding a patch of images to the 32-bit pre-trained network and calculating the mismatch in average/deviation of each layer. Given the $i$-th feature statistics $\tilde{X}_i$ from the pre-trained network, the mismatch of $i$-th convolutional layer is formulated as follows: $$m_i^\mu = \sigma(\mu_c(\tilde{X}_i)) \quad \text{and} \quad m_i^\sigma = \sigma(\sigma_c(\tilde{X}_i)),$$ where $\mu_c(\cdot)$ and $\sigma_c(\cdot)$ respectively calculate the channel-wise mean and standard deviation of a feature and $\sigma(\cdot)$ calculates the standard deviation. After all $m_i^\mu$'s and $m_i^\sigma$'s ($i = 1, \cdots, \#\text{layers}$) Algorithm 1 Quantization-aware training process of ODM Input: Pre-trained 32-bit network $ P $, distribution offset ratio $ p $. Output: Quantized network $ Q $. Using $ P $, obtain $ m^\mu_i $ and $ m^\sigma_i $ ($ i = 1, \cdots, \# \text{layers} $) using Eq. 6 for $ i = 1, \cdots, \# \text{layers} $ do Initialize quantization range parameters $ \alpha_\ell, \alpha_u $ for feature $ X_i $ of $ Q $ using percentile if $ m^\mu_i $ is in top-$ p $ ratio among $ m^\mu $ then Shift $ X_i $ with $ s^\mu $ using Eq. 7 if $ m^\sigma_i $ is in top-$ p $ ratio among $ m^\sigma $ then Scale $ X_i $ with $ s^\sigma $ using Eq. 8 Given $ X_i $, obtain variance regularization loss $ L_V(X_i) $ using Eq. 3 Given $ L_V $ and $ L_1 $, update parameters of $ Q $ using Eq. 5 are collected, we apply additional scaling offsets to top-$ p $ layers with high $ m^\sigma_i $ value and shifting offsets to top-$ p $ layers with high $ m^\mu_i $ value. The shifting and scaling process for feature $ X_i $ of the $ i $-th convolutional layer is formulated as follows: \[ X_i^* = X_i + s^\mu, \quad \text{if } m^\mu_i \in \text{top-}p(m^\mu), \] \[ X_i^* = X_i \cdot s^\sigma, \quad \text{if } m^\sigma_i \in \text{top-}p(m^\sigma), \] where $ s^\mu, s^\sigma \in \mathbb{R}^C $ are learnable parameters, $ \text{top-}p(\cdot) $ constructs a set that contains values greater than the $ 100(1 - p)$-percentile value of the given set. $ p $ is the hyperparameter that determines the ratio of layers to apply distribution offsets, which we set to 0.3 in our experiments. Moreover, both offsets $ s^\mu $ and $ s^\sigma $ are quantized to low-bit, 4-bit in our experiments, to minimize the computational overhead. Consequently, the offsets additionally incur only 0.02% overhead to the network storage size for EDSR. The offsets further relieve distribution mismatch with minimal overhead. 3.5 Overall training Alg. 1 summarizes the overall pipeline for our framework, ODM. To update the parameters of the quantized network, including the selective offsets, we follow the common practice (Li et al., 2020a; Zhong et al., 2022) to use $ L_1 $ and $ L_{SKT} $ for the reconstruction loss as follows: \[ L_R = L_1 + \lambda L_{SKT}, \] where $ L_1 $ loss indicates the $ l_1 $ distance between the reconstructed image and the ground-truth HR image, and $ L_{SKT} $ loss is the $ l_2 $ distance between the structural features of the quantized network and the 32-bit pre-trained network. The structural features are obtained from the last layer of the high-level feature extractor. The balancing weight $ \lambda $ is set as 1000 in our experiments. Also, the weight $ \lambda_V $ to balance $ L_V $ and $ L_R $ in Eq. 3 is set differently depending on the mismatch severeness of the SR architecture. We provide detailed settings in the supplementary materials. 4 Experiments The efficacy and adaptability of the proposed quantization framework ODM are assessed through its application to several SR networks. The experimental settings are outlined (Sec. 4.1), and both quantitative (Sec. 4.2) and qualitative (Sec. 4.3) evaluations are conducted on various SR networks. Ablation experiments are conducted to examine each component of the framework (Sec. 4.4). 4.1 Implementation details Models and training. The proposed framework is applied directly to existing representative SR networks that produce satisfactory SR results but with heavy computations: EDSR (baseline) (Lim et al., 2017), RDN (Zhang et al., 2018b), and SRRResNet (Ledig et al., 2017). Following existing works on SR quantization (Li et al., 2020a; Ma et al., 2019; Xin et al., 2020; Hong et al., 2022b; Zhong et al., 2022; Hong et al., 2022a), weights and activations of the high-level feature extraction module are quantized which is the most computationally-demanding. Training and validation are done with DIV2K (Agustsson & Timofte, 2017) dataset. ODM trains the network for 60 epochs, with $1 \times 10^{-4}$ initial learning rate that is halved every 15 epochs and with a batch size of 8. All our experiments are implemented with PyTorch and run using a single RTX 2080Ti GPU. Evaluation. We evaluate our framework on the standard benchmark (Set5 (Bevilacqua et al., 2012), Set14 (Ledig et al., 2017), BSD100 (Martin et al., 2001), and Urban100 (Huang et al., 2015)). We report peak signal-to-noise ratio (PSNR) and structural similarity index (SSIM (Wang et al., 2004)) to evaluate the SR performance. To evaluate the computational complexity of our framework, we measure the BitOPs and storage size. BitOPs is the number of operations that are weighted by the bit-widths of the two operands. Storage size is the number of stored parameters weighted by the precision of each parameter value. ### 4.2 Quantitative Results To evaluate the effectiveness of our proposed scheme, we compare the results with existing SR quantization works PAMS (Li et al., 2020a), DAQ (Hong et al., 2022b), and DDTB (Zhong et al., 2022) using the official code. To make a fair comparison with the existing works, we reproduce the results of other methods using the same training epochs. As shown in Table 1, our framework ODM outperforms other methods largely for all 4, 3, and 2-bit, and notably, the improvement is significant for 2-bit quantization. Also, 4-bit EDSR-ODM achieves closer accuracy to the 32-bit EDSR, where the margin is 0.07dB for Set5. This indicates that ODM can effectively bridge the gap between the quantized network and the floating-point network. Also, Table 2 compares the results on RDN. The results show that ODM achieves consistently superior performance on 4, 3, and 2-bit quantization. Furthermore, we evaluate our framework on SRResNet which is shown in Table 3. SRResNet architecture includes BN layers and thus the distribution mismatch problem is not as severe as in EDSR. #### Table 1: Quantitative comparisons on EDSR of scale 4. \| Model \| Bit \| Set5 PSNR \| SSIM \| Set14 PSNR \| SSIM \| B100 PSNR \| SSIM \| Urban100 PSNR \| SSIM \| \|----------------\|-----\|-----------\|------\|------------\|------\|-----------\|------\|--------------\|------\| \| EDSR \| 32 \| 32.10 \| 0.894\| 28.58 \| 0.781\| 27.56 \| 0.736\| 26.04 \| 0.785\| \| EDSR-PAMS \| 4 \| 31.59 \| 0.885\| 28.20 \| 0.773\| 27.32 \| 0.728\| 25.32 \| 0.762\| \| EDSR-DAQ \| 4 \| 31.85 \| 0.887\| 28.38 \| 0.776\| 27.42 \| 0.732\| 25.73 \| 0.772\| \| EDSR-DDTB \| 4 \| 31.85 \| 0.889\| 28.39 \| 0.777\| 27.44 \| 0.732\| 25.69 \| 0.774\| \| EDSR-ODM (Ours)\| 4 \| 32.03 \| 0.891\| 28.48 \| 0.779\| 27.49 \| 0.735\| 25.79 \| 0.778\| #### Table 2: Quantitative comparisons on RDN of scale 4. \| Model \| Bit \| Set5 PSNR \| SSIM \| Set14 PSNR \| SSIM \| B100 PSNR \| SSIM \| Urban100 PSNR \| SSIM \| \|----------------\|-----\|-----------\|------\|------------\|------\|-----------\|------\|--------------\|------\| \| RDN \| 32 \| 32.24 \| 0.896\| 28.67 \| 0.784\| 27.63 \| 0.738\| 26.29 \| 0.792\| \| RDN-PAMS \| 4 \| 30.44 \| 0.862\| 27.54 \| 0.753\| 26.87 \| 0.710\| 24.52 \| 0.726\| \| RDN-DAQ \| 4 \| 31.91 \| 0.889\| 28.38 \| 0.775\| 27.38 \| 0.733\| 25.81 \| 0.779\| \| RDN-DDTB \| 4 \| 31.97 \| 0.891\| 28.49 \| 0.780\| 27.49 \| 0.735\| 25.90 \| 0.783\| \| RDN-ODM (Ours) \| 4 \| 32.03 \| 0.892\| 28.51 \| 0.780\| 27.54 \| 0.736\| 25.92 \| 0.784\| The results show that ODM achieves consistently superior performance on 4, 3, and 2-bit quantization. \| Model \| Bit \| Set5 PSNR \| SSIM \| Set14 PSNR \| SSIM \| B100 PSNR \| SSIM \| Urban100 PSNR \| SSIM \| \|------------------------\|-----\|-----------\|------\|------------\|------\|-----------\|------\|--------------\|------\| \| SRResNet \| 32 \| 32.07 \| 0.893\| 28.50 \| 0.780\| 27.52 \| 0.735\| 25.86 \| 0.779\| \| SRResNet-PAMS \| 4 \| 31.88 \| 0.891\| 28.41 \| 0.777\| 27.45 \| 0.732\| 25.68 \| 0.773\| \| SRResNet-DAQ \| 4 \| 31.85 \| 0.889\| 28.41 \| 0.777\| 27.45 \| 0.732\| 25.70 \| 0.772\| \| SRResNet-DDTB \| 4 \| 31.97 \| 0.892\| 28.46 \| 0.778\| 27.48 \| 0.733\| 25.77 \| 0.776\| \| SRResNet-ODM (Ours) \| 4 \| 32.00 \| 0.892\| 28.46 \| 0.778\| 27.48 \| 0.734\| 25.77 \| 0.776\| \| SRResNet-PAMS \| 3 \| 31.68 \| 0.888\| 28.27 \| 0.774\| 26.79 \| 0.709\| 25.46 \| 0.765\| \| SRResNet-DAQ \| 3 \| 31.81 \| 0.889\| 28.35 \| 0.776\| 27.40 \| 0.733\| 25.63 \| 0.772\| \| SRResNet-DDTB \| 3 \| 31.85 \| 0.890\| 28.39 \| 0.776\| 27.44 \| 0.731\| 25.64 \| 0.770\| \| SRResNet-ODM (Ours) \| 3 \| 31.86 \| 0.890\| 28.39 \| 0.776\| 27.44 \| 0.732\| 25.65 \| 0.771\| \| SRResNet-PAMS \| 2 \| 30.25 \| 0.861\| 27.36 \| 0.750\| 26.79 \| 0.709\| 24.19 \| 0.713\| \| SRResNet-DAQ \| 2 \| 31.57 \| 0.886\| 28.19 \| 0.773\| 27.30 \| 0.729\| 25.39 \| 0.765\| \| SRResNet-DDTB \| 2 \| 31.51 \| 0.887\| 28.23 \| 0.773\| 27.33 \| 0.728\| 25.37 \| 0.762\| \| SRResNet-ODM (Ours) \| 2 \| 31.59 \| 0.887\| 28.27 \| 0.773\| 27.36 \| 0.729\| 25.44 \| 0.765\| Table 3: Quantitative comparisons on SRResNet of scale 4. \| Model \| Bit \| Storage size \| BitOPS \| PSNR \| SSIM \| \|------------------------\|-----\|--------------\|--------\|------\|------\| \| EDSR \| 32 \| 1517.6K \| 527.1T \| 32.10\| 0.894\| \| EDSR-PAMS \| 2 \| 411.7K \| 215.1T \| 29.51\| 0.835\| \| EDSR-DAQ \| 2 \| 411.7K \| 215.6T \| 31.01\| 0.871\| \| EDSR-DDTB \| 2 \| 413.4K \| 215.1T \| 30.97\| 0.876\| \| EDSR-ODM (Ours) \| 2 \| 411.7K \| 215.1T \| 31.49\| 0.883\| \| RDN \| 32 \| 22271.1K \| 6032.9T\| 32.24\| 0.896\| \| RDN-PAMS \| 2 \| 1715.9K \| 236.6T \| 29.54\| 0.838\| \| RDN-DAQ \| 2 \| 1715.9K \| 287.7T \| 30.33\| 0.858\| \| RDN-DDTB \| 2 \| 1769.7K \| 236.6T \| 30.57\| 0.867\| \| RDN-ODM (Ours) \| 2 \| 1727.9K \| 236.6T \| 30.98\| 0.871\| \| SRResNet \| 32 \| 1546.8K \| 588.8T \| 32.10\| 0.894\| \| SRResNet-PAMS \| 2 \| 440.9K \| 276.9T \| 30.25\| 0.861\| \| SRResNet-DAQ \| 2 \| 440.9K \| 279.0T \| 31.57\| 0.886\| \| SRResNet-DDTB \| 2 \| 442.3K \| 276.9T \| 31.51\| 0.887\| \| SRResNet-ODM (Ours) \| 2 \| 441.4K \| 276.9T \| 31.59\| 0.887\| Table 4: Computational complexity comparison with SR quantization methods on EDSR (×4). or RDN. Nevertheless, ODM is also proven effective for quantizing SRResNet on all bit settings, showing slightly better performance than the existing quantization methods. Additional experiments that further demonstrate the applicability of ODM are provided in the supplementary materials. Along with the SR accuracy, we also compare the computational complexity of our framework in Table 4. We calculate the BitOPS for generating a 1920×1080 image. Overall, our framework ODM achieves higher accuracy (PSNR/SSSIM) with similar or less computational resources. As reported in Table 5, the distribution offsets incur minimal computational overhead. On EDSR, distribution offsets involve an additional 0.02% storage size and the offset scaling and shifting involves 0.005% additional BitOPS. In particular, RDN-ODM involves ×4 smaller storage size overhead than RDN-DDTB, and ×800 fewer BitOPS overhead than RDN-DAQ, while the PSNR gap is 0.41dB or higher. Although the PSNR gap is smaller on SRResNet, ODM still achieves higher PSNR with fewer computations than DAQ and DDTB. Compared to existing works that utilize dynamic adaptation, the computational overhead is ×30 smaller in storage size than DDTB and ×100 smaller in BitOPS than DAQ. Compared to PAMS, our framework incurs additional storage size on RDN (0.7%) and SRResNet (0.1%), but the accuracy gap with PAMS is significant (~1.4 dB). ### 4.3 Qualitative Results Figure 4 provides qualitative results and comparisons with the output images from quantized EDSR and RDN. Our method, ODM, produces a further visually clean output image compared to existing quantization methods. In contrast, existing methods, especially PAMS, suffer from blurred lines Figure 4: Qualitative results on Urban100 with EDSR and RDN-based models. \| Model \| Coop. \| Var. Reg. \| Sel. Off. \| Storage size \| BitOPs \| PSNR \| SSIM \| \|-------------\|-------\|-----------\|-----------\|--------------\|--------\|------\|------\| \| EDSR-PAMS \| - \| - \| - \| 411.7K \| 215.0T \| 29.51\| 0.835\| \| (a) \| - \| ✓ \| - \| - \| - \| 31.08\| 0.872\| \| (b) \| ✓ \| ✓ \| - \| - \| - \| 31.31\| 0.879\| \| (c) \| - \| - \| ✓ \| +0.08K (+0.02%) \| +0.01T (+0.005%) \| 31.40\| 0.880\| \| EDSR-ODM \| ✓ \| ✓ \| ✓ \| +0.08K (+0.02%) \| +0.01T (+0.005%) \| 31.49\| 0.883\| Table 5: Ablation study on each attribute of our framework. Var. Reg. refers to the variance regularization loss and Coop. denotes whether the cooperative variance regularization is utilized or not, and Sel. Off. refers to the selective distribution offsets. Percentage in brackets denotes the additional computation compared to the baseline. or artifacts. The qualitative results stress the importance of alleviating the distribution mismatch problem in SR networks. More results are provided in the supplementary materials. 4.4 ABLATION STUDY In Table 5, we verify the importance of each attribute of our framework: cooperative variance regularization and distribution offsets. According to the results, cooperative variance regularization and distribution offsets respectively improve the baseline accuracy. Compared to using variance regularization directly (a), our cooperative scheme (b) improves the SR accuracy (+0.23dB). Although leveraging distribution offsets (c) incurs additional computations, it largely increases the accuracy while the computational overhead is minimal. Notably, when two components are jointly used, the accuracy increases compared to using each component separately, although the increase is relatively minor. This indicates that there is an overlapping effect of selective offsets and variance regularization on reducing the mismatch. Still, both components contribute to reducing the mismatch, resulting in a further accurate quantized SR network. 5 CONCLUSION SR networks suffer accuracy loss from quantization due to the inherent distribution mismatch of the features. Instead of adapting resource-demanding dynamic modules to handle distinct distributions during test time, we introduce a new quantization-aware training technique that relieves the mismatch problem via distribution optimization. We leverage variance regularization loss that updates the SR network towards being quantization-friendly and also accurately super-resolving images. Also, through analysis of the distribution mismatch of different layers, we find that applying additional shifting offsets to layers with a large mismatch in terms of shift and scaling offsets to the layers with a large scaling mismatch can further reduce the distribution mismatch issue with minimal computational overhead. 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LWDRiFzbHQ	Can you combine / learn different similarity functions and transformations? If the impact of the different image transformations is limited (as we observe in Table 2), can we (and should we) select them arbitrarily? Is there a reason some transformations might be more beneficial than others and in what settings would this be the case? How should these design decisions be made in a general setting?	VICINAL ASSESSMENT OF MODEL GENERALIZATION Anonymous authors Paper under double-blind review ABSTRACT This paper studies how to assess the generalization ability of classification models on out-of-distribution test sets without relying on test ground truths. Existing works usually compute an unsupervised indicator of a certain model property, such as confidence and invariance, which is correlated with out-of-distribution accuracy. However, these indicators are generally computed based on a single test sample in isolation (and subsequently averaged over the test set), and thus are subject to spurious model responses, such as excessively high or low confidence. To address this issue, we propose to integrate model responses of neighboring test samples into the correctness indicator of the every test sample. Intuitively, if a model consistently demonstrates high correctness scores for nearby samples, it becomes more likely that the target sample will also be correctly predicted, and vice versa. This score is finally averaged across all test samples to indicate model accuracy holistically. This strategy is developed under the vicinal risk formulation, and, since its computation does not rely on labels, is called vicinal risk proxy (VRP). We show that VRP methodologically can be applied to existing generalization indicators such as average confidence and effective invariance and experimentally brings consistent improvements to these baselines. That is, stronger correlation with model accuracy is observed, especially on severe out-of-distribution test sets. 1 INTRODUCTION Because of the ubiquitous existence of distributional shifts in real-world systems, it is important to evaluate the generalization capacity of trained models on out-of-distribution (OOD) test data. In practical OOD scenarios, because obtaining test ground truths is expensive, model evaluation techniques that do not rely on test labeled are attracting increasing attention. For this problem, unsupervised risk measurements are introduced that capture useful model properties, such as confidence and invariance. These measurements serve as indicators of a model’s generalization ability. Importantly, for a sample of interest without ground-truth, these methods compute a risk proxy merely using this sample alone. For example, Hendrycks and Gimpel (2017) use the maximum Softmax value of a test sample itself as its confidence score, and show that once averaged over the OOD test set, it serves as a reliable indicator. The Effective Invariance score (Deng et al., 2022) is computed as the prediction consistency between this sample and its transformed version (e.g., rotation and grey-scale). Because these indicators measure model properties that underpin its generalization ability, they generally exhibit fair correlation with the model’s out-of-distribution accuracy. However, we find this isolated way of measuring model effectiveness for a sample of interest suffers from spurious model responses. In Fig. 1, we show two examples where an incorrect prediction on a test sample may have a much higher confidence score than a correct prediction. In other words, a high (resp. low) confidence or invariance score sometimes does not mean a correct (resp. incorrect) prediction. These erroneous scores, once accumulated in the test set, would compromise the effectiveness of risk measurements. To address this problem, when computing the risk proxy for a test sample, we propose to integrate into its risk proxy the model behaviour on its adjacent samples, where such integration is performed using the vicinal distribution for the test sample (Fig. 1). Intuitively, if neighboring samples generally exhibit high risks (e.g., low confidence), the center sample with excessively low risk will be assigned an increased risk score, and vice versa. Here, the contribution of each neighboring sample to the center sample is proportional to their similarity. This strategy allows model responses (risk proxy Figure 1: Examples of spurious model responses and an intuitive illustration how our method corrects them. Here we use confidence as model generalization indicator and aim to improve it. $x_1$ and $x_2$, from ImageNet-R (Hendrycks et al., 2021a), are classified correctly and incorrectly, respectively. However, their confidence scores are excessively high (0.991) and excessively low (0.139), respectively, indicating spuriousness. More reasonable scores (0.431 and 0.816) are given by the proposed vicinal method, which is a similarity weighted sum of confidence. For example, the vicinal score of $x_1$ is computed as: \[ 0.431 = \frac{0.653 \times 0.207 + 0.766 \times 0.217 + 0.815 \times 0.721 + 0.785 \times 0.525}{0.653 + 0.766 + 0.815 + 0.785} \] score) to better indicate prediction correctness for the test sample, as shown in Fig. 2(b) and (d). To indicate the overall generalization ability of models, we further compute the vicinal risk proxy (VRP) as the average individual vicinal score over the entire test set. Another advantage of this vicinal assessment scheme is that it can be applied on top of various risk proxies based on individual test samples, including confidence, invariance and their variants. Our experiments show that VRP brings consistent improvements to them: stronger correlation between risk proxies rectified by VRP and model OOD accuracy over 200 different classifiers is generally observed on 9 benchmarks. In summary, this paper has the following main points. - We examine existing methods in OOD generalization assessment in the lens of risk estimation. - We propose to integrate vicinal distribution of a sample into its risk estimate, to inhibit spurious model responses. - The proposed vicinal risk proxy (VPR), when applied to existing risk proxies, brings consistent improvement: stronger correlation is observed between vicinal proxies and model OOD accuracy. 2 RELATED WORK Data-centric model generalization assessment aims to predict the accuracy of a given model on various unlabeled test sets. Average model confidence (Hendrycks and Gimpel, 2017; Tu et al., 2023) on the testing samples is a simple and useful indicator of model accuracy. Guillory et al. (2021) propose using the confidence discrepancy between the validation and test sets to correct the confidence-based accuracy. Deng et al. (2021) tackle this challenge by comparing models based on their accuracy in self-supervised tasks. Garg et al. (2022) predict accuracy by using the percentage of testing samples exceeding a threshold learned from a validation set in the source domain. In addition to confidence, domain shift can also be used as a cue to predict model accuracy on the target set (Guillory et al., 2021; Deng and Zheng, 2021). This paper does not focus on this setup and only provides some results in the supplementary materials. Model-centric generalization assessment. Some works focus on in-distribution generalization (Garg et al., 2021; Jiang et al., 2021; Negrea et al., 2020; Zhou et al., 2020). This paper instead studies OOD generalization. In this problem, we train a variety of models on a training set and predict and compare their performance on an unlabeled out-of-distribution test set. Deng et al. (2022) propose effective invariance (EI) to measure the consistency between model predictions on the original image and its transformed versions. It is also feasible to use data-centric indicators such as average confidence (Hendrycks and Gimpel, 2017; Tu et al., 2023). However, we find these methods Figure 2: Examples how vicinal risks of individual samples better separate models making a correct/incorrect prediction. We use a single test sample from ImageNet-R and 140 models trained on the ImageNet training set. From (a) to (d), we use confidence (Tu et al., 2023) [Hendrycks and Gimpel, 2017], confidence + our method, EI (Deng et al., 2022), and EI+our method, as risk estimate, respectively, and draw its distribution across the 140 models. Because confidence and EI leverage the sample in isolation, we observe spurious model responses: in (a) and (c) many models making incorrect predictions may give excessively high confidence/EI, while many making correct predictions give unexpectedly low confidence/EI. In (b) and (d), our method effectively rectifies the erroneous risk estimates, so that risk estimates of the individual sample better separate good models from poor ones. As such, the vicinal risk proxy averaged over the entire OOD test set is more indicative of model accuracy. More examples are shown in the supplementary material. sometimes give excessively high (low) scores to incorrect (correct) samples, which compromise performance assessment; we show this problem can be alleviated by the proposed method. Vicinal risk was originally introduced in the vicinal risk minimization (VRM) principle (Chapelle et al., 2000). In VRM, each training sample defines a vicinal distribution, and accordingly, model risk is evaluated based on these distributions instead of individual training samples. VRM is widely reflected in data augmentation methods (Chapelle et al., 2000) [Cao and Rockett, 2015] [Ni and Rockett, 2015] [Zhang et al., 2017] [Yun et al., 2019] [Qin et al., 2020] [Kim et al., 2020]. For example, MixUp (Zhang et al., 2017) generates samples from vicinal distributions by mixing two images and their corresponding labels with random mixing weights. Other examples are CutMix (Yun et al., 2019), ResizeMix (Qin et al., 2020), and PuzzleMix (Kim et al., 2020). While these methods reflect vicinal risks on labeled training data, we apply this idea to unlabeled test data. Our strategy smooths out spurious proxy scores and allows for better approximation to model OOD accuracy. 3 PRELIMINARIES 3.1 RISK AND ACCURACY IN SUPERVISED EVALUATION We consider a model $f$ belonging to a class of models $\mathcal{F}$ and a target distribution $P(x, y)$. Model risk can be formulated as the expectation of a given loss function $\ell(f(x), y)$ on the distribution $P$: $$R(f) = \int \ell(f(x), y) dP(x, y). \quad (1)$$ In practice, since distribution $P(x, y)$ is unknown, Eq. 1 cannot be directly computed. Standard practice thus approximates the test risk by replacing $P(x, y)$ with an empirical distribution $P_{emp}(x, y)$, formed by assembling Dirac delta functions (Dirac, 1981) centered at each sample in a given test set $\mathcal{D} := \{(x_i, y_i)\}_{i=1}^n$: $$dP_{emp}(x, y) = \frac{1}{n} \sum_{i=1}^{n} \delta_{x_i}(x) \cdot \delta_{y_i}(y). \quad (2)$$ Substituting Eq. 2 into Eq. 1, the empirical risk under empirical target distribution $P_{emp}$ becomes: $$R_{emp}(f) = \frac{1}{n} \sum_{i=1}^{n} \ell(f(x_i), y_i). \quad (3)$$ By convention, accuracy can be viewed as a type of empirical risk with the accuracy loss: $$\ell_{acc} = \begin{cases} 0, & \text{if } \hat{y} = y \\ 1, & \text{if } \hat{y} \neq y \end{cases},$$ where $ \hat{y} $ is the predicted class with maximum (Softmax) confidence in $ f(x) $. ### 3.2 Vicinal Risk Minimization In training, the risk (loss values) of individual training samples may not fairly reflect the true generalization ability of a model on this sample. That is, the model may trivially minimize $ R_{emp}(f) $ in Eq. 3 by mere memorization of the samples (Zhang et al., 2017, 2021), rather than learn effective patterns. To address this problem, vicinal risk minimization (Chapelle et al., 2000) suggests to replace the Dirac delta function $ \delta_{x_i}(x) $ and $ \delta_{y_i}(y) $ in Eq. 2 by some density estimates of the vicinity of point $(x_i, y_i)$: \[ dP_v(x, y) = \frac{1}{n} \sum_{i=1}^{n} dv(x, y \| x_i, y_i), \] where $ dv(x, y \| x_i, y_i) $ is the vicinal density function describing the probability of finding point $(x, y)$ in the vicinity of $(x_i, y_i)$, and $ P_v $ is a mixture distribution of $ n $ vicinal distributions $ v $. The expectation of the vicinal risk of model $ f $ is now the mean of risk expectation in each vicinal distribution $ v $: \[ R_v(f) = \int \ell(f(x), y) dP_v(x, y) = \frac{1}{n} \sum_{i=1}^{n} \int \ell(f(x), y_i) dv(x, y \| x_i, y_i). \] The uniform vicinal distribution (Chapelle et al., 2000), Gaussian vicinal distribution (Chapelle et al., 2000) and mixup vicinal distribution (Zhang et al., 2017) are well-known vicinities in the risk minimization task (Zhang et al., 2018). The success of vicinal risk minimization in model training (Chapelle et al., 2000; Cao and Rockett, 2015; Ni and Rockett, 2015; Hai-Yan and Hua, 2010; Dong et al., 2022; Zhang et al., 2017) demonstrates the effectiveness of this idea, which inspires us to enhance the risk estimation in the unsupervised evaluation problem. ### 4 Proposed Method #### 4.1 From Risk to Risk Proxy in Unsupervised Evaluation In unsupervised evaluation, it is infeasible to use $ \ell_{acc} $ defined in Eq. 3 because of the absence of ground-truths. To still be able to indicate model risk, existing methods typically design a proxy loss $ \hat{\ell} $ and compute its expectation on the unlabeled distribution $ P(x) $: \[ \hat{R}(f) = \int \hat{\ell}(f, x, \varphi) dP(x), \] where $ \hat{R}(f) $ is defined as risk proxy. $ \hat{\ell} $, usually reflecting crucial model properties (e.g., confidence and invariance), is computed based on the response of model $ f $ to input $ x $ and additional knowledge $ \varphi $ of the model. In average confidence (Tu et al., 2023) and EI (Deng et al., 2022), $ \hat{\ell} $ takes the confidence of $ x $ or its transformation, so $ \varphi = \emptyset $. In DoC (Guillory et al., 2021), $ \varphi $ means model accuracy and average confidence evaluated on a validation set. In ATC (Garg et al., 2022), $ \varphi $ is a model-specific confidence threshold learned from a validation set. In practice, on a test set with $ n $ test samples, $ \hat{R}(f) $ is approximated by the empirical risk proxy: \[ \hat{R}_{emp}(f) = \frac{1}{n} \sum_{i=1}^{n} \hat{\ell}(f, x_i, \varphi). \] #### 4.2 Proposed Vicinal Risk Proxy Issues of empirical risk proxies. Existing methods in unsupervised evaluation assume that the designed risk proxy $ \hat{R}(f) $ well correlates with model generalization on the target OOD distribution. However, this assumption can be tenuous when we zoom in individual samples. As depicted in Fig. 2 (a-b), many models correctly classifying a sample have unexpected low confidence/invariance scores, and many incorrectly model predictions have excessively high confidence/invariance scores. The presence of such spurious model responses in individual samples introduces noise to the empirical risk proxy defined in Eq. [7] rendering it less effective in assessing model generalization capabilities. Note that a dual problem that exists in training is described in Section 3.2. Solution. Given an unlabeled test set $ D := \{ (x_i) \}_{i=1}^n $, we propose to compute the vicinal risk proxy on $ D $ as an unsupervised indicator of the accuracy of model $ f $ on this test set. We first define vicinal distribution $ \mu $ for each test sample $ x_i $ below: \[ \mu(x, y \| f, x_i). \] The probability density function for $ \mu $ in this paper is defined as: \[ d\mu(x, y \| f, x_i) = \begin{cases} s(f(x'), f(x_i)), & \text{if } y = \hat{y}_i \\ 0, & \text{if } y \neq \hat{y}_i, \end{cases} \] where $ x' $ is a transformed view of $ x $, and $ \hat{y}_i $ is the predicted class of $ x_i $. There are different choices of image transformations in practice and we empirically choose rotation. $ s(\cdot, \cdot) $ computes the similarity between the outputs given by model $ f $, where we empirically compute the dot product between the Softmax vectors. Intuitively, $ \mu $ is the probability distribution of finding pair $(x, y)$ in the vicinity of $ x_i $, and $ d\mu $ is its probability density function. Integrating such vicinal assessment into the point-wise empirical risk proxy, Eq. [7] can be updated as the vicinal risk proxy: \[ \hat{R}_v(f) = \frac{1}{n} \sum_{i=1}^n \int \hat{\ell}(f, x, \varphi) d\mu(x, y \| f, x_i). \] Essentially, instead of merely using $ x_i $ itself for risk estimation, we also use its neighboring samples. A sample with higher similarity with $ x_i $ contributes more to the risk. We find that spurious model responses on $ x_i $ can be effectively inhibited by its vicinal risks. For example, in Fig. 2(c-d), for a test sample, models making correct and incorrect predictions are better separated. Quantitative analysis will be provided in Section 5. In practice, we approximate the expectation of $ \hat{\ell} $ within the $ i $-th distribution $ \mu $ as: \[ \hat{\ell}(f, x, \varphi) = \frac{\sum_{j=1}^m \hat{\ell}(f, x_j, \varphi) d\mu(x_j, \hat{y}_j \| f, x_i)}{\sum_{j=1}^m d\mu(x_j, \hat{y}_j \| f, x_i)}, \] where $ \hat{y}_j $ is the predicted class of $ x_j $ in $ D $, and $ m $ is the number of samples in a vicinal distribution. Intuitively, Eq. [11] gives an empirical estimation of the risk proxy considering the vicinal distribution of $ x_i $ and the probability density defined in Eq. [9] for each vicinal sample. For individual samples, the vicinal score allows correct and incorrect model predictions to be better separated (refer Fig. 2 for an example). Collectively on the test set, models making more correct predictions (higher accuracy) will receive higher vicinal scores than models make less correct prediction (lower accuracy). An illustrative derivation is presented in the supplementary material. ### 4.3 Apply Vicinal Risk Proxy to Existing Risk Proxies. In Eq. [6] and Eq. [7] of Section 4.1, we show that some existing approaches in unsupervised evaluation can be seen as unsupervised proxies for the empirical risk on the test set. Moreover, Eq. [10] means that the proposed vicinal risk proxy marries existing risk proxies with vicinal distribution. In other words, the idea of considering vicinal samples can be applied to various proxy loss functions $ \hat{\ell} $. For example, when $ \hat{\ell} $ is sample confidence, or equivalently, $ R_{emp}(f) $ is the test average confidence (Tu et al., 2023; Hendrycks and Gimpel, 2017) computed empirically, also called empirical risk proxy (ERP), $ \hat{R}_v(f) $ is the vicinal average confidence under our vicinal assessment, so is called vicinal risk proxy (VRP). By default, we search for neighboring samples for each vicinal distribution throughout the entire dataset. Samples with similarities greater than 0 are used to approximate the VRP score in Eq. [11] for this vicinity of interest. ### 4.4 Discussions Why are there spurious model responses? One possible reason for excessively high confidence for incorrect predictions is the over-confidence problem (Guo et al., 2017). This problem, when happening in in-distribution (IND) test sets, can be rectified by model calibration, but there still lack adaptive solutions to different OOD datasets. As for excessively low confidence for correct OOD prediction, we are yet to find a convincing solution, and it would be an interesting problem to study in the future. Effectiveness of vicinal assessment under in-distribution test sets. For IND data, models generally have good behaviours, so the effectiveness of vicinal assessment may be limited, but applying it does not compromise the system (partially demonstrated in Table I). On OOD data, on the other hand, vicinal assessment will be very useful as model responses are much less indicative of its correctness. Vicinal assessment for data-centric unsupervised evaluation. When we assume fixed classifier and training data and vary the test data (Deng et al., 2021; Garg et al., 2022), technically vicinal assessment can still be applied. However, because it is designed to differentiate models (see Fig. 2) rather than differentiate test sets, it does not give noticeable improvement under the data-centric setup, shown in our supplementary material. ## 5 EXPERIMENTS ### 5.1 DATASETS AND EVALUATION METRICS ImageNet-1k setup. 1. Model. We use 140 models that have been trained or fine-tuned using the ImageNet-1k (Deng et al., 2009) training set. We source these models from the models zoo Timm (Wightman, 2019). As suggested by Deng et al. (2022), these models exhibit a diverse range of architectures, training strategies, and pre-training settings. 2. Data. (1) ImageNet-A(dversarial) (Hendrycks et al., 2021b) comprises natural adversarial examples that are unmodified and occur in the real-world. (2) ImageNet-S(ketch) (Wang et al., 2019) contains images with a sketch-like style. (3) ImageNet-R(edition) (Hendrycks et al., 2021a) comprises of 30,000 images that exhibit diverse styles. (4) ImageNet-Blur (Hendrycks and Dietterich, 2019) was produced by applying a Gaussian function to blur the images from ImageNet-Val. We use blur with highest severity. (5) ObjectNet (Barbu et al., 2019) is a real-world set for object recognition with control where object backgrounds, rotations, and imaging viewpoints are random. (6) ImageNet-V2 (Recht et al., 2019) is a reproduced ImageNet dataset, whose distribution is similar to the ImageNet dataset. CIFAR10 setup. 1. Model. We use 101 models in this set sup. We follow the practice in Deng et al. (2022) to access model weights. 2. Data. (1) CINIC-10 (Darlow et al., 2018) is a fusion of CIFAR-10 and ImageNet-C (Hendrycks and Dietterich, 2019) image classification datasets. It contains the same 10 classes as CIFAR-10. (2) CIFAR-10.1 (Recht et al., 2018) is produced with almost the same distribution as CIFAR-10. iWildCam setup. 1. Model. We use 35 models trained on the iWildCam (Beery et al., 2020) training set. 2. Data. iWildCam-OOD test set contains animal pictures captured by camera traps in the wild. Evaluation metrics. We use the same evaluation metrics as (Deng et al., 2022), i.e., Pearson’s Correlation coefficient ($\gamma$) (Cohen et al., 2009) and Spearman’s Rank Correlation coefficient ($\rho$) (Kendall, 1948). They assess the degree of linearity and monotonicity between risk proxies and OOD accuracy, respectively. The values of both coefficients fall between -1 and 1. A coefficient being close to -1 or 1 indicates a robust negative or positive correlation. Conversely, a value of 0 denotes no correlation (Cohen et al., 2009). Following Deng et al. (2022), we use top-1 classification accuracy as a metric to measure model generalization. ### 5.2 EXISTING RISK PROXIES AS BASELINES We evaluate the effectiveness of vicinal assessment in enhancing the following risk proxies in unsupervised generalization prediction. 1) Average Confidence (AC) (Tu et al., 2023; Hendrycks and Gimpel, 2017). The mean of the softmax confidence for each samples on the test set. 2) Effective Insurance (EI) (Deng et al., 2022) is the multiplication between the confidence of the image and a different view of it (e.g., rotation) if the predicted class of them is the same. Otherwise, it is equal to zero for this sample. 3) Consistency Invariance (CI) (Aithal et al., 2021) is the predicted probability of the transformed view affecting the predicted class of the original image. 4) Difference of Confidence (DoC) (Guillory et al., 2021) is obtained by using the accuracy on the held-out validation Figure 3: Comparing existing risk estimates and their vicinal improvements. We first estimate the distributions of risk estimate scores for correct and incorrect model predictions (140 in total) for each test sample. Then, the overlap of the two distributions for each sample is computed and finally averaged over the entire test set. All models are trained on ImageNet. In each figure, we use four test sets, ImageNet-A (A), ImageNet-R (R), ImageNet-S (S), and ObjectNet (O). From (a) to (e), EI, AC, CI, DoC, and ATC are used as baselines, respectively. A smaller value indicates lower overlap or higher separability. We clearly observe that vicinal risk scores statistically better differentiate models making correct and incorrect predictions by better separating their scores. set to subtract the gap between the AC on the validation set and the AC on the test set. 5) Average Thresholded Confidence (ATC) (Garg et al., 2022) quantifies the proportion of samples that have a softmax confidence score exceeding the threshold learned from the validation set. 5.3 Main Observations Statistically, vicinal risk scores better differentiate correct and incorrect model predictions. Apart from the example shown in Fig. 2, we further provide statistical evidence of the working mechanism of our method. Specifically, we employ the Gaussian kernel density estimation (KDE) to estimate the distributions of proxy scores of models that make correct and incorrect predictions for each sample. Then, we use numerical integration to calculate the their overlap, known as the overlap coefficient. In Fig. 2, we present the average coefficient across each test set. Results demonstrate that vicinal assessment has lower overlap coefficient than the baseline proxies, making it easier to differentiate between correct and incorrect predictions at the individual sample level. Vicinal assessment consistently improves existing risk proxies on OOD test sets. In Table 1, we compare five existing risk proxies and their vicinal versions on OOD test sets. Each experiment is repeated three times to show statistical significance. We observe a consistent improvement in the strength of correlation between accuracy and risk proxy. For example, on ImageNet-A, vicinal assessment brings about 4.8%, 3.9%, 10.5%, 3.6%, and 4.5% improvement in the Spearman’s coefficient over EI, AC, CI, DoC, and ATC, respectively. These results indicate the effectiveness of the proposed method. The improvements can be illustrated by two examples in Fig. 4, where the ranks of the majority models have been adjusted to be closer to the actual rank. Vicinal assessment is neither beneficial nor detrimental on near-OOD test sets. When test data are near OOD or even IND, the use of vicinal assessment does not have noticeable performance improvement or compromise. For example, on ImageNet-V2, we observe slight improvement for EI, CI, DoC and ATC, and slight decrease for AC. On the CIFAR10.1 test set, similar observations are made. Further given its effectiveness in OOD scenarios, this allows for safe deployment of vicinal assessment in practice. 5.4 Further Analysis of Vicinal Risk Proxy Comparing different similarity measurements in Eq. 9. Dot product is used in Eq. 9 to compute similarity between a sample of interest and a sample in its vicinity. Here, we compare it with other options including random value (giving random similarity values), equal similarity (i.e., uniform vicinal distribution (Chapelle et al., 2000)), and the Gaussian kernel function (i.e., Gaussian vicinal distribution (Chapelle et al., 2000)). Results of ranking the 140 models on ImageNet-A are summarized in Table 2. We find that dot product generally has similar performance with Gaussian similarity, and both are much better than random similarity and equal similarity. It illustrates the benefit of letting closer sample to contribute more to the score of the sample of interest. Table 1: Comparing vicinal risk proxies (VRP) and empirical risk proxies (ERP) on various test sets. For each test set, results in the first row is from ERP and the second is VRP. $\gamma$ and $\rho$ represent the Pearson’s coefficient and the Spearman’s correlation coefficient, respectively. $$ means near IND test sets. The notations $\uparrow$ ($\downarrow$) on our score means the correlation coefficient of VRP is higher (lower) than that of ERP with statistical significance (p-value < 0.05) based on the two-sample t-test. Otherwise, their difference is not statistical significant. \| Train \| Test \| EI \| AC \| CI \| DoC \| ATC \| \|----------------\|--------------\|------\|------\|------\|------\|------\| \| \| $\gamma$ \| $\rho$ \| $\gamma$ \| $\rho$ \| $\gamma$ \| $\rho$ \| \| ImageNet-A \| \| 0.882 \| 0.645 \| 0.581 \| 0.464 \| 0.856 \| 0.617 \| 0.877 \| 0.761 \| 0.851 \| 0.436 \| \| \| \| 0.900 \| 0.692 \| 0.624 \| 0.503 \| 0.905 \| 0.722 \| 0.908 \| 0.797 \| 0.866 \| 0.481 \| \| ImageNet-R \| \| 0.914 \| 0.814 \| 0.736 \| 0.625 \| 0.873 \| 0.729 \| 0.898 \| 0.862 \| 0.937 \| 0.887 \| \| \| \| 0.956 \| 0.931 \| 0.818 \| 0.736 \| 0.931 \| 0.854 \| 0.905 \| 0.894 \| 0.967 \| 0.946 \| \| ImageNet-S \| \| 0.893 \| 0.853 \| 0.742 \| 0.711 \| 0.868 \| 0.820 \| 0.911 \| 0.919 \| 0.948 \| 0.915 \| \| \| \| 0.920 \| 0.871 \| 0.763 \| 0.728 \| 0.878 \| 0.840 \| 0.926 \| 0.931 \| 0.954 \| 0.953 \| \| ObjectNet \| \| 0.961 \| 0.949 \| 0.788 \| 0.777 \| 0.958 \| 0.946 \| 0.819 \| 0.834 \| 0.841 \| 0.860 \| \| \| \| 0.975 \| 0.972 \| 0.838 \| 0.814 \| 0.969 \| 0.962 \| 0.849 \| 0.868 \| 0.857 \| 0.876 \| \| ImageNet-Blur \| \| 0.870 \| 0.831 \| 0.711 \| 0.730 \| 0.824 \| 0.793 \| 0.781 \| 0.776 \| 0.882 \| 0.867 \| \| \| \| 0.907 \| 0.857 \| 0.737 \| 0.741 \| 0.829 \| 0.802 \| 0.821 \| 0.821 \| 0.912 \| 0.890 \| \| ImageNet-V2 \| \| 0.889 \| 0.884 \| 0.609 \| 0.501 \| 0.882 \| 0.870 \| 0.982 \| 0.979 \| 0.993 \| 0.990 \| \| \| \| 0.895 \| 0.881 \| 0.613 \| 0.513 \| 0.886 \| 0.887 \| 0.990 \| 0.984 \| 0.995 \| 0.993 \| \| CIFAR10 \| \| 0.913 \| 0.936 \| 0.978 \| 0.887 \| 0.834 \| 0.876 \| 0.985 \| 0.953 \| 0.983 \| 0.937 \| \| \| \| 0.954 \| 0.956 \| 0.979 \| 0.889 \| 0.875 \| 0.929 \| 0.985 \| 0.956 \| 0.982 \| 0.942 \| \| CIFAR10.1* \| \| 0.886 \| 0.905 \| 0.982 \| 0.972 \| 0.804 \| 0.811 \| 0.992 \| 0.985 \| 0.991 \| 0.982 \| \| \| \| 0.883 \| 0.886 \| 0.982 \| 0.972 \| 0.813 \| 0.855 \| 0.992 \| 0.985 \| 0.991 \| 0.982 \| \| iWildCam \| \| 0.337 \| 0.362 \| 0.635 \| 0.445 \| 0.268 \| 0.258 \| 0.547 \| 0.532 \| 0.509 \| 0.526 \| \| iWildCam-OOD \| \| 0.402 \| 0.393 \| 0.635 \| 0.495 \| 0.208 \| 0.180 \| 0.556 \| 0.592 \| 0.518 \| 0.595 \| Figure 4: Correlation between Effective Invariance (EI) and accuracy. In each figure, every dot represents a model, and straight lines are fitted using a robust linear fit. A blue dots represent the rectified score (VRP score) of this models brings its rank closer to the actual accuracy rank. On the other hand, the rank of red models deviates further from the real accuracy when using the VRP paradigm. The rank of black models remains unchanged. $\rho$ and $\gamma$ have the same meaning as those in Table 1. The shaded region in each figure represents a 95% confidence region for the linear fit, calculated from 1,000 bootstrap samples. We observe that the VRP paradigm can effectively rectify the proxy score for the majority of models in both the ImageNet-R and ObjectNet datasets. Comparing different image transformations. In Eq. 9, we define the probability density function for vicinal distribution using transformed images. Here we compare the effectiveness of different transformations, and results on ImageNet-A are presented in Table 2. We find that there is no significant performance difference between rotation, gray-scale transformation and color jitters. Impact of the number of neighbors. The number of neighbors $m$ is an important hyper-parameter used in Eq. 10. To evaluate system sensitivity to $m$, we experiment with ImageNet-R as OOD test set and five baseline risk proxies setting $m = 25, 50, 75, 100, 125, 150$. We repeated each experiment three times and reported the mean and standard deviation. From Fig. 5, we have two observations. 1Because of the limited size of ImageNet-R, 150 is the maximum value we can set $m$ to. Table 2: Comparison of variants of vicinal risk proxy. (Top): various similarity metrics that can be used in Eq. 9. (Bottom): various image transformations that can be used in Eq. 9. Bold numbers denote the best one across compared settings. \| Settings \| EI γ \| EI ρ \| AC γ \| AC ρ \| CI γ \| CI ρ \| DoC γ \| DoC ρ \| ATC γ \| ATC ρ \| \|-------------------\|------\|------\|------\|------\|------\|------\|-------\|-------\|-------\|-------\| \| Random \| 0.003\| 0.197\| 0.072\| 0.133\| 0.216\| 0.350\| 0.073 \| 0.160 \| 0.066 \| 0.055 \| \| Equal \| 0.883\| 0.659\| 0.554\| 0.446\| 0.857\| 0.624\| 0.864 \| 0.757 \| 0.837 \| 0.424 \| \| Gaussian kernel \| 0.876\| 0.675\| 0.611\| 0.498\| 0.887\| 0.706\| 0.887 \| 0.797 \| 0.881 \| 0.532 \| \| Dot product \| 0.903\| 0.713\| 0.605\| 0.489\| 0.903\| 0.731\| 0.901\| 0.801\| 0.867 \| 0.490 \| \| None \| 0.901\| 0.705\| 0.564\| 0.480\| 0.881\| 0.669\| 0.907 \| 0.806 \| 0.863 \| 0.487 \| \| Grey-scale \| 0.898\| 0.686\| 0.617\| 0.503\| 0.879\| 0.655\| 0.917 \| 0.812\| 0.889\| 0.514\| \| Color jitters \| 0.897\| 0.674\| 0.632\| 0.512\| 0.861\| 0.642\| 0.919\| 0.811 \| 0.874 \| 0.482 \| \| Rotation \| 0.903\| 0.713\| 0.605\| 0.489\| 0.903\| 0.731\| 0.901 \| 0.801 \| 0.867 \| 0.490 \| Figure 5: Impact of the number of neighbors $m$ on the correlation between proxy scores and accuracy. We use five existing proxies as baselines and report the mean and standard deviation for each data point. We observe that vicinal assessment is consistently beneficial under various $m$ values and yields stronger correlation with $m$ increases. First, using more neighboring samples is generally beneficial, evidenced by the increasing correlation strength. It is probably because a larger $m$ allows for better approximation of the true vicinal distribution. We set $m = 150$ by default. Second, when using fewer neighbors, e.g., $m = 25$, vicinal assessment is still beneficial, yielding stronger correlation than existing risk proxies. Impact of test set size. Table 3 presents our method under varying test set sizes using ImageNet-R as an OOD test set. We observe that the performance of all compared methods drops under smaller test sets. Nevertheless, the use of vicinal assessment consistently improves the correlation strength of the baselines under each test set size, demonstrating the effectiveness of our method. Table 3: Impact of the test set size. We evaluate our method on the ImageNet-R set with different number of test samples. \| # samples \| 3,000 \| 6,000 \| 12,000 \| 18,000 \| 24,000 \| \|-----------\|-------\|-------\|--------\|--------\|--------\| \| DoC \| 0.856 \| 0.848 \| 0.877 \| 0.874 \| 0.874 \| \| DoC + Ours\| 0.873\| 0.874\| 0.892\| 0.895\| 0.896\| 6 CONCLUSION In this paper, we propose the vicinal assessment strategy to improve existing risk proxies computed based on a single test sample. We demonstrate that existing point-wise methods are prone to erroneous model responses, a problem that can be alleviated by considering the responses of adjacent test samples. Inspired by the philosophy of vicinal risk minimization, we design a vicinal risk proxy. 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WbR415lO2L	Wordnet is used for token substitutions in this work. My understanding is that Wordnet could be quite noisy, containing words and tokens that are very infrequent in human-written text. As a result, it is surprising to see that Wordnet substitutions do not result in decreases in readability. I wonder if the authors might have conducted some filtering/selection of Wordnet to mitigate these issues.	Large Language Models can be Guided to Evade AI-Generated Text Detection Anonymous authors Paper under double-blind review Abstract Large language models (LLMs) have shown remarkable performance in various tasks and have been extensively utilized by the public. However, the increasing concerns regarding the misuse of LLMs, such as plagiarism and spamming, have led to the development of multiple detectors, including fine-tuned classifiers and statistical methods. In this study, we equip LLMs with prompts, rather than relying on an external paraphraser, to evaluate the vulnerability of these detectors. We propose a novel Substitution-based In-Context example Optimization method (SICO) to automatically construct prompts for evading the detectors. SICO is cost-efficient as it requires only 40 human-written examples and a limited number of LLM inferences to generate a prompt. Moreover, once a task-specific prompt has been constructed, it can be universally used against a wide range of detectors. Extensive experiments across three real-world tasks demonstrate that SICO significantly outperforms the paraphraser baselines and enables GPT-3.5 to successfully evade six detectors, decreasing their AUC by 0.5 on average. Furthermore, a comprehensive human evaluation as well as a validation experiment in the wild show that the SICO-generated text achieves human-level readability and task completion rates. Finally, the strong performance of SICO exhibits its potential as a reliable evaluation tool for future detectors. 1 Introduction The rapid advancement of large language models (LLMs), such as GPT (Brown et al., 2020), PaLM (Chowdhery et al., 2022), and LLaMa (Touvron et al., 2023), has led to a largely-increased capacity for generating high-quality human-like text. However, there are also growing concerns surrounding the misuse of these models, including generating fake product reviews (Adelani et al., 2020; Lin et al., 2022) and misinformation (Lin et al., 2022), enabling academic dishonesty (Stokel-Walker, 2022), and producing misleading answers on websites (StackOverflow, 2023). In response to these challenges, several methods for detecting AI-generated text have been proposed recently, ranging from fine-tuned classifiers (Guo et al., 2023; Solaiman et al., 2019), statistical methods (Mitchell et al., 2023), to watermarking (Kirchenbauer et al., 2023). There are also online detection services provided by companies such as GPTzero (Tian, 2023). However, the robustness of these detection methods has not been thoroughly evaluated. Recent studies (Krishna et al., 2023; Sadasivan et al., 2023) have shown the vulnerability of these detectors to the so-called paraphrase attacks, which adopt an external paraphraser to rewrite the text generated by LLMs to evade detectors. In this work, rather than relying on an external paraphraser, we explore equipping LLMs with carefully constructed prompts to evade detectors. The intuition is that, given the remarkable capabilities of LLMs, appropriate prompts can guide these models to potentially achieve and even exceed the evasion performance level of smaller external paraphrasers. We propose SICO, a Substitution-based In-Context example Optimization method, to automatically construct such prompts based on human-generated examples. Specifically, SICO iteratively substitutes words and sentences within the in-context examples to provide more representative demonstrations for LLMs to generate text that cannot be detected, where the substitution procedure is directed by a proxy detector (see Figure 1 for an overview of SICO). We assess the evasion performance of SICO across three real-world tasks that are susceptible to the misuse of LLMs, i.e., academic essay writing, open-ended question answering, and fake review... generation. The results demonstrate that SICO consistently outperforms the paraphraser baselines, leading to a decrease in AUC by approximately 0.5 on average for six existing detectors. Additionally, a comprehensive human evaluation involving 600 examples shows that the SICO-generated text is comparable to, and in some cases even better than, human-written text in terms of readability and task completion rates. To further evaluate the practical utility of SICO, we deploy it on Reddit, an online social platform, to generate responses for users’ questions. The high percentage of generated responses that are liked by Reddit users shows that SICO is capable of generating human-approved content while being barely identified as AI. In addition to its strong evasion performance, SICO is also cost-efficient and easy to use. Unlike paraphraser-based methods that often require extensive computational resources – as evidenced by the fine-tuning of a 13B model on a large dataset (Krishna et al., 2023) – SICO only requires 40 human-generated examples and a limited number of LLM inferences (e.g., costing approximately 1 USD using the GPT-3.5 API). Besides, once a task-specific prompt has been constructed by SICO, it can be universally used against a wide range of detectors. Considering the importance of detecting AI-generated text to avoid their misuse, the results presented in this work certainly reveal the vulnerability of the existing detectors. Besides, this work presents the first empirical evidence that LLMs can evade detectors through a prompt-guided approach. Finally, the strong evasion performance of SICO suggests that it can be used as a standard evaluation tool for any future AI-generated text detectors. We hope that these findings can better facilitate the research concerning the responsible use of LLMs. To summarize, our main contributions are: • We introduce SICO, a novel in-context example learning method, to automatically construct prompts that can guide LLMs to evade detectors. • With low cost, SICO achieves strong performance in evading six existing detectors across three tasks, significantly outperforming the paraphraser baselines. • A comprehensive human evaluation, as well as a validation experiment in the wild, verifies that the SICO-generated text achieves human-level readability and task completion rates. 2 RELATED WORKS 2.1 AI-GENERATED TEXT DETECTION In recent years, the research community has developed a wide range of detectors for AI-generated contents. In general, these detectors can be classified into three categories: training-based, statistical, and watermarking methods. Training-based methods treat the detection problem as a binary classification task, where neural networks are trained using AI-generated text and human-written text. Early studies utilized classifiers to identify fake reviews (Hovy, 2016) and fake news (Zellers et al., 2019). More recently, researchers have trained classifiers using text generated by LLMs, such as the GPT-3.5 detector (Guo et al., 2023) and GPT-2 detector (Solaiman et al., 2019). Statistical methods, on the other hand, focus on zero-shot detection without any additional training overhead. These methods seek to distinguish between human-written text and AI-generated text based on the statistical characteristics of text, such as the statistical irregularities in measures like entropy (Lavergne et al., 2008), perplexity (Beresneva, 2016) and token rank (Gehrmann et al., 2019). A recent method, DetectGPT (Mitchell et al., 2023), exploits the phenomenon that AI-generated text tends to lie in the negative curvature regions of log probability of text. The watermarking methods involve modifying the LLM’s text generation process to imprint specific patterns on the generated text, such that it can be detected (Abdelnabi & Fritz, 2021; Grinbaum & Adomaitis, 2022; Kirchenbauer et al., 2023). Although the proposed method SICO primarily focuses on the first two types of detection methods, it can also help evade watermarking when acted as an external paraphraser, as shown in Appendix G. 2.2 IN-CONTEXT LEARNING With the increasing scales of models and corpora (Devlin et al., 2019; Radford et al., 2019; Chowdhery et al., 2022; Gou et al., 2022), LLMs have demonstrated the in-context learning (ICL) ability, allowing them to perform tasks with only a few examples provided as demonstrations (Brown et al., 2020). Recent studies have focused on designing demonstrations during inference, which can be divided into Figure 1: Illustration of how SICO generates prompts for the question answering task. $ P_{AI} $ is the probability predicted by the proxy detector that the given text is AI-generated. Demonstration selection, ordering, and formatting (Dong et al., 2022). Specifically, demonstrations can be selected based on unsupervised metrics or supervised strategies (Kim et al., 2022; Gonen et al., 2022; Rubin et al., 2022). For ordering, Liu et al. (2021) sorted examples by their distances to the input. Regarding demonstration formatting, Wei et al. (2022) proposed the so-called chain-of-thoughts (COT) format, and subsequent works have developed automatic COT (Zhang et al., 2022). In contrast to these works, we focus on iteratively optimizing demonstrations through substitutions. In principle, the proposed method SICO can be used in combination with the above-mentioned methods, potentially leading to improved performance. 3 Substitution-based In-context Example Optimization (SICO) The illustration of SICO is presented in Figure 1. First, LLM is asked to extract language features of human-written text. Then, the in-context examples are initialized and optimized. The final prompt is composed of the feature, task instruction, and optimized in-context examples. Below, we first describe how to evaluate a prompt during its optimization and then elaborate all the steps of SICO. 3.1 Prompt Evaluation Given a natural language processing task, denote the task input as $ x $. To assess the utility of a prompt $ p $, we first collect a set of task inputs, $ X_{eval} $. For each input $ x \in X_{eval} $, $ p $ and $ x $ are first concatenated (denoted by $ p \oplus x $) and fed into the LLM, whose output text (denoted by $ \text{LLM}(p \oplus x) $) is then classified by a proxy detector. Let $ P_{AI} $ be the predicted probability of $ \text{LLM}(p \oplus x) $ to be AI-generated, then the utility score of prompt $ p $, denoted by $ U(p) $, is defined as one minus the averaged predicted probability across $ X_{eval} $ (the higher $ U $, the better): \[ U(p) = 1 - \frac{1}{\|X_{eval}\|} \sum_{x \in X_{eval}} P_{AI}(\text{LLM}(p \oplus x)). \] (1) 3.2 Prompt Construction Data collection We first collect a set of $ K $ triplets, i.e., $ D = \{(x^k_{ic}, y^k_{AI}, y^k_{human})\}_{k=1}^{K} $, where $ x^k_{ic} $ is a task input and $ y^k_{AI}, y^k_{human} $ are the corresponding outputs generated by the LLM and humans, respectively. Note $ D $ is used for prompt construction and it is independent of $ X_{eval} $ which is used for prompt evaluation. Algorithm 2 Substitution-based in-context example optimization (SICO) Require: large language model LLM, prompt utility function $ U(\cdot) $, $ D = \{(x_{ic}^k, y_{AI}^k, y_{human}^k)\}_{k=1}^K $, $ X_{eval} $, total iteration number $ N $ 1: Extract language feature $ t_{feature} $ using $ \{(y_{AI}^k, y_{human}^k)\}_{k=1}^K $ and LLM 2: Construct in-context outputs $ \hat{y}_{ic}^k = \text{LLM}(t_{feature} \oplus p_{para} \oplus y_{AI}^k), \forall k \in \{1, ..., K\} $ 3: Initialize $ p^* \leftarrow t_{feature} \oplus p_{task} \oplus \{(x_{ic}^k, \hat{y}_{ic}^k)\}_{k=1}^K $ 4: for $ n = 1 $ to $ N $ do 5: for $ k = 1 $ to $ K $ do 6: Generate sentence-level / word-level substitutions $ C^k $ of $ y_{ic}^k $, switching based on $ n $ 7: Optimize $ \hat{y}_{ic}^k $ using Algorithm 1 $ \hat{y}_{ic}^k \leftarrow \text{GreedyOPT}(y_{ic}^k, C^k) $ end for 9: Construct new prompt $ \hat{p} : \hat{p} \leftarrow t_{feature} \oplus p_{task} \oplus \{(x_{ic}^k, \hat{y}_{ic}^k)\}_{k=1}^K $ 10: if $ U(\hat{p}) > U(p^) $ then 11: Update in-context examples $ y_{ic}^k \leftarrow \hat{y}_{ic}^k $ and update the best prompt $ p^ \leftarrow \hat{p} $ 12: end if end for 13: return $ p^* $ Feature extraction This step involves the $ K $ pairs of AI-generated and human-written outputs from $ D $, denoted by $ \{(y_{AI}^k, y_{human}^k)\}_{k=1}^K $. We provide LLM with these pairs and ask LLM to extract the distinct linguistic features of human-written text, denoted as $ t_{feature} $. In-context example optimization We initialize the in-context examples as $ (x_{ic}^k, y_{ic}^k) $, where $ y_{ic}^k $ is generated by paraphrasing $ y_{AI}^k $. More specifically, the text feature $ t_{feature} $ is concatenated with a paraphrasing instruction to instruct LLM to paraphrase $ y_{AI}^k $ to generate $ y_{ic}^k $. Then the in-context output $ y_{ic}^k $ is iteratively optimized to be less AI-like, which is directed by the proxy detector. By presenting more and more representative in-context demonstrations to LLM, it is expected to understand how to generate human-like outputs. Formally, the optimization goal can be expressed as: \[ \hat{y}_{ic}^k = \arg\min_{y_{ic}^k \in \text{SIM}(y_{ic})} P_{AI}(y_{ic}^k), \] where $ \text{SIM}(y_{ic}) $ denotes the set of text that is semantically similar to $ y_{ic} $. The goal of setting such semantic restriction is to maintain the usability of the text during optimization. In SICO, we generate semantically similar text by replacing words with their synonyms and rephrasing sentences. This is explained in detail below. Substitution type To generate $ y_{ic}^k $ that is semantically similar to $ y_{ic} $, we employ substitution at word level and sentence level in turn. For word-level substitution, we use WordNet [Miller 1998], a lexical database of English words, to construct a synonym substitution set. We restrict substitutions to content words that carry meanings and ensure that the substitution would not change the part-of-speech tags. For sentence-level substitution, we utilize the LLM and a paraphrasing instruction $ p_{para} $ to generate paraphrases of each sentence in $ y_{ic} $. Algorithm 1 Greedy text optimization (GreedyOPT) Require: Text $ y $, substitutions $ C $ of $ y $, proxy detector $ P_{AI} $ 1: $ C_{i,} = \arg\min_{C_{i,j}} P_{AI}(y_{(i,j)}), \forall y_i \in y $ where $ y_{(i,j)} = \text{SUB}(y_i, C_{i,j}) $ 2: for each $ y_i $ in $ y $ do 3: $ y \leftarrow \text{SUB}(y_i, C_{i,}) $ end for 5: return $ y $ Algorithm As illustrated in Algorithm 2, SICO would optimize $ \{y_{ic}^k\}_{k=1}^K $ for $ N $ iterations (lines 4-17). At each iteration, each $ y_{ic}^k $ would be optimized by greedy substitution (line 11), as presented in Algorithm 1. Specifically, for the $ i $-th original word/sentence $ y_i $ in the text $ y $, let $ C_{i,j} $ denote its $ j $-th synonym/paraphrase, and let $ \text{SUB}(y_i, C_{i,j}) $ denote the new text resulting from substituting $ y_i $ with $ C_{i,j} $. For each $ y_i $, SICO finds the best synonym/paraphrase $ C_{i,} $ by checking which $ C_{i,j} $ gives the lowest AI-probability when substituting $ y_i $ (Line 1 in Algorithm 1). After obtaining the optimized in-context output $ \hat{y}_{ic} $, the new prompt is constructed as $ \hat{p} = t_{feature} \oplus p_{task} \oplus \{(x_{ic}^k, \hat{y}_{ic}^k)\}_{k=1}^K $, where $ p_{task} $ is the task instruction, as illustrated in Figure 1. Then $ \hat{p} $ would be compared with the current best prompt $p^$ based on their utility scores as defined in Eq. (1). If $\hat{p}$ scores higher, SICO replaces $p^$ with it. After $N$ iterations, $p^$ is returned as the final prompt. More implementation details of SICO are shown in Appendix A. 3.3 SICO FOR PARAPHRASING The approach described above directly generates the task output to evade detectors. We refer to this direct approach as SICO-Gen. Alternatively, SICO can be easily adapted for paraphrasing, which we term as SICO-Para. Instead of direct generation, SICO-Para evades detectors in two steps. Initially, LLM produces an intermediate task output, typically incapable of evading detectors. Then, this output is paraphrased using SICO-Para to successfully evade detectors. Switching from SICO-Gen to SICO-Para requires only two adjustments: (1) the task input $x$ is set to the AI-generated output text in $D$ and $X_{eval}$; (2) task instruction $p_{task}$ is modified to paraphrasing instruction. 4 EXPERIMENTS 4.1 EXPERIMENTAL SETUP Tasks & datasets We consider three real-world tasks that are susceptible to the misuse of LLMs, i.e., academic essay writing (Writing), open-ended question answering (QA), and fake review generation (Review). We use GPT-3.5, one of the most powerful LLMs, to complete the tasks and generate text in our experiments. For academic writing, we employ Wikipedia paragraphs from SQuAD dataset [Rajpurkar et al., 2016] as human-written text. Following the approach in [Mitchell et al., 2023], we use the first 30 words of these paragraphs as task inputs and ask GPT-3.5 to complete the rest. For open-ended question answering, we sample questions from Eli5 [Fan et al., 2019] dataset and ask GPT-3.5 to generate answers, following [Krishna et al., 2023]. For fake review generation, we first instruct GPT-3.5 to extract the business name and five keywords from human-written reviews from Yelp dataset [Zhang et al., 2015], and then generate fake reviews based on the extracted information with specified sentiment. For each task, we collect 200 examples from GPT-3.5 (called original AI-generated text) and 200 human-written examples. More details about dataset can be found in Appendix E. Detectors Six representative detectors belonging to three different types are considered. Details of these detectors can be found in Appendix B. Training-based methods. (i) GPT-3.5 Detector (GPT3-D) [Guo et al., 2023]: a RoBERTa model [Liu et al., 2019] fine-tuned on examples generated by GPT-3.5. (ii) GPT2 Detector (GPT2-D) [Solaiman et al., 2019]: a RoBERTa detector officially released by OpenAI, fine-tuned on GPT2-generated text. Statistical methods. (i) DetectGPT [Mitchell et al., 2023] evaluates the variation in a language model’s log probability by introducing minor perturbations to the detected text. (ii) Log-Rank [Mitchell et al., 2023] is a statistical method that employs a language model to compute the mean prediction rank of each token in a text, given its preceding context. We utilize a relatively smaller language model, GPT2-medium [Radford et al., 2019], for both methods. Because [Mireshghallah et al., 2023] find that smaller language models have better detection performance than larger ones. APIs (i) GPTzero [Tian, 2023] is a widely-used commercial detector, cooperated with many academic organizations; (ii) OpenAI Detector (OpenAI-D) [OpenAI, 2023] is officially offered by OpenAI, fine-tuned from a language model. Baselines We consider four paraphrasing baselines that evade detectors by paraphrasing the original AI-generated text. Specifically, two recently proposed methods are considered: (1) Parrot [Sadasivan et al., 2023] and (2) DIPPER [Krishna et al., 2023]. Both methods employ an external neural network specifically trained for paraphrasing. In addition, we include two prompting baselines to instruct GPT-3.5 to paraphrase the original AI-generated text: (3) GPT-Para that uses the straightforward instruction “Paraphrase this” to assess the capabilities of GPT-3.5 without intricate prompt engineering, and (4) Human Prompt that utilizes a human-designed prompt. More details can be found in Appendix A.2. 1We consider the API versions of May 15, 2023. For OpenAI-D, we follow the implementation of [Krishna et al., 2023]. Table 1: AUC scores of detectors on text generated by different methods. “–” refers to the detector’s AUC score on the original AI-generated text, without applying any evasion methods. Symbol ‘’ represents that SICO uses GPT3-D as the proxy detector for prompt construction. For each detector, the lowest AUC score is indicated in bold, and the second-lowest is underlined. \| Dataset \| Method \| GPT3-D \| GPT2-D \| GPTzero \| OpenAI-D \| DetectGPT \| Log-Rank \| \|---------\|------------\|---------\|--------\|---------\|----------\|-----------\|----------\| \| Writing \| – \| 0.908 \| 0.848 \| 0.779 \| 0.789 \| 0.834 \| 0.914 \| \| \| Parrot \| 0.666 \| 0.645 \| 0.632 \| 0.744 \| 0.502 \| 0.577 \| \| \| DIPPER \| 0.736 \| 0.907 \| 0.689 \| 0.750 \| 0.550 \| 0.684 \| \| \| GPT-Para \| 0.879 \| 0.623 \| 0.631 \| 0.690 \| 0.569 \| 0.713 \| \| \| Human Prompt \| 0.852 \| 0.560 \| 0.491 \| 0.655 \| 0.676 \| 0.759 \| \| \| SICO-Para \| 0.239 \| 0.332 \| 0.290 \| 0.488 \| 0.149 \| 0.147 \| \| \| SICO-Gen \| 0.242 \| 0.099 \| 0.184 \| 0.311 \| 0.441 \| 0.318 \| \| QA \| – \| 0.981 \| 0.906 \| 0.923 \| 0.781 \| 0.876 \| 0.956 \| \| \| Parrot \| 0.922 \| 0.837 \| 0.849 \| 0.698 \| 0.689 \| 0.806 \| \| \| DIPPER \| 0.888 \| 0.962 \| 0.869 \| 0.722 \| 0.604 \| 0.782 \| \| \| GPT-Para \| 0.956 \| 0.797 \| 0.811 \| 0.699 \| 0.640 \| 0.782 \| \| \| Human Prompt \| 0.912 \| 0.625 \| 0.791 \| 0.656 \| 0.662 \| 0.757 \| \| \| SICO-Para \| 0.407 \| 0.576 \| 0.572 \| 0.541 \| 0.178 \| 0.183 \| \| \| SICO-Gen \| 0.668 \| 0.489 \| 0.494 \| 0.524 \| 0.497 \| 0.535 \| \| Review \| – \| 0.925 \| 0.952 \| 0.939 \| 0.960 \| 0.808 \| 0.982 \| \| \| Parrot \| 0.871 \| 0.934 \| 0.913 \| 0.882 \| 0.654 \| 0.893 \| \| \| DIPPER \| 0.875 \| 0.984 \| 0.888 \| 0.824 \| 0.515 \| 0.814 \| \| \| GPT-Para \| 0.899 \| 0.851 \| 0.833 \| 0.925 \| 0.542 \| 0.864 \| \| \| Human Prompt \| 0.839 \| 0.610 \| 0.856 \| 0.858 \| 0.619 \| 0.851 \| \| \| SICO-Para \| 0.465 \| 0.264 \| 0.599 \| 0.540 \| 0.270 \| 0.300 \| \| \| SICO-Gen \| 0.455 \| 0.619 \| 0.399 \| 0.607 \| 0.485 \| 0.583 \| Evaluation metrics We use the area under the ROC curve (AUC) to measure the performance of detectors. The ROC curves are also illustrated to show the detection performance under different classification thresholds. For each task, we evaluate AUC score using 200 human-written text and 200 original or paraphrased AI-generated text. For each task input, we run each evasion method only once, instead of repeating multiple times until successful evasion, to simulate real-world scenarios where the target detector is inaccessible. Experimental settings We set $\|X_{\text{eval}}\| = 32$, $K = 8$, $N = 6$, and use GPT-3.5, specifically gpt-3.5-turbo-0301, as the LLM, where the inference parameters are kept in default. And we use GPT3-D as the proxy detector. Experiments using other LLMs and proxy detectors are presented in Section 5.2. ### 4.2 Evasion Performance and Analysis Table 1 presents the performance of SICO and other baselines against six detectors in AUC score. SICO consistently outperforms other baselines by a substantial margin in all cases. Notably, in most cases, SICO reduces the AUC score to less than 0.5, equivalent to the expected performance of a random classifier. Figure 2 shows the ROC curves of evasion methods on academic writing task. ![Figure 2: ROC curves of six detectors on the text generated by different evasion methods on academic writing task.](image-url) can clearly observe that SICO curves lie below others along different thresholds, often lower than the random classifier curve. More evasion results including detection rates are shown in Appendix H. One interesting trend is that SICO-Para consistently outperforms SICO-Gen against statistical detectors, i.e., DetectGPT and Log-Rank. We speculate this performance difference comes from the varying influence of the prompt on the generated text between the two methods. In SICO-Para, the distribution of generated text is largely influenced by the original AI-generated text, which is in the prompt. However, in SICO-Gen, the distribution of generated text depends more on the previously generated text. Given that statistical detectors have access to the newly generated text but not the prompt, their estimation of token probability becomes less accurate for SICO-Para text, thus misleading the detection. It might also explain why GPT-Para can reduce the performance of statistical detectors. 4.3 Human Evaluation From the users’ perspective, using AI-generated text goes beyond evading detection systems; the usability of text is equally critical. For example, for academic writing task, users expect the text to be readable, properly formatted, and relevant to the given topic. Therefore, we evaluate the usability of text based on two criteria: readability and task completion rate. For each task, we randomly sample 200 examples generated by four methods (50 per method), including human-written text. Then we ask three human annotators to rate the readability of text on a scale from 1 to 5, and judge if the text accomplishes the task’s goal. More details and results of human evaluation are shown in Appendix C. As shown in Table 2, both SICO-Gen and SICO-Para demonstrate superior performance over DIPPER in terms of task completion and readability over three tasks. Furthermore, SICO-generated text performs competitively compared with human-written text in both metrics, with a negligible difference less than 0.1. In contrast, DIPPER exhibits inferior performance relative to human-written text, particularly with a notable 0.27 decline in readability. Table 2: Human evaluation results. “Avg.D.” represents the average difference between the results achieved by the evasion method and by human-written text across three tasks. The best is set bold. \| Method \| Readability ↑ \| Task Completion Rate % ↑ \| \|--------------\|---------------\|--------------------------\| \| \| Writing QA Review Avg.D. ↑ \| Writing QA Review Avg.D. ↑ \| \| DIPPER \| 3.52 4.12 3.42 -0.27 \| 70.6 100 61.6 -13.3 \| \| SICO-Para \| 3.68 4.36 3.58 -0.09 \| 82.0 100 72.4 -5.9 \| \| SICO-Gen \| 3.84 4.28 3.70 -0.02 \| 93.6 100 69.6 -2.9 \| \| Human-Written \| 3.92 4.36 3.60 - \| 98.2 100 73.8 - \| 4.4 Real-life Experiments To further assess the applicability of SICO in real-world, we simulate one potential misuse case of LLM, where SICO is deployed as an automatic reply bot on Reddit, a popular online social platform. We wrote a script to monitor the new posts submitted in the community for asking questions and used GPT-3.5 equipped with SICO-Para to automatically reply them. The prompt we used is trained for question answering task. On Reddit, except for giving comments, users can express their approval of other’s responses by clicking the “like” or “dislike” button. To minimize the social impact, we limit the number of responses to 40 and deleted them after collecting results. The quantitative results in Table 3 demonstrate that users generally react positively to the text from SICO. Specifically, 40% of the responses from SICO receive “likes” from Reddit users, significantly higher than the 2.5% that are disliked. The remaining 57.5% of responses go unnoticed, which is common in social media. Besides, in 7.5% of cases, users suspect that the response is generated by --- 2https://www.reddit.com/ AI, as evidenced by comments such as “Are you ChatGPT?”. Additionally, Figure 3 presents two SICO responses that got approval from users, as indicated by “likes” and comments. 4.5 Cost Efficiency In terms of the data prerequisite, SICO only needs $K + \|X_{\text{eval}}\|$ human-written input-output examples to build prompt, which is $8 + 32 = 40$ in the experiments. Furthermore, SICO offers the advantage of low cost for prompt construction. Based on three repeated runs, the actual USD costs of SICO-Para are $1.04 \pm 0.04$, $1.08 \pm 0.05$, and $0.75 \pm 0.04$ for Writing, QA, Review tasks, respectively. 5 Further Experiments 5.1 Ablation Study We conducted an ablation study over academic writing task to evaluate the contribution of individual components within the SICO framework. “Human-ICE” denotes the approach where human-written text is directly utilized as the in-context example for constructing the prompt. “w/o feature” and “w/o ICE” refer to the prompts without feature text and the optimized in-context examples, respectively. “w/o OPT” represents the initial prompt before optimization (see Line 3 in Algorithm 2). “–” indicates the case where no evasion method is used. In our experiment, we explore SICO-Para on three types of detectors: GPT3-D, OpenAI-D and DetectGPT. The AUC scores are averaged across these detectors. Results in Table 4 shows that directly using human-written text is ineffective, even making the detection more accurate. We speculate that the human-written examples are too heterogeneous and characterized in multiple ways, so LLM cannot effectively learn their attributes. Besides, the importance of feature text is comparatively less than that of optimized in-context examples. Furthermore, the result reveals the significant role of the optimization step in SICO. Using in-context examples that are not optimized is essentially equivalent to not using any in-context examples. 5.2 SICO with Different Proxy Detectors and LLMs As described in Section 3, SICO requires a proxy detector and a LLM to construct a prompt. In this experiment, we explore the performance of SICO-Para on writing task, using three types of proxy detectors: (1) training-based model GPT-3.5 detector, (2) API detector GPTzero, and (3) statistical method DetectGPT. For different LLMs, we adopt Vicuna-13B (Chiang et al., 2023), an open-source \| Method \| AUC \| \|--------------\|-------\| \| – \| 0.844 \| \| Human-ICE \| 0.863 \| \| SICO-Para \| 0.292 \| \| w/o feature \| +0.076\| \| w/o ICE \| +0.301\| \| w/o OPT \| +0.293\| Table 4: Ablation study of SICO. chatbot fine-tuned from LLaMa (Touvron et al., 2023). Results in Table 5 show that SICO maintains a high degree of detection evasion performance, regardless of proxy detectors or LLMs. In most cases, SICO manages to reduce the AUC of detectors by approximately 0.4. Table 5: The AUC scores of SICO using different proxy detectors and LLMs on writing task. The first line indicates the performance without applying any evasion method. \| LLM \| Proxy Detector \| GPT3-D \| GPT2-D \| GPTzero \| OpenAI-D \| DetectGPT \| Log-Rank \| \|-----------\|----------------\|--------\|--------\|---------\|----------\|-----------\|----------\| \| - \| - \| 0.908 \| 0.848 \| 0.779 \| 0.789 \| 0.834 \| 0.914 \| \| GPT-3.5 \| GPT3-D \| 0.239 \| 0.332 \| 0.290 \| 0.488 \| 0.149 \| 0.147 \| \| GPT-3.5 \| GPTzero \| 0.549 \| 0.143 \| 0.299 \| 0.458 \| 0.429 \| 0.493 \| \| GPT-3.5 \| DetectGPT \| 0.291 \| 0.361 \| 0.244 \| 0.483 \| 0.174 \| 0.173 \| \| Vicuna-13B\| GPT3-D \| 0.393 \| 0.777 \| 0.380 \| 0.690 \| 0.337 \| 0.375 \| 5.3 EXAMPLES OF THE GENERATED TEXT Table 6 presents the fake reviews generated by SICO-Gen and SICO-Para. The generated text shows high readability and fulfill the task’s goal, successfully mentioning all keywords and generating positive reviews of the specified object. The AI probability, denoted as $P_{AI}$ in the table, is determined by the GPT3-D. More examples are shown in Appendix I. Table 6: Fake reviews generated by SICO. The first line shows the task input of review generation. Object: Sushi Station Keywords: revolving sushi bar, affordable, krab mix, creamy sauce, fresh menu. Sentiment: Positive AI: If you’re a sushi lover, you have to check out Sushi Station! This revolving sushi bar is not only affordable, but it offers a fresh and delicious menu that will leave you craving more. My personal favorite is the krab mix sushi with its creamy sauce - it’s simply heavenly. The staff is friendly and attentive, and the atmosphere is perfect for a casual night out with friends or family. Don’t hesitate to give Sushi Station a try - you won’t be disappointed! $P_{AI} = 0.99$ SICO-Para: Sushi enthusiasts, please take note: Sushi Station provides an unbeatable revolving sushi bar experience at wallet-friendly prices. Their menu is of the freshest quality and tastes nothing short of delicious. My top pick has to be the krab mix sushi; its creamy sauce is pure bliss. Additionally, their friendly staff provided excellent service, while the casual ambiance proved ideal for a relaxed night out with friends or family. Don’t hesitate to give Sushi Station a try, you won’t regret it! $P_{AI} = 0.00$ SICO-Gen: I recently tried out Sushi Station’s revolving sushi bar and I must say, it was an amazing experience. The affordable pricing along with the wonderful krab mix and creamy sauce they offer is what really made it stand out from the rest. The menu was constantly updated with fresh and delicious options making it hard to choose just one. If you’re looking for a great sushi experience, Sushi Station is definitely worth a visit. $P_{AI} = 0.04$ 6 CONCLUSION AND LIMITATION In conclusion, we have proposed a novel in-context learning approach, SICO, designed to guide LLMs in generating text that can effectively evade detectors. Our extensive experiments on evasion demonstrate the superior performance of SICO, which significantly reduces the detection capabilities of existing AI text detectors across three tasks. A comprehensive human evaluation shows SICO text can achieve human-level readability and task completion rates. The experiment in the wild shows the functionality of SICO in real life. One limitation of SICO is the homogeneity in the style of the text generated from a single constructed prompt. This uniformity makes it easier for people to memorize the style and identify the text as AI afterward. One simple solution is to train multiple prompts for one task. 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wk77w7DG1N	Given that the reference and candidate documents might have varying numbers of sentences, how do you handle sentence-level comparisons? Specifically, do you employ any sentence matching techniques, and if so, how are they implemented?	Evaluating and Improving Generation Consistency of Large Language Models via A Divide-Conquer-Reasoning Approach Anonymous authors Paper under double-blind review Abstract Evaluating the quality and variability of text generated by Large Language Models (LLMs) poses a significant, yet unresolved research challenge. Traditional evaluation methods, such as ROUGE and BERTScore, which measure token similarity, often fail to capture the holistic semantic equivalence. This results in a low correlation with human judgments and intuition, which is especially problematic in high-stakes applications like healthcare and finance where reliability, safety, and robust decision-making are highly critical. This work proposes an automated framework for evaluating the consistency of LLM-generated texts using a divide-and-conquer strategy. Unlike existing LLM-based evaluators that operate at the paragraph level, our method employs a divide-and-conquer evaluator (DCE) that breaks down the comparison between two generated responses into individual sentences, each evaluated based on predefined criteria. To facilitate this approach, we introduce an automatic metric converter (AMC) that translates the output from DCE into an interpretable numeric score. Beyond the consistency evaluation, we further present a reason-assisted improver (RAI) that leverages the analytical reasons with explanations identified by DCE to generate new responses aimed at reducing these inconsistencies. Through comprehensive and systematic empirical analysis, we show that our approach outperforms state-of-the-art methods by a large margin (e.g., +19.3% and +24.3% on the SummEval dataset) in evaluating the consistency of LLM generation across multiple benchmarks in semantic, factual, and summarization consistency tasks. Our approach also substantially reduces nearly 90% output inconsistencies, showing promise for effective hallucination mitigation and reduction. 1 Introduction Large language models (LLMs) such as GPT-4 and PaLM 2 (Yang et al., 2023; Bubeck et al., 2023) have demonstrated impressive performance on a variety of natural language generation (NLG) tasks, including summarization (Tam et al., 2022), open-book question-answering (QA) (Kamalloo et al., 2023), and retrieval-augmented generation (RAG) (Lewis et al., 2020; Liu et al., 2023a). However, conventional evaluation methods, such as BARTScore (Yuan et al., 2021) and BERTScore (Zhang et al., 2020), which rely on token-level comparison, are inadequate for accurately and reliably measuring the quality of generated content, particularly in complex scenarios with long paragraphs (Liu et al., 2023b; Hama & Bojar, 2021). To address this issue, LLM-based evaluators such as G-Eval (Liu et al., 2023b) and GPTScore (Jinlan et al., 2023) have proposed a new framework that evaluates texts via paragraph-level comparison. While these evaluators show promise for certain tasks, their scores often fail to achieve high concordance with human judgments of semantic equivalence. Furthermore, as only numeric scores are provided with no explanation, it can be challenging for humans to trust or reason about these scores, particularly when using LLMs that are known to hallucinate (Li et al., 2023; Ji et al., 2023; Rawte et al., 2023). Assessing the consistency of LLMs is more broadly connected to AI safety and has become a critical step in improving the reliability of these systems by preventing the generation of misinformation and harmful content. Wang et al. (2022) demonstrates that consistency checking can significantly enhance the chain of thought reasoning in LLMs. Similarly, Kuhn et al. (2023) leverages semantic... consistency for uncertainty estimation in NLG. Recent studies employ consistency checking to detect hallucinations based on pre-trained LLMs (Manakul et al., 2023) and instruction-tuned LLMs (Mündler et al., 2023). Although these methods exhibit promising results on several specific tasks, including mathematical reasoning and factual assessment, the potential failures of self-consistency are often overlooked. This is essentially due to a lack of a generic, automatic, and reliable strategy that assesses the consistency of two responses, let alone remediating such inconsistency after identifying them. ![Figure 1](image) Figure 1: (a) Overview of the proposed DCR framework. The first two components (DCE-AMC) aim at providing a better strategy for evaluating and quantifying semantic consistency to best match human judgments. Building on this, a third component RAI further utilizes analytical reasoning to iteratively improve the consistency of LLM-generated content w.r.t. the reference by minimizing hallucinations. (b) The combination of DCE and AMC (DCE-AMC-4) significantly outperforms the baseline methods in terms of correlations with human ratings. (c) RAI substantially reduces output inconsistencies by ~90% through a single improvement iteration on SummEval and QAGS benchmarks. In this paper, we introduce a novel framework, called Divide-Conquer-Reasoning (abbreviated as DCR hereafter), for developing an automatic and reliable consistency evaluation method. Our approach capitalizes on the intuition that human evaluators typically assess consistency by comparing the semantic meaning of the generated text to the reference text sentence-by-sentence, and then combining the analysis to make a holistic judgment of the complete concept. Unlike existing metrics that rely on either token-level or paragraph-level checks, our approach is rooted in the sentence level and is better aligned with human judgments. This approach avoids confusing LLM by either providing too much information at once or zooming in too narrowly. Additionally, our approach does not rely on LLMs, which are prone to hallucination, to output numeric scores without justification. Another advantage of our approach is its ability to mitigate inconsistencies after identifying them. The DCR framework is composed of three components, each executed by an LLM agent, as shown in Fig. 1. Given the reference and candidate, the Divide-Conquer Evaluator (DCE) realizes the notion of divide-conquer to determine whether the candidate is semantically equivalent to the reference at a sentence level. DCE automatically partitions the candidate paragraph into sentences (divide), evaluates each sentence against the reference paragraph based on pre-defined semantic-level consistency criteria (conquer), and generates a list of reasons that explain why each sentence is or is not consistent with the reference paragraph. Next, the Auto-Metric Converter (AMC) which builds upon DCE, converts the reasons (with explanations) into a numeric score system that is more intuitive for humans to comprehend and evaluate the performance of DCE. The numeric score can be used to evaluate consistency in various tasks, such as summarization, factual assessment, and hallucination detection. Our DCR framework not only evaluates consistency but also enhances it through the Reason-Assisted Improver (RAI), a third LLM agent that utilizes the outputs of DCE to generate new candidate sentences. By incorporating the explanations provided by DCE with the original context, RAI produces sentences that mitigate inconsistencies (hallucinations). This improvement process can be iteratively applied by utilizing the re-evaluation produced by DCE to ultimately achieve a candidate response that is fully aligned with the reference text. We conducted an evaluation of our approach on three different NLG tasks, including semantic, summarization, and factual consistency evaluations. Our results demonstrate that DCR significantly outperforms all existing baseline methods as a consistency evaluator, with improvements of up to 19.3% and 24.3% compared to G-Eval on the SummEval dataset. Additionally, our approach achieved high correlations with human judgment on all three benchmarks. Notably, we observed highly promising results in consistency improvement rate (from 86.71% to 91.11%) at a substantially lower effort and cost due to its multi-thread parallel implementation. 2 PRELIMINARIES Black-Box LLM Evaluation. One of the drawbacks of current grey-box LLM evaluations is that they require output token-level probabilities (Jiang et al., 2023). However, prominent LLMs such as GPT-3.5, GPT-4, PaLM 2, and Claude 2, are only available through restricted API calls. Therefore, such token-level information might not be available. By contrast, in this paper, we focus on the design of a black-box approach that remains applicable even when only text-based responses are available from the LLM; that is, we only have access to the model output. Problem Formulation. Given a user query $Q$ and LLM model $M$, let $C$ refer to the candidate response drawn from $C = M(Q)$. LLM-generated responses are commonly evaluated using some reference texts, denoted by $R$, for instance, human writing samples for generation tasks and original content for summarization tasks. The objective of consistency evaluation is to build a function $f$ that quantitatively measures the semantic equivalence $S$ between the generated candidates $C$ and reference $R$ as $S = f(R, C\|Q, M)$ where $S$ could be binary decision, such as “Yes” or “No”, “Consistent” or “Not Consistent”, or numeric score, e.g., [-1, +1]. However, it is worth noting that our evaluation can be generally used to check consistency between two candidates where both are generated by LLMs. In that scenario, we only need to assume one candidate as the reference for self-check consistency. Limitation of Existing Methods. The conventional metrics, such as BERTscore and BARTscore, rely on a token-level comparison using n-gram or contextual embedding to calculate cosine-similarity. However, this approach fails to capture the overall semantic meaning as it directly aggregates token-level similarities. To address this issue, leveraging the power of LLMs for self-evaluation has been proposed. G-Eval (Liu et al., 2023b) and GPT-Eval (Jiang et al., 2023) evaluate consistency at a paragraph-level by prompting LLMs to compare two candidates as a whole. However, these approaches have a major drawback as the generated verbal scores by LLMs are prone to hallucinations, resulting in abnormally higher ratings for LLM-generated content that diverge from human judgment (Liu et al., 2023b). Such methods also generate no actionable insight to justify the score or mitigate inconsistencies after identifying them. 3 DIVIDE-CONQUER-REASONING To overcome the aforementioned limitations, we propose to evaluate and improve the consistency of LLM output via a Divide-Conquer-Reasoning approach, referred to as DCR. The approach comprises three key components, as illustrated in Fig. 1: (1) DCE, which disassembles the candidate paragraph and scrutinizes semantic inconsistencies sentence-by-sentence, (2) AMC, which converts sentence-level inconsistency/consistency reasons into numeric scores for quantitative interpretation, and (3) RAI, which conducts analytical reasoning to improve consistency through candidate regeneration. Essentially, our approach involves a combination of sentence-level analysis, semantic consistency checking, and causal analysis, making it an ideal evaluation metric for a diverse range of NLG tasks that require comparison to reference texts, such as summarization, open-book question-answering (QA), and retrieval-augmented generation. Moreover, DCR not only evaluates but also improves the consistency of generated text through analysis and reasoning, which aligns with human intuition. Fig. 2 provides an example of how DCR can evaluate and enhance the consistency of candidate text. In the following sections, we will discuss each component in detail. 3.1 Divide-Conquer Evaluator (DCE) The Divide-Conquer Evaluator (DCE) is an LLM Agent designed to perform semantic consistency checks between the reference and the candidate using a sentence-by-sentence strategy. This agent accepts a reference paragraph and a candidate paragraph as inputs, and employs a divide-conquer strategy to break down the entire paragraph into multiple individual sentences (divide) and then assess each sentence against the reference (conquer). More specifically, given the input reference $ R = \langle s_1^r, ..., s_l^r \rangle $ and candidate $ C = \langle s_1^c, ..., s_k^c \rangle $, we build a DCE agent $ L_{\text{DCE}} $ using the LLM model $ M $ (e.g., GPT-3.5/4) with an instructed prompt $ P_{\text{DCE}} $ as: \[ \{\gamma_1, \gamma_2, ..., \gamma_k\} = L_{\text{DCE}}(\langle s_1^c, s_2^c, ..., s_k^c \rangle, R \| M, P_{\text{DCE}}). \] Eq.1 generates reasons, denoted as $ \Gamma = \{\gamma_1, \gamma_2, ..., \gamma_k\} $, which is a list of reasons explaining why each sentence $ s_i^c (i = 1, 2, ..., k) $ is or is not consistent against the entire reference paragraph $ R $. It’s important to note that the reasons $ \gamma_i $ might comprise a short paragraph containing multiple explanation sentences. We can tailor instruction prompts by defining task-specific criteria to accommodate different tasks. Table 1 provides an example of a prompt example with pre-defined criteria for the summarization consistency task. Table 1: Summarization Consistency Divide-Conquer Evaluator Prompt Your task is to evaluate whether the summary is consistent with the article. You will evaluate it by going through each sentence of the summary and check against the following procedures: - Understands all the aspects of the sentence, and compare if each aspect exists in the article - If it does, compare if the information in this sentence is consistent with what is in the article - Compare if all the information in this sentence can be directly inferred or entailed from what is in the article. It is OK that not all information from the article exists in this summary 3.2 Auto-Metric Converter (AMC) The Auto-Metric Converter (AMC) is an LLM Agent that aims to quantitatively measure the consistency evaluation derived from the Divide-Conquer Evaluator (DCE) by converting the reasons from DCE into a numeric score system. This is accomplished by introducing an LLM agent, denoted as $ L_{\text{AMC}} $, which takes reasons $ \{\gamma_1, \gamma_2, ..., \gamma_k\} $ with an instructed prompt $ P_{\text{AMC}} $ as inputs: \[ \{z_1, z_2, ..., z_k\} = L_{\text{AMC}}(\{\gamma_1, \gamma_2, ..., \gamma_k\} \| M, P_{\text{AMC}}). \] The LLM Agent $ L_{\text{AMC}} $ functions as a binary sentiment classifier that classifies the reasons $ \{\gamma_1, \gamma_2, ..., \gamma_k\} $ to be either positive (marked by “+1” if the sentence is consistent), or negative (marked by “-1” otherwise). As a result, AMC outputs an array of scores $ \{z_1, z_2, ..., z_k\}, z_i \in \{-1, +1\} $ for each sentence $ \langle s_1^c, s_2^c, ..., s_k^c \rangle $ in the candidate $ C $. We then utilize this score array to calculate a comprehensive score $ Z $ to evaluate how consistent the candidate (paragraph) is against the reference (paragraph): \[ Z = \left( \sum_{i=1}^{k} z_i + \alpha \right) / (k + \beta), \quad \hat{Z} = (Z + 1)/2, \quad \hat{Z} \in [0, 1] \] where $ k $ is the length of the score array, i.e., the number of sentences in the candidate paragraph. Depending on the prompt, the reasons output by DCE may not all be on the sentence level. To ensure that the score calculated is solely generated by sentence-level reasons, we introduce $ \alpha $ and $ \beta $ in Eq.3 as explained in detail in Appendix A.4. Finally, we rescale $ Z $ to obtain the final score $ \hat{Z} $ that is typically between 0 (completely inconsistent) and 1 (completely consistent). The closer this score $ \hat{Z} $ is to 0, the more inconsistent the candidate $ C $ is against the reference $ R $. 3.3 Reason-Assisted Improver (RAI) The Reason-Assisted Improver (RAI) is an LLM Agent that focuses on improving the consistency of candidate sentences by reasoning through the inconsistent explanations generated by the Divide-Conquer Evaluator (DCE). To achieve this goal, we propose an LLM agent $ L_{\text{RAI}} $ to generate new candidate sentences $\langle \hat{s}_1^c, \hat{s}_2^c, ..., \hat{s}_k^c \rangle$ based on the collected reasons $\{\gamma_1, \gamma_2, ..., \gamma_k\}$ and original sentences $\langle s_1^c, s_2^c, ..., s_k^c \rangle$: $$\langle \hat{s}_1^c, \hat{s}_2^c, ..., \hat{s}_k^c \rangle = L_{RAI}(\{\gamma_1, \gamma_2, ..., \gamma_k\}, \langle s_1^c, s_2^c, ..., s_k^c \rangle, R \| M, P_{RAI}).$$ The core task of $L_{RAI}$ is to rewrite the original sentence $s_i^c$ if $s_i^c$ is inconsistent with the reference $R$ and return a new generated $\hat{s}_i^c (\hat{s}_i^c \neq s_i^c)$, otherwise retain $s_i^c$. The newly generated responses $\hat{C} = \langle \hat{s}_1^c, \hat{s}_2^c, ..., \hat{s}_k^c \rangle$ can be considered as the consistency-improved candidate, which can be re-evaluated by DCE to check if $\hat{C}$ mitigates inconsistencies against the reference $R$. The improved candidate $\hat{C}$ in Eq.4 can be directly fed to the DCE agent in Eq.1 after the first-round DCR, i.e., DCE → AMC → RAI. A straightforward extension is multi-round consistency improvement, where the consistency is iteratively improved until reaching the maximum number of rounds $m$. Algorithm 1 illustrates the workflow of the DCR framework, which consists of three core components: DCE, AMC, and RAI. Algorithm 1 Proposed Divide-Conquer-Reasoning (DCR) framework 1: Requirements: Candidate $C$, Reference $R$, LLM model $M$, LLM agents $L_{DCE}, L_{AMC}, L_{RAI}$ with instructed prompts $P_{DCE}, P_{AMC}$ and $P_{RAI}$, and the maximum number of rounds $m$ 1: for rounds $r = 1, ..., m$ do 2: Disassemble candidate $C$ into sentences $\langle s_1^c, s_2^c, ..., s_k^c \rangle$, evaluate sentence-level consistency against reference $R$, and return the reasons $\{\gamma_1, \gamma_2, ..., \gamma_k\} \leftarrow L_{DCE}(\langle s_1^c, s_2^c, ..., s_k^c \rangle, R \| M, P_{DCE})$ in Eq.1 3: Transform reasons into numeric scores $\{z_1, z_2, ..., z_k\} \leftarrow L_{AMC}(\{\gamma_1, \gamma_2, ..., \gamma_k\} \| M, P_{AMC})$ in Eq.2 4: Calculate the final consistency evaluation score $\hat{Z}$ based on $\{z_1, z_2, ..., z_k\}$ using Eq.3 5: Generate improved candidate $\langle \hat{s}_1^c, \hat{s}_2^c, ..., \hat{s}_k^c \rangle \leftarrow L_{RAI}(\{\gamma_1, \gamma_2, ..., \gamma_k\}, \langle s_1^c, s_2^c, ..., s_k^c \rangle, R \| M, P_{RAI})$ 6: Update the candidate $\langle \hat{s}_1^c, \hat{s}_2^c, ..., \hat{s}_k^c \rangle \leftarrow \langle \hat{s}_1^c, \hat{s}_2^c, ..., \hat{s}_k^c \rangle$ and return Step 2 7: return $\hat{Z}, \langle \hat{s}_1^c, \hat{s}_2^c, ..., \hat{s}_k^c \rangle$ ## 4 EXPERIMENTS ### 4.1 Benchmarks and Implementation Details We utilize GPT-3.5 (gpt-3.5-turbo) and GPT-4 (gpt-4) as our LLM agents, and the evaluations are carried out using the Azure OpenAI API. We set the temperature to 0.0 to generate responses via the greedy algorithm. The specific prompts used for each LLM agent are detailed in the Appendix (from Table 7 to Table 12). All experiments are conducted on our local machine (Macbook-Pro with M1 chip) without the need for GPU resources. In our experimental setup, we set both $\alpha$ and $\beta$ in Eq.3 to 0. We employ four datasets to evaluate DCR where QQP and PAWS are binary datasets, as well as SummEval and QAGS have numeric scores representing human judgments. - QQP and PAWS: Quora Question Pair corpus [Iyer et al., 2017] and the Paraphrase Adversaries from Word Scrambling dataset [amd Jason Baldridge & He, 2019] contain pairs of sentences labeled to indicate whether they are paraphrases or not, while PAWS specifically focuses on the adversarial paraphrases. Following the guidance of BERTScore [Zhang et al., 2020], we are using the PAWSQQP development set and the first 5000 from the training set of QQP. - SummEval [Fabbri et al., 2021] is a standard dataset that assesses various summarization evaluation techniques. It gathers human ratings in various aspects and is built on the CNN/DailyMail dataset [Hermann et al., 2015]. In this study, we mainly focus on the consistency evaluation. - QAGS [Wang et al., 2020] serves as a benchmark for assessing hallucinations in summarization tasks. Its objective is to evaluate the consistency aspect of summaries across two distinct summarization datasets: QGS-CNN and QAGA-XSUM. ### 4.2 Baselines We evaluate DCR against a variety of evaluation metrics and LLM-based evaluators that have achieved state-of-the-art performance. - ROUGE [Lin, 2004] is widely used evaluation metric with three different variants ROUGE-1, ROUGE-2, and ROUGE-L. We are using ROUGE-2 and ROUGE-L as comparisons in our study. - BERTScore [Zhang et al., 2020] calculates the similarities between two pieces of text using the contextualized embedding derived from the BERT model [Devlin et al., 2019]. It operates as a similarity-based assessment tool, which has been widely used for various applications. • MoverScore (Zhao et al., 2019) enhances BERTScore by incorporating soft alignments and introducing new aggregation techniques to provide a more robust similarity assessment. • BARTScore (Yuan et al., 2021) is a comprehensive evaluator that uses the average likelihood of the model’s output as its measurement criteria. • UniEval (Zhong et al., 2022) is a consolidated evaluator capable of assessing various elements of text generation as QA tasks. It manages diverse evaluation tasks by modifying the question format. • GPTScore (Jinlan et al., 2023) is an LLM-based evaluator that assesses texts using pre-training models, e.g., GPT-3, and is designed to provide a higher likelihood to high-quality generated text. • G-Eval (Liu et al., 2023b) is another LLM evaluator that utilizes LLMs with a chain-of-thoughts (CoT) approach with a form-filling paradigm to evaluate the quality of NLG outputs. ### 4.3 Main Results on Consistency Evaluation (DCE-AMC) #### Semantic Consistency Evaluation. Table 2 shows the Area Under the ROC curve (AUROC) for automatic baseline metrics and our method, following the practice of BERTScore (Zhang et al., 2020). We note that while most metrics from BERTScore perform acceptably on QQP, they exhibit a significant performance drop on PAWSQQP. This suggests that these baseline metrics struggle to detect the challenging adversarial examples from a semantic consistency perspective. In contrast, our method, whether implemented with GPT-3.5 or GPT-4, outperforms all the baseline metrics on both QQP and PAWSQQP, without a significant drop. Notably, DCE-AMC-4 demonstrates superior robustness in adversarial paraphrase classification (semantic consistency) achieving a relatively large improvement (+4.6% in QQP and +9.9% in PAWSQQP) compared to BERTScore. #### Factual Consistency Evaluation. While advanced NLG models are capable of generating high-quality responses, LLMs are known to occasionally produce non-factual statements or hallucinate facts, which can undermine trust in their output. Recent work (Manakul et al., 2023) has been conducted to identify such inconsistencies in terms of factuality. To verify the effectiveness of our method in evaluating hallucination, we test it on the QAGS benchmark, which includes two summarization datasets: QAGS-CNN and QAGS-XSUM. Table 4 provides a comprehensive comparison of various metrics based on Pearson, Spearman, and Kendall-Tau correlations. We observe that BARTScore performs competitively on the extractive subset (QAGS-CNN) but fails to demonstrate a high correlation on the abstractive subset (QAGS-XSUM). UniEval exhibits a better correlation than G-Eval-3.5 but is comparable to G-Eval-4. Our proposed DCE-AMC-4 outperforms all the baseline methods on both subsets, particularly by a significant margin on QAGS-XSUM. Unlike the G-Eval method, which shows a larger gap between GPT-3.5 and GPT-4, our DCE-AMC method remains relatively stable when switching between LLMs. It’s crucial to note that QAGS-XSUM is an abstractive dataset, and its summaries are typically one sentence long. This contrasts with the extractive database of QAGS-CNN, where summaries are composed of multiple sentences. Consequently, our method operates at a sentence level for QAGS-XSUM, and our final score is always either 0 or 1. Furthermore, the binary label in QAGS-XSUM implies that we achieve the same correlation score using different correlation methods. #### Summarization Consistency Evaluation. We follow the setting of previous work (Zhong et al., 2022) to evaluate different summarization consistency using summary-level Spearman ($\rho$) and Kendall-Tau ($\tau$) correlation. As shown in Table 3, baseline metrics using semantic similarity, such as ROUGE and BERTScore, perform poorly on consistency evaluations. While LLM-based evaluators like GPT-Score and G-Eval exhibit higher correlations, they still underperformed compared to \| Metrics \| QQP \| PAWSQQP \| \|---------\|-----\|---------\| \| BLEU \| 0.707 \| 0.527 \| \| METEOR \| 0.755 \| 0.532 \| \| ROUGE-L \| 0.740 \| 0.536 \| \| CHRF++ \| 0.577 \| 0.608 \| \| BEER \| 0.741 \| 0.564 \| \| EED \| 0.743 \| 0.611 \| \| CharaCTER \| 0.698 \| 0.650 \| \| BERTScore \| 0.777 \| 0.693 \| \| DCE-AMC-3.5 \| 0.788 \| 0.770 \| \| DCE-AMC-4 \| 0.823 \| 0.792 \| \| Metrics \| SummEval-Consistency \| \|---------\|----------------------\| \| \| Spearman ($\rho$) \| Kendall-Tau ($\tau$) \| \| ROUGE-2 \| 0.187 \| 0.155 \| \| ROUGE-L \| 0.115 \| 0.092 \| \| BARTScore \| 0.382 \| 0.315 \| \| BERTScore \| 0.110 \| 0.090 \| \| MoverScore \| 0.152 \| 0.127 \| \| UniEval \| 0.446 \| 0.371 \| \| GPT-Score \| 0.449 \| - \| \| G-Eval-3.5 \| 0.386 \| 0.318 \| \| G-Eval-4 \| 0.507 \| 0.425 \| \| DCE-AMC-3.5 \| 0.592 \| 0.563 \| \| DCE-AMC-4 \| 0.700 (+19.3%) \| 0.668 (+24.3%) \| our proposed method. DCE-AMC-4 achieves much higher human correspondence compared to DCE-AMC-3.5 on both Spearman and Kendall-Tau correlation, which indicates that the larger size of GPT-4 model is beneficial for sentence-level consistency checking in summarization tasks. DCE-AMC-4 with stronger correlations of $\rho = 0.7$ and $\tau = 0.668$, substantially improves upon the G-Eval-4 baseline by a large margin (+19.3% and +24.3% respectively). Table 4: Pearson ($r$), Spearman ($\rho$), and Kendall-Tau ($\tau$) correlations of different baseline metrics on QAGS-CNN and QAGS-XSUM benchmark. \| Metrics \| QAGS-CNN \| QAGS-XSUM \| \|---------------\|---------------------------\|---------------------------\| \| \| Pearson ($r$) \| Spearman ($\rho$) \| Kendall-Tau ($\tau$) \| Pearson ($r$) \| Spearman ($\rho$) \| Kendall-Tau ($\tau$) \| \| ROUGE-2 \| 0.459 \| 0.418 \| 0.333 \| 0.097 \| 0.083 \| 0.068 \| \| ROUGE-L \| 0.357 \| 0.324 \| 0.254 \| 0.024 \| -0.011 \| -0.009 \| \| BARTScore \| 0.735 \| 0.680 \| 0.557 \| 0.184 \| 0.159 \| 0.130 \| \| BERTScore \| 0.576 \| 0.505 \| 0.399 \| 0.024 \| 0.008 \| 0.006 \| \| MoverScore \| 0.414 \| 0.347 \| 0.271 \| 0.054 \| 0.044 \| 0.036 \| \| UniEval \| 0.682 \| 0.662 \| 0.532 \| 0.461 \| 0.488 \| 0.399 \| \| G-Eval-3.5 \| 0.477 \| 0.516 \| 0.410 \| 0.211 \| 0.406 \| 0.343 \| \| G-Eval-4 \| 0.631 \| 0.685 \| 0.591 \| 0.558 \| 0.537 \| 0.472 \| \| DCE-AMC-3.5 \| 0.699 \| 0.648 \| 0.596 \| 0.573 \| 0.573 \| 0.573 \| \| DCE-AMC-4 \| 0.782 \| 0.760 \| 0.706 \| 0.602 \| 0.602 \| 0.602 \| 4.4 Results for Consistency Improvement (RAI) After implementing DCE and AMC, we can quantitatively determine whether each candidate is consistent (score = 1) to the reference or not (score <1). Table 5 offers a statistical analysis of the number of inconsistent data after evaluations (DCE-AMC), revealing 286, 111, and 86 inconsistent candidates for the SummEval, QAGS-CNN, and QAGS-XSUM benchmarks respectively. Identifying these inconsistent candidates is valuable but the more critical objective is how to improve these responses to align with the references. To achieve this goal, we re-generate a new response by implementing RAI based on the reasons provided by DCE, and then use DCE to re-evaluate these improved responses. We observe a significant improvement, with most inconsistencies corrected, specifically 84 out of 86 examples on the QAGS-XSUM benchmark. The rate of consistency improvement is 86.71%, 88.29%, and 97.67% on SummEval, QAGS-CNN and QAGS-XSUM respectively. These impressive results demonstrate that our reasoning approach RAI not only provides better consistency evaluation metrics that align more closely with human judgments, but also sheds light on improving consistency beyond evaluation. This finding is particularly crucial for mitigating hallucination once we detect non-factual statements via consistency checks. It’s worth noting that our reasoning method RAI is a generic component that can also be applied directly at the paragraph level, and the improvement in this context is significant as well, as illustrated in Table 5. Table 5: Performance of consistency improvement with RAI on three benchmark datasets. \| Dataset (size) \| SummEval (1600) \| QAGS-CNN (236) \| QAGS-XSUM (239) \| \|---------------\|-----------------\|----------------\|-----------------\| \| \| Sentence \| Paragraph \| Sentence \| Paragraph \| Sentence \| Paragraph \| \| Inconsistent data \| 286 \| 209 \| 111 \| 68 \| 86 \| 90 \| \| Corrected data with RAI \| 248 \| 198 \| 89 \| 64 \| 84 \| 82 \| \| Consistency improvement rate \| 86.71% \| 94.73% \| 88.29% \| 94.11% \| 97.67% \| 91.11% \| 4.5 Analysis Why DCR Prefers Sentence-level Evaluation? To further assess the potential advantage of the sentence-level approach in consistency checking, we employed the same logic of outputting decisions and reasons as used in DCE, and developed an evaluator at the paragraph level, with prompts provided in Appendix (Table 11). The comparative results between paragraph level and sentence level can be viewed in Fig. 3. While the recall of paragraph-level evaluation is higher on SummEval and QAGS-CNN benchmarks, its overall performance in terms of the F1 score and precision is lower than that of sentence-level evaluations, particularly on the QAGS benchmark. This combination of higher recall and lower precision implies that more candidates are incorrectly marked as consistent. In the task of consistency checking, a metric with low recall and high precision (sentence level) is preferable as it contributes to higher safety compared to a metric with high recall and low precision (paragraph level), erring on the side of caution. Figure 3: F1 score, precision, and recall performance of our method on sentence-level and paragraph-level evaluations. We also verify the effectiveness of the auto-metric converter. In addition to superior accuracy, sentence-level evaluations can facilitate more thorough inconsistency remediation when integrating with RAI. We compared the performance improvement between our sentence level DCE and paragraph level, as indicated in Table 5. Despite the higher recall of the paragraph-level approach, fewer items are flagged as inconsistent, resulting in fewer candidates being corrected, even though the improvement rate is higher. In fact sentence level DCE leads to 25.25% and 39.05% more corrections compared to the paragraph-level approach in SummEval and QAGS-CNN respectively. Therefore, our sentence-level approach not only outperforms in terms of F1 score and precision during consistency checks, but also facilitates comprehensive improvements through RAI. Is Auto-metric Converter Necessary? We present a comparison of our method, both with and without AMC, as shown in Fig. 3. We observe that our method with only the DCE (red bar) performs marginally better on the SummEval dataset but underperforms DCE-AMC (orange bar) on all other benchmarks. Although DCE plays a key role in our method, the AMC component is still desirable and highly necessary not only because it shows better performance, but also because it facilitates the conversion of reasons outputted by DCE to a numeric system. This conversion is both user-friendly and practical, making it easy for humans to understand and apply. Furthermore, it provides a straightforward means of evaluating the effectiveness of the DCE component. Multi-round Consistency Improvement. Table 5 showcases encouraging results on consistency improvement via RAI. This naturally leads to the question: can we further enhance the consistency through multiple rounds of RAI? Fig. 4 shows our investigation on multi-round consistency improvement by iteratively applying RAI. It is noteworthy that the convergence of consistency improvement is remarkably swift, achieving nearly 100% in just two rounds. The convergence rate on the QAGS datasets is highly consistent across both subsets, slightly surpassing SummEval due to its high initial rate after the first round of RAI. This is also corroborated by the frequency distribution of the consistency score (Fig. 4(right)). As the number of rounds increases, the lower consistency scores (<1) gradually decrease, and more inconsistent candidates tend to be consistent, where the score is 1. The Effect of LLM models. We evaluated the performance of our method using different LLMs across all three benchmarks. It is noteworthy that DCE-AMC-4 generally outperforms DCE-AMC-3.5 across most datasets. The performance gap between the two LLM models is relatively minor in terms of semantic consistency (QQP and PAWSQP in Table 2), and the abstractive subset (QAGS-XSUM in Table 4) in factual consistency, but a significant difference is observed in summarization consistency in Table 5. This suggests that GPT-4 can further enhance performance, especially for more complex evaluation tasks. As such, we applied RAI with GPT-4 directly to verify its superior capability in consistency improvement. Nonetheless, the benefits of GPT-3.5, such as higher computational efficiency and lower API costs, should not be overlooked. Computational Cost. We assessed the computational cost of our method based on wall-clock time, which is primarily consumed by LLMs inference. However, the divide-conquer strategy we employed is scalable and easily implemented in parallel. Fig. 5 illustrates the computational cost of GPT-3.5 and GPT-4 with varying numbers of threads on the QAGS-CNN benchmark. A clear reduction in computational cost is observed as the number of threads increases. It’s important to note that the decrease in time is more significant when transitioning from a single thread to four threads, but tends to plateau as more threads are utilized. While GPT-3.5, being the smaller LLM, is a more efficient option, GPT-4 often delivers better performance. 5 RELATED WORK LLM-based Evaluations. Unlike conventional evaluating metrics leveraging token-level or similarity embeddings, such as BLEU (Papineni et al., 2002), ROUGE (Lin, 2004), or BERTScore (Zhang et al., 2020), recent proposed LLM-based evaluators (Wang et al., 2023), such as GPTScore (Jinlan et al., 2023) and G-Eval (Liu et al., 2023b), have demonstrated competitive performance on multiple NLG tasks. Their idea is to utilize the LLMs to score the candidate output under the assumption that the LLMs have learned to assign higher probabilities to fluent and high-quality contexts. However, these LLM evaluators often exhibit lower correlations with human judgments, and their reliability, robustness, and validity remain under-explored (Liu et al., 2023b). Specifically, LLM evaluators may pose potential risks of producing hallucinated or overconfidence scores if the LLM model is not well calibrated for complex tasks (Kadavath et al., 2022; Zhou et al., 2023). This results in limited confidence in using LLM evaluators to directly evaluate paragraph-level responses. Our proposed DCR framework addresses these challenges through a divide-conquer strategy (DCE) coupled with a numeric score system (AMC). Our method quantitatively evaluates paragraphs sentence-by-sentence and does not rely on LLMs to directly output numeric scores, thus providing a more accurate and comprehensive score that better aligns with human feedback. Consistency Evaluations. Consistency checking plays an essential role in a wide range of NLG tasks, including question-answering (Durmus et al., 2020; Wang et al., 2020), factual knowledge extraction (Elazar et al., 2021), summarization (Durmus et al., 2020) and hallucination detection (Manakul et al., 2023). However, due to various limitations of existing methods, such as reliance on additional pre-trained models or question sets (Durmus et al., 2020), it is highly desirable to develop a unified and automatic consistency metric (Wang et al., 2022). Our proposed framework successfully fills this gap and demonstrates superior performance compared to state-of-the-art baselines (Jinlan et al., 2023; Liu et al., 2023b; Wang et al., 2023). More importantly, our proposed RAI enables consistency improvement where the re-generated candidate response significantly helps mitigate LLM hallucinations (Dhuliawala et al., 2023; Mündler et al., 2023; Zhang et al., 2023) in summarization, and open-book QA tasks (Li et al., 2023). 6 CONCLUSION AND DISCUSSION We proposed a general evaluation framework based on a divide-and-conquer strategy for assessing the consistency between the LLM-generated output and the reference texts across various NLG tasks. Moreover, the proposed method can leverage analytical reasoning to generate revised text with improved consistency. Through comprehensive and systematic empirical study across multiple benchmarks in semantic, factual, and summarization consistency tasks, we demonstrated that our approach significantly outperforms existing methods in evaluating and enhancing the consistency of LLM-generated content. Despite these advancements, we acknowledge several potential limitations of our proposed method: Not a Silver Bullet. While our sentence-level approach (DCE-AMC) excels in evaluating consistency and detecting hallucination, it may not be universally effective for all dimensions of text evaluation, even with updated criteria in prompts. For instance, dimensions such as coherence, which pertains to the collective quality of all generated sentences, or relevance, which involves selecting important information and eliminating redundant content from the reference text, require a holistic focus on the entire candidate. These dimensions may not be ideally suited for our DCE-AMC approach. However, if a different evaluator that outputs reasons for action is used, our AMC and RAI could still be employed to quantify and improve performance on such dimensions. Garbage in, Garbage Out. The DCR framework requires two inputs: a reference paragraph and a candidate paragraph. As we use the reference paragraph as the target for consistency and hallucination checks, any non-factual statements present in the reference paragraph would not be detected by our method. Therefore, for tasks such as retrieval-augmented generation (RAG), the accuracy of our method is inherently limited by the correctness of the input paragraphs. 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D9SA02esgh	The volume bound has to be selected a-priori for the dataset. This approach seems to not be extensible for non-local morphologies (e.g. considering long range axons would require looking at the entire brain volume)	MORPHOCC: AN IMPLICIT GENERATIVE MODEL OF NEURONAL MORPHOLOGIES Anonymous authors Paper under double-blind review ABSTRACT Understanding the diversity and complexity of the morphology of different types of neurons is important for understanding neural circuits. We need quantitative, unbiased methods to capture the structural and morphological features of neurons. With the advent of large-scale structural datasets, this analysis becomes feasible using data-drive approaches. Existing generative models are limited to modeling dendritic and axonal skeleton graphs, without considering the actual 3D shape. In this work, we propose MORPHOCC, a model that represents the diversity of neurons in mouse primary visual cortex (V1) in a single neural network by encoding each neuron’s morphology into a low-dimensional embedding. From this embedding the 3d shape can be reconstructed. We train our model on 797 dendritic shapes of V1 neurons. The learned embedding captures morphological features well and enables cell type classification into known cell types. Interpolating between samples in embedding space generates new instances of neurons without supervision. MORPHOCC has the potential to improve our understanding of neurons in the brain by facilitating large-scale analysis and providing a model for representing neuronal morphologies. 1 INTRODUCTION The diversity of neuronal morphologies has fascinated researchers for over 100 years (Ramón y Cajal, 1911). Understanding a neuron’s structure is important, because it constraints the functions the neuron can implement. For example, the length and branching patterns of dendrites and axons affects the way that neurons receive and transmit signals (Goldberg et al., 2004; Hill et al., 2012; Oberlaender et al., 2012). Neurons vary significantly in their morphology: Some neurons have long, massively branching dendrites, which allow them to receive and integrate information from many other cells. Other neurons have a more compact structure, with dendrites and axons that are shorter and less branched (Markram et al., 2015; DeFelipe et al., 2013). Starting with early work by Cajal (Ramón y Cajal, 1911), we have learned a great deal about the different morphological cell types in the brain and some of the core principles of their morphological organization. However, much of this knowledge is based on visual inspection (Ramón y Cajal, 1911; DeFelipe et al., 2013) or manually defined features such as the length and branching patterns of their dendrites and axons (Scorcioni et al., 2008; Armañanzas & Ascoli, 2015; Wang, 2018; Kanari et al., 2019; Gouwens et al., 2019), but does not describe the full heterogeneity and morphological diversity of cell types. To capture this full diversity, we would need generative models that can sample realistic instances of neurons. Previous work on generative models exist, for instance using biologically motivated growth rules (van Pelt & Schierwagen, 2004; Eberhard et al., 2006), manipulating shape templates until they approximately match the observed data (Cuntz et al., 2011; Farhoodi & Kording, 2018) or 3D random walks (Laturmus & Berens, 2021). However, all of these methods generate tree-like representations – skeletons of neurons – and therefore do not generate details beyond the skeletal graph. We propose MORPHOCC, an implicit model for neuronal morphologies that allows clustering and neuron generation at the same time. Our model captures the diversity of cells in a single network and embeds the 3D shapes into low dimensional latent vectors. These latent codes – or “bar codes” – are used to cluster cell types and retrieve neurons. We further use the latent codes to reconstruct the neurons. By interpolating between two neurons’ latent codes, we generate new morphologies. that resemble the previously seen neurons. The analysis on 797 neurons shows that our clustering is consistent with existing knowledge on cell types and MORPHOCC has the potential to reveal new findings in neuroscience. 2 RELATED WORK 3D objects are represented in various ways. We differentiate between explicit and implicit methods. The most common explicit representations are meshes, point clouds and voxels. Meshes describe 3D objects with faces and vertices. Point clouds are a natural choice for representing 3D data acquired from scanning sensors such as LiDAR or depth cameras. Voxels represent 3D objects in a grid-like structure of values. Additionally, neurons tend to be skeletonized into graph-like structures to reduce data complexity. Those skeletons consist of nodes and edges with features, e.g., Cartesian coordinates as node features. There exist several approaches on generating skeletonized neuronal morphologies. One approach is growing tree-like structures based on biologically motivated growth rules (van Pelt & Schierwagen 2004; Memelli et al. 2013; Torben-Nielsen & De Schutter 2014; Koene et al. 2009; Palombo et al. 2019). Ascoli et al. (2001) and Eberhard et al. (2006) developed software tools (L-Neuron and NeuGen) to generate morphologies based on recursive and descriptive, iterative rules, respectively, to model the growth of dendritic patterns of neurons. Such methods are limited to generating neurons according to known rules, but by definition cannot discover new cell types or principles of morphological organization from data. Kanari et al. (2022) introduce a topology-guided synthesis algorithm that generates neurons where they sample topology values and then use a dendritic growth algorithm. Other approaches manipulate shapes until they approximately match the observed data, i.e., by first sampling points or morphologies followed by iterative perturbation (Cuntz et al. 2011; Farhoodi & Kording 2018). MorphVAE (Laternus & Berens 2021) generates neural morphologies using a sequence-to-sequence variational autoencoder that operates on 3D-walks within the tree structure of a neuron and then heuristically combines the random walks into a complete neuron morphology. However, all those methods generate tree-like representations – skeletons – of neurons which do not contain any detail of the neuron, such as the thickness or local curvature of the dendrites. In contrast to relying on explicit data structures to representing signals, implicit methods represent signals by parameterizing a mapping $f(x)$ where $x$ is a location in space (and potentially time) and $f(x)$ are signal properties at $x$. In the context of 3D shapes, $x$ is a point in $\mathbb{R}^3$ and $f(x)$ indicates the location of a point relative to the surface of the object. This results in continuous, memory-efficient representations of the 3D geometry of neurons without topological restrictions. Traditionally, implicit approaches address representing only a single object or scene (Sitzmann et al. 2020b; Takikawa et al. 2021; Martel et al. 2021; Müller et al. 2022). While early approaches used simple MLPs (Mescheder et al. 2019; Chen & Zhang 2018), more recent work incorporated mechanisms to increase detail. This involves Fourier features and periodic activation functions (Sitzmann et al. 2020b). Other approaches are based on a distributed feature volume, which can be generated by encoding an image (Saito et al. 2019; Xu et al. 2019), be distributed across an octree (Takikawa et al. 2021) or an implicit grid (Jiang et al. 2020). These approaches do not fit our purpose as they only represent a single sample. Furthermore, we require the presence of a representative vector (shape code) for each object to enable clustering. Multi-shape representation has been shown by occupancy networks (Mescheder et al. 2019) and IM-Net (Chen & Zhang 2018). However, the methods only work on relatively simple shapes, such as objects in ShapeNet (Chang et al. 2015) and have not been shown to represent fine details of the objects. DeepSDF (Park et al. 2019) introduce a latent code-conditioned auto-decoder that represents a space of shapes. Its representation quality was improved by Duan et al. (2020) through curriculum learning. Next, we discuss approaches closest to ours. MetaSDF (Sitzmann et al. 2020a) uses a meta network to predict the weights of an implicit SIREN network representing each shape. Wiesner et al. (2022) combine DeepSDF and SIREN to model the temporal evolution of growing and dividing C. elegans and lung cancer cells, which are morphologically much less complex than cortical neurons. De Luigi et al. (2023) train an individual, SIREN-based model for each object in the dataset and use its weights to predict a latent code, which forms the contextual input to an implicit decoder. This approach is impractical for large-scale datasets due to its high memory and compute costs. Figure 1: MORPHOCC. a 3D reconstructed mesh of a neuron in V1. Zoom ins show intricate details like spines on its dendrites. b Model architecture of MORPHOCC. The point cloud is encoded into a 64-dim latent vector. The decoder is an implicit model that predicts the occupancy of a sample in 3D space conditioned on the latent vector. The model is trained by optimizing the binary cross entropy between the predicted and the ground truth occupancy. 3 MORPHOCC MORPHOCC is an implicit generative model to represent neuronal morphologies. The architecture consists of an encoder and a decoder (Figure 1). The encoder $g$ is a small PointNet (Qi et al., 2016), which encodes the input point cloud $P$ into an embedding vector $z = g(P)$. The input to the decoder is a 3D coordinate $x$ concatenated with the neurons embedding $z$, and it outputs the probability of the point $x$ being inside the neuron’s volume. Note, for a given neuron the decoder can be queried multiple times but the latent code $z$ needs to be computed only once. We define the occupancy for a point $x \in \mathbb{R}^3$ by the function $\varphi$ where $\varphi(x) = 1$ if $x$ is on the surface of the neuron (occupied) and $\varphi(x) = 0$ if it is outside. We approximate the occupancy function $\varphi$ by a neural network $f$: $$\hat{\varphi}(x) = f(x, z) = f(x, g(P)).$$ Once trained, the surface of the neuron is implicitly represented by the zero iso-surface of $f(x, z)$. 3.1 Architecture The encoder $g$ is PointNet (Qi et al., 2016) consisting of four encoder layers with 128, 256 and 512 units in the hidden layers and 64-dimensional output $z$. It uses batch normalization and ReLU activation functions. The decoder $f$ follows SIREN (Sitzmann et al., 2020b) and is an MLP with eight hidden layers, each with 512 hidden units. The network uses sine activation functions as nonlinearities, except in the last layer, where it uses a sigmoid function to predict the occupancy probability. 3.2 Sampling points for the implicit decoder of MORPHOCC For training our model, we sample multiple 3D coordinates $x$ as input for each neuron in a batch. In each minibatch, we sample 5,000 points randomly from the surface of each neuron in this minibatch. In addition, we sample 5,000 off-surface points. These are composed of 2,000 points drawn uniformly from within the volume containing all cells in the dataset. An additional 2,000 points are sampled uniformly within the tight bounding box of the neuron. The remaining 1,000 points are hard negatives, i.e. points that are close to the surface of the neuron, but outside of it. This way, our model learns the decision boundary between surface and non-surface of the neuron. To generate hard negatives, we sample a non-negative distance along the direction of the surface normals. The distance $d$ is defined as $d = \gamma \Delta + 10^{-3}$, where $\Delta$ is drawn from a log-normal distribution $\Delta \sim \text{LogNormal}(0.002, 1)$ and $\gamma$ is a pre-factor that gets adjusted over training (next paragraph). 3.3 Hard-negative-based Curriculum Learning For training on hard negatives we use a curriculum strategy, where we progressively increase the level of difficulty. Specifically, we decrease the distance of the hard negatives to the neuron’s surface by adjusting the factor $\gamma$ from initially 0.1 to 0.05 over the course of training. The parameters were chosen such that the distances $d$ approximately align with the typical thickness of a neuron’s dendrites. 4 Experiments 4.1 Dataset We base our work on the MICrONS dataset (MICrONS Consortium et al., 2021), a $1.3 \times 0.87 \times 0.82$ mm$^3$ volume of tissue from the visual cortex of an adult P75–87 mouse. The volume has been densely reconstructed using serial section electron microscopy and has been further segmented into individual cells. It includes non-neuronal types and more than 54,000 neurons whose soma was located within the volume. It spans primary visual cortex (V1) and two higher visual areas, antero-lateral area (AL) and rostro-lateral area (RL). We restrict ourselves to roughly 100 $\mu$m column across all cortical layers located in V1 that has been manually proofread and corrected for segmentation errors (Schneider-Mizell et al., 2023). For this subset there are manual cell type labels available. We use these labels only to evaluate our model. We are not using them during training. We refer to the original papers on the dataset (MICrONS Consortium et al., 2021; Schneider-Mizell et al., 2023) and Appendix A.2 for further details on the identification and morphological reconstruction of individual neurons. To generate the input to the encoder, we use the Trimesh library (Dawson-Haggerty et al., 2019) to sample points on the surface of the neuron. We sample twice as many points as there are vertices in the neuron’s mesh (range ca. 152k–2.24M). For each point, we calculate the surface normal vector. We model only the dendritic morphology and remove the axons, because they are not reconstructed accurately for all neurons in this dataset. Preprocessing of the neurons’ point clouds includes centering each neuron on its soma position and scaling (isotropically) by a constant factor across neurons such that all neurons lie within the unit cube $[-1, 1]^3$. We split the dataset into training ($n = 767$), validation ($n = 15$) and test set ($n = 15$). We use the test set for the neuron retrieval task (Subsection 5.3). 4.2 Training We use the Adam optimizer (Kingma & Ba, 2014) and a learning rate of $10^{-5}$. We train for 5,000 epochs with a minibatch size of 24 neurons. In each iteration, we sample 5,000 random points on the surface of the neuron and 5,000 off-surface points (details on the sampling procedure below). The on-surface points form the input to the PointNet encoder; both on- and off-surface points are used to train the implicit decoder. The weights of the encoder are initialized uniformly using Kaiming initialization. For the decoder, the weights $W$ of the first layer are initialized uniformly between $\pm 1$, all others uniformly between $\pm \sqrt{6}/(\omega_0 n)$, where $n$ is the number of inputs and $\omega_0 = 30$ (see Sitzmann et al., 2020b). The loss function for training is the binary cross entropy on the occupancy predictions of the decoder. 4.3 Baselines We compare our reconstruction results to other model architectures that have been used to learn to represent a shape space of objects, namely DeepSDF (Park et al., 2019), Occupancy network (OccNet) (Mescheder et al., 2019) and the model proposed by Wiesner et al. (2022). In addition, we compare different encoders: SIREN (Sitzmann et al., 2020b), DGCNN (Wang et al., 2019) and Point-MAE (Pang et al., 2022) in combination with our implicit decoder. We focus on a comparison of encoder and implicit decoder architectures. All baselines are trained with the same cross-entropy occupancy loss. Figure 2: Latent codes. - a Distribution of cell types in the subvolume of MICrONS Minnie. - b Classifier trained on latent vectors of MORPHOCC. - c-e Confusion matrices of classifier predictions on test sets of cross-validation for c coarse cell type, d layer and e cell type. - f-h t-SNE embeddings (perplexity=30) of the latent codes colored by f coarse cell type, g layer of excitatory neurons and h cell type of excitatory neurons. ### 4.4 Surface reconstruction and visualization We reconstruct an explicit surface mesh from our implicit model using the Marching Cubes algorithm on the predicted occupancy of our decoder, decoding a 3D grid conditioned on the neuron’s latent code of arbitrary resolution. To enhance the quality of our reconstructed meshes used for visualization, we remove small components using a greedy algorithm that progressively adds components until at least 75% of the vertices are included. ### 4.5 Evaluation We evaluate our model and the baselines using the established metrics and procedures by Mescheder et al. (2019). The following metrics are calculated based on the ground truth (GT) mesh and the predicted mesh. To evaluate the reconstruction results of our model, we calculate three metrics: volumetric Intersection over Union (IoU), mean Chamfer-$L_1$ distance (CD) and normal consistency (NC). Volumetric IoU is defined as the intersection of the predicted and GT volume, divided by their union. We sample 100k points, 50k points on the GT surface and 50k in the unit cube, and determine whether the points lie inside or outside the volume of each mesh. Chamfer distance is defined as the mean distance of the points in the predicted mesh to their nearest neighbors in GT. To estimate it, we sample 100k points from both meshes and estimate the distances between nearest neighbors using a KD-tree. Normal consistency measures how well the surface normals of both meshes align by calculating the mean absolute dot product of the normals and compare them to the normals of the corresponding nearest neighbors in the other mesh. The IoU metric is not very expressive in our context, because neurons occupy only an extreme small fraction of the volume compared to typical ShapeNet objects, for which the IoU metric was initially proposed (Mescheder et al., 2019). We therefore developed another metric, which we refer to as “local IoU”, by sampling 100k points that are close to the neuron’s volume. To this end, we first sample a point on the surface of the GT mesh and then add isotropic Gaussian noise. Half of these points are sampled in close vicinity of the neuron (SD 0.001), the other half further away (SD 0.01). The metric is then defined as the IoU of the predicted and GT occupancy of these points. 5 RESULTS 5.1 MORPHOCC’S LEARNED EMBEDDINGS CAPTURE MORPHOLOGICAL FEATURES WELL The encoder embeds the full 3D shape of a neuron into a compact latent vector $ \mathbf{z} $. We start by evaluating the nature of this learned latent space. The dataset contains 797 neurons, the majority of which are seven pyramidal neuron types, along with four types of interneurons (Figure 2a). Qualitatively, the latent space captures the different cell types’ morphological features when reduced to two dimensions using t-distributed stochastic neighbor embedding (t-SNE) (van der Maaten & Hinton, 2008) (Figure 2f-h). Inhibitory cells are predominantly grouped in the top-left region. There is a noticeable gradient from layer 2/3 to layer 4 pyramidal cells, which continues with layers 5 and then 6. Some of the layer 6 cells are more dispersed as they morphologically resemble inhibitory and more superficial cells, and the model is not provided with the laminar location. Layers 5 and 6 are further grouped into distinct cell types, which also cluster in latent space; except for 6P-IT cells, which are morphologically diverse and more dispersed. To quantify the representative power of the latent space, we train a support vector machine (SVM) classifier on the latent codes to predict cell types and layer (Figure 2b). We followed the procedure described in (Weis et al., 2022) to find the best hyperparameters for our classifiers using ten-fold cross-validation on a grid search (details in A.3). Excitatory and inhibitory cells can be classified almost perfectly using a combination of our latent codes and the spine density as an additional input feature (Figure 2c and Table 1), substantially outperforming earlier work using the self-supervised features and spine density (Weis et al., 2022). The latent codes also effectively capture the relative depth towards the pia, leading to accurate classification of excitatory neurons into specific layer boundaries (Figure 2d). The predictions for layer 2/3 and layer 4 exhibit high accuracy with only a few errors (Figure 2e). When it comes to layer 5 and 6 neurons, some degree of confusion arises. Layer 5 comprises three distinct cell types that share greater similarities among themselves than with neurons from other layers. Layer 6 IT cells are highly dissimilar, they differ in size and therefore some resemble more L6-CT neurons, the smaller ones more the layer 2/3 cells. Since the model has no information about the cortical depth, this result is reasonable. We further demonstrate representational power of the latent codes on predicting the manual labels of coarse cell type (Inhibitory/Excitatory, I/E), cell type and layer and compare the results to GraphDINO, a recent representation model on neuronal morphologies (Weis et al., 2022) (Table 1). We observe distinct strengths in their respective latent embeddings. Specifically, MORPHOCC’s latent codes demonstrate superior performance in distinguishing between inhibitory and excitatory cells, displaying a substantial +7% gain in balanced accuracy in this regard, while being inferior to GraphDINO’s latent codes on classification into cell type and layer. While the performance of our model is roughly in the same ballpark, this result suggests that some of the details about a neuron’s shape that are contained in our model’s embeddings are not directly helpful for cell type classification. This is perhaps not surprising, as reconstruction and classification are two very different objectives. \| I/E \| cell type \| layer \| \|-----\|-----------\|-------\| \| GraphDINO \| 92% \| 85% \| 89% \| \| MORPHOCC \| 99% \| 74% \| 84% \| 5.2 RECONSTRUCTIONS SHOW REPRESENTATIONAL POWER OF MORPHOCC An important capability of our model is reconstruction, enabling the visualization of neurons’ crucial structural and morphological features. We now turn to evaluating the reconstruction performance of our approach against a number of baselines. All models capture the rough outline and size of the neurons (Figure 3). However, the DeepSDF architecture fails to represent details in the individual dendrites of the neurons. The model of (Wiesner et al., 2022), which utilizes sine activation functions to represent high-frequency values, reconstructs significantly more details and fine-grained structures. OccNet, while capable of generating reasonable meshes, still falls short in terms of capturing intricate details. MORPHOCC stands out by capturing the most details in the shapes of individual dendrites within the neurons. While this achievement underscores our model’s ability Table 2: Quantitative 3D reconstruction measured using normal consistency (NC), Chamfer-$L_1$ distance (CD) in $\mu$m, volumetric intersection over union (IoU) and localized (volumetric) IoU. \| \| NC ↑ \| CD ($\mu$m) ↓ \| IoU ↑ \| localized IoU ↑ \| \|----------------------\|------\|---------------\|-------\|-----------------\| \| DeepSDF (Park et al., 2019) \| 0.5630 \| 21.18 \| 0.9971 \| 0.22 \| \| Wiesner et al. (2022) \| 0.5523 \| 13.76 \| 0.9987 \| 0.23 \| \| OccNet (Mescheder et al., 2019) \| 0.5747 \| 7.52 \| 0.9948 \| 0.25 \| \| no encoder \| 0.5999 \| 2.37 \| 0.9875 \| 0.41 \| \| Siren encoder (Sitzmann et al., 2020b) \| 0.5970 \| 4.91 \| 0.9862 \| 0.42 \| \| DGCNN encoder (Wang et al., 2019) \| 0.6014 \| 3.48 \| 0.9905 \| 0.40 \| \| Point-MAE encoder (Pang et al., 2022) \| 0.5208 \| 20.65 \| 0.9820 \| 0.28 \| \| MORPHOCC \| 0.6021 \| 4.44 \| 0.9997 \| 0.33 \| to preserve fine structural details during the reconstruction process, there is still clearly room for improvement when comparing to the ground truth. Our qualitative observations are supported by quantitative findings (Table 2): Different variants of our model achieve the best metrics. MORPHOCC with the simple PointNet encoder achieves the highest normal consistency and Intersection over Union (IoU). In terms of Chamfer distance, more sophisticated encoders or directly learning the embeddings (auto decoding) achieve more fine-grained reconstruction than the simple PointNet encoder. The IoU metric proposed by Mescheder et al. (2019) is not very informative in our setting, because neurons are extremely fine structures that occupy only a very small fraction of the volume, rendering almost all off-surface points far away from the surface and all models are above 0.98. We therefore computed an additional localized IoU, which focuses on points close to the neuron. This metric confirms that MORPHOCC outperforms the baselines and that the stronger DGCNN encoder or learned embeddings improve reconstruction quality. However, because the PointNet encoder resulted in qualitatively the best embeddings and is the simplest, we chose to use this version for further analysis despite its somewhat weaker reconstruction quality. Directly learning the embeddings (no encoder) produces the most accurate reconstructions, but the embedding space was not organized semantically at all – the model essentially learned a lookup table and completely overfitted on the samples in the training set. Figure 3: Qualitative 3D reconstruction results. First column: ground truth, following columns: reconstructions of various baselines, last column: reconstruction of our model. 5.3 Infer latent codes for unknown neurons We use MORPHOCC to infer latent codes for unseen neurons. The encoder outputs a latent code which we use to classify the unseen neuron (test set). This functionality is valuable when new neuron shapes become available, enabling us to classify them without the necessity of retraining the entire model. Moreover, this process serves as a testament to the model’s generalization capabilities, as it effectively handles out-of-distribution samples. Here we shown results for neuron retrieval. Given the latent code of an unseen neuron, we calculate its similarity to the latent codes of known neurons and retrieve the five nearest neighbors (NN) along with their respective labels. The label for the unseen neuron is assigned through majority voting among these retrieved neighbors. In Figure 4, we visualize three instances of neuronal retrievals, each presented in a row. Within this context, the blue box represents the unknown neuron, while the neurons in the respective row are the five nearest neighbors, accompanied by their labels below. The first row is labeled as layer 2/3 pyramidal neuron, because four out of five neighbors are identified as 23P cells. Only one cell is a layer 4 pyramidal neuron. Yet, it is crucial to note the discernible similarity between this L4 neuron and the unknown neuron. Remarkably, in subsequent examples, all retrieved neurons share the same label. 5.4 Generation of neuronal morphologies We use our model to generate new neurons based on latent codes. Generation essentially reflects our comprehension of the fundamental attributes that define a neuron. Our approach to generating neuronal morphologies involves interpolation between two distinct neurons. To delve deeper into this process, we interpolate between the latent codes $x_{n_0}$ and $x_{n_1}$ of two neurons $n_0$ and $n_1$, effectively generating intermediate latent codes that lie in between $$x_\alpha = \alpha x_{n_0} + (1 - \alpha) x_{n_1},$$ (2) where $\alpha \in [0.2, 0.4, 0.6, 0.8]$. By querying the decoder with these interpolated latent codes $x_\alpha$ together with samples in a 3D grid, we obtain the predicted occupancy for these samples. We reconstruct the mesh as described in 4.5. This method enables us to not only generate new neurons but also to explore the continuum of neuronal morphologies lying between the two original examples. Figure 4: Neuron retrieval. Test set neuron (blue shaded) with inferred label along with five retrieved neurons based on the MORPHOCC embedding. Figure 5: Interpolation series. First and last neuron are reconstructions, the four shapes in between are neurons generated by interpolating between those neurons. Table 3: Ablation study on sampling strategy, curriculum learning and network architecture. \| Sampling strategy \| NC ↑ \| CD (µm) ↓ \| IoU ↑ \| local IoU ↑ \| \|-------------------------------------------------------\|--------\|-----------\|---------\|-------------\| \| w/o perturbed \| 0.5952 \| 4.67 \| 0.9998 \| 0.30 \| \| w/o uniform & perturbed \| 0.5865 \| 4.49 \| 0.9951 \| 0.31 \| \| w/o restricting to bounding box & perturbed only on surface (training diverged) \| 0.5426 \| 12.04 \| 0.9981 \| 0.25 \| \| w/o curriculum learning \| 0.5999 \| 5.24 \| 0.9991 \| 0.34 \| Network architecture \| ReLU in decoder \| 0.5267 \| 34.93 \| 0.9807 \| 0.22 \| \| shape dim = 32 \| 0.5979 \| 4.61 \| 0.9997 \| 0.34 \| \| shape dim = 32 & hidden layers = 12 \| 0.6008 \| 4.47 \| 0.9995 \| 0.33 \| \| MORPHOCC \| 0.6021 \| 4.44 \| 0.9997 \| 0.33 \| Figure 5 shows a series of interpolations. In each row, the first shape represents the reconstruction of a neuron from our training dataset, while the last shape represents the reconstruction of its neighboring neuron. In between the generated neurons that exemplify a continuum of morphological changes. In the first row, the shape of the neuron changes gradually from one to the other, characterized by the dissipation and regrowth of dendrites in alignment with the neighboring neuron. Notably, the basal dendrites become progressively denser throughout this transformation. The second row depicts a transition from an attuned neuron morphology to one with a small tuft. Finally, the third row showcases a similar transition, but in this case, the neuron undergoes significant shortening. Within this interpolation, the one oblique dendrite in the original neuron gradually dissipates, while multiple new obliques form below. 5.5 Ablation Study Finally, we test how different components of our model, point sampling strategy and training procedure influence the reconstruction performance of our model. All four aspects of our sampling strategy are necessary to achieve the best performance (Table 3), as does introducing the curriculum on the hard negatives during training. In terms of network architecture, we restricted the ablation on the top three architectures of an extensive hyperparameter search. The sine activation function in the decoder is very helpful, as expected from previous work [Sitzmann et al., 2020b]. Reducing the dimensionality of the embedding decreased performance only mildly, as did at the same time increasing the depth of the decoder (Table 3). 6 Conclusion In this paper we introduced MORPHOCC, a model that learns vector representations of 3D neuronal morphologies while also being able to generate new morphologies. Our experiments demonstrate that the model enables the classification of neuronal morphologies into cell types based on the low-dimensional embeddings learned by the model. We show that the embeddings can be used to retrieve similar neurons and apply cell type labels to new neurons. Competitive methods learn the shapes of the neurons but fail to reconstruct the individual dendrites with their relative depth and curvature, while MORPHOCC succeeds in this task. However, our model is not yet able to reconstruct fine-grained details like spines and synapses. In the future, this limitation could be addressed by using a hierarchical, multi-scale model. 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jJvXNpvOdM	The graph-based state representation requires shortest-path computation for all fully connected edges, but this seems quite computationally heavy, especially when we have a large number of nodes, leading to a drastically increasing number of edges.	Task Planning for Visual Room Rearrangement under Partial Observability Karan Mirakhor, Sourav Ghosh, Dipanjan Das & Brojeshwar Bhowmick Visual Computing and Embodied Intelligence Lab TCS Research, Kolkata, India {karan.mirakhor, g.sourav10, dipanjan.da, b.bhowmick}@tcs.com Abstract This paper presents a novel modular task planner under partial observability that empowers an embodied agent to use visual input to efficiently plan a sequence of actions for simultaneous object search and rearrangement in an untidy room, to achieve a desired tidy state. The paper introduces (i) a novel Search Network that utilizes commonsense knowledge from large language models to find unseen objects, (ii) a Deep RL network trained with proxy reward, along with (iii) a novel graph-based state representation to produce a scalable and effective planner that interleaves object search and rearrangement to minimize the number of steps taken and overall traversal of the agent, as well as to resolve blocked goal and swap cases, and (iv) a sample-efficient cluster-biased sampling for simultaneous training of the proxy reward network along with the Deep RL network. Furthermore, the paper presents new metrics and a benchmark dataset - RoPOR, to measure the effectiveness of rearrangement planning. Experimental results show that our method significantly outperforms the state-of-the-art rearrangement methods [Weih et al., 2021; Gadre et al., 2022; Sarch et al., 2022; Ghosh et al., 2022]. 1 Introduction Tidying a disordered room based on user specifications is a challenging task as it involves addressing issues related to perception, planning, navigation, and manipulation [Batra et al., 2020]. An agent performing an embodied room rearrangement must use the sensor observations and a prior knowledge to produce a long horizon plan for generating a sequence of object movements to achieve the tidy goal state. This goal state is specified through geometry, images, language, etc. [Batra et al., 2020]. Majority of the existing research on room rearrangement emphasizes on perception and commonsense reasoning while assuming navigation and manipulation abilities, without incorporating efficient planning. Based on the goal state definition, they broadly fall into two categories; (i) Commonsense based reasoning without a predefined goal state: The methods [Kant et al., 2022; Sarch et al., 2022] in this category utilize image or language-based commonsense reasoning to identify if an object is misplaced from the correct receptacles in their egoview followed by rearranging them using a sub-optimal heuristic planner. Moreover, utilizing text or semantic relation-based anomaly detectors to identify misplaced objects does not resolve blocked goal or swap cases, where an object’s goal position is obstructed by another misplaced object or vice versa. (ii) User-specific room rearrangement with a pre-defined tidy goal state: In this setting, the rearrangement is done based on explicit user specification. Methods like [Weih et al., 2021; Gadre et al., 2022] focus on egocentric perception and use image or image feature-based scene representation to identify misplaced objects and a greedy planner to sequence actions for rearrangement. [Sarch et al., 2022] also performs a user-specific room rearrangement by using semantic relations to identify misplaced objects in agent’s egoview, and then rearrange them as they appear without planning. Methods such as [Kant et al., 2022; Sarch et al., 2022; Gadre et al., 2022] explicitly explore the room to find objects that are initially outside the agent’s egoview, since it only provides a partial information about the room. However, these approaches incur a significant traversal cost due to exploration. Additionally, these methods employ non-optimal planning that does not optimize the number of steps or overall traversal. In contrast, efficient planning makes rearrangement more effective by optimizing the sequence of actions and minimizing the time and effort required to achieve the goal state. [Ghosh et al., 2022], These authors contributed equally. Figure 1: (a) shows the top down view of our Rearrangement task and (b) is the agent’s initial egocentric view in the untidy current state for the same setup. The solid 2D bounding boxes indicate the desired goal state for all objects, while the dashed ones show the initial positions of visible objects in the untidy current state. The dotted 2D bounding boxes represent initial positions of unseen objects in the untidy current state. The sponge (magenta), an unseen object, is in a drawer near the stove, while the tomato (green), another unseen object, is on a stool behind the countertop. There are two scenarios: a blocked goal case with the lettuce (blue) and kettle (yellow) and a swap case between the bread (dark magenta) and pot (dark cyan). addresses the rearrangement task planning problem by assuming the complete visibility of the room, through the bird’s eye view. Their method addresses some planning problems, such as the combinatorial expansion of rearrangement sequencing, and blocked goal and swap cases without explicit buffer. However, the approach does not minimize overall agent traversal during the planning, and its state representation is not scalable to large numbers of objects. Moreover, their reliance on the ground truth object positions in both the current and goal states is impractical in real-life. Our aim is directed towards a novel and more practical aspect of the room rearrangement problem through efficient task planning under partial observability of a room using agent’s egocentric camera view. The major challenges associated with efficient task planning for room rearrangement under partial observability, as shown in Fig. 1, are (i) uncertainty over the location of unseen objects due to partial observability (objects outside the agent’s field of view presently which are visible from a different perspective, or objects placed within a closed receptacle e.g., spoon in drawer), (ii) scalability to a large number of objects, (iii) combinatorial expansion of sequencing due to simultaneous object search (for unseen objects) and rearrangement, (iv) minimizing the overall traversal during simultaneous object search and rearrangement, (v) blocked goal and swap cases without explicit buffer. In this paper, we propose a novel modular method for a task planner to address the aforementioned challenges. At the beginning, our agent captures the goal state by exploring the room to record the semantic and the geometric configuration Batra et al. (2020) of objects and receptacles through egocentric perception. Once the goal state is captured, the objects in the room are shuffled. In the untidy current state, our method partitions the task planning problem into two parts; object search and planning, with the aim of minimizing the overall agent traversal during simultaneous object search and rearrangement. First, we propose a novel commonsense knowledge based Search Network using large language models (LLMs) Liu et al. (2019); Kant et al. (2022) that leverages the object-receptacle semantics to predict the most probable receptacle for an unseen object in the egoview. Second, we use a Deep RL network with hybrid action space Ghosh et al. (2022) to plan our action sequence for simultaneous object search and rearrangement by resolving blocked goal and the swap cases. To this extent, we define the Deep RL state space with a novel graph-based state representation for the current and the goal state that incorporates geometric information about objects. This representation compactly encodes the scene geometry that aids in rearrangement planning and makes the Deep RL state space scalable to a large number of objects and scene invariant. In addition, we present a novel, sample-efficient cluster-biased sampling for simultaneous training of the proxy reward network Ren et al. (2022) and Deep RL to get a better estimate of the problem’s true objective from the episodic reward than the dense reward in Ghosh et al. (2022). The judicious combination of all the The aforementioned components effectively tackle the challenging combinatorial optimization problem in rearrangement as detailed in Sec. 3.6. The major contributions of this paper are: 1. To the best of our knowledge, this is the first end-to-end method to address the task planning problem for room-rearrangement from an egocentric view under partial observability, using a user-defined goal state. 2. A novel Search Network that leverages object-receptacle semantics using the commonsense knowledge from LLMs to predict the most probable receptacle for an unseen object. 3. Use of Deep RL based planner trained with proxy reward to overcome combinatorial expansion in rearrangement sequencing and, to optimize the overall traversal and the number of steps taken. 4. A new Graph-based state representation for the current and goal state to include geometric information about objects, making the Deep RL state space scalable to large numbers of objects and scene-invariant. 5. Introduction of a novel, sample-efficient cluster-biased sampling for simultaneous training of the proxy reward network and the Deep RL network. 6. We introduce a new set of metrics in Sec. 3.4 to obtain a thorough assessment of the rearrangement planner’s effectiveness by not only evaluating the success of the rearrangement, but also taking into account the number of steps taken and the overall agent traversal. 7. To address the inadequacies in existing benchmarks (Weihs et al., 2021) for evaluating task planning under partial observability, we introduce the RoPOR-Benchmark Dataset. We plan to openly release the dataset to enable further research in this domain. 2 METHODOLOGY In our room-rearrangement setup, the agent explores the room to capture the tidy user-specified goal state. During this exploration, the agent creates a 2D occupancy map $M^{2D}$ for the agent’s navigation while, 3D map $M^{3D}$ is utilized to augment the detection of 3D object and receptacle centroids to a fixed global reference frame ($\mathbb{R}^3$). Additionally, we generate an object list $O = \{[W_i, P_i], i = 1, 2, ..., N\}$ and a receptacle list $R = \{[W_i^R, P_i^R], i = 1, 2, ..., N_R\}$. Here, $N$, $W$ and $P \in \mathbb{R}^3$ are the total numbers of objects, their semantic labels, and 3D object centroids, respectively. While $N_R$, $W^R$ and $P^R \in \mathbb{R}^3$ are the total numbers of receptacles, their semantic labels including the room name from Ai2Thor (Kolve et al., 2017), and the 3D receptacle centroids respectively. Then, we randomly shuffle a few objects from the goal state to make the room untidy and fork the agent at a random location in the room. In this untidy current state, the agent’s knowledge is limited to the visible part of the room in its egocentric view. In the agent’s egocentric perception, only a set of objects $O^V = \{[W_i^V, P_i^V], i = 1, 2, ..., N_V\}$ are visible. $N_V$, $W^V$ and $P^V \in \mathbb{R}^3$ are the number of visible objects, their semantic labels, and their 3D object centroids respectively in the current state. Comparing $O$ in the goal state with $O^V$ in the current state allows for determining only the semantics... of unseen objects $ O_{\tilde{V}} = \{ W_{i}^{\tilde{V}}, i = 1, 2, ..., N_{\tilde{V}} \} $, where $ N_{\tilde{V}} $ is the number of unseen objects and $ W_{i}^{\tilde{V}} $ their semantic labels. To plan efficiently and achieve the goal state, the agent must know the positions of all objects in the current state. This involves optimizing the search for unseen objects based on the object-receptacle semantics and simultaneously rearranging visible objects based on their positions in the current and goal state. To this end, we present a modular approach for task planner, as shown in Fig. 2 with: (i) Search network, (ii) Graph-based state representation, (iii) Deep RL network trained with proxy reward. The objective of our task planner is to minimize the number of steps and the agent’s overall traversal by simultaneously sequencing high-level actions to either pick-place misplaced objects or search for unseen objects at predicted receptacles. 2.1 BACKGROUND The agent maps the room in the goal state using an exploration strategy [Sarch et al., 2022] and receives RGB-D images and egomotion information at each step from Ai2Thor [Kolve et al., 2017]. The agent constructs $ M_{2D} $ and $ M_{3D} $ of the environment using the RGB-D input and egomotion. A d-DETR [Zhu et al., 2021] detector is used on the RGB images to obtain 2D bounding boxes and semantic labels for objects and receptacles, and the corresponding 3D centroids are obtained using depth input, camera intrinsic and extrinsic. Finally, the agent has $ O, R, M_{2D}, $ and $ M_{3D} $ from the goal state. In the current state, the agent uses d-DETR detector [Zhu et al., 2021] along with $ M_{3D} $ to obtain $ O_{\tilde{V}} $. The agent uses the Dijkstra path planner on $ M_{2D} $ to navigate and execute high-level actions by assuming perfect motion and manipulation capabilities. 2.2 SEARCH NETWORK We present a novel LLM-based Search Network to reliably predict the receptacles for $ O_{\tilde{V}} $. In case the predicted receptacle is articulated, the agent opens it and looks for the object. The agent uses the predicted receptacle’s position from the goal state to be the probable location for $ O_{\tilde{V}} $ in the current state, since receptacles are static in the room. To this end, we finetune the RoBERTa embeddings to exploit the commonsense knowledge in LLM and learn the semantic relationship between $ O_{\tilde{V}} $ and $ R $. Fine-tuning LLM embeddings is essential because LLMs, being trained on large data corpus, may not necessarily produce human-commonsense compliant predictions for untidy scenes (see the Appendix for more details). Our Search Network (SN) consists of two parts: the Sorting Network (SRTN) and the Scoring Network (SCN). We use RoBERTa-Large model [Liu et al., 2019] to generate pairwise embeddings $ (E_{\tilde{R}}^{V}) $ for $ \{ W_{i}^{\tilde{V}} \}_{i=1,2,...,N_{\tilde{V}}} $ and $ \{ W_{i}^{R} \}_{i=1,2,...,N_{R}} $ in the current state. Therefore, there are $ N_{E} = N_{\tilde{V}} \times N_{R} $ number of embeddings for all the object-room-receptacle (ORR) pairs. Each ORR embedding is classified into one of the 3 classes, based on the probability $ \{ p_{i} \}_{i=1,2,3} $ from the Sorting Network. The ground truth class labels $ \{ Y_{i} \}_{i=1,2,3} $ for each ORR in the dataset (Sec. 3.1) is based on the probability to find an object at that room-receptacle, where $ \{ i = 1 : \text{Most Probable Class}, 2 : \text{Less Probable Class}, 3 : \text{Implausible Class} \} $. SRTN filters out the room-receptacles, where there is a negligible chance of finding the misplaced object. For instance, even in an untidy room, it is nearly impossible to find a cup in the bathtub of a bathroom. This sorting step reduces the scoring network’s computation and minimizes the chances of erroneous scoring of an implausible ORR. We train a fully connected MLP in SRTN using the Cross-Entropy Loss ($ L_{CE} $) as shown in Eq. (1). The Scoring Network estimates probability scores $ \{ \hat{\chi}_{i} \}_{i=1,2,...,N_{SR}} $ for embeddings of higher probability classes, with $ N_{SR} $ representing the total number of such embeddings. SCN provides a probability score metric, to choose the most probable receptacle for $ O_{\tilde{V}} $. For training the fully connected MLP in SCN, we calculate the MSE Loss ($ L_{MSE} $) of probability scores, as in Eq. (2), with respect to the ground truth probability scores $ \{ \chi_{i} \}_{i=1,...,N_{SR}} $. Finally, we get the position $ (P_{i}^{\tilde{V}R})_{i=1,...,N_{\tilde{V}}} $ of the unseen objects as the position of their most probable receptacle. \[ L_{CE} = -\frac{1}{N_{E}} \sum_{i=1}^{N_{E}} \sum_{j=1}^{3} Y_{ij} \log p_{ij} \] \[ L_{MSE} = \frac{1}{N_{SR}} \sum_{i=1}^{N_{SR}} (\hat{\chi}_{i} - \chi_{i})^2 \] To prevent fruitless searches, we implement simple strategies. If the agent cannot find the unseen object at the predicted receptacle, the Search Network identifies the next most probable room-receptacle, and the prior prediction is discarded before re-planning a new sequence. Additionally, if the agent encounters a receptacle on its path that does not contain any unseen objects, it is removed from future searches. The agent updates $ O^V $ whenever it detects an unseen object in its egoview. If the agent locates the unseen object it is searching for before arriving at the predicted receptacle, it updates $ O^V $ and re-plans a new sequence. Refer appendix for more details on the re-planning strategy. ### 2.3 Graph-Based State Representation For our task planning algorithm, we create a spatial graph $(G = \{V, E\})$ representation of the current and the goal state namely $ G_c = \{V_c, E_c\} $ and $ G_g = \{V_g, E_g\} $ respectively. The nodes $ V_c = \{O^V\} $ and $ V_g = \{O\} $. The fully connected edges of the graph contain the path length as edge features, where $ E_c = \{\mathcal{D}(P_i^V, P_j^V)\}_{i \neq j} $ and $ E_g = \{\mathcal{D}(P_i, P_j)\}_{i \neq j} $. The path length $ \mathcal{D}(A_i, A_j)_{i \neq j} $ is the length of the shortest collision free path, computed using Dijkstra, between the 2D projections of $ A_i, A_j \in \mathbb{R}^3 $ on $ M^{2D} $. For unseen objects in the current state, the object nodes and edges in $ G_c $ are augmented with $ P^{\hat{V}R} $ from the search network as $ V_c = V_c \cup \{O^{\hat{V}}, P^{\hat{V}R}\} $ and $ E_c = \{\mathcal{D}(\overline{P}_i, \overline{P}_j)\}_{i \neq j} $, where $ \overline{P} = P^V \cup P^{\hat{V}R} $. This graph representation helps the Deep RL state space to understand the semantic and geometric information of the current and the goal state. We use a novel Graph Representation Network (GRN) with an encoder-decoder to generate meaningful embeddings from $ G_c $ and $ G_g $ for Deep RL state space to incorporate the residual relative path length notion between every pair of current and goal state nodes. GRN consists of two major blocks, the Graph Siamese Encoder Network (GSEN) and the Residual Geodesic Distance Network (RGDN). GSEN uses a Graph Convolution Network (Gao et al., 2020) to encode the graphs $ G_c $ and $ G_g $ and produce the graph embeddings $ Z_c $ and $ Z_g $ respectively. These graph embeddings are concatenated to get the final embeddings $ Z_p = Z_c \cup Z_g $. RGDN acts as a decoder and predicts the residual relative path length $ \tau_p $ between the two graphs. This network is trained in a supervised way as in Eq. (3), using the Graph Dataset (Sec. 3.1), which contains the ground truth relative path length ($ \tau $) between the two graphs. This graph embedding makes the Deep RL state space invariant to a large number of objects and the scene. This compact representation concisely encodes the pairwise distance between the source and target nodes which aids in the reduction of the combinatorial expansion of rearrangement sequencing. \[ \tau_p = \text{GRN}(G_c, G_g) \] \[ L_{GRN} = \|\|\tau - \tau_p\|\|^2 \] ### 2.4 Deep RL Based Planner Our task planner needs to select the objects or the probable receptacles for the unseen objects in an efficient manner, to minimize the overall traversal of the agent to simultaneously search the unseen objects and rearrange the visible ones. Moreover, the planner needs to identify free locations, when selecting objects with swap cases. #### 2.4.1 Parameterized Deep-Q Network In order to achieve the aforementioned goals, we implement a Parameterized Deep-Q Network with hybrid action space, similar to Ghosh et al. (2022). We define a binary Collision vector $(C_N \times 1)$, that signifies the objects with a blocked goal or swap case. The Deep RL state space defined as $ s = Z_p \cup C $. Each action $\{a_i = (k, p_k)\}$ in our sequence of actions $\{a_i\}_{i=1,2,...,K}$ of length $ K $ is made up of a discrete action $ k $, denoting the index of the selected object or the probable receptacle, followed by a continuous parameter $ p_k $ which signifies the location for object placement or receptacle search. We use a Parameter network $(\Phi_P)$ and the Q-network $(\Phi_Q)$ to generate a continuous parameter $ p_k $ and a discrete action $ k $ respectively, similar to Ghosh et al.. According to a Markov Decision Process (MDP), our method receives a reward $ r(s, a) $ at each time step $ t $, for choosing an action $ a $, that advances the agent from the current state $ s $ to the next state $ \bar{s} $. Inspired by the work in Ghosh et al. (2022); Bester et al. (2019), we define the Q-values as a function of the joint continuous action parameter $ p = [p_k]_{k=1,2,...,K} $ instead of updating the Q-values with its corresponding continuous parameter sample $ p_k $. The modified Bellman equation is shown in Eq. (4). This prevents our method from producing degenerate solutions by incorporating the effect of other parameters for updating the Q-values. \[ Q(s, k, p) = \mathbb{E}_{r, \bar{s}}[r + \gamma \max_{k \in K} Q(\bar{s}, \bar{k}, \Phi_P(\bar{s}))\|s, k, p] \] The loss function $L_P(\Phi_P)$ and $L_Q(\Phi_Q)$ for the parameter network($\Phi_P$) and the Q network($\Phi_Q$), is given by Eq. (5) $$L_P(\Phi_P) = - \sum_{k=1}^{K} \sum_{r=1}^{R_B} Q(s, k, \Phi_P(s); \Phi_Q)$$ $$L_Q(\Phi_Q) = E_{(s,k,p,r,\bar{s}) \sim R_B} \left[ \frac{1}{2}(y - Q(s, k, p; \Phi_Q))^2 \right]$$ Here, $y = r + \gamma \max_{k \in K} Q(\bar{s}, \bar{k}, p(\bar{s}; \Phi_P); \Phi_Q)$ is the updated target from Eq. (4), and $R_B$ is the replay buffer. $L_P(\Phi_P)$ indicates how the $p$ must be updated to increase the Q-values. Here $\Phi_Q$ works as critic to $\Phi_P$. For Long Horizon planning, the sparse reward is not sampling efficient for training the Deep RL [Gehring et al. (2021)]. Hence, we use step-wise environmental feedback based on the hierarchical dense reward similar to Ghosh et al.. The detailed reward structure is explained in the Appendix. This reward structure provides per-step feedback, but we need episodic reward-based feedback to improve RL policy generalization [Amodei et al. (2016), Dewey (2014)]. Thus, for every episode ($\Lambda$), we calculate the episodic reward ($R_{ep}$) using the step-wise hierarchical dense reward ($r$) and overall episodic path length ($L$) as in Eq. (6), and save the reward and each step $(s, a, \bar{s})$ of the episode into the replay buffer ($R_B$). As this episodic reward is sparse, we use a proxy reward network to generate per-step dense Markovian reward with an episodic notion. ### 2.4.2 Proxy Reward Network Our proxy reward network is trained on the sampled experience data from the replay buffer, to give our agent a notion of the overall objective of the episode. The random return decomposition (RRD) method used in Ren et al. (2022), trains a proxy reward network by randomly sampling steps from an episode. This training method is not sample efficient because it uniformly samples the steps without considering the reward distribution in the episode. To this end, we propose a novel cluster-biased return reward decomposition (CB-RD) to train our proxy reward network. We cluster the per-step reward for the episode into 3 clusters each of size $T_j$, where $j \in \{1, 2, 3\}$, using the c-means clustering. These clusters represent the reward distribution in an episode. This information helps us to efficiently sample $N_s$ number of steps from the episode. We randomly sample $U_j = \{(s_{ij}, a_{ij}, \bar{s}_{ij})\}_{i=1}^{N_j}$ from each cluster $j$, such that $N_j = N_s \times T_j/N_{ep}$. Using $\{U_j\}_{j=1,2,3}$, we estimate the learned episodic reward ($R_{ep,\theta}$) from the proxy reward network ($r_\theta(s, a, \bar{s})$), where $\theta$ is the learned weight. $$R_{ep} = \frac{N_{ep}}{L} \sum_{i=1}^{N_{ep}} r_i$$ $$R_{ep,\theta} = \sum_{j=1}^{3} p_j \frac{T_j}{N_j} \sum_{i=1}^{N_j} r_\theta(s_{ij}, a_{ij}, \bar{s}_{ij})$$ $$L_{CBRD} = \frac{1}{M} \sum_{i=1}^{M} \left[ (R_{ep,i} - R_{ep,\theta,i})^2 \right]$$ Here, $M$ is the number of episodes sampled, $N_{ep}$ is the number of steps in an episode and $p_j = T_j/N_{ep}$ is the uniform probability of choosing a sample from the episode that belongs to cluster $j$. We simultaneously train our Deep RL using Eq. (5) and proxy reward network using Eq. (8) as shown in Algorithm 1. Fig. 3 shows that CB-RD provides effective feedback to our Deep RL method to achieve a higher average return in a lesser number of steps during training. Hence, CB-RD makes our Deep RL method more sample efficient compared to RRD, hierarchical dense reward and sparse reward. We use an off-policy method with a replay buffer to train our Deep RL method with a diverse set of rearrangement configurations, similar to the work proposed by Kalashnikov et al. (2018). We use the $ \epsilon $-greedy method (Kalashnikov et al., 2018) to strike a balance between exploration and exploitation. We stabilize our Deep RL training using target networks for $ \Phi_Q $ and $ \Phi_p $, and update the weights of target networks using polyak (Lillicrap et al., 2015) averaging similar to Bester et al. (2019); Ghosh et al. (2022). Our ablation study in Appendix, shows that the selection of $ \epsilon $ has a significant impact on the solution. 3 EXPERIMENTS In this section, we describe the datasets, metrics, and detailed results of our proposed method and its modules, in addressing the room-rearrangement problem. 3.1 DATASET Graph Dataset: We generate this dataset to train GRN using Ai2Thor (Kolve et al., 2017), by randomly placing objects for two types of rearrangement scenarios: (i) 40% without goal occupied rearrangement: by placing the objects in free spaces and (ii) goal occupied rearrangement: by placing the object in another object’s target. Search Network Dataset: The AMT dataset in Kant et al. (2022) contains 268 object categories in 12 different rooms and 32 receptacle types. Each object-room-receptacle (ORR) pair is ranked by 10 annotators in 3 classes: correct (positively ranked), misplaced (negatively ranked), and implausible (not ranked). For our problem statement, the misplaced class is of utmost importance. Hence, we rename the classes as (i) misplaced class → most probable class, (ii) correct class → less probable class, and (iii) implausible class remains the same. We find the ground truth score values for each ORR as the mean inverse of the ranks. 3.2 BENCHMARK DATASET FOR TESTING The existing benchmark dataset, RoomR (Weihs et al., 2021), has limitations as it only allows up to 5 objects, no object placement within another receptacle, and no blocked goal or swap cases. Thus, it cannot fully evaluate planning aspects such as the number of steps taken, agent traversal, blocked goal, or swap cases. To address this, we introduce RoPOR, a new benchmark dataset for testing task planners in Ai2Thor. It includes a diverse range of rooms (120) and object-receptacle pairs (118), allowing for a wide variety of rearrangement scenarios with up to 20 objects and random partial observability cases, object placement within receptacles in the current state, and blocked goal and swap cases. Moreover, object placement configurations in RoPOR affect sub-optimal planning policies in terms of agent traversal. The mean room dimensions along x-axis and y-axis are 3.12m and 5.80m, respectively. Refer Appendix for details on the distribution of objects, rooms and receptacles. 3.3 TRAINING The training details of our Search network, Graph-based state Representation Network, Deep RL planner, and proxy reward network are available in the Appendix. 3.4 METRICS Metrics in Weihs et al. (2021) do not highlight the efficacy of a task planner to judge efficient sequencing to reduce the number of steps taken or the agent traversal during rearrangement. For a fair evaluation of our method, and comparison against the existing methods and ablations, we define new metrics: - SNS: Success measured by the inverse Number of Steps uses a binary success rate ($ S $) to evaluate the successful completion of a rearrangement episode along with the number of steps ($ N_T $) taken by \| Number of Objects \| Visible Objects \| Unseen Objects \| Swap Case \| Ours-GT \| Ours \| Weih et al. \| Gadre et al. \| Sarch et al. \| Ghosh et al. \| \|------------------\|-----------------\|----------------\|-----------\|--------\|------\|------------\|-------------\|--------------\|--------------\| \| 5 \| 5 \| 0 \| 0 \| 0 \| 138 \| NC \| 12.57 \| 0.74 \| NC \| \| \| 5 \| 0 \| 0 \| 2 \| 0.76 \| NC \| 23.36 \| 0.53 \| NC \| \| \| 3 \| 2 \| 0 \| 0 \| 0.81 \| 0.61 \| 12.93 \| 0.60 \| 0.48 \| \| \| 3 \| 0 \| 2 \| 0 \| 0.79 \| 0.60 \| 13.39 \| 0.58 \| 0.47 \| \| 10 \| 10 \| 0 \| 0 \| 4 \| 0.70 \| NC \| 24.63 \| 0.52 \| NC \| \| \| 10 \| 0 \| 0 \| 6 \| 0.84 \| 0.69 \| 23.78 \| 0.64 \| 0.53 \| \| \| 6 \| 4 \| 0 \| 0 \| 0.84 \| 0.69 \| 23.78 \| 0.64 \| 0.53 \| \| \| 6 \| 0 \| 4 \| 0 \| 0.84 \| 0.69 \| 23.78 \| 0.64 \| 0.53 \| \| 20 \| 20 \| 0 \| 0 \| 8 \| 0.70 \| NC \| 45.32 \| 0.52 \| NC \| \| \| 12 \| 8 \| 0 \| 0 \| 0.87 \| 0.75 \| 41.29 \| 0.67 \| 0.58 \| \| \| 12 \| 0 \| 8 \| 0 \| 0.87 \| 0.74 \| 42.13 \| 0.66 \| 0.57 \| Table 1: (OOF : Objects outside agent’s field of view initially, which are visible from a different perspective, OPR : Objects placed inside closed receptacles, NC : Not computable). When there are no unseen objects, the ENR is NC. Similarly, when SNS is zero, ENR and ATC are NC. Weih et al., Gadre et al., and Sarch et al. do not handle 20 objects and cannot resolve swap cases without explicit buffer or OPR cases (SNS = 0). Ghosh et al. shows a slight decline in performance as the number of objects increase under complete visibility and swap cases, but fails to account for unseen objects. In comparison, Ours significantly outperforms Weih et al., Gadre et al. and Sarch et al. in terms of SNS, ENR, and ATC for visible objects, unseen objects, and swap cases without explicit buffer. Similarly, ours-GT performs better than Ghosh et al. in terms of SNS and ATC under complete visibility and swap cases without explicit buffer. an agent to rearrange a given number of objects $ N $. $ S $ is 1 if all object positions in the current and goal state are approximately equal. Higher the SNS implies a lower $ N_T $ for a given $ N $, indicating more efficient and successful rearrangement episode. $ (SNS = S \times N/N_T) $ • ENR: Efficiency in Number of Re-plans during object search by taking the ratio of the number of unseen objects initially ($ N_{\bar{V}} $) with respect to the number of attempts to search ($ N_{S\bar{V}} $). A higher ENR shows a lower $ N_{S\bar{V}} $ for a given $ N_{\bar{V}} $ indicating a more efficient search to find unseen objects. $ (ENR = N_{\bar{V}}/N_{S\bar{V}}) $ • Absolute Traversal Cost (ATC)*: The metric shows the overall distance traversed by the agent during the successful completion of a rearrangement episode. In an identical test configuration, a lower ATC indicates a more efficient rearrangement sequencing. ### 3.5 Ablation We ablate our task planner against ground-truth perception, various methods for object search and a dense reward structure. To study the effect of erroneous perception on our task planner, we assume the availability of Ground-Truth object detection labelling and 3D centroid localisation from Ai2Thor (Ours-GT). To understand the importance of our Search Network in planning, we replace it by a (i) Random Search policy (Ours-RS), which predicts probable receptacles for unseen objects with uniform probability and a (ii) Greedy Exploration strategy (Ours-GE) Chaplot et al. (2020) that optimizes for map coverage to discover all the unseen objects. To highlight the generalisation of proxy reward network to the overall objective of the rearrangement episode, we replace it with a hierarchical Dense Reward structure Ghosh et al. (2022) (Ours-DR). Please refer to the appendix to find the results for the ablations, along with the analysis for the choice of hyper-parameters for each of our learning based modules. ### 3.6 Quantitative Results We evaluate our approach along with the existing methods on RoPOR - Benchmark Dataset in Ai2Thor. Tab. 1 indicates that our method is scalable to large number of objects, as demonstrated by the consistent value of SNS despite the increasing number of objects across complete visibility, partial observability, and swap cases without an explicit buffer. The gradual increase in ENR with the increase in number of objects can be attributed to the fact that rearrangement of visible objects and the search for some unseen objects, indirectly aids in finding other unseen objects. Comparing our method against Housekeep Kant et al. (2022) would be unfair because it does not perform a user-specific room-rearrangement with a pre-defined goal state. Instead, we have compared our method to previous works such as Weih et al. (2021), Gadre et al. (2022), Sarch et al. (2022) and Ghosh et al. (2022), all of which have demonstrated results for a user-specific room-rearrangement. For a fair comparison with Weih et al., we have used their best performing model - RN18+ANM, PPO+IL. Since, Ghosh et al., uses groundtruth object positions in the current and the goal state, we compare it with our ablation method Ours-GT. Without erroneous perception, Ours-GT demonstrates efficient planning, by performing significantly better than all the existing methods; Weih et al. (2021), Gadre et al. (2022), Sarch et al. (2022); Ghosh et al. (2022), including Ours, in terms of SNR, ENR and ATC. Under complete visibility, ours significantly outperforms Weihs et al., Gadre et al. and Sarch et al. in terms of SNS and ATC. Similarly, Ours-GT significantly outperforms Ghosh et al. in terms of ATC. The improvement over Weihs et al., Gadre et al. and Sarch et al. shows their heuristic planner is neither scalable nor does it optimize the overall agent traversal or the number of rearrangement steps. In contrast, our method leverages compact graph-based scene geometry capable of addressing large numbers of objects, and robust Deep RL makes our planner efficient in reducing the redundant traversal of the agent. Our method uses path length cost and proxy reward with the episodic notion, which helps to improve the overall traversal of the agent to produce lower ATC. In comparison, Ghosh et al. uses greedy Euclidean distance based reward without having an episodic notion, thus failing to optimize overall traversal. Moreover, Ghosh et al. shows a drop in performance on the RoPOR dataset as compared to their results evaluated on RoomR [Weihs et al. (2021)], due to the variations in the testing scenarios in RoPOR that significantly impact agent traversal for sub-optimal rearrangement policies. Under partial observability, there are two cases - (i) OOF: Objects located outside the field of view initially which are visible from a different perspective and (ii) OPR: Objects placed inside closed receptacles. In the case of OOF, our method substantially outperforms Weihs et al., Gadre et al. and Sarch et al. in terms of SNS, ENR and ATC. All these above methods use greedy sub-optimal planners and employ explicit scene exploration to find objects outside the field of view, incurring huge traversal cost as indicated by their ATC. To gauge the performance of the exploration strategy for object search in terms of ENR, we consider each newly generated location or a set of navigational steps from the exploration policy as a search attempt. Our approach’s significantly higher ENR shows that the Search Network outperforms the exploration policies of [Weihs et al. (2021); Gadre et al. (2022); Sarch et al. (2022)] in terms of the number of attempts to find unseen objects. Ghosh et al. does not address any case of partial observability. While Weihs et al., Gadre et al. and Sarch et al. do not solve the case of OPR, which involves object placement inside receptacles (SNS = 0). However, our approach performs equally well in both cases of partial observability due to our search network’s ability to comprehend a commonsense based semantic relationship between an object and any type of receptacle - rigid or articulated. Swap cases without an explicit buffer are not handled by Weihs et al., Gadre et al. and Sarch et al., which is evident from SNS = 0. Ours, Ours-GT and Ghosh et al. can effectively resolve an increasing number of swap cases without an explicit buffer using the hybrid action space [Ghosh et al. (2022)] in the Deep RL network. However, Ours-GT performs better than Ghosh et al. in terms of ATC due to a novel collision resolution reward that optimizes the agent’s traversal. To ground the values of our RoPOR dataset, we show the results for Ours, the ablation methods and the SOTA in the test set of RoomR in the Appendix. Moreover, additional results for individual methods in our pipeline can be found in the Appendix. 3.7 QUALITATIVE RESULTS To show the results of our method in room-rearrangement, we have created videos in a number of test scenarios to highlight the robustness of our method. We also test our method in a new environment - Habitat, as demonstrated in our supplementary video. This transfer does not require any additional training for our Search Network, Graph-based State Representation or Deep RL planner. This shows the capability of our method for seamless sim-to-sim transfer, further emphasizing its suitability for real-world deployment. Please refer the supplementary video. 4 LIMITATIONS Our approach is not capable of identifying unseen objects that are occluded due to clutter on receptacles (for e.g. a spoon may become occluded, if bread, box, lettuce etc. is placed before it). Our method also assumes the availability of perfect motion planning and manipulation capabilities. 5 CONCLUSION This paper presents an innovative task planner designed for organizing rooms under conditions of partial observability. Our approach minimizes agent traversal and step count during both object search and rearrangement by leveraging a Search Network followed by a Deep RL-based planner. By utilizing a graph-based state representation and episodic proxy reward, our method exhibits versatility and applicability across a range of scenarios. The RoPOR benchmark dataset facilitates additional research in the realm of Embodied AI-based rearrangement. Future endeavors will concentrate on deploying our approach in real-world settings. 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9ztL7Trdnx	If TAFS employs an MLP to approximate the optimal activation function, the performance of activation function would also depend on the number of layers in the MLP and the non-linear transformation functions. This has not been discussed in this paper.	TAFS: Task-aware Activation Function Search for Graph Neural Networks Anonymous authors Paper under double-blind review Abstract Since the inception of Graph Neural Networks (GNNs), extensive research efforts have concentrated on enhancing graph convolution, refining pooling operations, devising robust training strategies, and advancing theoretical foundations. Notably, one critical facet of current GNN research remains conspicuously underexplored—the design of activation functions. Activation functions serve as pivotal components, imbuing GNNs with the essential capacity for non-linearity. Yet, the ubiquitous adoption of Rectified Linear Units (ReLU) persists. In our study, we embark on a mission to craft task-aware activation functions tailored for diverse GNN applications. We introduce TAFS (Task-aware Activation Function Search), an adept and efficient framework for activation function design. TAFS leverages a streamlined parameterization and frames the problem as a bi-level stochastic optimization challenge. To enhance the search for smooth activation functions, we incorporate additional Lipschitz regularization. Our approach automates the discovery of the optimal activation patterns, customizing them to suit any downstream task seamlessly. Crucially, this entire process unfolds end-to-end without imposing significant computational or memory overhead. Comprehensive experimentation underscores the efficacy of our method. We consistently achieve substantial improvements across a spectrum of tasks, including node classification over diverse graph data. Moreover, our approach surpasses state-of-the-art results in the realm of link-level tasks, particularly in biomedical applications. 1 Introduction Graph Neural Networks (GNN) have demonstrated their prowess in modeling relationships within graph-structured data, as evidenced by their superior performance in various domains (Kipf & Welling, 2017; Velickovic et al., 2017; Hu et al., 2020; Xu et al., 2019). They have excelled in applications spanning biomedicine (Wu et al., 2023; Jiang et al., 2021), physical simulation (Sanchez-Gonzalez et al., 2020), material design (Reiser et al., 2022), sustainability (Donon et al., 2020), social network (Fan et al., 2019), transportation (Li et al., 2018b), recommendation (Wu et al., 2019), and more. Consequently, GNN models continue to captivate the attention of researchers across diverse scientific communities (Shi et al., 2020; Wang et al., 2022; Seo et al., 2020). Despite the extensive body of literature, we must highlight a significant gap in current research, specifically the design of activation functions, a fundamental component used in nearly every GNN model. While Rectified Linear Unit (ReLU) (Nair & Hinton, 2010) is a prevalent choice for activation, it often falls short, as illustrated in Figure 1. Regrettably, GNN studies have hardly explored alternative activation functions. This oversight is critical, as the activation function plays a pivotal role in introducing non-linearity to GNNs. Without it, GNNs merely perform linear transformations on raw graph features. In contrast, the Computer Vision community has spent decades exploring a wide array of manually designed activation functions such as Sigmoid (LeCun et al., 1998), Tanh, ReLU (Nair & Hinton, 2010), and improved variants of ReLU (He et al., 2015; Clevert et al., 2016; Maas et al., 2013). However, transferring these manually crafted functions to different tasks poses challenges, and customizing new ones is a labor-intensive process. Furthermore, the marginal performance gains from human-designed functions diminish rapidly. To address this, researchers have proposed automated methods to discover tailored activation functions, which have demonstrated notable improvements in other network architectures like Convolutional Neural Networks (CNNs) or Recurrent Neural Networks (RNNs) (Ramachandran et al., 2018; Eger et al., 2018; Farzad et al., 2019). Hence, our research question is how can we design GNN activation functions to adapt effectively to various graph-based tasks, creating task-aware activation functions? Addressing this question poses two primary challenges. Challenge #1: Existing search algorithms are inefficient. Current activation function search methods suffer from over-parameterization and heavy computation. For example, APL (Agostinelli et al., 2015) introduces additional parameters for each neuron, which usually leads to at least ten times more parameters upon any model, thus significantly increasing the model complexity. Swish (Ramachandran et al., 2018) requires training a full network until convergence for each iteration, making it computationally burdensome. These issues render current search algorithms inefficient and less effective. Challenge #2: Current search methods lack support for non-differentiable objectives. GNN methods have wide applications regarding tasks of different levels (node, link, graph), many of which are evaluated by non-differentiable metrics. In the case of drug interaction prediction, we would like to know a certain positive (synergy) or negative (confliction) interaction exist. In fact, most drug pairs does not have positive or negative interactions. Receiver Operating Characteristic curve (ROC) which is not differentiable, is what we should use instead of accuracy. Similar application cases can be found in hit ratio of recommendation, latency optimization, hardware resources constraint, etc. Supporting these non-differentiable objectives would broaden the applicability of activation function search in diverse GNN tasks. In this study, we embark on a systematic exploration of GNN activation function search—a first of its kind. We frame this search as a bi-level optimization problem, with the inner level optimizing GNN parameters and the outer level optimizing activation function parameters. We propose an efficient search algorithm that navigates through a compact search space. This space is characterized by universal approximators with additional smoothness constraints, facilitating the rapid discovery of high-quality functions, thereby addressing Challenge #1. Additionally, we tackle Challenge #2 by jointly considering non-differentiable objectives and potential activation function constraints. We incorporate these elements into a stochastic relaxation of the outer level optimization, removing the need to compute gradients for non-differentiable metrics used in GNN tasks. Our algorithm undergoes extensive experimentation across various GNN models, datasets, and objectives, consistently outperforming existing activation functions. By overcoming Challenges #1 and #2, our algorithm achieves task-awareness in GNN activation function design. Our contributions can be summarized in three key points: 1. To the best of our knowledge, we are the first to propose activation function search in the context of Graph Neural Networks. Our work serves as a catalyst, drawing attention to this critical aspect of GNN model design and paving the way for future investigations. 2. We propose TAFS (Task-aware Activation Function Search), a probabilistic search algorithm capable of efficiently exploring a regularized functional space to discover novel activation functions tailored for diverse downstream tasks. 3. Through comprehensive evaluations spanning node and link level tasks, we demonstrate that our algorithm enhances activation function design without requiring extensive manual effort and excels in optimizing non-differentiable objectives. We also conduct ablation studies to examine the searched activation functions, the impact of design choices, and algorithm efficiency. 2 RELATED WORKS Graph Neural Networks. GNN is a power model in capturing relational information including typically GCN (Kipf & Welling, 2017), GAT (Velickovic et al., 2017), GIN (Xu et al., 2019), etc. Mathematically, in the context of GNN with a given network $G = \{V, E\}$ containing node set $V$ and edge set $E$, the problem is formulated below: \[ z_u^{(l+1)} = \text{UPDATE}^{(l+1)} \left( h_u^{(l)}, \text{AGGREGATE}^{(l)} \left( \{ h_v^{(l)}, \forall v \in N(u) \} \right) \right), \tag{1} \] where $ h_u^{(l)} $ is the latent representation of node $ u $ at layer $ l $ and $ z_u $ is the pre-activation of node $ u $. AGGREGATE (abbr. Agg) and UPDATE (abbr. Up) are core modules of GNN, denoting the different message passing operations used across the model for collecting and updating representations. The latent representation (respectively pre-activation) for all nodes constitute $ H $ (resp. $ Z $) and then we have the activation transformation: $ H^{(l+1)} = \sigma^{(l+1)}(Z^{(l+1)}) $, where $ Z $ is activated by function $ \sigma $ at the $(l + 1)$th layer. Numerous studies have been devoted to ameliorate it through different aspects. For example, GCN (Kipf & Welling, 2017) simplifies graph convolution and derives performant GCN networks. GAT (Velickovic et al., 2017) proposes graph attention as a replacement to model global features. GNN-Pretrain (Hu et al., 2020) studies pretraining strategies at the node level and graph level to make GNN model work for transferable tasks. GIN (Xu et al., 2019) understands the fundamental question of graph expressiveness by discriminating Weisfeiler-Lehman graph isomorphism. GNN Co-training (Li et al., 2018a) connects Laplacian smoothing with graph convolution and studies the problem of oversmoothing. However, almost every GNN model uses ReLU as the activation function (Kipf & Welling, 2017; Velickovic et al., 2017; Xu et al., 2018; Huang et al., 2020; KC et al., 2022; Xu et al., 2019), leaving GNN activation function a missing research piece. Activation Function Design. Since the early application of Sigmoid in Le-Net, activation functions have been considered as an important component until today (Hayou et al., 2019). In 2012, ReLU (Nair & Hinton, 2010) was proposed to train Boltzman Machines and soon extensively adopted in every neural network models. The study of activation function design happens mostly in CNN community, where a couple of milestone works include Swish (Ramachandran et al., 2018; Eger et al., 2018) and APL (Agostinelli et al., 2015). Swish proposes a Reinforcement Learning (RL)-based search algorithm to find appropriate activation functions in a discrete space. APL (Adaptive Piecewise Linear) uses linear hinge functions to approximate target patterns in a differentiable way. Comprehensive surveys of manual designed and parametric activation functions can be found in (Apicella et al., 2021; Dubey et al., 2022). Another notable work related to our research question is GReLU (Zhang et al., 2022), which tries to make GNN activation function adaptive by including graph convolution into the activation function. However, such design is not a typical univariate activation function. As a result, no work yet has proposed novel activation functions designed under the context of GNN. ### 3 PROBLEM FORMULATION AND CHALLENGES Our research problem requires to propose a systematic way of designing adaptive activation functions that can be effectively integrated into GNN for downstream applications. Similar to Neural Architecture Search (NAS) (Liu et al., 2019), the activation function design could be modeled into a bi-level optimization problem: \[ \min_{\alpha} M(w^(\alpha), \alpha; D_{\text{val}}) \quad \text{s.t.} \quad w^(\alpha) = \arg \min_w L(w, \alpha; D_{\text{train}}), \tag{2} \] where the inner level optimization learns $ w $ weight of GNN and the outer level optimization learns $ \alpha $ weights of activation function. Both levels may have different objective metrics $ M, L $ depending on downstream applications. Previous activation function search methods suffer from low efficiency (Challenge #1) and poor support of non-differentiable metrics (Challenge #2). On one hand, the efficiency bottleneck lies in the search space choice and search strategy design. The search space is crucial for search efficiency and requires careful consideration. The space should be proper both in candidate function number and effectiveness, making it a trade-off between quantity and quality. Then, the search strategy should be able to discover as quickly as possible the most suitable function candidate in the space. On the other hand, diverse GNN applications requires that the algorithm is able to tackle any downstream target metric, whether or not differentiable. All these issues prevent us from using off-the-shelf algorithms for GNN activation function search. To this end, we need a novel parameterization of the search space, that is jointly designed with search strategy to allow efficient search, and we need to deal with differentiable and non-differentiable target metrics at the same time to enable general applications in all kinds of GNN tasks. Figure 2: Algorithm framework. We replace the activation function with sampled weights from Gaussian distributions. The inner level optimization is the learning of distribution parameter $\theta^{(1)}, \ldots, \theta^{(L)}$ and the outer level optimization is the learning of GNN weights. Both levels are iterated for faster convergence. 4 THE PROPOSED METHOD 4.1 ALGORITHM FRAMEWORK As illustrated in Figure 2 and Table 1, we propose TAFS to solve both challenges in a unified way. We follow the bi-level optimization formulation and search activation function represented by learnable parameters. Specifically, a typical GNN network of $L$ layers can be represented as below: $$\sigma^{(L)} \circ GL^{(L)} \cdots \sigma^{(2)} \circ GL^{(2)} \circ \sigma^{(1)} \circ GL^{(1)}(X),$$ where Graph Layer (GL) denotes all the AGGREGATE (Agg) and UPDATE (Up) operations related to graph. $X$ is the initial graph features. Note that different layers could use different activation functions whereas current GNN models tend to fix the same ReLU for every layer. Denote by $w_\sigma$ all the parameters of activation functions, i.e. $\sigma^{(1)}, \ldots, \sigma^{(L)}$, and denote by $\bar{w}_\sigma$ all the parameters of GNN, i.e., parameters of $GL^{(1)}, \ldots, GL^{(L)}$. We propose a continuous implicit functional space to parameterize $w_\sigma$. This search space is expressive yet compact, with smoothness regularization induced by human prior. The parameter update process is stochastic to deal with any downstream objective especially non-differentiable metrics. The search algorithm is bi-level and end-to-end trained. The optimization step of the outer level (learning $w_\sigma$) and the inner level optimization (learning $\bar{w}_\sigma$) are iterated. In the following parts, we explain in sequence the design of search space, stochastic relaxation and search algorithm. 4.2 IMPLICIT FUNCTIONAL SEARCH SPACE In order to facilitate activation function search, we propose a continuous implicit functional space that parameterizes the search space by universal approximators. This implicit functional space could be implemented by Multi-Layer Perception (MLP) to approximate target function. As in Figure 2, activation function parameters is equivalent to the parameters of MLP, denoted by $w_\sigma$. It’s worth noting that we employ MLP as a representative example of universal approximators, chosen for its simplicity, while retaining generality. However, it’s crucial to emphasize that alternative implementations, such as Gaussian Mixtures or Radial Basis Functions (RBF), are entirely feasible. In addition, we focus on the smooth functions such that the searched activation functions will not change dramatically if the pre-activation value $Z$ is slightly perturbed. Smooth functions are bounded by Lipschitz constant $c$, i.e. $\|f(x) - f(y)\| \leq c\|x - y\|$. As a result, the functional search space is regularized by smoothness constraint $ \{w_\sigma \| w_\sigma \in R^{\|w_\sigma\|}, c < \gamma\} $, where $ c $ is the Lipschitz constant of the function parameterized by $ w_\sigma $ and $ \gamma $ is the hard limit on $ c $. Here, we model the constraint on Lipschitz constant as a regularization term denoted by $ R(w_\sigma) $. Many references design Lipschitz constant as additional soft metric to be trained together with any loss (Hoffman et al., 2019; Weng et al., 2018; Liu et al., 2022). We use Jacobian regularization (Hoffman et al., 2019) without loss of generality $ R(w_\sigma) = \|\|J(x)\|\|_F = \left\{ \sum_{i,j} [J_{i,j}(x)]^2 \right\} $, where $ J_{i,j}(x) = \frac{\partial h_i}{\partial x_j}(x) $ is the Jacobian matrix. This design of our functional space encourages the discovery of smooth functions characterized by small Lipschitz constants. Notably, this characteristic aligns with existing manually designed functions, such as ReLU, Tanh, Sigmoid, and Swish, all of which exhibit 1-Lipschitz properties. ### 4.3 STOCHASTIC RELAXATION GNN applications are diverse that in many cases, the preferred evaluation metrics may not be differentiable. As mentioned in Table 1 and related works, APL could not deal with non-differentiable metrics but Swish could due to the RL-based search algorithm. However, the search efficiency of Swish is far from satisfactory. We propose to use stochastic relaxation that re-parameterizes the search space $ (w_\sigma) $ with a Gaussian distribution $ p_{\theta_\sigma}(w_\sigma) $. The Gaussian distribution has its own parameters $ \theta_\sigma $ and we sample the parameters of activation function $ w_\sigma $ from the probability $ p_{\theta_\sigma}(w_\sigma) $ and optimize the probability parameters $ \theta_\sigma $ instead of $ w_\sigma $. Following (4), we replace $ \sigma $ by $ w_\sigma $ to emphasize the parameters on activation function; we replace $ w $ by $ \overline{w}_\sigma $ to denote the rest parameters of GNN model. Task objective $ M $ is jointly integrated into stochastic relaxation with space regularization $ R $. The ultimate problem is formulated as below: \[ \begin{align} \theta_\sigma^ &= \arg\min_{\theta_\sigma} \left\{ J(\theta_\sigma) \equiv \mathbb{E}_{w_\sigma \sim p_{\theta_\sigma}(w_\sigma)}[M(w_\sigma, \overline{w}_\sigma^; D_{val}) + \eta R(w_\sigma)] \right\}, \\ \text{s.t. } \overline{w}_\sigma^ &= \arg\min_{\overline{w}_\sigma} L(\overline{w}_\sigma, w_\sigma; D_{train}), \end{align} \] where $ w_\sigma $ denotes the parameters of activation functions $ \sigma $ with a regularization term $ R $ weighted by $ \eta $, $ \overline{w}_\sigma $ represents GNN parameters, $ L $ is downstream task criterion of interest, $ M $ is upstream task criterion of interest, probably non-differentiable, $ \theta_\sigma $ represents the re-parameterization of $ w_\sigma $ through Gaussian distribution, then the whole learning problem is optimized in a stochastic way. To compute the target loss gradient with respect to probability parameters $ \nabla_{\theta_\sigma} J(\theta_\sigma) $, we have the following proposition. The proof is given in Appendix A. Proposition 1* Let $ w_\sigma \sim p_{\theta_\sigma}(w_\sigma) $ represent that the weights of activation functions are sampled from $ p_{\theta_\sigma} $. We have \[ \nabla_{\theta_\sigma} J(\theta_\sigma) = \nabla_{\theta_\sigma} \mathbb{E}_{w_\sigma \sim p_{\theta_\sigma}(w_\sigma)}[M(w_\sigma, \overline{w}_\sigma^; D_{val}) + \eta R(w_\sigma)] = \mathbb{E}_{w_\sigma \sim p_{\theta_\sigma}(w_\sigma)}[(M(w_\sigma, \overline{w}_\sigma^; D_{val}) + \eta R(w_\sigma)) \nabla_{\theta_\sigma} \log p_{\theta_\sigma}(w_\sigma)] \] With the help of stochastic relaxation, the previously needed derivation of $ M $ is replaced by a multiplication between forward pass of $ M $ and a gradient of probability loss. In practice, this gradient expectation could be further approximated by Monte Carlo samplings, i.e. $ \nabla_{\theta_\sigma} J(\theta_\sigma) \approx \sum_{i=1}^{K} \nabla_{\theta_\sigma} \log p_{\theta_\sigma}(w^i_\sigma)[M(w^i_\sigma, \overline{w}_\sigma^; D_{val}) + \eta R(w^i_\sigma)], K $ is the sample number that we use to approximate the gradient. As a result, the differentiability requirement of $ M $ is removed. ### 4.4 SEARCH STRATEGY According to (4), the learning is divided in two levels. The outer level optimizes probability parameters $ \theta_\sigma $ on validation dataset with (non-differentiable) metric $ M $. Every time $ K $ number of samples are generated from the probability distribution (such as Gaussian). Each sample is forwarded and calculated according to (Prop. 1), whose average is an approximation of outer level loss gradient. The optimization of outer level parameters $ \theta_\sigma $ influences directly the value of activation function weights since the weights are sample from the updated probability every time a forward pass is needed. The inner level optimizes GNN parameters $ \overline{w}_\sigma $ on training dataset with metric $ L $. It is similar to a normal training epoch of any network. The outer and inner levels are interplayed to accelerate convergence. Algorithm 1 TAFS: Task-aware Activation Function Search 1: Initialize $\theta^0 = 1$, initialize $w_\sigma$ by Xavier initialization and $w_\sigma$ randomly sampled from $p_{\theta^0}(w_\sigma)$. 2: for $m = 0, \ldots, M - 1$ do 3: // Outer level optimization 4: Freeze GNN parameters $w_\sigma$; 5: for $k = 0, \ldots, K - 1$ do 6: Sample activation functions weights $w^k_\sigma$ from $p_{\theta^m}(w_\sigma)$; 7: Forward inference of the whole network and accumulate stochastic loss $J(\theta_\sigma)$ as in Prop 1; 8: end for 9: Obtain $\nabla_{\theta_\sigma} J(\theta_\sigma)$ by automatic differentiation and update $\theta^m$; 10: // Inner level optimization 11: Sample activation function parameters $w_\sigma$ from distribution $p_{\theta^m}(w_\sigma)$ and freeze $w_\sigma$; 12: Forward inference of the whole network to obtain loss $L$; 13: Update GNN parameters $w_\sigma$ by automatic differentiation; 14: end for 15: Training until convergence and obtain the final model parameter $w^_\sigma$ and dist. parameter $\theta_\sigma$; 16: return Final model parameter $w^_\sigma$ and distribution parameter $\theta_\sigma$. Table 1: Our proposed TAFS (Task-aware Activation Function Search) enables efficient differentiable search through a flexible and powerful MLP functional space. TAFS supports non-differentiable objective metrics in diverse GNN applications. \| Search Method \| Search Efficiency \| Non-Differentiable Metric \| \|---------------\|-------------------\|---------------------------\| \| Swish \| Discrete template choice \| Reinforcement Learning \| Applicable \| \| APL \| Explicit piecewise linear \| Differentiable \| Not applicable \| \| TAFS (ours) \| Continuous implicit MLP \| Differentiable \| Applicable \| The complete TAFS algorithm is given in Algorithm 1. We also compare in Table 1 our proposed TAFS and literature methods. From the time efficiency perspective, Swish is the slowest in Table 1 because it optimizes a new network until convergence before the learning of RL controller. APL on the other hand, has a number of parameters dependent of base models due to its adaptability per neuron. As a result, TAFS enjoys a compact search space without over-parameterization and has superior efficiency in searching. Empirical results are given in Table 4. 5 EXPERIMENTS In this section, we experiment our methods on diverse GNN applications including node classification and link prediction, in order to fully evaluation the methods on differentiable and non-differentiable metrics. Later, we provide detailed analysis on search efficiency and hyperparameter impact. All our experiments are run on single NVIDIA RTX 3090. 5.1 NODE CLASSIFICATION Datasets. We experiment on diverse graph datasets for node tasks, including Cora and DBLP for paper classification based on reference network, Cornell and Texas for webpage classification from university network, and Chameleon for wikipedia page classification based on hyperlink network. Statistics are in Appx. E. The task metric here is classification accuracy. Baselines. To fairly compare different activation functions, we compare our searchable activation functions with manually designed ones or previously searched function. Some of these activation functions are visualized in Figure 3(a). For each dataset and baseline chosen, we evaluate on two aggregation layers (GCN and GraphSage) and five network connection topologies (stack, residual, dense, jump knowledge, mixhop). Each model has four layers of aggregation layers. The model is trained for 400 epochs. Table 2: Overall node classification improvement of different models on different datasets. Metric is classification accuracy. Avg. Imp. is the improvement of TAFS with respect to the other choices averaged over all the datasets. \| Model \| Activation \| Cora \| DBLP \| Cornell \| Texas \| Chameleon \| Avg. Imp. \| \|-----------\|------------\|------------\|-----------\|-----------\|-----------\|-----------\|-----------\| \| GCN \| \| \| \| \| \| \| \| \| Stack \| ReLU \| 83.06±0.66 \| 84.63±0.21\| 56.76±5.92\| 60.54±6.42\| 61.60±1.75\| ↑ 2.8% \| \| \| Tanh \| 84.82±0.51 \| 85.58±0.15\| 56.49±5.19\| 57.84±5.01\| 61.51±1.88\| ↑ 3.2% \| \| \| L-ReLU \| 84.57±0.93 \| 84.50±0.40\| 57.38±2.16\| 60.54±7.37\| 61.95±2.18\| ↑ 2.1% \| \| \| Swish \| 83.88±0.81 \| 84.89±0.34\| 57.30±3.97\| 58.65±5.55\| 58.33±1.68\| ↑ 4.1% \| \| \| TAFS \| 89.08±0.48 \| 86.24±0.17\| 57.37±4.37\| 62.11±5.48\| 62.31±1.82\| - \| \| Residual \| ReLU \| 85.13±0.95 \| 84.45±0.34\| 57.84±4.43\| 57.84±5.95\| 66.93±2.17\| ↑ 3.2% \| \| \| Tanh \| 86.02±0.55 \| 85.63±0.14\| 58.38±4.39\| 57.57±5.93\| 68.86±1.84\| ↑ 2.0% \| \| \| L-ReLU \| 86.60±0.72 \| 84.97±0.33\| 55.68±8.30\| 57.84±6.75\| 67.50±1.48\| ↑ 3.3% \| \| \| Swish \| 85.86±0.64 \| 84.67±0.19\| 56.22±6.14\| 60.54±7.66\| 66.29±2.12\| ↑ 2.8% \| \| \| TAFS \| 88.16±0.58 \| 86.29±0.18\| 58.20±4.80\| 60.22±5.51\| 70.49±1.64\| - \| \| JKNet \| ReLU \| 86.86±0.71 \| 84.99±0.25\| 76.49±7.36\| 77.57±7.36\| 58.18±1.63\| ↑ 3.8% \| \| \| Tanh \| 86.41±0.57 \| 85.57±0.20\| 68.92±6.76\| 65.95±9.22\| 60.20±2.19\| ↑ 9.1% \| \| \| L-ReLU \| 87.45±0.51 \| 85.04±0.15\| 74.05±5.57\| 76.49±8.80\| 57.98±2.36\| ↑ 4.1% \| \| \| Swish \| 86.34±0.92 \| 84.95±0.28\| 77.03±5.57\| 78.11±6.99\| 57.00±2.54\| ↑ 4.7% \| \| \| TAFS \| 88.84±0.56 \| 87.07±0.22\| 81.35±6.40\| 81.08±5.01\| 60.21±2.04\| - \| \| Mixhop \| ReLU \| 85.31±0.64 \| 85.10±0.18\| 73.78±5.55\| 74.05±9.53\| 51.64±2.24\| ↑ 2.7% \| \| \| Tanh \| 85.15±0.67 \| 85.12±0.30\| 72.97±7.55\| 76.76±6.86\| 50.59±2.60\| ↑ 2.6% \| \| \| L-ReLU \| 86.38±0.50 \| 85.01±0.17\| 72.43±6.14\| 72.70±5.05\| 51.36±2.80\| ↑ 3.4% \| \| \| Swish \| 86.21±1.03 \| 85.43±0.25\| 72.34±8.18\| 74.86±6.51\| 51.89±2.10\| ↑ 2.5% \| \| \| TAFS \| 88.77±0.57 \| 86.18±0.17\| 75.14±5.38\| 78.43±5.28\| 52.17±1.97\| - \| \| GraphSage \| Stack \| 83.06±0.66 \| 83.67±0.41\| 58.11±6.19\| 70.00±6.78\| 47.02±4.20\| ↑ 12.1% \| \| \| ReLU \| 84.82±0.51 \| 84.90±0.19\| 68.65±6.75\| 71.89±7.85\| 53.50±1.68\| ↑ 4.3% \| \| \| Tanh \| 84.57±0.65 \| 84.16±0.23\| 62.16±5.92\| 68.11±7.23\| 49.21±3.02\| ↑ 9.8% \| \| \| L-ReLU \| 81.53±0.74 \| 83.62±0.50\| 57.03±6.45\| 68.65±6.19\| 48.42±2.17\| ↑ 13.0% \| \| \| Swish \| 87.08±0.48 \| 85.22±0.30\| 72.43±7.23\| 74.51±6.92\| 58.57±1.20\| - \| \| \| TAFS \| 89.10±0.385\| 85.22±0.17\| 73.38±6.63\| 77.03±6.86\| 58.62±2.08\| - \| \| Residual \| ReLU \| 84.11±0.82 \| 83.05±0.33\| 65.95±6.64\| 73.51±6.71\| 55.02±2.73\| ↑ 6.1% \| \| \| Tanh \| 85.62±0.52 \| 85.22±0.17\| 72.43±5.97\| 78.11±9.00\| 59.17±1.80\| ↑ 0.6% \| \| \| L-ReLU \| 85.63±0.42 \| 84.05±0.21\| 71.89±3.67\| 74.86±5.80\| 55.86±1.83\| ↑ 3.1% \| \| \| Swish \| 84.97±0.79 \| 84.17±0.43\| 71.08±6.02\| 75.41±7.09\| 54.17±1.44\| ↑ 3.9% \| \| \| TAFS \| 89.51±0.66 \| 86.73±0.20\| 81.79±5.08\| 82.10±5.16\| 59.37±1.53\| - \| \| JKNet \| ReLU \| 85.29±0.56 \| 83.97±0.15\| 80.00±6.07\| 81.62±5.10\| 56.78±1.62\| ↑ 3.0% \| \| \| Tanh \| 86.01±0.51 \| 85.25±0.18\| 77.03±5.57\| 78.92±6.14\| 57.68±1.92\| ↑ 3.8% \| \| \| L-ReLU \| 85.90±0.42 \| 85.01±0.25\| 80.27±8.84\| 81.35±4.75\| 57.41±2.01\| ↑ 2.5% \| \| \| Swish \| 85.56±0.61 \| 84.71±0.22\| 77.13±5.30\| 81.06±5.43\| 55.00±1.93\| ↑ 4.5% \| \| \| TAFS \| 87.77±1.40 \| 85.30±0.24\| 77.77±4.39\| 83.70±4.05\| 55.07±0.57\| - \| Results. We provide in Table 2 the results of node classification tasks. The improvement of TAFS with respect to the other function choices is significant. Note that the improvements are observable across different graph data and GNN models, showing that TAFS is task-aware to graphs in citation, university webpage, wikipedia link graph, etc. 5.2 Molecule and Protein Interaction Prediction Datasets. Biomedical graph are one of the most active and effective application areas of GNN. Biomedical GNN has accelerated important studies in protein prediction, molecule generation, gene expression, etc. We consider the link prediction that is a typical task in molecule and protein interaction prediction. Specifically, we consider Drug-Drug Interaction (DDI), Drug-Target Interaction (DTI), Protein-Protein Interaction (PPI) and Disea-Gene Association (DGA). The statistics of the four datasets are provided in Appx. Table 6. Baselines.* We adopt two biomedical graph baselines SkipGNN (Huang et al., 2020) and HOGCN (KC et al., 2022). SkipGNN proposes a general GNN architecture to model molecular interactions and works well on all these biomedical tasks. We use both as base model and experiment TAFS to replace the activation functions. The training hyperparameters are the same as in the original work. Table 3: Drug and protein interaction predictions. \| Task \| Model \| Activation \| ROCAUC \| PRAUC \| \|-----------------------------\|-----------\|---------------------\|----------\|---------\| \| Drug-Target Interaction \| SkipGNN \| ReLU \| 0.922±0.004 \| 0.928±0.006 \| \| \| \| TAFS w.o. relaxation \| 0.933±0.002 \| 0.934±0.001 \| \| \| \| TAFS \| 0.952±0.001 \| 0.954±0.001 \| \| \| HOGCN \| ReLU \| 0.927±0.001 \| 0.929±0.001 \| \| \| \| TAFS w.o. relaxation \| 0.923±0.002 \| 0.922±0.001 \| \| \| \| TAFS \| 0.943±0.002 \| 0.940±0.001 \| \| Drug-Drug Interaction \| SkipGNN \| ReLU \| 0.886±0.003 \| 0.866±0.006 \| \| \| \| TAFS w.o. relaxation \| 0.890±0.002 \| 0.874±0.001 \| \| \| \| TAFS \| 0.911±0.002 \| 0.898±0.003 \| \| \| HOGCN \| ReLU \| 0.898±0.002 \| 0.881±0.003 \| \| \| \| TAFS w.o. relaxation \| 0.897±0.002 \| 0.901±0.002 \| \| \| \| TAFS \| 0.917±0.002 \| 0.901±0.001 \| \| Protein-Protein Interaction \| SkipGNN \| ReLU \| 0.917±0.004 \| 0.921±0.003 \| \| \| \| TAFS w.o. relaxation \| 0.920±0.001 \| 0.922±0.002 \| \| \| \| TAFS \| 0.927±0.001 \| 0.937±0.002 \| \| \| HOGCN \| ReLU \| 0.919±0.001 \| 0.922±0.002 \| \| \| \| TAFS w.o. relaxation \| 0.919±0.002 \| 0.924±0.001 \| \| \| \| TAFS \| 0.923±0.003 \| 0.929±0.002 \| \| Disease-Gene Association \| SkipGNN \| ReLU \| 0.912±0.004 \| 0.915±0.003 \| \| \| \| TAFS w.o. relaxation \| 0.916±0.001 \| 0.920±0.001 \| \| \| \| TAFS \| 0.930±0.001 \| 0.940±0.001 \| \| \| HOGCN \| ReLU \| 0.927±0.001 \| 0.934±0.001 \| \| \| \| TAFS w.o. relaxation \| 0.929±0.002 \| 0.933±0.001 \| \| \| \| TAFS \| 0.933±0.001 \| 0.942±0.002 \| Results. We provide in Table 3 the results of four link prediction tasks. Again, both SkipGNN and HOGCN use ReLU by default. With TAFS, SkipGNN and HOGCN has gained significant performance evaluated in ROCAUC and PRAUC, two non-differentiable metrics. Furthermore, when TAFS is integrated with SkipGNN, a model from 2020, it outperforms HOGCN, the state-of-the-art model from 2022. This underscores the significance of activation function search, which has hitherto been overlooked in the GNN community. ### 5.3 Ablation Study To further analyze our proposed TAFS algorithm, we provide additional experiments to illustrate the search results, search efficiency and hyperparameter impact. Visualization of activation function search. We show in Figure 3 the searched activation functions from the literature methods and TAFS. It can be observed that TAFS could find diverse activation functions different than manually design ones or the searched ones by Swish and APL. Moreover, TAFS learns layer-wise activation function, leading to different behaviours of functions in different layers as in Figure 3(b)(c). Deeper layers’ activation functions are smoother than shallow layers. Search efficiency The modeling differences between literature search methods and TAFS are given in Table 1. In this part, we provide more empirical details of the search efficiency comparison in Table 4. TAFS has a significantly smaller consumption of extra memory and shorter running time. This huge efficiency improvement is credited to TAFS’ compact MLP functional search space and differentiable search strategy, making TAFS’ extra parameters independent of base models, whichever dataset or GNN model (as long as the model has the same number of activation functions), while APL models each neuron with a piecewise linear unit, leading to over 2000 times more parameters than TAFS. Hyperparameters impact. The choice of hyperparameters significantly affects performance. TAFS introduces two sets of hyperparameters: the number of samples (K) and the selection of the MLP architecture. We present their effects in Figure 4. The number of samples (K) in stochastic optimization exhibits a consistent increasing trend, representing a trade-off between accuracy and computational time. Regarding MLP hyperparameters, we analyze their impact on two node tasks, DBLP and Cornell, using nine different configurations: depths ranging from two to four layers and widths spanning from 10 to 1000 neurons. It is evident that a very small MLP (e.g., two layers with 10 neurons) is inadequate for modeling adaptive activation functions. However, the distinctions between other choices are negligible. Given that deeper and wider MLPs require significantly more parameters, we opt for a two-layer MLP with 100 hidden units in all other experiments. \| Dataset \| Model \| Parameters \| Time(min) \| \|------------\|-------\|------------------\|-----------\| \| DBLP \| Base \| 420K \| 0.15 \| \| \| Swish \| +340K (+82%) \| 350 \| \| \| APL \| +2400K (+575%) \| 4 \| \| \| TAFS \| +1.3K (+0.3%) \| 1 \| \| Chameleon \| Base \| 315K \| 1.2 \| \| \| Swish \| +420K (+108%) \| 2990 \| \| \| APL \| +1760K (+558%) \| 33 \| \| \| TAFS \| +1.3K (+0.4%) \| 11 \| \| Ogbg-Molhiv\| Base \| 27M \| 1020 \| \| \| Swish \| - \| > 70 days \| \| \| APL \| OOM (+150M) \| - \| \| \| TAFS \| +12K \| 1380 \| Figure 4: Hyperparameter impact of number of samples in stochastic relaxation and the impact of MLP dimensions. In (b)(c), deeper color means better performance. 6 CONCLUSION In a word, we achieve a task-aware activation function search in GNN through an expressive and compact representation of search space, stochastic relaxation with reparameterization, which are carefully co-designed with search strategy. Our search space is inclusive and parameter efficient, including appropriate number of high-quality functions. The search strategy is end-to-end trained and every operation of the framework is differentiable. Finally, the stochastic relaxation is capable of dealing with any metric of interest, closing the optimization gap. REFERENCES Forest Agostinelli, Matthew D. Hoffman, Peter J. Sadowski, and Pierre Baldi. Learning activation functions to improve deep neural networks. In International Conference on Learning Representations, ICLR Workshop, 2015. Andrea Apicella, Francesco Donnarumma, Francesco Isgrò, and Roberto Prevete. A survey on modern trainable activation functions. 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Kexin Huang, Cao Xiao, Lucas M Glass, Marinka Zitnik, and Jimeng Sun. Skipgnn: predicting molecular interactions with skip-graph networks. Scientific reports, 10(1):1–16, 2020. Dejun Jiang, Zhenxing Wu, Chang-Yu Hsieh, Guangyong Chen, Ben Liao, Zhe Wang, Chao Shen, Dongsheng Cao, Jian Wu, and Tingjun Hou. Could graph neural networks learn better molecular representation for drug discovery? a comparison study of descriptor-based and graph-based models. Journal of Cheminformatics, 2021.
oYjPk8mqAV	Would it be possible for the authors to expand on their improvement on the miniF2F benchmark? Specifically, I would be interested to know what are the specific unsolved theorems that Magnushammer was able to solve, and how complex are those proofs.	MAGNUSHAMMER: A TRANSFORMER-BASED APPROACH TO PREMISE SELECTION Maciej Mikula∗ Google DeepMind† Szymon Tworkowski∗ xAI† Szymon Antoniak∗ Mistral AI† Bartosz Piotrowski IDEAS NCBR Albert Qiaochu Jiang University of Cambridge Jin Peng Zhou Cornell University‡ Christian Szegedy xAI‡ Łukasz Kuciński IDEAS NCBR, IMPAN Piotr Miłoś IDEAS NCBR, IMPAN Yuhuai Wu xAI‡ ABSTRACT This paper presents a novel approach to premise selection, a crucial reasoning task in automated theorem proving. Traditionally, symbolic methods that rely on extensive domain knowledge and engineering effort are applied to this task. In contrast, this work demonstrates that contrastive training with the transformer architecture can achieve higher-quality retrieval of relevant premises, without the engineering overhead. Our method, Magnushammer, outperforms the most advanced and widely used automation tool in interactive theorem proving called Sledgehammer. On the PISA and miniF2F benchmarks Magnushammer achieves 59.5% (against 38.3%) and 34.0% (against 20.9%) success rates, respectively. By combining Magnushammer with a language-model-based automated theorem prover, we further improve the state-of-the-art proof success rate from 57.0% to 71.0% on the PISA benchmark using 4x fewer parameters. Moreover, we develop and open source a novel dataset for premise selection, containing textual representations of (proof state, relevant premise) pairs. To the best of our knowledge, this is the largest available premise selection dataset, and the first one for the Isabelle proof assistant. 1 INTRODUCTION Automating mathematical reasoning has been a central theme of artificial intelligence since its earliest days [De Bruijn 1970]. Recently, machine learning has led to significant advancements in both informal [Lewkowycz et al. 2022] and formal mathematical reasoning [Kaliszyk and Urban 2015b; Alemi et al. 2016; Polu and Sutskever 2020; Han et al. 2022]. The latter approach, adopted in this paper, allows mechanical verification of proofs by proof assistants. Modern mathematics development is gradual: it feeds upon a huge body of already established knowledge and constantly adds to it. Proving a mathematical statement requires retrieval of facts from the knowledge base that can advance the proof. In automated reasoning literature, this retrieval process is known as premise selection. Figure 1: Proof success rate for varying computational budget for Magnushammer, Sledgehammer, and BM25. Magnushammer shows remarkable scalability. See Sections 5.1 for the definition of computational budget and Section 5.2.1 for configurations depicted in this figure. ∗Equal contribution. †Work performed while at the University of Warsaw. ‡Work performed while at Google Research. Many tools have been developed to tackle premise selection (Alama et al., 2011; Kühlewein et al., 2012; Kaliszyk et al., 2017; Bansal et al., 2019), including a broad class known as “hammers,” which leverage powerful automated theorem provers (ATPs) to determine useful premises (Paulson and Blanchette, 2012; Gauthier and Kaliszyk, 2015; Kaliszyk and Urban, 2015a; Czajka and Kaliszyk, 2018). One such tool, Sledgehammer (SH) (Paulson and Blanchette, 2012), has gained prominence with Isabelle (Paulson, 1993), where it helped to create a significant portion of Isabelle’s proof corpus. Hammers are not yet available in all proof assistants (Ebner, 2020): implementing them is challenging due to the complex techniques required for different logics and type systems. There is a need for an effective premise selection tool that requires less adaptation to work for different proof assistants. In this study, we provide a generic, data-driven, transformer-based (Vaswani et al., 2017) premise selection tool: Magnushammer. It constitutes a novel way to tackle the premise selection task, effective while requiring little domain-specific knowledge. Magnushammer is trained contrastively to perform premise retrieval in two stages: in the SELECT stage, it retrieves the most relevant 1024 premises (measured by the cosine similarity of their embeddings to that of the current proof state) from tens of thousands (the database contains 433K premises in total and typically 30K–50K are available in each proof state); in the RERANK stage, the retrieved premises are re-ranked with proof-state-aware scores: tokens of the proof state directly attend to tokens of the premise, giving a more contextualized relevance score. An overview of Magnushammer’s architecture is shown in Figure 2b. Magnushammer can prove 59.5% of the theorems on the PISA benchmark (Jiang et al., 2021), a substantial improvement over Sledgehammer’s 38.3%. We demonstrate that this dominance is consistent with varying controlled compute budgets, shown in Figure 1. Furthermore, we replace the premise selection component (Sledgehammer) in a neural-symbolic model Thor (Jiang et al., 2022a) with Magnushammer and improve the state-of-the-art proof success rate on PISA from 57% to 71%. To train Magnushammer, we extracted a premise selection dataset from the Isabelle theorem prover and its human proof libraries. The dataset consists of 4.4M premise selection instances, with 433K unique premises. To the best of our knowledge, this is the largest open-sourced premise selection dataset, and the first one of this kind for Isabelle. We find Magnushammer to be data efficient, outperforming Sledgehammer with only 4K training examples (0.1% of the training data available). The main contributions of this work are the following: • We propose the use of transformers trained contrastively as a novel way of addressing the premise selection problem. Our method, Magnushammer, achieves a 59.5% proof rate on the PISA benchmark, significantly improving the 38.3% proof rate of Sledgehammer, the most powerful general-purpose automation tool for Isabelle. • We extract and open source the largest, to the best of our knowledge, premise selection dataset. It consists of 4.4M premise selection examples and 433K unique premises. • We analyze how Magnushammer’s performance depends on the model size, dataset size, and the inference-time compute budget. We show its superiority with moderate resources. 2 BACKGROUND: PROOF ASSISTANTS, ISABELLE, AND SLEDGEHAMMER Proof assistants (aka interactive theorem provers, or ITPs) such as Isabelle (Paulson, 1993), Lean (de Moura et al., 2015), Coq (Bertot, 2008), HOL Light (Harrison, 1996), or Mizar (Grabowski et al., 2010), are software tools designed to assist the development of formal proofs. They provide expressive language for the formalization of mathematical statements and proofs while verifying them formally. In Isabelle, theorems are proved sequentially: an initial proof state is obtained after the theorem is stated, and the proof state changes when the user provides a valid proof step (see Appendix A.1 for an example). Proof states contain information about the already established facts and the remaining goals to prove. Proof steps consist of tactics, which are optionally parametrized by premises. Tactics are theorem-proving procedures and can complete some proofs in one step provided with relevant premises. However, finding these premises is difficult: one needs to select a handful of relevant facts from the current proof context, which typically contains tens of thousands of them. Sledgehammer (Paulson and Blanchette, 2012; Blanchette et al., 2013) is a powerful automated reasoning tool for Isabelle. It belongs to a broader class of tools known as “hammers,” which integrate automated theorem provers (ATPs) into proof assistants. The goal of these tools is to support the process of finding and applying proof methods. Sledgehammer has become an indispensable tool for Isabelle practitioners (Paulson and Blanchette, 2012). It allows for closing low-level gaps between subsequent high-level steps of proof without the need to memorize entire lemma libraries. Sledgehammer is designed to first pre-select a number of relevant facts heuristically, translate them together with a conjecture to simpler logic, and try to prove the conjecture using strong, external ATPs like E (Schulz [2004]), SPASS (Weidenbach [2001]), Vampire (Kovács and Voronkov [2013]), Z3 (de Moura and Björner [2008]), or cvc5 (Barbosa et al. [2022]). If successful, these provers generate complete proofs. They are, however, not trusted by Isabelle. Instead, the facts used in the external proofs are extracted and used to produce a proof inside Isabelle using its native methods. Up to this last step, known as proof reconstruction, Sledgehammer is essentially used as a precise premise selection tool. See Figure 2a depicting the whole process. While immensely useful, Sledgehammer comes with several limitations. First, increasing computational power for Sledgehammer brings quickly diminishing returns (Böhme and Nipkow [2010]). Second, the logic projection and proof reconstruction in a hammer are not straightforward for type systems other than higher-order logic (Czajka and Kaliszyk [2018]). Finally, Sledgehammer’s performance hinges on the relevance filtering scheme, a suite of methods based on handcrafted heuristics (Meng and Paulson [2009]) or classical machine learning (Kühnwein et al. [2013]). Such approaches are unlikely to efficiently utilize the constantly growing body of proof data. We argue that all these limitations can be overcome with deep-learning-based approaches. Neural networks have shown remarkable effectiveness in end-to-end problem solving with little or no feature engineering (Krizhevsky et al. [2012]; Brown et al. [2020]). Adopting textual representations with generic neural solutions removes the need for logic projection, ATP solving, and proof reconstruction. Moreover, large language models have recently displayed impressive scaling properties with respect to both model size (Kaplan et al. [2020]) and data (Hoffmann et al. [2022]). 3 MAGNUSHAMMER The goal of premise selection is to find relevant mathematical facts for a given proof state. We focus on selecting premises with a neural model informed by their textual representations instead of relying on fact structures like Sledgehammer (see Section 2). The core idea of Magnushammer is to combine fast retrieval based on representational similarity (SELECT) with a more accurate re-ranking (RERANK), as outlined in Algorithm 1. Our method closely follows those of Nogueira and Cho (2019) and Izacard et al. (2021). This hierarchical approach is scalable to large formal libraries containing hundreds of thousands of facts. Below we describe the two-stage Magnushammer approach. SELECT leverages representation similarity and is based on batch-contrastive learning similar to the methods of Alemi et al. (2016), Bansal et al. (2019), Han et al. (2021), or Radford et al. (2021). SELECT embeds premises and proof states into a common latent space and uses cosine similarity to determine their relevance. During inference, it requires only one pass of a neural network to compute the proof state embedding and dot product with cached premise embeddings. SELECT is hence fast and scalable to large sets of premises. In our experiments, there are between 30K and 50K premises in a typical proof state context, from which we select $K_S = 1024$ most relevant ones. RERANK scores the relevance of the $K_S$ selected premises for the current proof state by analyzing the $(\text{proof\_state}, \text{premise})$ pairs. RERANK is trained to output the probability of the premise being relevant to the proof_state. The $K_S$ premises retrieved by SELECT are re-ranked with respect to these probabilities, and the final list comprises of the top $K_R$ premises (we set $K_R = K_S$). Having both the premise and the proof state in a single input allows RERANK to be more accurate. However, at the same time, it is much slower, as each pair must be scored individually. Algorithm 1 Premise selection with Magnushammer. Require: - $\text{proof\_state}, \text{premises}$ ▷ proof state to retrieve premises for and database of available premises - $K_S, K_R$ ▷ number of premises to retrieve with SELECT and RERANK, respectively 1: $\text{state\_embedding} \leftarrow \text{get\_embeddings}(\text{proof\_state})$ ▷ SELECT stage starts 2: $\text{premises\_embeddings} \leftarrow \text{get\_embeddings}(\text{premises})$ 3: $\text{Cache}(\text{premises\_embeddings})$ 4: $\text{sim\_scores} = \text{state\_embedding} \cdot \text{premises\_embeddings}$ 5: $\text{selected} = \text{premises}[\text{argsort}(-\text{sim\_scores})[:K_S]]$ 6: $\text{batch} = []$ ▷ RERANK stage starts 7: for premise in selected do 8: $\text{batch}.append((\text{proof\_state}, \text{premise}))$ 9: $\text{rerank\_scores} \leftarrow \text{get\_rerank\_scores}(\text{batch})$ 10: $\text{top\_premises} = \text{selected}[\text{argsort}(-\text{rerank\_scores})[:K_R]]$ 11: return $\text{top\_premises}$ Training We train Magnushammer using two alternating tasks: SELECT is trained with a modified InfoNCE loss (van den Oord et al., 2018), and RERANK is trained with the standard binary cross-entropy loss. The architecture of Magnushammer shares a transformer backbone with specialized linear projections on top (see Figure 2b). The backbone is pre-trained with a language modeling task on the GitHub and arXiv subsets of the Pile dataset (Gao et al., 2021). For training, we use datasets consisting of $(\text{proof\_state}, \text{premise})$ pairs extracted with a procedure described in Section 4. During SELECT’s training, each batch consists of $N$ proof states, $N$ positive premises (one for each proof state), and additional $M$ negative premises sampled from available facts that are not ground truth premises for any of the selected proof states. This gives $N - 1 + M$ negatives per proof state in one batch. We typically use $M = 3N$, which differs from standard batch-contrastive learning (Radford et al., 2021), in which $M = 0$ and negatives are only the other $N - 1$ premises in the batch. RERANK is trained using a binary classification objective. For each positive $(\text{proof\_state}, \text{premise})$ pair in the dataset, we construct 15 negatives from the most likely false positives returned by SELECT. Specifically, all the premises $M$ that are facts that were never used as a premise for $\text{proof\_state}$, are first chosen. Then, the top 1024 of $M$ according to SELECT are selected, and 15 are sampled from them to construct negative training pairs. See Appendix B for complete training details. Evaluation in Isabelle We outline how premises chosen by Magnushammer are used to prove theorems in Isabelle. Given a proof state, a list of the $k$ most relevant premises $P$ is retrieved. We construct proof steps consisting of a tactic $t$ and a subset of premises $S \subseteq P$. Such proof steps are executed in parallel, with a timeout of 2 seconds. The evaluation is successful if any of these proof steps completes the proof. For $S$, we pick the top $i$ of $P$, where $i$’s are consecutive powers of 2 up to $2^{10}$, or 0 for tactics that do not accept premises. More details, including the set of tactics used, are presented in Appendix D. An example of a proof with tactics and premises is given in Appendix A.3. Note that the procedure of trying multiple different subsets of premises is commonly applied in the context of automated theorem proving (Urban et al., 2008; Kühlwein et al., 2012) and similar to the technique implemented in Sledgehammer (Paulson and Blanchette, 2012). The rationale behind this is that the proof procedures implemented in ATPs and high-level ITPs’ tactics perform combinatorial search, and providing them with fewer premises to restrict their search space is beneficial. 4 DATASETS We created and released[^1] a comprehensive dataset of textual representations for Isabelle’s proof states and premises. To the best of our knowledge, this is the first high-quality dataset of this kind for Isabelle, and also the largest premise selection dataset overall. We used the two largest collections of Isabelle theories to create the dataset: the Archive of Formal Proofs[^2] and the Isabelle Standard library[^3]. For every proof step in every proof from these collections, we extracted the preceding proof state and the set of premises used in the proof step; this was turned into $(\text{proof\_state}, \text{premise})$ pairs constituting training data points. We call this the HUMAN PROOFS LIBRARY (HPL) dataset. In addition, we used Sledgehammer to generate proofs that are different from the human ones by using potentially alternative premises. We refer to this as the SH partition, and its union with HPL constitutes the MACHINE-AUGMENTED PROOFS LIBRARY (MAPL) dataset. Statistics for all these datasets are given in Table 1[^4]. Note that MAPL grosses over 4M data points. Below we describe in more detail how data points are extracted from a proof step. An Isabelle’s proof is a sequence of $(\text{proof\_state}, \text{proof\_step})$ pairs: \text{proof\_state} has the state information, and \text{proof\_step} is a tactic application that advances the proof. A \text{proof\_step} may use \text{premises}: theorems, lemmas, or definitions established previously. Suppose a \text{proof\_step} contains $n$ premises: $p_1, p_2, \ldots, p_n$. We then extract $n$ data points: $(\text{proof\_state}, p_1), \ldots, (\text{proof\_state}, p_n)$. Executing Sledgehammer on the \text{proof\_state} may result in multiple different synthetic \text{proof\_steps}, and data points can be extracted from each in the same way (see Appendix [A.2] for details). Mining the HPL partition took 10K CPU hours, and mining the SH partition took 150K CPU hours (17 CPU years) on a distributed system. Our datasets have 2 distinguishing features: 1. The human-originating dataset is augmented by alternatives generated with Sledgehammer, which results in a significantly larger and more diverse dataset. This also decreases the probability of sampling false negatives while training contrastively: a negative example $(\text{proof\_state}, \text{premise})$ may in fact be positive, but we just have not seen an alternative proof using \text{premise}. Generating multiple alternative proofs partially remedies this problem. 2. Both \text{proof\_states} and \text{premises} are represented as “high-level” Isabelle’s text instead of “low-level” logical formalism like, e.g., TPTP[^5] used by Alama et al. (2014). This makes the dataset more suitable for language models, decreases the need for feature engineering, and facilitates cross-proof-assistant pre-training (Conneau and Lample, 2019). 5 EXPERIMENTS We evaluate Magnushammer on the PISA and miniF2F theorem proving benchmarks using proof success rate as a metric. Our main result is that Magnushammer outperforms Sledgehammer by a large margin and, combined with Thor (Jiang et al., 2022a), sets a new state of the art on the PISA benchmark (71.0% from 57.0%). Through ablations, we study the effectiveness of Magnushammer and the contribution of its components. Additional results and details can be found in Appendix E. 5.1 EXPERIMENTAL DETAILS Benchmarks For evaluation, we use PISA (Jiang et al., 2021) and miniF2F (Zheng et al., 2022) benchmarks. PISA contains problems randomly selected from the Archive of Formal Proofs[^6]; we use the same 1000 problems as Jiang et al. (2022a) for our evaluations. miniF2F consists of 488 high-school competition-level problems, split into validation and test set, each with 244 problems. [^1]: https://huggingface.co/datasets/Simontwice/premise_selection_in_isabelle [^2]: When training on data from the Archive of Formal Proofs, we remove the subset of it appearing in PISA. Table 1: Statistics of MAPL and both its partitions: HPL (coming from human-written proofs) and SH (coming from Sledgehammer-generated proofs). The data points are of the form of $(\text{proof\_state}, \text{premise})$ pairs. \| Dataset \| HPL \| SH \| MAPL \| \|---------\|-----\|----\|------\| \| Data points \| 1.1M \| 3.3M \| 4.4M \| \| Unique proof states \| 570K \| 500K \| 570K \| \| Unique premises \| 300K \| 306K \| 433K \| Table 2: Proof rates on the PISA benchmark. On the single-step task, Magnushammer outperforms both Sledgehammer and BM25 by a wide margin. On the multi-step task, Magnushammer combined with Thor achieves the state-of-the-art proof rate of 71.0%. \| Task \| Method \| Proof rate (%) \| \|------------\|-------------------------------\|----------------\| \| Single-step\| BM25 \| 30.6 \| \| \| TF-IDF \| 31.8 \| \| \| OpenAI embed. (Neelakantan et al., 2022) \| 36.1 \| \| \| Sledgehammer \| 38.3 \| \| \| Magnushammer (ours) \| 59.5 \| \| Multi-step \| LISA (Jiang et al., 2021) \| 33.2 \| \| \| Thor (Jiang et al., 2022a) \| 57.0 \| \| \| Thor + Magnushammer (ours) \| 71.0 \| Table 3: Proof rates on the miniF2F benchmark. On the single-step task, Magnushammer outperforms Sledgehammer and its variant with additional heuristics (Jiang et al., 2022b). On the multi-step task, Thor+Magnushammer obtains competitive results, significantly outperforming Thor+Sledgehammer. \| Task \| Method \| Valid (%) \| Test (%) \| \|------------\|-------------------------------\|-----------\|----------\| \| Single-step\| Sledgehammer \| 9.9 \| 10.4 \| \| \| Sledgehammer + heuristics \| 18.0 \| 20.9 \| \| \| Magnushammer (ours) \| 33.6 \| 34.0 \| \| Multi-step \| Thor + Sledgehammer (Jiang et al., 2022a) \| 28.3 \| 29.9 \| \| \| Thor + Sledgehammer + auto (Wu et al., 2022a) \| 37.3 \| 35.2 \| \| \| Thor + Magnushammer (ours) \| 36.9 \| 37.3 \| \| \| DSP (Jiang et al., 2022b) \| 43.9 \| 39.3 \| Metric and evaluation setups To evaluate the performance, we measure proof success rate: the percentage of successful proofs. A proof is successful if it is formally verified by Isabelle. We distinguish single-step and multi-step settings. In the single-step setting, we check if the theorem can be proven in one step by applying premises retrieved by the evaluated premise selection method (e.g., Magnushammer). In the multi-step scenario, we perform a proof search using a language model following Thor (Jiang et al., 2022a). Thor + Magnushammer uses Magnushammer instead of Sledgehammer as the premise selection component. A further explanation is given in Section 5.2. Evaluation protocol and computational budget Algorithm 3 (in Appendix D) details the evaluation of Magnushammer in the single-step setting. It generates $\|\mathcal{T}\| \times \|K\|$ proof steps by combining each tactic $t \in \mathcal{T}$ with top $k$ premises from a ranking provided by Magnushammer, where $\mathcal{T}$ is a prescribed set of tactics, $k \in K$, and $K$ is a list of integers. Such constructed proof steps are then executed in Isabelle. We define the computational budget for such an evaluation as $C = \|\mathcal{T}\| \times \|K\| \times T$, where $T$ is a timeout expressed in seconds (we use $T = 2$ s as we observed little benefit from increasing it). Estimating the computational budget for Sledgehammer is difficult due to its complex internal architecture. We approximate it by $C = S \times T$, where $S$ is the ‘number of CPU cores’ (corresponding to steps executed in parallel) and $T$ is the timeout. We use $S = 10$ for our calculations. See Appendix A.4 for more details. Architecture and training details For our main experiments, we pre-train standard decoder-only transformer models with 38M and 86M non-embedding parameters and fine-tune them for downstream tasks of premise selection or proof step generation. Full details are given in Appendix C. In our experiments, we use the Portal-to-ISAbelle API (Jiang et al., 2021) to interact with Isabelle. 5.2 Results on PISA and miniF2F benchmarks Our main empirical results, summarized in Table 2 and Table 3, were obtained with the 86M parameter model. Figure 1 and Section 5.2.1 deepen this study, showing that Magnushammer outperforms Sledgehammer across a broad spectrum of computational budgets. Performance on the single-step task In the single-step setting, Magnushammer outperforms Sledgehammer by a wide margin on both PISA (59.5% vs. 38.3%) and miniF2F (34.0% vs. 20.9%). Additionally, on PISA, Magnushammer outperforms TF-IDF and BM25: text-based, non-trainable retrieval methods (Robertson and Zaragoza [2009]), which are strong baselines in common retrieval benchmarks (Thakur et al. [2021]). This suggests that Magnushammer is able to learn more than just superficial text similarity. In all these experiments we used the same evaluation protocol (following Algorithm 2) and computational budget of 1000 as detailed in Appendix D.1. Interestingly, retrieval based on the generic OpenAI embeddings (Neelakantan et al. [2022]) (specifically: text-embedding-ada-002) yields reasonable performance comparable to Sledgehammer. This confirms the potential of neural premise selection to replace traditional symbolic methods. There is, however, a large gap to match Magnushammer. This shows that contrastive fine-tuning on our dataset provides non-trivial gains and supports our hypothesis that Magnushammer learns more than just mere textual similarity exploited by the general purpose method. Performance on the multi-step task Neural theorem provers utilize language models to generate proof steps, following the approach proposed by Polu and Sutskever (2020). This allows for the creation of more complex, multi-step proofs. The proof generation involves sampling a proof step from the language model, verifying it, and repeating this process until the proof is closed or the computational budget is exceeded. The best-first search algorithm is often used to explore the most promising proof steps. Thor (Jiang et al. [2022a]) augments neural theorem provers with premise-selection capabilities. To this end, Thor allows the model to generate proof steps using Sledgehammer, which we replace with Magnushammer (see Appendix D.2 for details). Thor + Magnushammer establishes a new state of the art on the PISA benchmark (71.0% vs. 57.0%). On miniF2F, our method also significantly outperforms Thor and achieves results competitive with the current state of the art. In these experiments, we give Magnushammer a computational budget of 200. It is important to note that other theorem-proving approaches in the multi-step section of Table 3 require much larger language models: for Thor it is 700M non-embedding parameters; DSP (Draft, Sketch, and Prove) by Jiang et al. (2022b) uses Minerva model (Lewkowycz et al. [2022]) with 62B parameters. Moreover, these other approaches rely on ideas orthogonal to premise selection. Specifically, Thor + auto (Wu et al. [2022a]) proposes a variation of Thor, involving expert iteration on auto-formalized data. DSP involves creating a high-level outline of a proof and uses Sledgehammer to solve the low-level subproblems. We hypothesize that both methods would perform even better when combined with Magnushammer. 5.2.1 Scaling computational budget In this section, we discuss how the quality of premise selection methods varies with the computational budget available during evaluation. Figure 1 shows the results, and the definition of the compute budget is provided in Section 5.1. Notably, Magnushammer outperforms Sledgehammer even with very limited computational resources, and it scales well, particularly within the medium budget range. For Magnushammer and BM25, we use Algorithm 3 (Appendix D) in various configurations (i.e., settings of $T$ and $K$). We start with one tactic, $T = \{\text{smt}\}$, and $K = [2^7]$, which yields $C = 2$ (recall that $T = 2$ s). We then gradually add more tactics to $T$ and more values to $K$. The final setup uses $\|T\| = 36$ and $K$ containing all powers of 2, from $2^0$ up to $2^{10}$, which yields $C \approx 800$. The details are provided in Appendix D. For Sledgehammer, we scale the timeout parameter $T$ up to 80 s. 5.3 Impact of training data We study how the amount and type of data impact the proof success rate by comparing HPL and MAPL datasets. For this comparison, we used models with 38M non-embedding parameters and a computational budget of 800. Dataset size Our method is data-efficient: see Figure 3a. We observe that Magnushammer fine-tuned on only 0.1% of MAPL – equivalent to approximately 4K samples – is already able to outperform Sledgehammer. This indicates that when starting from a pre-trained model, Magnushammer is a promising approach for addressing premise selection in theorem-proving environments with limited training data. The effect of pre-training diminishes as the amount of training data increases. Dataset type Fine-tuning on MAPL or HPL leads to subtle differences (56.3% vs. 54.0% when the whole datasets are used). This outcome may be attributed to the impact of model pre-training and the fact that the HPL dataset is rich enough to obtain good performance on the PISA benchmark (as observed in the previous paragraph). We speculate that the bigger MAPL dataset might be essential for future harder benchmarks and scaling up the model size. (a) We randomly sample fractions of MAPL or HPL datasets and use them for training Magnushammer. Even 0.1% of the MAPL dataset allows pre-trained Magnushammer to outperform the Sledgehammer and BM25 baselines. See Table 4 for numerical data. (b) We train Magnushammer of different sizes. Even with a one-layer transformer, Magnushammer outperforms Sledgehammer. We observe consistent performance gains with increasing model sizes. Pre-trained models perform better. See Table 5 for numerical data. Figure 3: Impacts of the training data quantity and the model parameters on the proof rate. The vertical axis is the proof rate in percentage. In Subfigure 3a, the horizontal axis is the fraction of training dataset used and in Subfigure 3b, it is the number of parameters in the model. 5.4 Ablations We use models trained on the MAPL dataset and evaluate them with a computational budget of 800. To study how the performance of our method depends on the model size, we vary the number of layers $ L $ and embedding dimension $ D $. A positive correlation between the model size and the proof rate is shown in Figure 3b. We observe that even a tiny model with 920K parameters ($ L = 1, D = 256 $) outperforms Sledgehammer (40.7% vs. 38.3%). We also note the benefit of pre-training and that scaling the number of layers is more beneficial than scaling the embedding dimension. The details can be found in Appendix C.1. The impact of re-ranking is studied in Appendix C.5. 6 Related Work Premise selection becomes a crucial task whenever proving theorems automatically within a large formal library. Moreover, this task has several unique aspects that are challenging from the perspective of learning-based approaches. Therefore, there exist multiple works that tackle learning premise selection (either explicitly or implicitly) applying various methods focusing on different aspects. Many works employ classical machine learning like Bayesian and kernel methods (Kühlwein et al., 2012; Alama et al., 2014), $ k $-NN (Blanchette et al., 2016), or decision trees (Piotrowski and Urban, 2018; Nagashima and He, 2018; Piotrowski et al., 2023). The common weakness of these approaches is the necessity of using hand-engineered features, whereas faster, simpler training is an advantage. Alemi et al. (2016) were the first to apply deep learning to premise selection, thus dispensing with the hand-designed features completely. Their approach was evaluated in an automated theorem proving setting and not in a proof assistant, as is Magnushammer. They also implicitly learn embeddings of conjectures and premises, which are concatenated and passed through a shallow network, whereas the training signal comes from the logistic loss. In contrast, Magnushammer demonstrated the strength of training with the contrastive loss, where the obtained embeddings just need to be passed through a simple cosine similarity measure to provide high-quality rankings. Most of the methods explicitly targeting the premise selection problem (including this work) retrieve a ranking of independently treated premises. In contrast, Piotrowski and Urban (2020) aimed at modelling the implicit dependencies between the premises and used LSTM-based language models to produce structured sequences of premises. However, the premises were treated there as opaque tokens, not giving the neural model the ability to inspect the statements of the premises. Effective deep learning approaches often leverage the explicit structure of mathematical expressions using graph neural networks (Wang et al., 2017; Paliwal et al., 2020; Goertzel et al., 2022). Our work uses the transformer architecture (Vaswani et al., 2017), which is highly scalable and capable of producing powerful representations of raw text data. Pre-trained transformer language models have been applied to various aspects of theorem proving, including autoformalization (Wu et al., 2022a; Jiang et al., 2022b), conjecturing (Urban and Jakubuv, 2020), and tactic prediction / proof step search (Yang and Deng, 2019; Polu and Sutskever, 2020; Han et al., 2022; Lample et al., 2022; Polu et al., 2023). The works from the last category often implicitly deal with premise selection by treating premises as names / tokens to be generated and not inspecting their statements. The application of generative language models to statement-aware premise selection has been limited, as the length of the possible premises often greatly exceeds the context of several thousand tokens that the models are designed to handle. Thor (Jiang et al., 2022a) circumvents the difficulty of premise selection by invoking Sledgehammer. In contrast, Magnushammer retrieves rather than generates to overcome the context length limitation. Therefore it can be used in tandem with other models (its combination with Thor is demonstrated in Section 5). Batch-contrastive learning is widely used in speech (van den Oord et al., 2018), text (Izacard et al., 2021), image (Chen et al., 2020) and image-text (Radford et al., 2021) representation learning. These methods have proven effective despite the possibility of false negatives occurring in contrastive batches (Robinson et al., 2021). The SELECT phase of our premise selection model relies on in-batch negative examples to train the retriever, similar to HOList (Bansal et al., 2019) and Contriever (Izacard et al., 2021). Like HOList, we mine additional negatives, which we found crucial for performance. The RERANK stage closely resembles (Nogueira and Cho, 2019), but instead of using BM25, we jointly train retrieval and re-ranking, utilizing premises retrieved by SELECT as hard negatives for RERANK training. Han et al. (2021) use contrastive learning in informal premise selection. Concurrently to our work, Yang et al. (2023) develop a premise selection method for Lean also using contrastive learning in a way similar to our SELECT method, but without the RERANK stage. There are multiple lines of work considering datasets based on formal theorem proving. These include benchmarks like ProofNet (Azerbayev et al., 2022) for Lean, and miniF2F (Zheng et al., 2022) that supports multiple ITPs. These datasets only focus on evaluation, not providing data for training the models. Another line of research focuses on benchmarking machine learning models’ reasoning capabilities while also providing training data (Bansal et al., 2019; Li et al., 2021; Han et al., 2022). Existing public datasets for premise selection include the ones introduced in (Alama et al., 2014; Piotrowski and Urban, 2020). In comparison to these works, we publish the data in high-level, textual format, as seen in Isabelle, instead of low-level, structured languages such as TPTP (Sutcliffe, 2017). There exists a rich body of work developing complex hammers systems for different proof assistants (Paulson and Blanchette, 2012; Kaliszyk and Urban, 2015a; Gauthier and Kaliszyk, 2015; Czajka and Kaliszyk, 2018). Unlike the traditional hammers, our method does not depend on external ATPs and requires little domain-specific knowledge. 7 LIMITATIONS AND FUTURE WORK Other proof assistants Magnushammer treats proof states and premises as text and makes no assumptions about their structure. As such, no feature engineering is needed to apply it to other proof assistants. We conjecture that Magnushammer can prove effective in other environments because it is agnostic to the logic or type system used. We plan to evaluate Magnushammer in Lean proof assistant on ProofNet (Azerbayev et al., 2022) and miniF2F (Zheng et al., 2022) benchmarks, using the recently published LeanDojo toolkit (Yang et al., 2023) that also provides baselines for comparison. Richer proof and premise representations Magnushammer utilizes the textual representation of the proof state given by Isabelle. This representation, however, does not provide complete semantic information about the referenced objects. Including function definitions and object types in the proof state representation might further improve performance. Modelling full proof steps Combining language models with external premise selection tools significantly improves their theorem-proving performance, as demonstrated by Jiang et al. (2022a) and our work. A natural step would be to further integrate premise selection with language models into a single model capable of generating proof steps containing relevant retrieved premises. A proof of concept of this idea was explored by Tworkowski et al. (2022). This would also allow to model existing implicit dependencies between returned premises, which was shown beneficial by Piotrowski and Urban (2020). We believe that recent advances in retrieval-augmented language models (Wu et al., 2022b; Borgeaud et al., 2022) could facilitate progress in this direction. ACKNOWLEDGEMENTS We gratefully acknowledge that our research was supported with Cloud TPUs from Google’s TPU Research Cloud (TRC). Moreover, Piotr Miłoś was supported by the Polish National Science Centre grant 2019/35/O/ST6/03464. REPRODUCIBILITY STATEMENT The data that were used for pre-training of the backbone transformer model of Magnushammer are freely available under this link: https://pile.eleuther.ai/ The Isabelle data used in training for the down-stream tasks are available under this link: https://huggingface.co/datasets/Simontwice/premise_selection_in_isabelle The benchmarks used for evaluation of Magnushammer are freely available on GitHub: - miniF2F: https://github.com/openai/miniF2F - PISA: https://github.com/albertqjiang/Portal-to-ISAbelle PISA also implements the interface for interacting with Isabelle that we used in our experiments. Appendix A.4 specifies the setup of Sledgehammer that we used in our comparisons. Appendices B and C detail the shape of the transformer architecture used, define the loss functions applied in the SELECT and RERANK stages, specify the hyperparameters used in pre-training and training for our down-stream tasks, and disclose the hardware used for training. Appendix D details the setup for evaluation of Magnushammer in Isabelle, in particular the list of tactics applied on top of the Magnushammer’s premise selection. REFERENCES Jesse Alama, Daniel Kühlwein, Evgeni Tsivtsivadze, Josef Urban, and Tom Heskes. Premise selection for mathematics by corpus analysis and kernel methods. CoRR, abs/1108.3446, 2011. URL http://arxiv.org/abs/1108.3446 Jesse Alama, Tom Heskes, Daniel Kühlwein, Evgeni Tsivtsivadze, and Josef Urban. Premise selection for mathematics by corpus analysis and kernel methods. J. Autom. Reason., 52(2): 191–213, 2014. doi: 10.1007/s10817-013-9286-5. URL https://doi.org/10.1007/s10817-013-9286-5 Alexander A. Alemi, François Chollet, Geoffrey Irving, Christian Szegedy, and Josef Urban. DeepMath – deep sequence models for premise selection. CoRR, abs/1606.04442, 2016. URL http://arxiv.org/abs/1606.04442 Zhangir Azerbayev, Bartosz Piotrowski, and Jeremy Avigad. ProofNet: A benchmark for auto-formalizing and formally proving undergraduate-level mathematics problems. In Advances in Neural Information Processing Systems 35, 2nd MATH-AI Workshop at NeurIPS 22, 2022. URL https://mathai2022.github.io/papers/20.pdf Kshitij Bansal, Sarah M. Loos, Markus N. Rabe, Christian Szegedy, and Stewart Wilcox. HOList: An environment for machine learning of higher order logic theorem proving. In Kamalika Chaudhuri and Ruslan Salakhutdinov, editors, Proceedings of the 36th International Conference on Machine Learning, ICML 2019, 9-15 June 2019, Long Beach, California, USA, volume 97 of Proceedings of Machine Learning Research, pages 454–463. PMLR, 2019. URL http://proceedings.mlr.press/v97/bansal19a.html Haniel Barbosa, Clark W. Barrett, Martin Brain, Gereon Kremer, Hanna Lachnitt, Makai Mann, Abdalrhman Mohamed, Mudathir Mohamed, Aina Niemetz, Andres Nötzli, Alex Ozdemir, Mathias Preiner, Andrew Reynolds, Ying Sheng, Cesare Tinelli, and Yoni Zohar. cvc5: A versatile and industrial-strength SMT solver. In Dana Fisman and Grigore Rosu, editors, Tools and Algorithms for the Construction and Analysis of Systems – 28th International Conference, TACAS 2022, Held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS
MBIGXMT0qC	Despite the possible data and information leakage in the experimental framework, the proposed method fails to outperform standard protein language models like ESM-2 in tasks such as contact and secondary structure prediction. Given that the model incorporates a broader range of pre-training loss than ESM-2, these results suggest that universal pre-training across both proteins and molecules may not offer advantages in protein-only or molecule-only tasks. This significantly weakens the paper's primary claim: the benefit of combining protein and molecule data for pre-training.	MULTI-SCALE PROTEIN LANGUAGE MODEL FOR UNIFIED MOLECULAR MODELING Anonymous authors Paper under double-blind review ABSTRACT Protein language models have shown great potential in protein engineering. However, the current protein language models mainly work in the residue scale, which cannot offer information in the atom scale. The strong power of protein language models could not be fully exploited to benefit the applications that cross protein and small molecules. In this paper, we propose $ms$-ESM (multi-scale ESM) to realize the multi-scale unified molecular modeling by pre-training on multi-scale code-switch protein sequence and describing relationships among residues and atoms with a multi-scale position encoding. Experimental results show that $ms$-ESM outperforms previous methods in protein-molecule tasks and is on par with the state-of-the-art in protein-only and molecule-only tasks. 1 INTRODUCTION Protein language models (PLMs) have shown great potential in protein engineering, which captures biochemical and co-evolutionary knowledge in the pre-training of large-scale protein sequences, and gives strong results in protein structure prediction $[Wu et al., 2022; Fang et al., 2022b]$, protein fitness prediction $[Mardikoraem & Woldring, 2023; Notin et al., 2022]$, protein design $[Zheng et al., 2023; Ferruz et al., 2022]$, etc. For example, several important models have been built upon ESM $[Rives et al., 2021; Lin et al., 2022c]$—a widely used PLM—including ESM-fold $[Lin et al., 2023]$ for accurate protein structure prediction and Im-design $[Verkuil et al., 2022; Hie et al., 2022]$ for designing proteins with specific functions. Current PLMs mainly work in the protein residue (amino acid) scale, which cannot offer information in the atom scale. In such a case, the strong power of PLMs could not be fully exploited to benefit the applications across macro-molecule (protein) and small molecules$^1$ and external small molecular models have to be included to deal with these applications. However, proteins are also composed of atoms, and modeling protein only in the residue scale could be of low resolution. Intuitively, extending PLMs to work in both residue and atom scales would make it applicable to a larger range of applications. However, developing multi-scale PLMs is non-trivial. First, the unified molecular modeling that works in both residue and atom scales is infeasible, due to the incompatible vocabularies used in the two different scales. A direct way to inject atomic information into residue scale PLMs is to represent and pre-train the proteins in the atom scale, in addition to the original residue scale pre-training. Nevertheless, a protein can consist of thousands of residues and thus contains hundreds of thousands of atoms, which is quite inefficient for modeling. Second, designing appropriate position encoding to accurately describe the relationships among residue and atoms in the same protein is also challenging, which is quite complex and involves relationships varying from residues to residues, residues to atoms, and atoms to atoms. To address the above challenges, in this paper, we propose $ms$-ESM (multi-scale ESM), which realizes the multi-scale unified molecular modeling by a) pre-training on multi-scale code-switch protein sequence and b) describing relationships among residues and atoms with a multi-scale position encoding. --- $^1$These applications widely exist in chemistry and biology and are always quite crucial for specific scientific discoveries. For example, drug discovery aims to find small molecules that can bind to protein pockets $[Anderson et al., 2003; Batool et al., 2019]$ and enzyme engineering searches enzymes (a special protein) that can catalyze molecular reactions efficiently $[Mazureenko et al., 2019; Kroll et al., 2023a]$. sequence and b) describing relationships among residues and atoms with a multi-scale position encoding. First, inspired by the idea of multi-lingual code-switching in machine translation (Yang et al., 2020; Li et al., 2022a), ms-ESM proposes to learn multi-scale knowledge by pre-training on the multi-scale code-switch protein sequences, which is obtained by randomly unzipping protein residues into their corresponding atoms. In such a case, ms-ESM can not only capture the multi-scale aligned knowledge but also efficiently deal with inputs in both residue and atom scale in the meantime. Second, ms-ESM employs a multi-scale position encoding for comprehensively distinguishing residues and atoms in the code-switch protein sequence. In the residue scale, we extend the original position encoding used in ESM to consist with the current best practice in pure residue situations and avoid ill-defined position information among atoms. In the atom scale, to distinguish the relation among unzipped atoms, we use the spatial distance matrix directly encoding their 3D positions. With the above approach, we can appropriately describe all the relationships of all objects in the code-switch sequence. We use three types of downstream tasks (protein-molecule tasks, protein-only tasks, and molecule-only tasks) to demonstrate the versatility and effectiveness of our proposed ms-ESM. In protein-molecule tasks, ms-ESM outperforms previous methods that model proteins and molecules separately instead of unified modeling as ms-ESM. In protein-only and molecule-only tasks, ms-ESM is on par with the state-of-the-art. Experiment results show that we successfully model proteins and molecules in a unified style without suffering from severe information interference. 2 PROPOSED METHOD: MS-ESM In this section, we describe our multi-scale pre-training model, i.e., ms-ESM, in detail. Intuitively, inspired by the idea of multi-lingual code-switching method, ms-ESM first creates multi-scale code-switch protein sequences by unzipping partial residues. Through training on such sequences with correctly designed multi-scale position encoding, ms-ESM can work well in both residue and atom scale. When dealing with protein-molecule tasks, ms-ESM does not need any extra models and can exert the maximum potential of pre-training. Specifically, in Section 2.1, we first introduce the overall objective of training ms-ESM. Then, in Section 2.2, we dive into the details about how we construct a code-switch protein sequence and implement the multi-scale pre-training. To describe the complicated position relationship in the code-switch sequence, we design a multi-scale position encoding in Section 2.3. In Section 2.4, we provide more details about ms-ESM, including an elaboration of its parameterization. 2.1 OVERVIEW We start with an overview of our multi-scale pre-training model, i.e., ms-ESM (see Figure 1). Briefly, the total objective of our pre-training can be written as the following loss function: \[ L_\theta = \sum_{X_i \in D} L_{MLM}(\bar{X}_i, E_i; \theta) + L_{PDR}(\bar{X}_i, E_i; \theta) \] \[ = \sum_{X_i \in D} L_{MLM}(\text{UNZIP}(X_i), \text{MSPE}(X_i); \theta) + L_{PDR}(\text{UNZIP}(X_i), \text{MSPE}(X_i); \theta) \] For each data $ X_i $ in dataset $ D $, we first create its code-switch sequence $ \bar{X}_i $ by unzipping partial residues. Based on the code-switch sequence, we use Masked Language Modeling (MLM) and Pair-wise Distance Recovery (PDR) as the pre-training tasks. We discuss the details of $ \bar{X}_i $, $ L_{MLM} $, and $ L_{PDR} $ in Section 2.2. As residues and atoms coexist in the sequence, we further design a Multi-Scale Position Encoding (MSPE) $ E_i $ to describe the complicated position relationship in $ \bar{X}_i $ (see Section 2.3). We show more details of ms-ESM, including the parameterization of $ \theta $ in Section 2.4. Notably, as we also use the molecule data in pre-training, ms-ESM can take proteins or molecules as input separately. --- 2 construct sentences that alternate between two or more languages 2.2 Multi-scale Pre-training In this section, we elaborate how we create a code-switch sequence $\bar{X}$ and adopt the pre-training tasks, i.e., MLM and PDR on it (see Figure 2). Code-Switch Protein Sequence Specifically, in residue scale, a protein $X$ can be seen as a sequence of $L$ residues, i.e., $X = (r_1, \cdots, r_i, \cdots, r_L)$. Each residue $r_i$ further consists of a specific set of $N$ atoms $A_i = \{a_{i1}, \cdots, a_{iN}\}$. To create a code-switch protein sequence $\bar{X}$, we first choose a group of residues and insert their corresponding atoms into $X$. Especially, when inserting the atoms, we first assign an order to them. For example, after inserting the atom set $A_i$ into $X$, we get a code-switch sequence $$\bar{X} = (r_1, \cdots, r_i, \text{ORDER}(A_i), \cdots, r_L)$$ $$= (r_1, \cdots, r_i, a_{i1}, \cdots, a_{iN}, \cdots, r_L)$$ $$= (h_1, \cdots, h_i, h_{i+1}, \cdots, h_{i+N}, \cdots, h_{L+N})$$ where $\text{ORDER}$ is the order that is assigned to atom set (see Appendix A). $h_i$ represents a single residue or atom in $\bar{X}$. We also denote all the atoms in $\bar{X}$ as $\bar{A}$ and all the residues as $\bar{R}$. Notably, when we insert the atom set $A_i$ of residue $r_i$, we still retain $r_i$. This allows the model can either attend to the corresponding residue scale information or to the surrounding atom scale information when predicting masked atoms, which encourages the model to align residue scale and atom scale representations, just like cross-lingual pre-training (Conneau & Lample, 2019). We show an illustration of the code-switch sequence in Figure 2. Masked Language Modeling After obtaining the code-switch sequence $\bar{X}$, we can do the MLM on it. Different from the MLM used in ESM, we ask models to predict not only masked residues but also masked atoms. Specifically, we first randomly mask part of atoms or residues in $\bar{X}$, and then ask the model to predict the original atoms or residues based on the context. $$L_{\theta_{MLM}} = - \sum_{h \in \text{MASK}(\bar{X})} \log p_\theta(h\|\bar{X} \setminus \text{MASK}(\bar{X}))$$ where $\text{MASK}$ is the set of masked atoms and residues, and $h$ is a single masked atom or residue. Figure 2b is the framework of MLM task. Pair-wise Distance Recovery We also use the PDR as another pre-training task. Briefly, we use corrupted atoms as model input and ask model to recover the correct Euclidean distances between these atoms. We corrupt the atoms by adding noises to their coordinates. Specifically, we use a random position which is around (Euclidean distances < $\epsilon$, Appendix A) the ground-truth coordinate to replace the ground-truth. Models need to recover the real distances based on the corrupted coordinates. $$L_{\theta_{PDR}} = \sum_{h_i, h_j \in \bar{A}, i \neq j} \\| \text{Dis}_\theta(c_i + \sigma_i, c_j + \sigma_j) - \text{Dis}(c_i, c_j) \\|_2$$ where $c_i = \text{COORD}(h_i)$ and $c_j = \text{COORD}(h_j)$. where $ D_{IS_\theta} $ is the recovered distance and $ DIS $ is the ground truth. $ \text{COORD} $ extracts coordinates from atoms. $ \sigma_i, \sigma_j $ are the corresponding noises added to atom coordinates $ c_i, c_j $. In more detail, these noises will affect the atom scale position encoding in Section 2.3. Figure 2 shows the framework of PDR task. Notably, when training $ ms\text{-ESM} $, we can mix up a protein dataset $ D_p $ and a molecule dataset $ D_m $ as the final dataset, i.e., $ D = D_p \cup D_m $. For a data $ X $ from $ D_m $, its corresponding $ \tilde{X} $ is the ordered set of all its atoms and its $ \tilde{A} = \tilde{X}, \tilde{R} = \emptyset $. ### 2.3 Multi-scale Position Encoding Encoding the position relationship in the code-switch sequence is challenging. As residues and atoms coexist in the code-switch sequence, a well-functioning position encoding needs to describe the position relationship from residues to residues, residues to atoms, and atoms to atoms (from the same residue or not). This situation is more complicated than pure residue ones. Because previous encoding in PLM is only designed for pure residue situations, they cannot describe the relationship from residues to atoms, and atoms to atoms. In this section, we design a Multi-Scale Position Encoding $ E $ to encode the position relationship in a code-switch sequence. Specifically, $ E $ contains a residue scale position encoding $ E^R $ and an atom scale position encoding $ E^A $, i.e., $ E = (E^R, E^A) $. For $ E^R $, we carefully extend an existing encoding method letting it can encode the relation from residues to atoms, while keeping consistent with the original encoding when dealing with pure residue situations. For $ E^A $, to capture the relationship among atoms, we directly encode their 3D position with the spatial distance matrix. The multi-scale encoding style makes sure that no ill-defined position relationship influences the pre-training letting $ ms\text{-ESM} $ work well in both scales. Figure 3 is the framework of our multi-scale position encoding. We elaborate each of them in the following paragraphs. #### Residue Scale Position Encoding We design the residue scale position encoding $ E^R $ following two principles: a) For encoding the relationship between two residues, $ E^R $ should be consistent with the mainstream encoding method. b) For atoms from the same unzipped residue, $ E^R $ should not provide any ill-defined position information. As previous PLMs show the success of the mainstream encoding method in dealing with the pure residue situation, it is wise for $ E^R $ to keep consistent with it. Moreover, when dealing with two atoms from the same residue, as we cannot define the residue scale position relationship inside the residue, $ E^R $ needs to avoid the effect of such ill-defined information. In particular, we use Rotary Position Embedding (RoPE) (Su et al., 2021), the original position encoding in ESM-2, to describe the position relationship among the residues in a code-switch sequence. When we need to assign a position encoding to the atom in the code-switch sequence, we reuse the position encoding of the residue that the atom belongs to. If we cannot find which residue that the atom comes from, we assign a fixed position encoding (RoPE(0) in our paper) to it. Formally, for a code-switch sequence $ \tilde{X} $, its residue scale position encoding $ E^R = (e^R_1, \ldots, e^R_{L+N}) $. can be obtained according to the following formulation: \[ e_i^R = \begin{cases} \text{RoPE}(j) & h_i \in \bar{R}, h_i = r_j \\ \text{RoPE}(k) & h_i \in \bar{A}, \exists k, h_i \in A_k \\ \text{RoPE}(0) & \text{otherwise} \end{cases} \] By adopting such encoding strategy, $ E^R $ satisfies the two principles aforementioned. Specifically, for pure residue situations, $ E^R $ is exactly RoPE. When dealing with atoms from the same residue, the relative nature of RoPE makes sure no ill-defined information will affect the pre-training model. We refer readers to Su et al. (2021) for more details of RoPE’s properties. Atom Scale Position Encoding Because $ E^R $ will not provide the position encoding for atoms from the same residue, we need an atom scale position encoding $ E^A $ to describe the relationship from atoms to atoms. As suggested by Zhou et al. (2023), we use Euclidean distance matrix and Gaussian kernel to encode the 3D position of atoms. For $ h_i, h_j \in \bar{X} $, their atom scale position encoding $ e_{ij}^A $ can be calculate as: \[ e_{ij}^A = \begin{cases} 0 & h_i \in \bar{R} \text{ or } h_j \in \bar{R} \\ \text{GAUSSIAN}(\text{DIS}(c_i, c_j)) & \text{otherwise}, c_i = \text{COORD}(h_i), c_j = \text{COORD}(h_j) \end{cases} \] We refer readers to Zhou et al. (2023) for more details of this 3D position encoding. ### 2.4 OTHER DETAILS OF ms-ESM We parameterize the $ \theta $ with a slight modification of the original Transformer (Vaswani et al., 2017). To be specific, we first use our residue scale position encoding $ E^R $ to replace the sinusoidal encoding in the Transformer. For the atom scale position encoding $ E^A $, we treat it as the bias term of self-attention layers. The self-attention in ms-ESM can be calculated like: \[ \text{ATTENTION}(Q, K, V; E^A) = \text{SOFTMAX}\left( \frac{QK^T}{\sqrt{d_k}} + E^A \right)V \] where $ Q, K, V $ is the Query, Key, and Value corresponding to $ \bar{X} $. We refer readers to Vaswani et al. (2017) for more details of the original Transformer. By only modifying the original Transformer slightly, ms-ESM can process residues and atoms at the same time, which makes it a versatile model for many downstream tasks. Moreover, ms-ESM shows great compatibility with existing pre-training model, e.g., ESM series, which allows us to build up a better model based on previous study more easily. ## 3 EXPERIMENTS To verify the effectiveness of our multi-scale pre-training, we primarily evaluated the model’s performance on various protein-molecule tasks (Section 3.1). In addition, to validate our model’s competitive performance against other baseline models in protein-only and molecule-only tasks, we also conducted experiments on multiple protein-only tasks (Section 3.2) and molecule-only tasks (Section 3.3). For each of them, we will provide the details about fine-tuning protocol, baseline methods and performance results in corresponding paragraphs. Besides, we also perform ablation studies which discuss how different position encoding strategies (Section 3.4) effect the performance of our model. The detailed pre-training configuration, including pre-training datasets and hyperparameters, e.g., unzip ratio of residues, can be found in Appendix A. Table 1: Performance comparison on enzyme-substrate affinity regression task. \| Method \| Protein Pre-training Model \| Molecule Pre-training Model \| MSE ↓ \| $R^2$ ↑ \| Pearson ↑ \| \|-----------------\|----------------------------\|-----------------------------\|-------\|---------\|-----------\| \| Gollub et al. (2023) \| / \| / \| 0.463 \| 0.680 \| \| \| Kroll et al. (2021) \| / \| / \| 0.653 \| 0.527 \| 0.728 \| \| XGBoost \| ESM-2 35M \| Uni-Mol 48M \| 0.652 \| 0.528 \| 0.727 \| \| ProSmith \| ESM-2 35M \| Uni-Mol 48M \| 0.642 \| 0.536 \| 0.733 \| \| XGBoost \| ms-ESM 35M \| ms-ESM 35M \| 0.623 \| 0.548 \| 0.742 \| \| ProSmith \| ms-ESM 35M \| ms-ESM 35M \| 0.599 \| 0.566 \| 0.753 \| Table 2: Performance comparison on drug-target affinity regression task. \| Method \| Protein Pre-training Model \| Molecule Pre-training Model \| MSE ↓ \| CI ↑ \| $r_m^2$ ↑ \| \|-----------------\|----------------------------\|-----------------------------\|-------\|------\|----------\| \| Oztürk et al. (2018) \| / \| / \| 0.261 \| 0.878 \| 0.630 \| \| Shin et al. (2019) \| / \| Molecule Transformer \| 0.245 \| 0.887 \| 0.665 \| \| Nguyen et al. (2021a) \| / \| / \| 0.229 \| 0.893 \| 0.685 \| \| Nguyen et al. (2021b) \| TAPE 38M \| / \| 0.228 \| 0.893 \| / \| \| Qiu et al. (2021) \| ProtBert 420M \| / \| 0.205 \| 0.896 \| 0.709 \| \| Kao et al. (2021) \| / \| / \| 0.202 \| 0.907 \| / \| \| Yuan et al. (2022) \| ESM-1b 650M \| / \| 0.208 \| 0.913 \| 0.743 \| \| Yang et al. (2022) \| / \| / \| 0.207 \| 0.900 \| 0.710 \| \| He et al. (2023) \| BiLSTM \| BiLSTM \| 0.196 \| 0.914 \| 0.744 \| \| XGBoost \| ESM-2 35M \| Uni-Mol 48M \| 0.261 \| 0.885 \| 0.652 \| \| ProSmith \| ESM-2 35M \| Uni-Mol 48M \| 0.219 \| 0.899 \| 0.711 \| \| XGBoost \| ms-ESM 35M \| ms-ESM 35M \| 0.248 \| 0.889 \| 0.668 \| \| ProSmith \| ms-ESM 35M \| ms-ESM 35M \| 0.191 \| 0.906 \| 0.759 \| ### 3.1 PROTEIN-MOLECULE TASKS Fine-tuning Protocol For protein-molecule tasks, we follow the benchmark protocol from Pro-Smith (Kroll et al., 2023b) to evaluate ms-ESM on three tasks, including enzyme-substrate affinity regression (ESAR), drug-target affinity regression, and enzyme-substrate pair classification. Specifically, each task provides the protein residue sequence and molecule SMILES string as input and asks models to tell whether the protein-molecule pair has high affinity. As our ms-ESM cannot process SMILES strings, we first use RDKit (Landrum et al., 2013) to generate corresponding molecule conformation according to the SMILES and then extract the atom sequence and atom-scale position encoding for ms-ESM. For more details of the fine-tuning, see Appendix[B.1]. Baselines We compare ms-ESM with multiple baselines on each task, including supervised and pre-training baseline. For each baseline, we list their protein pre-training model and molecule pre-training model in corresponding tables. More details of each baseline can be seen in corresponding papers. As only ProSmith (Kroll et al., 2023b) provides a framework to combine protein pre-training model and molecule pre-training model, we follow their framework by substituting both protein model and molecule model to ms-ESM for fair comparison. We also provide an XGBoost (Chen & Guestrin, 2016) variant of ProSmith, which takes the concatenation of protein and molecule representation as features and can directly evaluates whether two representations can work well together. Results Table 1, Table 2, and Table 3 show the experiment results of ms-ESM and competitive baselines on three tasks. From the results, we can get the summarization as follows: (1) ms-ESM achieves the SOTA result on most metrics. (2) ProSmith and XGBoost based on our ms-ESM are always better than the version that combines two separate pre-training models. (3) ms-ESM can beat the methods which based on much larger pre-training models. These phenomena obviously indicate that pre-training proteins and molecules in one model can further release the power of pre-training technique on protein-molecule tasks. Fusing two separate pre-training models can be a sub-optimal for such tasks and the problem can not be fixed by using larger pre-training models. Table 3: Performance comparison on enzyme-substrate pair classification task. \| Method \| Protein Pre-training Model \| Molecule Pre-training Model \| ACC ↑ \| MCC ↑ \| ROC-AUC ↑ \| \|-----------------\|----------------------------\|----------------------------\|-------\|-------\|-----------\| \| Kroll et al. (2023b) \| ESM-1b 650M \| / \| 91.5% \| 0.780 \| 0.956 \| \| XGBoost \| ESM-2 35M \| Uni-Mol 48M \| 89.9% \| 0.729 \| 0.941 \| \| ProSmith \| ESM-2 35M \| Uni-Mol 48M \| 90.8% \| 0.754 \| 0.943 \| \| XGBoost \| ms-ESM 35M \| ms-ESM 35M \| 90.6% \| 0.750 \| 0.943 \| \| ProSmith \| ms-ESM 35M \| ms-ESM 35M \| 91.8% \| 0.781 \| 0.954 \| Table 4: Performance comparison on the contact prediction task. \| Method \| Short Range ↑ \| Medium Range ↑ \| Long Range ↑ \| \|--------------\|---------------\|----------------\|--------------\| \| \| P@L \| P@L/2 \| P@L/5 \| P@L \| P@L/2 \| P@L/5 \| P@L \| P@L/2 \| P@L/5 \| \| TAPE 38M \| 0.28 \| 0.35 \| 0.46 \| 0.19 \| 0.25 \| 0.33 \| 0.17 \| 0.20 \| 0.25 \| \| ResNet 38M \| 0.25 \| 0.34 \| 0.46 \| 0.18 \| 0.25 \| 0.35 \| 0.10 \| 0.13 \| 0.17 \| \| ESM-2 35M \| 0.20 \| 0.29 \| 0.46 \| 0.22 \| 0.32 \| 0.45 \| 0.30 \| 0.39 \| 0.49 \| \| ms-ESM 35M \| 0.21 \| 0.31 \| 0.48 \| 0.23 \| 0.32 \| 0.45 \| 0.29 \| 0.38 \| 0.48 \| 3.2 Protein-only tasks Fine-tuning Protocol We use protein-only tasks to evaluate whether ms-ESM still has good understanding of proteins. Specifically, we follow TAPE (Rao et al., 2019) and use the tasks secondary structure prediction and contact prediction to judge the ability of protein pre-training models in protein structure understanding. To perform secondary structure prediction, models need to understand the local structure of proteins, e.g., helix and strand. For the task contact prediction, it requires models to have a good understanding of proteins more globally. As ms-ESM supports protein residue sequences, we follow TAPE’s protocol strictly. For a fair comparison, we remove the test data that appears in the pre-training data, and the proportion of this part of the data is less than 4%. For more details of the fine-tuning protocol, readers can find them in Appendix B.2. Baselines For the protein-only benchmark, we chose several popular protein pre-training models as our baselines. TAPE (Rao et al., 2019) and ResNet (Rao et al., 2019) use a Transformer (Vaswani et al., 2017) and a dilated residual network (Yu et al., 2017) as the backbone network to train a masked language model (MLM) respectively. Because ms-ESM loads a checkpoint from ESM-2 as the parameter initialization, we also include the ESM-2 model (Lin et al., 2023) in our comparison. Results We report the results of contact prediction and secondary structure prediction in Table 4 and Table 5 respectively. Although ms-ESM does not achieve the best performance among comparing methods. However, as shown in the tables, ms-ESM performs very similarly to ESM-2 on both secondary structure prediction and contact prediction, which indicates that we do preserve the local and global understanding of proteins originally from ESM-2. Promisingly, ms-ESM can have a better protein understanding by simply using a larger ESM-2 as the parameter initialization. We leave it as the future work. Table 5: Performance comparison on secondary structure prediction task. \| Method \| SS3(ACC) ↑ \| SS8(ACC) ↑ \| \|--------------\|------------\|------------\| \| \| cb513 \| ts115 \| casp12 \| cb513 \| ts115 \| casp12 \| \| TAPE 38M \| 0.73 \| 0.77 \| 0.71 \| 0.59 \| 0.64 \| 0.59 \| \| ResNet 38M \| 0.75 \| 0.78 \| 0.72 \| 0.58 \| 0.64 \| 0.58 \| \| ESM-2 35M \| 0.80 \| 0.82 \| 0.74 \| 0.65 \| 0.70 \| 0.61 \| \| ms-ESM 35M \| 0.79 \| 0.81 \| 0.74 \| 0.63 \| 0.69 \| 0.60 \| 3.3 Molecule-only tasks Table 6: Performances on molecular property classification and regression tasks. \| Method \| Reg. (MAE) ↓ \| Cls. (AUC,%) ↑ \| \|-----------------\|--------------\|---------------\| \| \| QM8 \| QM9 \| HIV \| MUV \| \| D-MPNN \| 0.0190 \| 0.00814 \| 77.1\| 78.6\| \| N-GramXBG \| 0.0215 \| 0.00964 \| 78.7\| 74.8\| \| GROVERlarge \| 0.0224 \| 0.00986 \| 68.2\| 67.3\| \| MolCLR \| 0.0178 \| / \| 78.1\| 79.6\| \| GEM \| 0.0171 \| 0.00746 \| 80.6\| 81.7\| \| Uni-MolH \| 0.0156 \| 0.00467 \| 80.8\| 82.1\| \| Uni-Molw/o H \| 0.0160 \| 0.00540 \| 78.3\| 72.0\| \| ms-ESMw/o H \| 0.0166 \| 0.00590 \| 74.9\| 72.6\| Fine-tuning Protocol We use molecule-only tasks to evaluate whether we successfully make a protein pre-training model (originally trained under pure protein situations) work well under pure molecule situations. As we use the molecule data from Uni-Mol [Zhou et al., 2023] to train our ms-ESM, we also adopt the fine-tuning protocol of Uni-Mol to evaluate the molecule understanding ability of our models. Specifically, we only use two molecule property regression tasks (QM8, QM9) and two molecule property classification tasks (HIV, MUV) in our comparison, because each of these tasks can provide a large dataset (> 10000 instances), which can avoid the over-fitting problems in the fine-tuning stage and give us a more stable experiment results. For more fine-tuning details and results on more molecule-only tasks, we refer readers to Appendix B.3 and Appendix D. Baselines Following Uni-Mol, we use multiple supervised and pre-training methods as our baselines. The details of each baseline model can be found in the Uni-Mol paper [Zhou et al., 2023]. Notably, according to whether hydrogen atoms are removed or not in pre-training, there are two versions of Uni-Mol, i.e., Uni-MolH and Uni-Molw/o H. We report the results of the two versions of Uni-Mol in Table 6 for fair comparison, because we remove hydrogen atoms when training ms-ESM. We only distinguish the two versions of Uni-Mol here. Without further explain, we refer Uni-Molw/o H to Uni-Mol. Results Table 6 shows the experiment results of both molecular property classification and regression tasks. Similar to protein-only tasks, ms-ESM is not the best method for molecular property prediction. Nevertheless, it is comparable to the Uni-Mol (without hydrogen atoms version) on most of tasks, which makes it still a strong method for molecule-only tasks. Considering that retaining hydrogen atoms on Uni-Mol can improve performance, we believe that we can further boost the ms-ESM’s performance by keeping hydrogen atoms in pre-training. In summary, the results on molecule-only tasks clearly demonstrate that we successfully make a protein pre-training model works well under pure molecule situations. ### 3.4 ABLATION Multi-scale Position Encoding To validate the effectiveness of multi-scale position Encoding, we conduct ablation tests under two conditions: one without using atom scale PE (ASPE) and another without providing residue scale PE (RSPE) to atoms. The task employed is the enzyme-substrate affinity regression task. As shown in Table 7, when atom scale PE is not used, the model’s performance suffered significantly, which is because the model fails to capture positional information of atoms without providing atom scale PE. On the other hand, when residue scale PE is not provided to atoms, the model’s performance remains nearly unchanged. This suggests that for atom-scale information, 3D structural information is more crucial, and since the mapping relationship from residues to atoms is straightforward, there may be no need to provide residue scale PE to atoms to distinguish their corresponding residues. ### 3.5 VISUALIZATION To provide a more intuitive illustration of the higher consistency in protein and small molecule representations learned by this multi-scale unified model ms-ESM, we conducted visual comparisons. between proteins’ and molecules’ features extracted by \textit{ms-ESM} and the proteins’ features extracted by ESM2 along with the molecules’ features extracted by Unimol in both the enzyme-substrate pair classification task and drug-target affinity regression task. As depicted in Figure 4, the proteins’ and molecules’ representations learned by \textit{ms-ESM} model are closer. This implies that \textit{ms-ESM} can construct a more unified semantic representation for both proteins and molecules data. ### 4 RELATED WORK Protein Pre-training Pre-training has been proved to be an efficient technique in many domains, like natural language processing and protein engineering. Existing work studies protein pre-training mainly in two ways: (1) Sequence-based methods learn protein primary sequences to capture the biochemical and co-evolutionary knowledge. ESM series models (Rives et al., 2021; Lin et al., 2022b, 2023) use vanilla masked language modeling to learn protein representations on evolutionary scale. Aiming at the specific contact prediction task, Rao et al. (2021) further extends the masked language modeling to multiple sequence alignment (MSA) data. Inspired by the large language model (LLM), ProtGPT2 (Ferruz et al., 2022), ProGen (Madani et al., 2023), and ProGen2 (Nijkamp et al., 2022) scale up the model size of protein language model and show promising results in protein generation tasks. (2) Structure-based methods directly learn protein structure in different levels. Gligorjević et al. (2021); Zhang et al. (2022); Xu et al. (2022) learn residues from a local part of protein structures. Jing et al. (2020); Zhang et al. (2023) try to capture atomic structure knowledge in proteins. We develop \textit{ms-ESM} based on ESM. Differently, \textit{ms-ESM} is a mixture of sequence and structure-based methods, which gives it the ability to process information from different scales and makes it a versatile model. Unified Molecular Modeling Because of the huge scale difference of proteins and small molecules, it is challenging to model both of them in a unified style. As far as we know, Uni-Mol (Zhou et al., 2023) is the only method that tries to process proteins and molecules uniformly. Uni-Mol realizes the uniformity by directly modeling proteins and molecules at atom scale. However, because an entire protein contains hundreds of thousands of atoms, Uni-Mol can only model a local structure of proteins, i.e., protein pocket. Unlike Uni-Mol, as \textit{ms-ESM} only unzips partial residues into their corresponding atoms, it can handle an entire protein efficiently. We also provide some discussions of molecular modeling in Appendix C. ### 5 CONCLUSIONS In this study, we propose a multi-scale protein language model \textit{ms-ESM}, which realizes multi-scale unified molecular modeling by pre-training on multi-scale code-switch protein sequence and describing relationships among residues and atoms with a multi-scale position encoding. 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ruGY8v10mK	In scenarios where there's a very good classifier and a high-quality dataset (like MNIST or CIFAR10, with low aleatoric uncertainty), there might be a significant imbalance between positive and negative samples. This could lead to the situation that the trained uncertainty measure is unreliable. How should one address this situation?	A DATA-DRIVEN MEASURE OF RELATIVE UNCERTAINTY FOR MISCLASSIFICATION DETECTION Eduardo Dadalto∗ Laboratoire des signaux et systèmes (L2S) Université Paris-Saclay CNRS CentraleSupélec Gif-sur-Yvette, France eduardo.dadalto@centralesupelec.fr Marco Romanelli∗ New York University New York, NY, USA mr6852@nyu.edu Georg Pichler∗ Institute of Telecommunications TU Wien 1040 Vienna, Austria georg.pichler@ieee.org Pablo Piantanida International Laboratory on Learning Systems (ILLS) Quebec AI Institute (MILA) CNRS CentraleSupélec - Université Paris-Saclay Montreal, Canada pablo.plantanida@cnrs.fr ABSTRACT Misclassification detection is an important problem in machine learning, as it allows for the identification of instances where the model’s predictions are unreliable. However, conventional uncertainty measures such as Shannon entropy do not provide an effective way to infer the real uncertainty associated with the model’s predictions. In this paper, we introduce a novel data-driven measure of uncertainty relative to an observer for misclassification detection. By learning patterns in the distribution of soft-predictions, our uncertainty measure can identify misclassified samples based on the predicted class probabilities. Interestingly, according to the proposed measure, soft-predictions corresponding to misclassified instances can carry a large amount of uncertainty, even though they may have low Shannon entropy. We demonstrate empirical improvements over multiple image classification tasks, outperforming state-of-the-art misclassification detection methods. 1 INTRODUCTION Critical applications, such as autonomous driving and automatic tumor segmentation, have benefited greatly from machine learning algorithms. This motivates the importance of understanding their limitations and urges the need for methods that can detect patterns on which the model uncertainty may lead to dangerous consequences (Amodei et al., 2016). In recent years, considerable efforts have been dedicated to uncovering methods that can deceive deep learning models, causing them to make classification mistakes. While these findings have highlighted the vulnerabilities of deep learning models, it is important to acknowledge that erroneous classifications can also occur naturally. The likelihood of such incorrect classifications is strongly influenced by the characteristics of the data being analyzed and the specific model being used. Even small changes in the distribution of the training and evaluation samples can significantly impact the occurrence of these misclassifications. A recent thread of research has shown that issues related to misclassifications might be addressed by augmenting the training data for better representation (Zhu et al., 2023a; Zhang et al., 2017; Pinto et al., 2022). However, in order to build misclassification detectors, all these approaches rely on some statistics derived from the soft-prediction output by the model, such as the entropy or related notions, interpreting it as an expression of the model’s confidence. We argue that relying on the assumption that the model’s output distribution is a good representation of the uncertainty of the model is inadequate. For example, a model may be very confident on a sample that is far from the training distribution and, therefore, it is likely to be misclassified, which undermines the effective use of the Shannon entropy as a measure of the real uncertainty associated with the model’s prediction. ∗Equal contribution. In this work, we propose a data-driven measure of relative uncertainty inspired by Rao (1982) that relies on negative and positive instances to capture meaningful patterns in the distribution of soft-predictions. For example, positive instances can be correctly classified samples for which the uncertainty is expected to be low while negative instances (misclassified samples) are expected to have high uncertainty. Thus, the goal is to yield high and low uncertainty values for negative and positive instances, respectively. Our measure is “relative”, as it is not characterized axiomatically, but only serves the purpose of measuring uncertainty of positive instances relative to negative ones from the point of view of a subjective observer $d$. We employ relative uncertainty to measure the overall uncertainty of a model, encompassing both aleatoric and epistemic uncertainty components. By learning to minimize the uncertainty in positive instances and to maximize it in negative instances, our metric can effectively capture meaningful information to differentiate between the underlying structure of distributions corresponding to two categories of data. Interestingly, this notion can be expanded to any binary detection tasks in which both positive and negative samples are available. Our contributions are three-fold: 1) We leverage a novel statistical framework for categorical distributions to devise a learnable measure of relative uncertainty (REL-U) for a model’s predictions, which induces large uncertainty for negative instances, even if they may lead to low Shannon entropy (cf. Section 3); 2) We propose a closed-form solution for training REL-U in the presence of positive and negative instances (cf. Section 4); 3) We report favorable and consistent results over different models and datasets, considering both natural misclassifications within the same statistical population, and in case of distribution shift, or mismatch, between training and testing distributions (cf. Section 5). 2 RELATED WORKS Misclassification detection aims to evaluate the reliability of decisions made by classifiers and determine whether they can be trusted or not. A simple baseline relies on the maximum predicted probability (Hendrycks & Gimpel, 2017), but state-of-the-art classifiers have shown to be overconfident in their predictions, even when they fail (Cobb & Looveren, 2022). Liang et al. (2017) proposes applying temperature scaling (Guo et al., 2017) and perturbing the input samples to the direction of the decision boundary to detect misclassifications better. A line of research trains auxiliary parameters to estimate a detection score (Corbière et al., 2019) directly, following the idea of learning to reject (Chow, 1970; Getifman & El-Yaniv, 2017). Exposing the model to outliers or severe augmentations during training has been explored in previous work (Zhu et al., 2023a) to evaluate if these heuristics are beneficial for this particular task apart from improving robustness to outliers. Granese et al. (2021) proposes a mathematical framework and a simple detection method based on the estimated probability of error. We show that their proposed detection metric is a special case of ours. Zhu et al. (2023b) study the phenomenon that calibration methods are most often useless or harmful for failure prediction and provide insights into why. Cen et al. (2023) discusses how training settings such as pre-training or outlier exposure impact misclassification and open-set recognition performance. Related sub-fields are out-of-distribution detection (Lee et al., 2018), open set recognition (Geng et al., 2021), novelty detection (Pimentel et al., 2014), anomaly detection (Chalapathy & Chawla, 2019), adversarial attacks detection (Akhtar & Mian, 2018), and predictive uncertainty estimation via Bayesian Neural Networks estimation (Gal & Ghahramani, 2016; Lakshminarayanan et al., 2016; Mukhoti et al., 2021; Einbinder et al., 2022; Snoek et al., 2019). We refer the reader to Vadera et al. (2020) as a survey in the topic. A different take on the problem of uncertainty in AI is conformal learning (Angelopoulos et al., 2021; Romano et al., 2020): in addition to estimating the most likely outcome, a conformal predictor provides a “prediction set” that provably contains the ground truth with high probability. Recently, considerable effort has been invested into quantifying uncertainty by disentangling and estimating two quantities: epistemic uncertainty, i.e. the uncertainty that can be decreased by adding new observation to the training set available to a model, and aleatoric uncertainty, which is fundamentally present in the data and cannot be reduced by adding training data. In general, these works rely on inducing higher sensitivity at the level of the network’s internal representation or on modifications to the training procedure or auxiliary models (Liu et al., 2020) proposes to embed a distance-awareness ability in a given neural network by adding spectral normalization to the weights during training so as to translate the distance in the data manifold into dissimilarity at the hidden representation level. Van Amersfoort et al. (2020) proposes a new loss function and centroid updating scheme to speed up the computation of RBF-based nets. Kotelevskii et al. (2022) proposes an approach that provides disentanglement of epistemic and aleatoric uncertainty by computing the agreement between a given model and the Bayes classifier based on their kernel estimate and measuring the point-wise Bayes risk. Finally, Mukhoti et al. (2023) utilizes spectral normalization during training and a feature-space density estimator after training to quantify the epistemic uncertainty disentangling it from the aleatoric one. 3 A DATA-DRIVEN MEASURE OF UNCERTAINTY Figure 1: Intuitive example illustrating the advantage of REL-U compared to entropy-based methods: REL-U (left-end side heatmap) captures the real uncertainty (central heatmap) much better than Doctor (Granese et al., 2021); a detailed analysis is provided in Section 5.3. Before we introduce our method, we start by stating basic definitions and notations. Then, we describe our statistical model and some useful properties of the underlying detection problem. Let $\mathcal{X} \subseteq \mathbb{R}^d$ be a (possibly continuous) feature space and let $\mathcal{Y} = \{1, \ldots, C\}$ denote the label space related to some task of interest. Moreover, we denote by $P_{XY}$ be the underlying joint probability distribution on $\mathcal{X} \times \mathcal{Y}$. We assume that a machine learning model is trained on some training data, which ultimately yields a model that, given samples $x \in \mathcal{X}$, outputs a probability mass function (pmf) on $\mathcal{Y}$, which we denote as a vector $\hat{p}(x)$. This may result from a soft-max output layer, for example. A predictor $f : \mathcal{X} \rightarrow \mathcal{Y}$ is then constructed, which yields $f(x) = \arg \max_{y \in \mathcal{Y}} \hat{p}(x)_y$. We note that we may also interpret $\hat{p}(x)$ as the probability distribution of $\hat{Y}$ on $\mathcal{Y}$, i.e., given $X = x$, $\hat{Y}$ is distributed according to $p_{\hat{Y}\|X}(y\|x) \triangleq \hat{p}(x)_y$. In statistics and information theory, many measures of uncertainty were introduced, and some were utilized in machine learning to great effect. Among these are Shannon entropy (Shannon, 1948 Sec. 6), Rényi entropy (Rényi, 1961), $q$-entropy (Tsallis, 1988), as well as several divergence measures, capturing a notion of distance between probability distributions, such as Kullback-Leibler divergence (Kullback & Leibler, 1951), $f$-divergence (Csiszár, 1964), and Rényi divergence (Rényi, 1961). These definitions are well motivated, axiomatically and/or by their use in coding theorems. While some measures of uncertainty offer flexibility by choosing parameters, e.g., $\alpha$ for Rényi $\alpha$-entropy, they are invariant w.r.t. relabeling of the underlying label space. In our case, however, this semantic meaning of specific labels can be important and we do not expect a useful measure of “relative” uncertainty to satisfy this invariance property. Recall that the quantity $\hat{p}(x)$ is the soft-prediction output by the model given the input $x$. The entropy measure of Shannon (Shannon, 1948 Sec. 6) $$H(\hat{Y}\|x) \triangleq - \sum_{y \in \mathcal{Y}} \hat{p}(x)_y \log (\hat{p}(x)_y)$$ and the concentration measure of Gini (Gini, 1912) $$s_{\text{gini}}(x) \triangleq 1 - \sum_{y \in \mathcal{Y}} (\hat{p}(x)_y)^2$$ have commonly been used to measure the dispersion of a categorical random variable $\hat{Y}$ given a sample $x$. It is worth to emphasize that either measure may be used to carry out an analysis of dispersion for a random variable predicting a discrete value (e.g., a label). This is comparable to the analysis of variance for the prediction of continuous random values. Regrettably, these measures suffer from two major inconveniences: they are invariant to relabeling of the underlying label space, and, more importantly, they lead to very low values for overconfident predictions, even if they are wrong. These observations make both Shannon entropy and the Gini coefficient unfit for our purpose, i.e., the detection of misclassification instances. Evidently, we need a novel measure of uncertainty that can operate on probability distributions $\hat{p}(x)$ and that allows us to identify meaningful patterns in the distribution from which uncertainty can be inferred from data. To overcome the aforementioned difficulties, we propose to construct a class of uncertainty measures that is inspired by the measure of diversity investigated in Rao (1982), defined as $$s_d(x) \triangleq \mathbb{E}[d(\hat{Y}, \hat{Y'})\|X = x] = \sum_{y \in Y} \sum_{y' \in Y} d(y, y')\hat{p}(x)_y\hat{p}(x)_{y'},$$ (3) where $d \in D$ is in a class of distance measures and, given $X = x$, the random variables $\hat{Y}, \hat{Y'} \sim \hat{p}(x)$ are independently and identically distributed according to $\hat{p}(x)$. The statistical framework we are introducing here offers great flexibility by allowing for an arbitrary function $d$ that can be learned from data, as opposed to fixing a predetermined distance as in Rao (1982). In essence, we regard the uncertainty in equation (3) as relative to a given observer $d$, which appears as a parameter in the definition. To the best of our knowledge, this is a fundamentally novel concept of uncertainty. 4 FROM UNCERTAINTY TO MISCLASSIFICATION DETECTION We wish to perform misclassification detection based on the statistical properties of soft-predictions of machine learning systems. In essence, the resulting problem requires a binary hypothesis test, which, given a probability distribution over the class labels (the soft-prediction), decides whether a misclassification event likely occurred. We follow the intuition that by examining the soft-prediction of categories corresponding to a given sample, the patterns present in this distribution can provide meaningful information to detect misclassified samples. For example, if a sample is misclassified, this can cause a significant shift in the soft-prediction, even if the classifier is still overconfident. From a broad conceptual standpoint, examining the structure of the population of predicted distributions is very different from the Shannon entropy of a categorical variable. We are primarily interested in the different distributions that we can distinguish from each other by means of positive (correctly classified) and negative (incorrectly classified) instances. 4.1 MISCLASSIFICATION DETECTION BACKGROUND We define the indicator of the misclassification event as $E(X) \triangleq 1[f(X) \neq Y]$. The occurrence of the “misclassification" event is then characterized by $E = 1$. Misclassification detection is a standard binary classification problem, where $E$ needs to be estimated from $X$. We will denote the misclassification detector as $g : X \rightarrow \{0, 1\}$. The underlying pdf $p_X$ can be expressed as a mixture of two random variables: $X_+ \sim p_{X\|E}(x\|0)$ (positive instances) and $X_- \sim p_{X\|E}(x\|1)$ (negative instances), where $p_{X\|E}(x\|1)$ and $p_{X\|E}(x\|0)$ represent the pdfs conditioned on the error event and the event of correct classification, respectively. Let $s : X \rightarrow \mathbb{R}$ be the uncertainty measure in (3) that assigns a score $s(x)$ to every sample $x$ in the input space $X$. We can derive a misclassification detector $g$ by fixing a threshold $\gamma \in \mathbb{R}$, $g(x; s, \gamma) = 1[s(x) \leq \gamma]$, where $g(x) = 1$ means that the input sample $x$ is detected as being $E = 1$. In Granese et al. (2021), the authors propose to use the Gini coefficient (2) as a measure of uncertainty, which is equivalent to the Rényi entropy of order two, i.e., $H_2(\hat{Y}\|x) = -\log \sum_{y \in Y} (\hat{p}(x)_y)^2$. 4.2 A DATA-DRIVEN MEASURE OF RELATIVE UNCERTAINTY FOR MODEL’S PREDICTIONS We first rewrite $s_d(x)$ (3) in order to make it amenable to learning the metric $d$. By defining the $C \times C$ matrix $D \triangleq (d_{ij})$ using $d_{ij} = d(i, j)$, we have $s_d(x) = \hat{p}(x) D \hat{p}(x)^\top$. For $s_d(x)$ to yield a good detector $g$, we design a contrastive objective, where we would like $\mathbb{E}[s_d(X_+)]$, which is the expectation over the positive samples, to be small compared to the expectation over negative samples, i.e., $ \mathbb{E}[s_d(X_-)] $. This naturally yields the following objective function, where we assume the usual properties of a distance function $ d(y, y) = 0 $ and $ d(y', y) = d(y, y') \geq 0 $ for all $ y, y' \in Y $. Definition 1. Let us introduce our objective function with hyperparameter $ \lambda \in [0, 1] $, \[ L(D) \triangleq (1 - \lambda) \cdot \mathbb{E} \left[ \hat{p}(X_+) D \hat{p}(X_+)^T \right] - \lambda \cdot \mathbb{E} \left[ \hat{p}(X_-) D \hat{p}(X_-)^T \right] \] (4) and for a fixed $ K \in \mathbb{R}^+ $, define our optimization problem as follows: \[ \begin{align} \text{minimize}_{D \in \mathbb{R}^{c \times c}} & \quad L(D) \\ \text{subject to} & \quad d_{ii} = 0, \quad \forall i \in Y \\ & \quad d_{ij} \geq 0, \quad \forall i, j \in Y \\ & \quad d_{ij} = d_{ji}, \quad \forall i, j \in Y \\ & \quad \text{Tr}(DD^T) \leq K \end{align} \] (5) The first constraint in equation (5) states that the elements along the diagonal are zeros, which ensures that the uncertainty measure is zero when the distribution is concentrated at a single point. The second constraint ensures that all elements are non-negative, which is a natural condition, so the measure of uncertainty is non-negative. The natural symmetry between two elements stems from the third constraint, while the last constraint imposes a constant upper-bound on the Frobenius norm of the matrix $ D $, guaranteeing that a solution for the underlying optimization problem exists. Proposition 1 (Closed form solution). The constrained optimization problem defined in (5) admits a closed form solution $ D^* = \frac{1}{Z} (d^_{ij}) $, where \[ d^_{ij} = \begin{cases} \text{ReLU} \left( \lambda \cdot \mathbb{E} \left[ \hat{p}(X_-)_i \hat{p}(X_-)_j \right] - (1 - \lambda) \cdot \mathbb{E} \left[ \hat{p}(X_+)_i \hat{p}(X_+)_j \right] \right) & i \neq j \\ 0 & i = j \end{cases}. \] (6) The multiplicative constant $ Z $ is chosen such that $ D^* $ satisfies the condition $ \text{Tr}(D^(D^)^T) = K $. The proof is based on a Lagrangian approach and relegated to Appendix A.1. Algorithm 1 in Appendix A.2 summarizes all the main steps for the empirical evaluation, including the data preparation and the computation of the matrix $ D^* $. Note that, apart from the zero diagonal and up to normalization, \[ D^* = \text{ReLU} \left( \lambda \cdot \mathbb{E} \left[ \hat{p}(X_-) \hat{p}(X_-)^T \right] - (1 - \lambda) \cdot \mathbb{E} \left[ \hat{p}(X_+) \hat{p}(X_+)^T \right] \right). \] (7) Finally, we define the Relative Uncertainty (REL-U) score for a given sample $ x $ as \[ s_{\text{REL-U}}(x) \triangleq \hat{p}(x) D^* \hat{p}(x)^T. \] (8) Remark. (2) is a special case of (8) when $ d_{ij} = 1 $ if $ i \neq j $ and $ d_{ii} = 0 $. Thus, $ s_{1-d}(x) = s_{\text{gini}}(x) $ when choosing $ d $ to be the Hamming distance, which was also pointed out in [Rao, 1982 Note 1]. ## 5 Experiments and Discussion In this section, validate our measure of uncertainty in the context of misclassification detection, considering both the case when the training and test distributions match (cf. Section 5.1), and the case in which the two distributions mismatch (cf. Section 5.2). Although our method requires additional positive and negative instances, we show that lower amounts are needed (hundreds or few thousands) compared to methods that involve re-training or fine-tuning (hundreds of thousands). ### 5.1 Misclassification Detection on Matched Data We designed our experiments as follows: for a given model architecture and dataset, we trained the model on the training dataset. We split the test set into two sets: one portion for tuning the detector (held out validation set) and the other for evaluating it. Consequently, we can compute all hyperparameters in an unbiased way and cross-validate performance over many splits generated from ten random seeds. For ODIN ([Liang et al., 2017] and Doctor ([Granese et al., 2021]), we found the best temperature ($T$) and input pre-processing magnitude perturbation ($\epsilon$). For our method, we tuned the best lambda parameter ($\lambda$), $T$, and $\epsilon$. For details on temperature and input pre-processing equations, see Appendix A.6. As of evaluation metric, we consider the false positive rate (fraction of misclassifications detected as being correct classifications) when 95% of data is true positive (fraction of correctly classified samples detected as being correct classifications), denoted as FPR at 95% TPR (lower is better). AUROC results are similar among methods (see Figure 6 in the appendix). Table 1 showcases the misclassification detection performance in terms of FPR at 95% TPR of our method and the strongest baselines (MSP (Hendrycks & Gimpel 2017), ODIN (Liang et al. 2017), Doctor (Granese et al. 2021)) on different neural network architectures (DenseNet-121 (Huang et al. 2017), ResNet-34 (He et al. 2016)) trained on different datasets (CIFAR-10, CIFAR-100 (Krizhevsky 2009)) with different learning objectives (Cross-entropy loss, LogitNorm (Wei et al. 2022), MixUp (Zhang et al. 2017), RegMixUp (Pinto et al. 2022), OpenMix (Zhu et al. 2023a)). Please refer to Appendix A.3 for details on the baseline methods. We observe that, on average, our method performs best 11/20 experiments and is equal to the second best in 4/9 out of the remaining experiments. It works consistently better on all the models trained with cross-entropy loss and the models trained with RegMixUp objective, which achieved the best accuracy among them. We observed some negative results when training with logit normalization, but also, the accuracy of the base model decreased. Results for Bayesian methods for uncertainty estimation such as Deep Ensembles (Lakshminarayanan et al. 2016) and MCDropout (Gal & Ghahramani 2016), as well as results for an MLP directly trained on the tuning data are reported in Table 3 in the Appendix A.6. We report superior detection capabilities for the task at hand. Table 1: Misclassification detection results across two different architectures trained on CIFAR-10 and CIFAR-100 with five different training losses. We report the average accuracy of these models and the detection performance in terms of average FPR at 95% TPR (lower is better) in percentage with one standard deviation over ten different seeds in parenthesis. \| Model \| Training \| Accuracy \| MSP \| ODIN \| Doctor \| Rel-U \| \|----------------\|--------------\|----------\|--------\|--------\|--------\|-------\| \| DenseNet-121 \| CrossEntropy \| 94.0 \| 32.7 (4.7) \| 24.5 (0.7) \| 21.5 (0.2) \| 18.3 (0.2) \| \| \| LogitNorm \| 92.4 \| 39.6 (1.2) \| 32.7 (1.0) \| 37.4 (0.5) \| 37.0 (0.4) \| \| \| Mixup \| 95.1 \| 54.1 (13.4) \| 38.8 (1.2) \| 24.5 (1.9) \| 37.6 (0.9) \| \| \| OpenMix \| 94.5 \| 57.5 (0.0) \| 53.7 (0.2) \| 33.6 (0.1) \| 31.6 (0.4) \| \| \| RegMixUp \| 95.9 \| 41.3 (8.0) \| 30.4 (0.4) \| 23.3 (0.4) \| 22.0 (0.2) \| \| DenseNet-121 \| CrossEntropy \| 73.8 \| 45.1 (2.0) \| 41.7 (0.4) \| 41.5 (0.2) \| 41.5 (0.2) \| \| \| LogitNorm \| 73.7 \| 66.4 (2.4) \| 60.8 (0.2) \| 68.2 (0.4) \| 68.0 (0.4) \| \| \| Mixup \| 77.5 \| 48.7 (2.3) \| 41.4 (1.4) \| 37.7 (0.6) \| 37.7 (0.6) \| \| \| OpenMix \| 72.5 \| 52.7 (0.0) \| 51.9 (1.3) \| 48.1 (0.3) \| 45.0 (0.2) \| \| \| RegMixUp \| 78.4 \| 49.7 (2.0) \| 47.5 (1.1) \| 43.5 (0.3) \| 40.9 (0.2) \| \| ResNet-34 \| CrossEntropy \| 95.4 \| 25.8 (4.8) \| 19.4 (1.0) \| 14.3 (0.2) \| 14.1 (0.1) \| \| \| LogitNorm \| 94.3 \| 30.5 (1.6) \| 26.0 (0.6) \| 31.5 (0.5) \| 31.3 (0.6) \| \| \| Mixup \| 96.1 \| 60.1 (10.7) \| 38.2 (2.0) \| 26.8 (0.6) \| 19.0 (0.3) \| \| \| OpenMix \| 94.0 \| 40.4 (0.0) \| 39.5 (1.3) \| 28.3 (0.7) \| 28.5 (0.2) \| \| \| RegMixUp \| 97.1 \| 34.0 (5.2) \| 26.7 (0.1) \| 21.8 (0.2) \| 18.2 (0.2) \| \| ResNet-34 \| CrossEntropy \| 79.0 \| 42.9 (2.5) \| 38.3 (0.2) \| 34.9 (0.5) \| 32.7 (0.3) \| \| \| LogitNorm \| 76.7 \| 58.3 (1.0) \| 55.7 (0.1) \| 65.5 (0.2) \| 65.4 (0.2) \| \| \| Mixup \| 78.1 \| 53.5 (6.3) \| 43.5 (1.6) \| 37.5 (0.4) \| 37.5 (0.3) \| \| \| OpenMix \| 77.2 \| 46.0 (0.0) \| 43.0 (0.9) \| 41.6 (0.3) \| 39.0 (0.2) \| \| \| RegMixUp \| 80.8 \| 50.5 (2.8) \| 45.6 (0.9) \| 40.9 (0.8) \| 37.7 (0.4) \| Figure 2 displays how the amount of data reserved for the tuning split impacts the performance of the best two detection methods. We demonstrate how our data-driven uncertainty estimation metric generally improves with the amount of data fed to it in the tuning phase, especially on a more challenging setup such as on the CIFAR-100 model. Training losses or regularization is independent of detection. Previous work highlights the independence of training objectives from detection methods, which challenges the meaningfulness of evaluations. In particular, we identify three major limitations in (Zhu et al. 2023a): The evaluation of post-hoc methods, such as Doctor and ODIN, lacks consideration of perturbation and temperature hyperparameters. Despite variations in accuracy and the absence of measures for coverage and risk, different training methods are evaluated collectively. Furthermore, the post-hoc methods are not assessed on these models. The primary flaw in their analysis stems from evaluating different detectors on distinct models, leading to comparisons between (models, detectors) tuples that have different misclassification rates. As a result, such an analysis may fail to determine the most performant detection method in real-world scenarios. Does calibration improve detection? There has been growing interest in developing machine learning algorithms that are not only accurate but also well-calibrated, especially in applications where reliable probability estimates are desirable. In this section, we investigate whether models with calibrated probability predictions help improve the detection capabilities of our method or not. Previous work (Zhu et al., 2023b) has shown that calibration does not particularly help or impact misclassification detection on models with similar accuracies, however, they focused only on calibration methods and overlooked detection methods. To assess this problem in the optics of misclassification detectors, we calibrated the soft-probabilities of the models with a temperature parameter (Guo et al., 2017). Note that this temperature is not necessarily the same value as the detection hyperparameter temperature. This calibration method is simple and effective, achieving performance close to state-of-the-art (Minderer et al., 2021). To measure how calibrated the model is before and after temperature scaling, we measured the expected calibration error (ECE) (Guo et al., 2017) before, with $ T = 1 $, and after calibration. We obtained the optimal temperature after a cross-validation procedure on the tuning set and measured the detection performance of the detection methods over the calibrated model on the test set. For the detection methods, we use the optimal temperature obtained from calibration, and no input pre-processing is conducted ($ \epsilon = 0 $), to observe precisely what is the effect of calibration. We set $ \lambda = 0.5 $. Table 2 shows the detection performance over the calibrated models. We cannot conclude much from the CIFAR benchmark as the models are already well-calibrated out of the training, with ECE of around 0.03. In general, calibrating the models slightly improved performance on this benchmark. However, for the ImageNet benchmark, we observe that Doctor gained a lot from the calibration, while REL-U remained more or less invariant to calibration on ImageNet, suggesting that the performance of REL-U is robust under the model’s calibration. Table 2: Impact of model probability calibration on misclassification detection methods. The uncalibrated and the calibrated performances are in terms of average FPR at 95% TPR (lower is better) and one standard deviation in parenthesis. \| Architecture \| Dataset \| ECE$_1$ \| ECE$_T$ \| Uncal. Doctor \| Cal. Doctor \| Uncal. REL-U \| Cal. REL-U \| \|----------------\|-------------\|-----------\|-----------\|---------------\|-------------\|--------------\|------------\| \| DenseNet-121 \| CIFAR-10 \| 0.03 \| 0.01 \| 31.1 (2.4) \| 28.2 (3.8) \| 32.7 (1.7) \| 27.7 (2.1) \| \| \| CIFAR-100 \| 0.03 \| 0.01 \| 44.4 (1.1) \| 45.9 (0.9) \| 45.7 (0.9) \| 46.6 (0.6) \| \| ResNet-34 \| CIFAR-10 \| 0.03 \| 0.01 \| 24.3 (0.0) \| 23.0 (1.4) \| 26.2 (0.0) \| 24.2 (0.1) \| \| \| CIFAR-100 \| 0.06 \| 0.04 \| 40.0 (0.3) \| 38.7 (1.0) \| 40.6 (0.7) \| 38.9 (0.9) \| \| ResNet-50 \| ImageNet \| 0.41 \| 0.03 \| 76.0 (0.0) \| 55.4 (0.7) \| 51.7 (0.0) \| 53.0 (0.3) \| ### 5.2 MISMATCHED DATA So far, we have evaluated methods for misclassification detection under the assumption that the data available to learn the uncertainty measure and that during testing are drawn from the same distribution. In this section, we consider cases in which this assumption does not hold true, leading to a mismatch between the generative distributions of the data. Specifically, we investigate two sources of mismatch: i) Datasets with different label domains, where the symbol sets and symbols cardinality are different in each dataset; ii) Perturbation of the feature space domain generated using popular distortion filters. Understanding how machine learning models and misclassification detectors perform under such conditions can help us gauge and evaluate their robustness. Mismatch from different label domains. We considered pre-trained classifiers on the CIFAR-10 dataset and evaluated their performance in detecting samples in CIFAR-10 and distinguishing them from samples in CIFAR-100, which has a different label domain. Similar experiments have been conducted in Ren et al. (2021); Fort et al. (2021); Zhu et al. (2023a). The test splits were divided into a validation set and an evaluation set, with the validation set consisting of 10%, 20%, 33%, or 50% of the total test split and samples used for training were not reused. For each split, we combine the number of validation samples from CIFAR-10 with an equal number of samples from CIFAR-100. In order to assess the validity of our results, each split has been randomly selected 10 times, and the results are reported in terms of mean and standard deviation in Figure 3. We observe how our proposed data-driven method performs when samples are provided to accurately describe the two groups. In order to reduce the overlap between the two datasets, and in line with previous work (Fort et al., 2021), we removed the classes in CIFAR-100 that most closely resemble the classes in CIFAR-10. For the detailed list of the removed labels, we refer the reader to Appendix A.7. Mismatch from feature space corruption. We trained a model on the CIFAR-10 dataset and evaluated its ability to detect misclassification on the popular CIFAR-10C corrupted dataset, which contains a version of the classic CIFAR-10 test set perturbed according to 19 different types of corruption and 5 levels of intensities. With this experiment, we aim to investigate if our proposed detector is able to spot misclassifications that arise from input perturbation, based on the sole knowledge of the misclassified patterns within the CIFAR-10 test split. Consistent with previous experiments, we ensure that no samples from the training split are reused during validation and evaluation. To explore the effect of varying split sizes, we divide the test splits into validation and evaluation sets, with validation sets consisting of 10%, 20%, 33%, or 50% of the total test split. Each split has been produced 10 times with 10 different seeds and the average of the results has been reported in the spider plots in Figure 4. In the case of datasets with perturbed feature spaces, we solely utilize information from the validation samples in CIFAR-10 to detect misclassifications in the perturbed instances of the evaluation datasets, without using corrupted data during validation. We present visual plots that demonstrate the superior performance achieved by our proposed method compared to other methods. Additionally, for the case of perturbed feature spaces, we introduce radar plots, in which each vertex corresponds to a specific perturbation type, and report results for intensity 5. This particular choice of intensity is motivated by the fact that it creates the most relevant divergence between the accuracy of the model on the original test split and the accuracy of the model on the perturbed test split. Indeed the average gap in accuracy between the original test split and the perturbed test split is reported in Table 5 in Appendix A.8. We observe that our proposed method outperforms Doctor in terms of AUROC and FPR, as demonstrated by the radar plots. As we can see, in the case of CIFAR-10 vs CIFAR-10C, the radar plots (Figure 4) show how the area covered by the AUROC values achieves similar or larger values for the proposed method, indeed confirming that it is able to better detect misclassifications in the mismatched data. Moreover, the FPR values are lower for the proposed method. For completeness, we report the error bar tables in Tables 6 and 7, Appendix A.8. Additionally, as a particular case of mismatch from feature space corruption, we have considered the task of detecting mismatch between MNIST and SVHN, the results are reported in Figure 7, Appendix A.8. ![Figure 4: CIFAR-10 vs CIFAR-10C, ResNet-34, using 10% of the test split for validation.](image) 5.3 Empirical Interpretation of the Relative Uncertainty Matrix. Figure 1 exemplifies the advantage of our method over the entropy-based methods in (1) and (2). In particular, the left-end side heatmap represents the $D$ matrix learned by optimizing (4) on CIFAR-10. Darker shades of blue indicate higher uncertainty, while lighter shades of blue indicate lower uncertainty. The central heatmap is the predictor’s class-wise true confusion matrix. The vertical axis represents the true class, while the horizontal axis represents the predicted class. For each combination of two classes $ij$, the corresponding cell reports the count of samples of class $j$ that were predicted as class $i$. The correct matches along the diagonal are dashed for better visualization of the mistakes. The confusion matrix is computed on the same validation set used to compute the $D$ matrix. Crucially, our uncertainty matrix can express different degrees of uncertainty depending on the specific combination of classes at hand. Let us focus for instance on the fact that most of the incorrectly classified dogs are predicted as cats, and vice-versa. The matrix $D$ fully captures this by assigning high uncertainty to the cells at the intersection between these two classes. This is not the case for the entropy-based methods, which cannot capture such a fine-grained uncertainty, and assign the same uncertainty to all the cells, regardless of the specific combination of classes at hand. 6 Summary and Concluding Remarks To the best of our knowledge, we are the first to propose Rel-U, a method for uncertainty assessment that departs from the conventional practice of directly measuring uncertainty through the entropy of the output distribution. Rel-U uses a metric that leverages higher uncertainty score for negative data w.r.t. positive data, e.g., incorrectly and correctly classified samples in the context of misclassification detection, and attains favorable results on matched and mismatched data. In addition, our method stands out for its flexibility and simplicity, as it relies on a closed form solution to an optimization problem. Extensions to diverse problems present both an exciting and promising avenue for future research. Limitations. We presented machine learning researchers with a fresh methodological outlook and provided machine learning practitioners with a user-friendly tool that promotes safety in real-world scenarios. Some considerations should be put forward, such as the importance of cross-validating the hyperparameters of the detection methods to ensure their robustness on the targeted data and model. As a data-driven measure of uncertainty, to achieve the best performance, it is important to have enough samples at the disposal to learn the metric from as discussed on Section 5.1. As every detection method, our method may be vulnerable to targeted attacks from malicious users. ACKNOWLEDGEMENTS This work has been supported by the project PSPC AIDA: 2019-PSPC-09 funded by BPI-France. This work was granted access to the HPC/AI resources of IDRIS under the allocation 2023 - AD011012803R2 made by GENCI. REFERENCES Naveed Akhtar and Ajmal Mian. Threat of adversarial attacks on deep learning in computer vision: A survey. CoRR, abs/1801.00553, 2018. Dario Amodei, Chris Olah, Jacob Steinhardt, Paul Christiano, John Schulman, and Dan Mané. Concrete problems in ai safety. arXiv preprint arXiv:1606.06565, 2016. Anastasios N. Angelopoulos and Stephen Bates. A gentle introduction to conformal prediction and distribution-free uncertainty quantification. CoRR, abs/2107.07511, 2021. Anastasios Nikolas Angelopoulos, Stephen Bates, Michael I. Jordan, and Jitendra Malik. Uncertainty sets for image classifiers using conformal prediction. In 9th International Conference on Learning Representations, ICLR 2021, Virtual Event, Austria, May 3-7, 2021. OpenReview.net, 2021. 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i4kDKfllrz	OE achieves remarkable performance by solely requiring a single network to concurrently manage classification and the rejection of unknowns. It's worth noting that this paper doesn't make any references to OE, and there is a noticeable lack of in-depth discussion or comparison concerning methods and experimental results.	Synergistic Classification and Unknown Discrimination for Open Set Recognition Anonymous authors Paper under double-blind review Abstract Deep learners tend to perform well when trained under the closed set assumption but struggle when deployed under open set conditions. This motivates the field of Open Set Recognition in which we seek to give deep learners the ability to recognize whether a data sample belongs to the known classes trained on or comes from the surrounding infinite world. Existing open set recognition methods typically rely upon a single function for the dual task of distinguishing between knowns and unknowns as well as making fine known class distinction. This dual process leaves performance on the table as the function is not specialized for either task. In this work, we introduce Synergistic Classification and unknown Discrimination (SCAD), where we instead learn specialized functions for both known/unknown discrimination and fine class distinction amongst the world of knowns. Our experiments and analysis demonstrate that SCAD handily outperforms modern methods in open set recognition when compared using AUROC scores and correct classification rate at various true positive rates. 1 Introduction Recent studies have demonstrated the capacity of deep learners to achieve or even surpass human-level performance, particularly in the image recognition domain (He et al., 2015). This performance is typically achieved under the closed set assumption, however, in which the classes used for training the model are fixed and the model should only make predictions on this predefined set of classes. In practicality, the model may actually be deployed under open set conditions where the classes used for training are only a subset of the infinite surrounding world and the model must be able to distinguish between these known, trained on classes and the encompassing open world. Conventionally, deep neural networks struggle under these open set conditions as they will confidently map unknown classes to the known class decision space (Nguyen et al., 2015; Hendrycks & Gimpel, 2017) as demonstrated in Figure 1a. This motivates the study of Open Set Recognition where we seek to discriminate between the world of knowns the model is trained on and the surrounding infinite unknown space. Open set recognition was first formalized in Scheirer et al. (2013) and has since inspired an entire subfield of research. One of the first lines of work focused on an analysis of test time softmax scores (Hendrycks & Gimpel, 2017) as classifiers trained under the closed set assumption tend to produce low softmax probabilities for samples belonging to the unknown space. Bendale & Boult (2016) take a similar route by extending the softmax layer to allow prediction of an unknown class. These softmax based methods still suffer in open set recognition due to the inherent limitations of training the networks under the closed set assumption (Chen et al., 2020). Other methods take a generative approach (Neal et al., 2018; Oza & Patel, 2019) in an attempt to generate samples belonging to the unknown world, or a distance-based approach (Mendes Júnior et al., 2017; Shu et al., 2020) by thresholding a distance to the nearest known class. While these methods perform better than traditionally used softmax score analysis, they still do not perform to their maximum capability as they have no true representation for what the world of unknowns may resemble. Additionally, most current open set methods operate under the proposed setup of Scheirer et al. (2013) in which a single function is given the task of distinguishing between knowns and unknowns. and additionally making fine distinction amongst the world of knowns (i.e., classification). This leads to a function that may perform relatively well for this joint task, but is not specialized for either task leaving performance on the table. To this end, we introduce our method Synergistic Classification and unknown Discrimination (SCAD) to better address these shortcomings. In SCAD, we hypothesize that the known and unknown classes should clearly separate in the embedding space as demonstrated in Figure 1B. This separation can be accomplished by training an embedding network with a representative set of the unknown world referred to as known unknowns as in [Scheirer et al., 2014]. Each embedding space can then be represented by its respective prototype for best separation. Furthermore, we train a classifier network under the closed set assumption for discrimination amongst the world of knowns. At test time, we can determine if a sample belongs to the world of knowns or unknowns by setting a threshold on the distance to the unknown prototype, and if a sample is deemed as known, we can query the classifier to determine its class. This formulation of two specialized decision functions allows each to be an expert in their respective task leading to higher performance when combined together. 2 RELATED WORK Open Set Recognition. The field of open set recognition can be traced back to decision theory where we attempt to instill a classifier with a reject option when a classifier’s confidence is low for a particular test sample [Bartlett & Wegkamp, 2008; Yuan & Wegkamp, 2010; Scheirer et al., 2013] first formalized the problem of open set recognition and explored the use of a “1-vs-all” SVM for unknown detection. Since then, deep learning methods have become the de facto method for open set recognition due to their great success. Bendale & Boult [2016] first introduce deep learning in the context of open set recognition by extending the softmax layer to modeling the distance of activation vectors based on extreme value theory. This approach was further extended by [Ge et al., 2017] where deep neural networks are trained with unknown samples coming from a generative model. Other generative approaches include image reconstruction methods such as [Oza & Patel, 2019] and [Yoshihashi et al., 2019] where unknown samples can be identified by poor reconstruction. More recently, prototype-based methods [Chen et al., 2020; Shu et al., 2020; Chen et al., 2021] have shown great success by representing knowns and unknowns with learned prototypes and proceed to identify test samples based on distance to each prototype. Out-of-Distribution Detection. Open set recognition is closely related to the field of out-of-distribution detection [Hendrycks & Gimpel, 2017] where we wish to identify if test samples come from a drastically different distribution. The key difference lies in open set methods’ ability to fur- ther distinguish fine labels amongst the world of knowns as mentioned in Boult et al. (2019), Liang et al. (2017) and Hsu et al. (2020) build upon the work of Hendrycks & Gimpel (2017) by performing post processing on the softmax confidence scores similar to the softmax method described above for open set recognition. Haroush et al. (2022) use hypothesis testing to generate p-values for each testing sample for determination of whether the sample comes from the in-distribution data. Zaeemzadeh et al. (2021) and Khalid et al. (2022) propose that learned features lie on a restricted low dimensional embedding space and the out-of-distribution data occupies the surrounding unrestricted space similar to the open set recognition methods Dhamija et al. (2018); Chen et al. (2020) and Chen et al. (2021). Our work draws inspiration from this described overlap region between open set recognition and out-of-distribution detection. 3 PRELIMINARIES We first establish the formalities of the open set recognition problem before formulating our proposed solution (Scheirer et al., 2013; Geng et al., 2020; Chen et al., 2020). Suppose we are given a dataset $ \mathcal{D}_{KK} $ of $ n $ labeled data points we will refer to as known knowns, namely $ \mathcal{D}_{KK} = \{(x_1, y_1), ..., (x_n, y_n)\} $ where $ y_i \in \{1, ..., C\} $ is the label for $ x_i $ for $ C $ unique class labels in $ \mathcal{D}_{KK} $. At test time, we will perform inference on the larger test data $ \mathcal{D}_T $ consisting of data from $ \mathcal{D}_{KK} $ as well as data from an unknown set $ \mathcal{D}_{UU} $, which we refer to as unknown unknowns, whose labels $ t_i \notin \{1, ..., C\} $. That is $ \mathcal{D}_T = \mathcal{D}_{KK} \cup \mathcal{D}_{UU} $. We denote the embedding space of known category $ k $ as $ S_k $ with corresponding open space $ O_k = \mathbb{R}^d - S_k $ where $ \mathbb{R}^d $ is the full embedding space consisting of known knowns and unknowns unknowns. We further define the positive open space from other known knowns as $ O_{k}^{pos} $ and the remaining infinite space consisting of unknown unknowns as the negative open space $ O_{k}^{neg} $, that is $ O_k = O_{k}^{pos} \cup O_{k}^{neg} $. We first introduce open set recognition for a single known class and then extend to the multi-class scenario. Given the data $ \mathcal{D}_{KK} $, let samples from known category $ k $ be positive training data occupying space $ S_k $, samples from other known classes be negative training data occupying space $ O_{k}^{pos} $, and all other samples from $ \mathbb{R}^d $ be unknown data, $ \mathcal{D}_{UU} $, occupying space $ O_{k}^{neg} $. Let $ \psi_k : \mathbb{R}^d \rightarrow \{0, 1\} $ be a binary measurable prediction function which maps the embedding $ x $ to label $ y $ with the label for the class of interest $ k $ being 1. In this 1-class scenario, we wish to optimize the discriminant binary function $ \psi_k $ by minimizing the expected error $ R_k $ as \[ \arg\min_{\psi_k} \{ R_k = R_o(\psi_k, O_{k}^{neg}) + \alpha R_e(\psi_k, S_k \cup O_{k}^{pos}) \} \] where $ R_o $ is the open space risk function, $ R_e $ is the empirical classification risk on the known data, and $ \alpha $ is a regularization parameter. We can extend to the multiclass recognition problem by incorporating multiple binary classification tasks and summing the expected risk category by category as \[ \sum_{k=1}^{C} R_o(\psi_k, O_{k}^{neg}) + \alpha \sum_{k=1}^{C} R_e(\psi_k, S_k \cup O_{k}^{pos}) \] leading to the following formulation \[ \arg\min_{f \in \mathcal{H}} \{ R_o(f, \mathcal{D}_{UU}) + \alpha R_e(f, \mathcal{D}_{KK}) \} \] where $ f : \mathbb{R}^d \rightarrow \mathbb{N} $ is a measurable multiclass recognition function. From this, we can see that solving the open set recognition problem is equivalent to minimizing the combination of the empirical classification risk on the labeled known data $ \mathcal{D}_{KK} $ and open space risk on the unknown data $ \mathcal{D}_{UU} $ simultaneously over the space of allowable recognition functions $ \mathcal{H} $. 4 METHODOLOGY 4.1 SYNERGISTIC CLASSIFICATION AND UNKNOWN DETECTION In the traditional formulation of the open set recognition problem as described above, we assume a singular embedding space $ \mathbb{R}^d $ consists of $ N $ discriminant spaces for all known categories with all remaining space being the open space consisting of infinite unknowns. In formulating the framework of SCAD, we instead postulate that the embedding space $\mathbb{R}^d$ is composed of two disjoint spaces, namely a known space $S_{known}$ and an unknown space $O_{unknown}$. That is to say that all of $D_{KK}$ belongs to the space $S_{known}$ and all of $D_{UU}$ belongs to the infinite surrounding open space $O_{unknown}$. Thus, the open space is formulated as $O_{unknown} = \mathbb{R}^d - S_{known}$. Under this new assumption of the embedding space, we can now pose a new formulation of the open set recognition problem by introducing a cascading optimization procedure where we wish to optimize both a binary prediction function $h : \mathbb{R}^d \rightarrow \{0, 1\}$ which maps the embedding of data $x$ to the label of known or unknown, and the classification function $f : x_i \rightarrow N$ which maps the known data $x_i$ to their respective target label $y_i \in \{1, ..., N\}$ as $$\arg\min_h \{R_o(h, \mathbb{R}^d)\}$$ (4a) $$\arg\min_f \{R_e(f, S_{known})\}$$ (4b) where $R_o$ is the open space risk and $R_e$ is the empirical classification risk. Based on this formulation we can see that the first optimization procedure leads to another binary prediction function $h$ similar to the traditional formulation while the second procedure leads to a multiclass prediction function $f$. All that remains now is to find a method that best creates the full embedding space $\mathbb{R}^d$ to give a simple discriminant function $h$ and obtain a high performing multiclass prediction function $f$. ### 4.2 Embedding Separation of Knowns and Unknowns We first focus on the discrimination between knowns and unknowns in the embedding space $\mathbb{R}^d$. A deep neural network $g_\theta : x \rightarrow \mathbb{R}^d$ is used as an embedding network to obtain embedding vectors for all data $x \in D_{KK} \cup D_{UU}$. In order to enforce the separation between the spaces $S_{known}$ and $O_{unknown}$, the triplet loss (Schroff et al., 2015) is a natural choice of loss function to use when training $g_\theta$. One could consider using other contrastive learning methods such as contrastive loss (Khosla et al., 2020) or tuple loss (Sohn, 2016), however, the choice to use triplet loss was made as contrastive loss only considers pairs and tuplet loss is a more general version of triplet loss. With the triplet loss, we can treat all training data in $D_{KK}$ as the positive samples. For negative samples, we now need to find a representation of $D_{UU}$ for modeling the space $O_{unknown}$. Of course this open space and therefore this dataset is infinite, but we can use a representative set of $D_{UU}$ we refer to as known unknowns, $D_{KU} \subseteq D_{UU}$, to train $g_\theta$ for embedding space separation of knowns and unknowns. The choice to use a representative training set $D_{KU}$ to represent the entire world of unknowns is taken from out-of-distribution detection literature (Liang et al., 2017; Lee et al., 2018; Haroush et al., 2022). Now armed with the known training set $D_{KK}$ and representative unknown training set $D_{KU}$, we can formalize use of the triplet loss to train $g_\theta$ as $$L_{g_\theta} = \sum_{i=1}^{n} \|\|g_\theta(x_i^K) - g_\theta(x_i^{KK})\|\|_2^2 - \|\|g_\theta(x_i^K) - g_\theta(x_i^{KU})\|\|_2^2 + \beta$$ (5) where $x_i^K$ is a known known anchor, $x_i^{KK}$ is a known known positive sample, $x_i^{KU}$ is a known unknown negative sample, and $\beta$ is a margin that is enforced between the positive and negative pairs. ### 4.3 Discrimination Between Knowns and Unknowns With a binary discriminant embedding space $\mathbb{R}^d$ now at hand, we must now develop the discriminant function $h$ to differentiate between knowns and unknowns. As such, we draw inspiration from Mensink et al. (2013); Ristin et al. (2014); Bendale & Boult (2016) by measuring the distance to the embedding prototypes for known/unknown discrimination. We represent each of the known and unknown clusters in the embedding space by their respective prototype determined by taking the means of the known knowns, $\mu_{KK}$, and known unknowns, $\mu_{KU}$, in the embedding space. We then measure the Euclidean distance to $\mu_{KU}$ and set a threshold for final determination of whether a test sample is known or unknown. Thus, the binary function $h$ takes the form $$ h = \begin{cases} \text{known} & \text{if } d(g_\theta(x_t), \mu_{KU}) > \tau \\ \text{unknown} & \text{if } d(g_\theta(x_t), \mu_{KU}) \leq \tau \end{cases} $$ (6) where $x_t$ is a test sample from $D_T$, $d(g_\theta(x_t), \mu_{KU}) = \|\|g_\theta(x_t) - \mu_{KU}\|\|_2^2$ is the Euclidean distance between the embedding of $x_t$ and the known unknown prototype $\mu_{KU}$ and $\tau$ is a threshold. ### 4.4 Management of Open Space Risk In theory, the open space $O_{unknown}$ is infinite making for difficult management of the open space risk $R_o$. We instead opt to indirectly bound this open space for easier management of $R_o$ as a direct bounding would be nearly impossible due to the infinite nature of $O_{unknown}$. By enforcing the distance between samples from $S_{known}$ and $O_{unknown}$ to be outside some predefined margin of separation we are able to indirectly bound $O_{unknown}$. This bounding procedure gives rise to Eq. (5) which enforces the distance between samples from the known knowns and known unknowns to be greater than or equal to the margin $\beta$. The use of $D_{KK}$ and $D_{KU}$ in the training of $g_\theta$ for embedding space separation gives rise to the bounding spaces $B_{known}$ and $B_{unknown}$ respectively. Ideally, these spaces would be completely separable in $\mathbb{R}^d$, but in practicality there will be some overlap in the margin region. By representing each bounding space by its prototype as described above, we are able to achieve greater separation in $\mathbb{R}^d$. As a result, training with triplet loss for separation between $B_{known}$ and $B_{unknown}$ and further representing each bounding region with its appropriate prototype for final binary prediction can be viewed as managing the open space risk $R_o(h, \mathbb{R}^d)$ in Eq. (4). ### 4.5 Distinction Amongst Knowns The last remaining step is now developing a way to best identify which known class a sample belongs to for reduction of the empirical classification risk $R_e$. In order to distinguish fine class labels amongst the world of knowns, we train a separate deep neural network $f_{\theta'}$ using cross-entropy loss in parallel with the embedding network $g_\theta$. As $f_{\theta'}$ is only concerned with classification of the knowns, we only use the data from $D_{KK}$ to train the classifier. Figure 2a shows the full training procedure for training the multiclass prediction function $f_{\theta'}$ and the embedding network $g_\theta$. At the inference stage, we only query $f_{\theta'}$ for a fine class label if the binary discriminant function $h$ predicts that a test sample $x_t$ belongs to the known space $S_{known}$. Otherwise, $x_t$ is assigned to the world of unknowns. Figure 2b gives an overview for the entire inference stage. 5 EXPERIMENTS AND RESULTS 5.1 EXPERIMENTAL SETUP Datasets. We test on four commonly used datasets in open set recognition literature. Each of the CIFAR datasets (Krizhevsky et al., 2009) is taken from either CIFAR10 or a combination of CIFAR10 and CIFAR100. For CIFAR10 experiments, all experiments are performed by treating the 6 non-vehicle classes as known classes and the remaining 4 vehicle classes as the unknown (i.e., open) classes. CIFAR+M experiments takes the 4 vehicle classes from CIFAR10 as known and randomly samples from M disjoint classes (i.e., non-vehicle classes) from the CIFAR100 dataset. Lastly, in Tiny-Imagenet experiments (Le & Yang, 2015) we randomly choose 20 classes as the known classes and treat all other 180 classes as unknown. Metrics. We use the standard area under the ROC curve (AUROC) as the main metric when evaluating the performance of all compared methods. The benefit of using AUROC is its threshold independent measure of the binary open set discriminator and its ability to summarize each method’s ability to distinguish between positive and negative instances across the various thresholds. A drawback of AUROC as commonly reported in open set trials, is it only takes into consideration known/unknown discrimination. A good open set recognizer should be able to additionally discriminate amongst the knowns given that a sample is predicted to be known. For this reason we additionally report the correct classification rate (CCR) at 95% true positive rate (TPR) of known detection similar to Dhamija et al. (2018). Compared Methods. We compare our method, SCAD, to four open set recognition methods that are most comparable in regards to methodology. Counter-factual images (Neal et al., 2018) uses a GAN (Goodfellow et al., 2014) to generate counter examples to the known class which are then treated as the unknown class and used to train a “$K + 1$” classifier where the $(K + 1)$th class is the unknown class. Class anchor clustering (CAC) (Miller et al., 2021) poses a new loss function to entice each of the distinct known classes to cluster around their respective standard basis vector so that the unknown classes will then occupy the remaining open space. A distance threshold is then used for distinct known or unknown discrimination similar to SCAD. Adversarial Reciprocal Point Learning + confusion samples (ARPL+CS) (Chen et al., 2021) learns reciprocal points for each known class open space while simultaneously using a generator to generate confusing training samples to encourage known class separation in the latent space and uses a distance measure to the furthest reciprocal point to obtain a probability of belonging to a particular known class. Lastly, Vaze et al. (2022) propose that the best open set recognition model is simply one that is a Good Classifier for the closed-set scenario. With this good closed-set classifier at hand, an analysis of the maximum logit score produced by a sample is used in the final determination of distinct known or unknown. Setup. For all methods, we train on the dataset splits described above. For neural network architectures, we use Resnet18 (He et al., 2016) in all tested methods for fairest comparisons except in counterfactual images and CAC. We keep the architectures unchanged in both of these methods as the former used a specific generator and discriminator for best GAN performance and the latter did not allow simplistic modulation with a Resnet encoder. Besides described architecture changes, all other hyperparameters for compared methods remain unchanged. All methods are trained via SGD with standard L2 regularization. For SCAD, the margin of separation $b$ in Eq. 5 is set to 0.5 and a combination of semihard and hard negative mining are used for finding triplets. Lastly, we use half of unknown classes for all datasets as the training set $D_{KU}$ in SCAD. 5.2 RESULTS COMPARISON. We first evaluate the performance of SCAD vs. all other compared methods from an AUROC standpoint. Table 1 shows AUROC results averaged across 3 runs for all methods and Figure 5 shows the respective ROC curves. We observe that SCAD outperforms all compared methods for all datasets handily. This can be attributed to SCAD’s specialized function $h$ for declaration of knowns and unknowns whereas all other methods use a singular function for both known/unknown discrimination and known class distinction as is commonly done in the traditional formulation of the open set recognition problem in Eq. 3. Additionally, SCAD’s $h$ discriminator is further assisted by clear known and unknown separation Table 1: Reported AUROC score means and standard deviations for each tested method for the various tested datasets averaged over 3 runs. \| Method \| CIFAR10 \| CIFAR+10 \| CIFAR+50 \| Tiny-Imagenet \| \|-------------------------\|------------------\|------------------\|-------------------\|-------------------\| \| Counter-Factual Images \| 0.6999 ± 0.006 \| 0.8251 ± 0.004 \| 0.8168 ± 0.001 \| 0.5734 ± 0.007 \| \| Class Anchor Clustering \| 0.7156 ± 0.002 \| 0.7425 ± 0.013 \| 0.7721 ± 0.002 \| 0.5452 ± 0.036 \| \| Good Classifier \| 0.7479 ± 0.008 \| 0.7734 ± 0.014 \| 0.7720 ± 0.002 \| 0.6291 ± 0.016 \| \| ARPL+CS \| 0.7813 ± 0.002 \| 0.8346 ± 0.005 \| 0.8241 ± 0.004 \| 0.6402 ± 0.023 \| \| SCAD (Ours) \| 0.9613 ± 0.01\| 0.9223 ± 0.023\| 0.9257 ± 0.014\| 0.6548 ± 0.0103\| Figure 3: Corresponding ROC curves for each tested method for the various tested datasets. in the embedding space $\mathbb{R}^d$ as initially hypothesized by means of the triplet loss. We can confirm this by analyzing the TSNE (Van der Maaten & Hinton [2008]) plot of the embeddings produced by $g_\theta$ as done in Figure 4 for the CIFAR10 data split. Of course, we observe an overlap region where discrimination between knowns and unknowns can prove challenging, but by representing each embedding cluster by its respective prototype, we are able to achieve better separation leading to a more favorable AUROC performance. We do note the performance of SCAD vs. that of ARPL+CS and Good Classifier for Tiny-Imagenet in Figure 3d. While SCAD maintains a favorable AUROC score, there is a small region where these other two methods actually perform better. This would suggest in scenarios where small false positive rate (FPR) is desirable, one may want to consider alternatives to SCAD. However, this small region of the ROC curve where SCAD is inferior is offset by the superior performance of SCAD in CCR elaborated on below. Figure 4: CIFAR10 TSNE plot of the embedding space. Table 2: Reported CCR at 95% TPR score means and standard deviations for each tested method for the various tested datasets averaged over 3 runs. \| Method \| CIFAR10 \| CIFAR+10 \| CIFAR+50 \| Tiny-Imagenet \| \|----------------------\|------------------\|------------------\|-------------------\|-------------------\| \| Class Anchor Clustering \| 0.688 ± 0.009 \| 0.8869 ± 0.004 \| 0.8805 ± 0.007 \| 0.3773 ± 0.038 \| \| Good Classifier \| 0.5650 ± 0.001 \| 0.5731 ± 0.012 \| 0.5694 ± 0.003 \| 0.5263 ± 0.002 \| \| ARPL+CS \| 0.6571 ± 0.002 \| 0.8233 ± 0.002 \| 0.5821 ± 0.004 \| 0.1732 ± 0.004 \| \| SCAD (Ours) \| 0.6962 ± 0.004 \| 0.8620 ± 0.002 \| 0.8611 ± 0.001 \| 0.6077 ± 0.028 \| Figure 5: Corresponding CCR vs. TPR curves for each tested method for the various tested datasets. We now evaluate the performance of SCAD against all other compared methods from a CCR standpoint. Table 2 reports the CCR at 95% TPR for all methods except Counter-Factual Images. We do not report results for Counter-Factual images due to the inherent nature of using a "$K + 1$" classifier (i.e., the "$K + 1$" classifier is not dependent on known/unknown discrimination as course distinction is based on discriminator scores and fine distinction amongst the "$K + 1$" classes is based on separate classifier scores). We overall observe that SCAD is mostly competitive with all other tested methods, but in particular performs exceptionally well on Tiny-Imagenet. The clear superiority of SCAD on Tiny-Imagenet can be attributed to having a specialized classifier $f_\theta$ capable of making fine distinction amongst knowns for challenging datasets. While SCAD remains competitive in all other datasets in regards to CCR at 95% TPR, we question if this is true for all operating TPRs. To answer this, we plot the CCR against various TPRs in Figure 5. From this, we make multiple interesting observations. Firstly, we can observe that SCAD is, in general, more stable than any of the compared methods. Again, this can be attributed to having a specialized classifier capable of consistent performance regardless of the number of known declarations. Secondly, we observe the CIFAR+10 and CIFAR+50 trials where SCAD is competitive, but not dominant in regards to CCR at 95% TPR. Figures 5b and 5c actually suggest that at nearly all other operating TPRs, SCAD is in fact superior. This would suggest that SCAD is the superior method in scenarios where higher TPRs can be waived. We note the unintuitive performance of CCR being greater than 0 when TPR is 0. All methods except Good Classifier are distance based methods to some anchor point (e.g., distance to standard basis vector in CAC and distance to prototype in SCAD). Upon further inspection of these scenarios, few test samples are being correctly declared as known while the overwhelming majority are declared unknown. This can be attributed to a small amount of samples being infinitesimally close to their respective anchor allowing for correct declaration as known and thus leading to a non-trivial CCR at 0% TPR. This same principle applies to Good Classifier but in the context of logit scores. 5.3 Performance on Known Unknowns vs. Unknown Unknowns We now turn our attention to analyzing the impact of using a representative set of the unknowns, $ D_{KU} $, when training the embedding space $ \mathbb{R}^d $ and how this might generalize to the entire world of unknowns, $ D_{UU} $. To do so, we partition the testing data into two disjoint testing sets with respect to the unknown data: one testing set contains only known unknowns while the other contains only unknown unknowns. We report the AUROC for each of these testing sets in Table 3 averaged over 3 runs. Table 3: Reported AUROC score means and standard deviations for each disjoint unknown test set for the various tested datasets averaged over 3 runs. \| Unknown Dataset \| CIFAR10 \| CIFAR+10 \| CIFAR+50 \| Tiny-Imagenet \| \|-----------------\|---------------\|---------------\|----------------\|---------------\| \| $ D_{KU} $ \| 0.970 ± 0.001 \| 0.925 ± 0.019 \| 0.952 ± 0.006 \| 0.640 ± 0.037 \| \| $ D_{UU} $ \| 0.9347 ± 0.024\| 0.8712 ± 0.001\| 0.944 ± 0.005 \| 0.6269 ± 0.033\| We observe that the difference in performance between $ D_{KU} $ and $ D_{UU} $ is relatively small. Even the isolated performance of $ D_{UU} $ still outperforms all other compared methods in Table 1 suggesting the the representative set $ D_{KU} $ allows the embedding model $ g_\theta $ to generalize well to the world of unknowns. Furthermore, we note the small disparity in AUROC scores for each of the unknown datasets in the CIFAR+50 and Tiny-Imagenet trials compared to that of CIFAR10 and CIFAR+10. Since we are using half of the entirety of unknown classes as the representative set $ D_{KU} $ in SCAD, this suggests that the larger we can make the representative training set, the better our ability to generalize to the entire world of unknowns will be. 6 Conclusion In this work, we introduce our method SCAD for open set recognition. SCAD benefits from having two specialized functions for known and unknown discrimination as well as fine class distinction amongst knowns. This allows each function to be an expert for their respective task allowing for top tier performance compared to that of traditional open set recognition methods where a single function is used for both known/unknown discrimination and fine class distinction. Additionally, by using a representative set of the unknowns termed known unknowns, we are able to train an embedding network for distinct separation between knowns and unknowns in the embedding space allowing for easy discrimination. 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4A5D1nsdtj	The design of heterophilic bases relies on the dataset's homophily rate, denoted as $h$ in Algorithm 1. I am concerned this approach is impractical due to obtaining the exact homophily rate $h$ from the training data is not feasible. It appears that the authors have directly utilized the entire dataset, including the labels of the test set. There are also methods to learn the homophily rate $h$ during the training process, but I think this process might affect the model's performance.	AN EFFECTIVE UNIVERSAL POLYNOMIAL BASIS FOR SPECTRAL GRAPH NEURAL NETWORKS Anonymous authors Paper under double-blind review ABSTRACT Spectral Graph Neural Networks (GNNs), also referred to as graph filters have gained increasing prevalence for heterophily graphs. Optimal graph filters rely on Laplacian eigendecomposition for Fourier transform. In an attempt to avert the prohibitive computations, numerous polynomial filters by leveraging distinct polynomials have been proposed to approximate the desired graph filters. However, polynomials in the majority of polynomial filters are predefined and remain fixed across all graphs, failing to accommodate the diverse heterophily degrees across different graphs. To tackle this issue, we first investigate the correlation between polynomial bases of desired graph filters and the degrees of graph heterophily via a thorough theoretical analysis. Afterward, we develop an adaptive heterophily basis by incorporating graph heterophily degrees. Subsequently, we integrate this heterophily basis with the homophily basis, creating a universal polynomial basis UniBasis. In consequence, we devise a general polynomial filter UniFilter. Comprehensive experiments on both real-world and synthetic datasets with varying heterophily degrees significantly support the superiority of UniFilter, demonstrating the effectiveness and generality of UniBasis, as well as its promising capability as a new method for graph analysis. 1 INTRODUCTION Spectral Graph Neural Networks (GNNs) (Kipf & Welling, 2017), known as graph filters have been extensively investigated in recent years due to their superior performance in handling heterophily graphs. Optimal graph filters conduct Laplacian eigendecomposition for Fourier transform. To bypass the computation complexity, existing graph filters leverage various polynomials to approximate the desired filters for graphs with varying heterophily degrees. For example, ChebNet (Defferrard et al., 2016) employs truncated Chebyshev polynomials (Mason & Handscomb, 2002; Hammond et al., 2011) and accomplishes localized spectral filtering. BernNet (He et al., 2021) utilizes Bernstein polynomials (Farouki, 2012) to acquire better controllability and interpretability. Later, Wang & Zhang (2022) propose JacobiConv by exploiting Jacobi polynomial bases (Askey, 1974) with improved generality. Recently, the state-of-the-art (SOTA) graph filter OptBasisGNN (Guo & Wei, 2023) orthogonalizes the polynomial basis to reach the maximum convergence speed. However, polynomial bases utilized in existing polynomial filters ignore the varying heterophily degrees underlying graphs. Among them, orthogonal bases are proved optimal in terms of convergence speed (Wang & Zhang, 2022; Guo & Wei, 2023). Yet, it demonstrates suboptimal empirical performance on node classification, especially on strong homophily graphs in our experiments (sections 5.1 and 5.3). This scenario arises from the lack of consideration of the homophily property in the construction of the orthonormal basis, rendering it inferior to strong homophily graphs. As we prove in Theorem 1, frequencies of signals filtered by optimal graph filters are proportional to the heterophily degrees. This suggests that ideal polynomial bases are obligated to provide adaptability to the diverse heterophily degrees. Ergo, a natural question to ask is: how can we design a universal polynomial basis that encapsulates the graph heterophily degrees? Inspired, we first establish the relation between the heterophily degree and the frequency of optimal filtered signals (Theorem 1). Subsequently, we explore how the distribution of polynomial bases in Euclidean space affects the basis spectrum (Theorem 3). Based on those insightful findings, we design an adaptive heterophily basis by incorporating het- erophily degrees of graphs. Eventually, we integrate the heterophily basis and the homophily basis into a universal basis denoted as UniBasis. Upon UniBasis, we devise a general polynomial filter called UniFilter. For a comprehensive evaluation, we compare UniFilter with 20 baselines on 6 real-world datasets and synthetic datasets with a range of heterophily degrees. The notably superior performance of UniFilter strongly confirms the effectiveness and generality of UniBasis, especially on heterophily graphs. Meanwhile, we demonstrate the spectrum distribution of trained UniBasis on each tested dataset (section 5.2). The experimental results explicitly support the promising capability of UniBasis as a new method for graph analysis with enriched interpretability. In a nutshell, our contribution can be summarized as: 1) We reveal that underlying polynomials of desired polynomial filters are meant to keep aligned with degrees of graph heterophily; 2) We design a universal polynomial basis UniBasis by incorporating graph heterophily degrees and devise a general graph filter UniFilter; 3) We evaluate UniFilter on both real-world and synthetic datasets against 18 baselines. The remarkable performance of UniFilter strongly confirms the effectiveness and generality of UniBasis, as well as its promising capability as a new method for graph analysis. 2 PRELIMINARIES 2.1 NOTATIONS AND DEFINITIONS We represent matrices, vectors, and sets with bold uppercase letters (e.g., $ \mathbf{A} $), bold lowercase letters (e.g., $ \mathbf{x} $), and calligraphic fonts (e.g., $ \mathcal{N} $), respectively. The $ i $-th row (resp. column) of matrix $ \mathbf{A} $ is represented by $ \mathbf{A}[i,\cdot] $ (resp. $ \mathbf{A}[\cdot,i] $). We denote $[n] = \{1, 2, \cdots, n\}$. Let $ G = (\mathcal{V}, \mathcal{E}) $ be an undirected and connected graph with node set $ \|\mathcal{V}\| = n $ and edge set $ \|\mathcal{E}\| = m $. Let $ \mathbf{X} \in \mathbb{R}^{n \times d} $ be the $ d $-dimension feature matrix. For ease of exposition, we employ node notation $ u \in \mathcal{V} $ to denote its index, i.e., $ \mathbf{X}_u = \mathbf{X}[u,\cdot] $. Let $ \mathbf{Y} \in \mathbb{R}^{n \times \|C\|} $ be the one-hot label matrix, i.e., $ \mathbf{Y}[u,i] = 1 $ if node $ u $ belongs to class $ C_i $, for $ i \in \{1, 2, \cdots, \|C\|\} $, where $ C $ is the set of node labels. The set of direct (one-hop) neighbors of node $ u \in \mathcal{V} $ is denoted as $ \mathcal{N}_u $ with degree $ d_u = \|\mathcal{N}_u\| $. The adjacency matrix of $ G $ is denoted as $ \mathbf{A} \in \mathbb{R}^{n \times n} $ that $ \mathbf{A}[u,v] = 1 $ if edge $ \langle u, v \rangle \in \mathcal{E} $; otherwise $ \mathbf{A}[u,v] = 0 $. $ \mathbf{D} \in \mathbb{R}^{n \times n} $ is the diagonal degree matrix of $ G $ with $ \mathbf{D}[u,u] = d_u $. Let $ \mathbf{L} $ be the normalized Laplacian matrix of graph $ G $ defined as $ \mathbf{L} = \mathbf{I} - \mathbf{D}^{-\frac{1}{2}} \mathbf{A} \mathbf{D}^{-\frac{1}{2}} $ where $ \mathbf{I} $ is the identity matrix and $ \tilde{\mathbf{L}} $ be the normalized Laplacian matrix of $ G $ with self-loops as $ \tilde{\mathbf{L}} = \mathbf{I} - \tilde{\mathbf{D}}^{-\frac{1}{2}} \tilde{\mathbf{A}} \tilde{\mathbf{D}}^{-\frac{1}{2}} $ where $ \tilde{\mathbf{D}} = \mathbf{D} + \mathbf{I} $ and $ \tilde{\mathbf{A}} = \mathbf{A} + \mathbf{I} $. 2.2 SPECTRAL GRAPH FILTERS In general, the eigendecomposition of the Laplacian matrix is denoted as $ \mathbf{L} = \mathbf{U} \Lambda \mathbf{U}^\top $, where $ \mathbf{U} $ is the matrix of eigenvectors and $ \Lambda = \text{diag}[\lambda_1, \cdots, \lambda_n] $ is the diagonal matrix of eigenvalues. Eigenvalues $ \lambda_i $ for $ i \in [n] $ mark the frequency and the eigenvalue set $ \{\lambda_1, \cdots, \lambda_n\} $ is the graph spectrum. Without loss of generality, we assume $ 0 = \lambda_1 \leq \lambda_2 \leq \cdots \leq \lambda_n \leq 2 $. When applying a spectral graph filter on graph signal $ \mathbf{x} \in \mathbb{R}^n $, the process involves the following steps. First, the graph Fourier operator $ \mathcal{F}(\mathbf{x}) = \mathbf{U}^\top \mathbf{x} $ projects the graph signal $ \mathbf{x} $ into the spectral domain. Subsequently, a spectral filtering function $ g_w(\cdot) $ parameterized by $ \mathbf{w} \in \mathbb{R}^n $ is employed on the derived spectrum. Eventually, the filtered signal is transformed back via the inverse graph Fourier transform operator $ \mathcal{F}^{-1}(\mathbf{x}) = \mathbf{Ux} $. The process is formally expressed as \[ \mathcal{F}^{-1}(\mathcal{F}(g_w) \odot \mathcal{F}(\mathbf{x})) = \mathbf{U} g_w(\Lambda) \mathbf{U}^\top \mathbf{x} = \mathbf{U} \text{diag}(g_w(\lambda_1), \cdots, g_w(\lambda_n)) \mathbf{U}^\top \mathbf{x}, \] where $ \odot $ is the Hadamard product. In particular, spectral graph filters enhance signals in specific frequency ranges and suppress the signals in the rest parts according to objective functions. For node classification, homophily graphs are prone to contain low-frequency signals whilst heterophily graphs likely own high-frequency signals. In order to quantify the heterophily degrees of graphs, numerous homophily metrics have been introduced, e.g., edge homophily (Zhu et al., 2020), node homophily (Pei et al., 2020), class homophily (Lim et al., 2021; Luan et al., 2021), and a recent adjusted homophily (Platonov et al., 2022). By following the literature of spectral graph filters (Zhu et al., 2020; Lei et al., 2022), we adopt edge homophily in this work, explained as follows. Table 1: Polynomial Graph Filters \| Method \| Basis \| Graph Filter $g_w(\lambda)$ \| Propagation Matrix $P$ \| \|-----------------\|-------------\|-----------------------------------------------------------------------------------------------\|------------------------\| \| ChebNet \| Chebyshev \| $\sum_{k=0}^{K} w_k T_k(\lambda)$ \| $2L/\lambda_{max} - I$ \| \| GPR-GNN \| Monomial \| $\sum_{k=0}^{K} w_k (1 - \lambda)^k$ \| $I - \hat{L}$ \| \| BernNet \| Bernstein \| $\sum_{k=0}^{K} \frac{w_k}{2^K} \binom{K}{k} (2 - \lambda)^{K-k} \lambda^k$ \| $I - \frac{L}{2}$ \| \| JacobiConv \| Jacobi \| $\sum_{k=0}^{K} w_k P_k^{(1)}(1 - \lambda)$ \| $I - L$ \| \| OptBasisGNN \| Orthonormal \| — \| $I - L$ \| Definition 1 (Homophily Ratio $h$) Given a graph $G = (V, E)$ and its label matrix $Y$, the homophily ratio $h$ of $G$ is the fraction of edges with two end nodes from the same class, i.e., $$h = \frac{\| \{(u,v) \in E : y_u = y_v \} \|}{\|E\|}.$$ Besides the homophily metrics for categorical node labels, the similarity of numerical node signals can also be measured via Dirichlet Energy (Zhou et al., 2021; Karhadkar et al., 2023). Specifically, we customize the metric to node signals $x \in \mathbb{R}^n$ and propose spectral signal frequency as follows. Definition 2 (Spectral Signal Frequency $f$) Consider a graph $G = (V, E)$ with $n$ nodes and Laplacian matrix $L$. Given a normalized feature signal $x \in \mathbb{R}^n$, the spectral signal frequency $f(x)$ on $G$ is defined as $f(x) = \frac{x^T L x}{2}$. By nature of Dirichlet energy, spectral signal frequency $f(x)$ quantifies the discrepancy of signal $x$ on graph $G$. For $f(x)$, it holds that Lemma 2.1 For any normalized feature signal $x \in \mathbb{R}^n$ on graph $G$, the spectral signal frequency $f(x) \in [0, 1]$ holds. 3 Revisiting Polynomial Graph Filters Optimal graph filters require eigendecomposition on the Laplacian matrix at the cost of $O(n^3)$. To bypass the high computation overhead, a plethora of polynomial graph filters (Defferrard et al., 2016; Chien et al., 2021; He et al., 2021; Wang & Zhang, 2022; He et al., 2022; Guo & Wei, 2023) have been proposed to approximate optimal graph filters by leveraging distinct polynomials. Table 1 summarizes several such polynomial graph filters, including adopted polynomials, graph filter functions, and propagation matrices if applicable. By identifying the appropriate matrix $P$, those polynomial filters applied on graph signal $x \in \mathbb{R}^n$ can be equally expressed as $$z = \sum_{k=0}^{K} w_k P^k \cdot x,$$ where $K$ is the length of polynomial basis, $w \in \mathbb{R}^{K+1}$ is the learnable weight vector, and $z \in \mathbb{R}^n$ is the final representation. For example, He et al. (2021) utilize Bernstein polynomial and propose polynomial filter BernNet as $z = \sum_{k=0}^{K} \frac{w_k}{2^K} \binom{K}{k} (2I - L)^{-k} L^k x$. By setting $P = I - \frac{L}{2}$ as the propagation matrix and rearranging the expression, an equivalent formulation is expressed as $z = \sum_{k=0}^{K} w_k (I - \frac{L}{2})^k x$ where $w_k = \sum_{i=0}^{k} w_{k-i} \binom{K}{K-i} (-1)^{k-i}$ is the learnable parameter. In particular, vectors $P^k x$ in Equation (2) for $k \in \{0, 1, \cdots, K\}$ collectively constitute a signal basis $\{P^0 x, P^1 x, \cdots, P^K x\}$. Spectral graph filters attempt to learn a weighted combination of signal bases, aiming to systematically produce node representations for nodes from graphs with varying heterophily degrees for label prediction. From the spectral perspective, spectral filters essentially execute filtering operations on the spectrum $\{f(P^0 x), f(P^1 x), \cdots, f(P^K x)\}$ in order to approximate the frequencies of label signals $Y$. Meanwhile, label signal frequencies are closely correlated with the homophily ratio $h$. To formally depict the correlation between the filtered signal $\sum_{k=0}^{K} w_k P^k x$ and homophily ratio $h$, we establish a theorem as follows. Theorem 1 Given a connected graph $G = (V, E)$ with homophily ratio $h$, consider an optimal polynomial filter $F(w) = \sum_{k=0}^{K} w_k P^k$ with propagation matrix $P$ and weights $w \in \mathbb{R}^{K+1}$ toward for node classification. Given a feature signal $ x \in \mathbb{R}^n $, the spectral frequency $ f(\sum_{k=0}^{K} w_k P^k x) $ is proportional to $ 1 - h $. Theorem 1 uncovers the critical role of graph homophily ratios when generating desired node representations. Intuitively, ideal signal bases are obligated to consider different heterophily degrees for various graphs. However, the majority of existing polynomial filters exploit predefined polynomials, ignoring the corresponding homophily ratios. 4 Universal Polynomial Basis for Graph Filters 4.1 Theoretical Analysis of Homophily Basis Conventional GNN models (Kipf & Welling, 2017; Hamilton et al., 2017; Klicpera et al., 2019a) employ homophily as a strong inductive bias (Lim et al., 2021). To aggregate information within $ K $ hops, graph signal $ x $ are propagated to $ K $-hop neighbors via propagation matrix $ P = I - L $, yielding homophily basis $ \{x, Px, \cdots, P^K x\} $. To elucidate how the homophily basis accommodates homophily graphs, we establish the following Theorem. Theorem 2 Given a propagation matrix $ P $ and graph signal $ x $, consider an infinite homophily basis $ \{x, Px, \cdots, P^K x, P^{K+1} x, \cdots\} $. It holds that (i) as the exponent $ k $ increases, the angle $ \arccos \left( \frac{P^K x \cdot P^{K+1} x}{\\|P^K x\\| \\|P^{K+1} x\\|} \right) $ is progressively smaller, and (ii) $ \lim_{K \to \infty} \arccos \left( \frac{P^K x \cdot \psi}{\\|P^K x\\| \\|\psi\\|} \right) = 0 $ where $ \psi = D^\frac{1}{2} 1 $. The homophily basis exhibits growing similarity and asymptotic convergence for the purpose of capturing homophily signals, thus resulting in the over-smoothing issue. For better visualization, Figure 1(a) simply illustrates the homophily basis $ \{h_0, h_1, h_2, \cdots, h_{K-1}, h_K, \cdots\} $ gradually converges to $ \psi $ in 3-dimension Euclidean space. 4.2 Adaptive Heterophily Basis As discussed aforementioned, desired signal bases are expected to conform to homophily ratios. A natural question is: how can we apply homophily ratios in a sensible manner when designing signal bases without involving graph signals or structures? To answer this question, we initially explore the correlation between the basis distribution in Euclidean space and the basis frequency on regular graphs. Theorem 3 Consider a regular graph $ G $, a random basis signal $ x \in \mathbb{R}^n $, and a normalized all-ones vector $ \phi \in \mathbb{R}^n $ with frequency $ f(\phi) = 0 $. Suppose $ \theta := \arccos(\phi \cdot x) $ denotes the angle formed by $ x $ and $ \phi $. It holds that the expectation of spectral signal frequency $ \mathbb{E}_{G \sim \mathcal{G}}[f(x)] $ over the randomness of $ G $ is monotonically increasing with $ \theta $ for $ \theta \in [0, \frac{\pi}{2}) $. Theorem 3 reveals the correlation between the expected frequency of the signal basis and its relative position to the 0-frequency vector $ \phi $ on regular graphs. This fact implicitly suggests that we may take the angles (relative position) between two basis vectors into consideration when aiming to achieve the desired basis spectrum on general graphs. Meanwhile, Theorem 2 discloses the growing similarity and asymptotic convergence phenomenon within the homophily basis. To mitigate this over-smoothing issue, we can intuitively enforce all pairs of basis vectors to form an appropriate angle of $ \theta \in [0, \frac{\pi}{2}] $. Pertaining to this, Theorem 1 proves the spectral frequency of ideal signals proportional to $ 1 - h $, aligning with the homophily ratios of the underlying graphs. By leveraging the monotonicity property proved in Theorem 3, we empirically set the $ \theta := \frac{\pi}{2}(1-h) $. Consequently, a signal basis capable of capturing the heterophily degrees of graphs is derived, formally denoted as heterophily basis. Consider to construct a heterophily basis with a length of $ K + 1 $. The procedure of computing heterophily basis is outlined in Algorithm 1 and illustrated in Figure 1b. To start with, we normalize the input signal $ x $ as the initial signal $ u_0 $ and set $ \theta := \frac{(1-h)\pi}{2} $. In order to manipulate the formed angles between signal vectors, we forge an orthonormal basis, denoted as $ \{v_0, v_1, \cdots, v_K\} $ where $ v_0 $ is initialized as $ u_0 $. In particular, at the $ k $-th iteration for $ k \in [1, K] $, we set $ v_k = Pv_{k-1} $ where $ P = I - L $ is the propagation matrix. Subsequently, $ v_k $ is calculated as $ v_k := v_k - (v_k^\top v_{k-1})v_{k-1} - (v_k^\top v_{k-2})v_{k-2} $ as per the three-term recurrence Theorem (Gautschi, 2004; Liesen & Strakoš, 2013; Guo & Wei, 2023). Meanwhile, signal vector $ u_k $ is set as $ u_k := s_{k-1} $ where $ s_{k-1} := \sum_{i=0}^{K-1} u_i $. Subsequently, $ u_k $ is updated as $ u_k := \frac{u_k + t_kv_k}{\\|u_k + t_kv_k\\|} $ where $ t_k $ is \[ t_k = \sqrt{\left(\frac{s_{k-1}^\top u_{k-1}}{k\cos(\theta)}\right)^2 - \frac{(k-1)\cos(\theta)+1}{k}}. \] As a result, the final vector set $ \{u_0, u_1, \cdots, u_K\} $ is returned as the heterophily basis. The desired property of the heterophily basis is proved in the following Theorem. Detailed proofs are presented in Appendix A.1. Algorithm 1: Heterophily Basis Input: Graph $ G $, propagation matrix $ P $, input feature signal $ x $, hop $ K $, estimated homophily ratio $ h $ Output: Heterophily basis $ \{u_0, u_1, \cdots, u_K\} $ 1. $ u_0 \leftarrow \frac{x}{\\|x\\|}, v_0 \leftarrow u_0, v_{-1} \leftarrow 0, s_0 \leftarrow u_0, \theta \leftarrow \frac{(1-h)\pi}{2}; $ 2. for $ k \leftarrow 1 $ to $ K $ do 3. $ v_k \leftarrow Pv_{k-1}; $ 4. $ v_k \leftarrow v_k - (v_k^\top v_{k-1})v_{k-1} - (v_k^\top v_{k-2})v_{k-2}; $ 5. $ v_k \leftarrow \frac{v_k}{\\|v_k\\|}, u_k \leftarrow \frac{s_{k-1}}{k}; $ 6. $ t_k $ is calculated as in Equation (3); 7. $ u_k \leftarrow \frac{u_k + t_kv_k}{\\|u_k + t_kv_k\\|}, s_k \leftarrow s_{k-1} + u_k; $ 8. return $ \{u_0, u_1, \cdots, u_K\} $; Theorem 4 Consider a heterophily basis $ \{u_0, u_1, \cdots, u_K\} $ constructed from Algorithm 1 for graphs with homophily ratio $ h $. It holds that $ u_i \cdot u_j = \begin{cases} \cos\left(\frac{(1-h)\pi}{2}\right) & \text{if } i \neq j \\ 1 & \text{if } i = j \end{cases} $ for $ \forall i, j \in \{0, 1, \cdots, K\} $. Homophily ratio estimation. The exact homophily ratio $ h $ relies on the label set of the entire graph and thus is normally unavailable. To address this issue, we estimate $ h $ through labels of training data, denoted as $ \hat{h} $. Appendix A.3 presents the experimental results of the homophily ratio estimation, which signifies that a qualified homophily ratio can be effectively estimated via training data. Time complexity. In the $ k $-th iteration, it takes $ O(m + n) $ to calculate the orthonormal basis and $ O(n) $ to update $ u_k $. Therefore, the total time complexity of Algorithm 1 is $ O(K(m + n)) $, i.e., linear to propagation hops and input graph sizes. ### 4.3 Universal polynomial basis and graph filter The heterophily basis employs fixed angle $ \theta := \frac{(1-\hat{h})\pi}{2} $ associated with heterophily degrees, effectively encapsulating the heterophily of graphs. However, it can be restrictive by nature when handling strong homophily graphs with homophily ratios $ h $ close to 1. To tackle graphs ranging from strong homophily to strong heterophily, we intuitively introduce a hyperparameter $ \tau \in [0, 1] $. and merge the homophily basis and heterophily basis into a universal polynomial $\tau P^K x + (1 - \tau) u_k$, referred to as UniBasis. As a consequence, a general polynomial filter UniFilter is proposed as $$z = \sum_{k=0}^{K} w_k (\tau P^K x + (1 - \tau) u_k)$$ with learnable weight vector $w \in \mathbb{R}^{K+1}$. Convergence Discussion. The convergence speed of $P^K x$ to $\psi$ in Theorem 2 is affected by the Cheeger constant (Chung & Graham, 1997) of the underlying graphs. In general, dense graphs with larger Cheeger constant exhibit more rapid convergence while it is contrary for sparse graphs. Meanwhile, the rate of basis approximation convergence is determined by the condition number of the Hessian matrix (Wright et al., 1999; Boyd et al., 2004). It is known that orthogonal polynomial bases achieve the maximum convergence rate (Wang & Zhang, 2022; Guo & Wei, 2023). Yet it is essential to emphasize that orthonormal bases do not consistently yield empirically superior node representations, as verified in Sections 5.1 and 5.3. ## 5 EXPERIMENTS Datasets. We evaluate the performance of UniFilter on 6 real-world datasets with varied homophily ratios. Specifically, the three citation networks (Sen et al., 2008), i.e., Cora, Citeseer, and Pubmed, are homophily graphs with homophily ratios 0.81, 0.73, and 0.80 respectively; the two Wikipedia graphs, i.e., Chameleon and Squirrel and the Actor co-occurrence graph from WebKB3 (Pei et al., 2020) are heterophily graphs with homophily ratios 0.22, 0.23, and 0.22 respectively. Dataset details are presented in Table 4 in Appendix A.2. Baselines. We compare UniFilter with 20 baselines in two categories, i.e., polynomial filters and model-optimized methods. Specifically, polynomial filters employ various polynomials to approximate the optimal graph filters, including monomial SGC (Wu et al., 2019), SIGN (Frasca et al., 2020), ASGC (Chanpuriya & Musco, 2022), GPR-GNN (Chien et al., 2021), and EvenNet (Lei et al., 2022). Chebyshev polynomial ChebNet (Defferrard et al., 2016) and its improved version ChebNetII (He et al., 2022), Bernstein polynomial BerniNet (He et al., 2021), Jacobi polynomial JacobiConv (Wang & Zhang, 2022), the orthogonal polynomial OptBasisGNN (Guo & Wei, 2023) and learnable basis Specformer (Bo et al., 2023). In contrast, model-optimized methods optimize the architecture for improved node representations, including GCN (Kipf & Welling, 2017), GCNII (Chen et al., 2020), GAT (Velickovic et al., 2018), MixHop (Abu-El-Haija et al., 2019), H2GCN (Zhu et al., 2020), LINKX (Lim et al., 2021), WRGAT (Suresh et al., 2021), ACM-GCN (Luan et al., 2022), and GloGNN++ (Li et al., 2022). Experiment Settings. There are two common data split settings 60%/20%/20% and 48%/32%/20% for train/validation/test in the literature. Specifically, the polynomial filters are mostly tested in the previous setting (He et al., 2021; Wang & Zhang, 2022; Guo & Wei, 2023; Bo et al., 2023) while the model-optimized methods are normally evaluated in the latter (Zhu et al., 2020; Li et al., 2022; Song et al., 2023). ### 5.1 NODE CLASSIFICATION PERFORMANCE Table 2 and Table 3 present the results of UniFilter compared with existing polynomial filters and model-optimized methods for node classification respectively. For ease of exposition, we highlight the highest accuracy score in bold and underline the second highest score for each dataset. As shown, our method UniFilter consistently achieves the highest accuracy scores on both the homophily datasets and heterophily datasets, except in one case on Actor in Table 2. UniFilter exhibits explicit performance advantages over both SOTA polynomial filter Specformer and SOTA model-optimized method GloGNN++ for the majority of cases. In particular, the performance improvements are remarkably significant on the two heterophily datasets Chameleon and Squirrel. Specifically, the corresponding performance gains reach up to 1.03% and 2.76% in Table 2 and 2.45% and 6.38% in Table 3, respectively. It is worth mentioning that the computation time of UniBasis is linear. --- 1 Please note that those model-optimized methods reuse the public data splits from Pei et al. (2020) which are actually in the splits of 48%/32%/20% in the implementation. Table 2: Accuracy (%) compared with polynomial filters \| Methods \| Cora \| Citeseer \| Pubmed \| Actor \| Chameleon \| Squirrel \| \|---------------\|----------\|----------\|----------\|----------\|-----------\|----------\| \| SGC \| 86.83 ± 1.28 \| 79.65 ± 1.02 \| 87.14 ± 0.90 \| 34.46 ± 0.67 \| 44.81 ± 1.20 \| 25.75 ± 1.07 \| \| SIGN \| 87.70 ± 0.69 \| 80.14 ± 0.87 \| 89.09 ± 0.43 \| 41.22 ± 0.96 \| 60.92 ± 1.45 \| 45.59 ± 1.40 \| \| ASGC \| 85.35 ± 0.98 \| 76.52 ± 0.36 \| 84.17 ± 0.24 \| 33.41 ± 0.80 \| 71.38 ± 1.06 \| 57.91 ± 0.89 \| \| GPR-GNN \| 88.54 ± 0.67 \| 80.13 ± 0.84 \| 88.46 ± 0.31 \| 39.91 ± 0.62 \| 67.49 ± 1.38 \| 50.43 ± 1.89 \| \| EvenNet \| 87.77 ± 0.67 \| 78.51 ± 0.63 \| 90.87 ± 0.34 \| 40.36 ± 0.65 \| 67.02 ± 1.77 \| 52.71 ± 0.85 \| \| ChebNet \| 87.32 ± 0.92 \| 79.33 ± 0.57 \| 87.82 ± 0.24 \| 37.42 ± 0.58 \| 59.51 ± 1.25 \| 40.81 ± 0.42 \| \| ChebNetII \| 88.71 ± 0.93 \| 80.53 ± 0.79 \| 88.93 ± 0.29 \| 41.75 ± 1.07 \| 71.37 ± 1.01 \| 57.72 ± 0.59 \| \| BernNet \| 88.51 ± 0.92 \| 80.08 ± 0.75 \| 88.51 ± 0.39 \| 41.71 ± 1.12 \| 68.53 ± 1.68 \| 51.39 ± 0.92 \| \| JacobitConv \| 88.98 ± 0.72 \| 80.78 ± 0.79 \| 89.62 ± 0.41 \| 41.17 ± 0.64 \| 74.20 ± 1.03 \| 57.38 ± 1.25 \| \| OptBasisGNN \| 87.00 ± 1.35 \| 80.58 ± 0.82 \| 90.30 ± 0.19 \| 42.39 ± 0.52 \| 74.26 ± 0.74 \| 63.62 ± 0.76 \| \| Specformer \| 88.57 ± 1.01 \| 81.49 ± 0.94 \| 87.73 ± 0.58 \| 41.93 ± 1.04 \| 74.72 ± 1.29 \| 64.64 ± 0.81 \| \| UniFilter \| 89.49 ± 1.35 \| 81.39 ± 1.32 \| 91.44 ± 0.50 \| 40.84 ± 1.21 \| 75.75 ± 1.65 \| 67.40 ± 1.25 \| Table 3: Accuracy (%) compared with model-optimized methods \| Methods \| Cora \| Citeseer \| Pubmed \| Actor \| Chameleon \| Squirrel \| \|---------------\|----------\|----------\|----------\|----------\|-----------\|----------\| \| GCN \| 86.98 ± 1.27 \| 76.50 ± 1.36 \| 88.42 ± 0.50 \| 27.32 ± 1.10 \| 64.82 ± 2.24 \| 53.43 ± 2.01 \| \| GCNII \| 88.37 ± 1.25 \| 77.33 ± 1.48 \| 90.15 ± 0.43 \| 37.44 ± 1.30 \| 63.86 ± 3.04 \| 38.47 ± 1.58 \| \| GAT \| 87.30 ± 1.10 \| 76.55 ± 1.23 \| 86.33 ± 0.48 \| 27.44 ± 0.89 \| 60.26 ± 2.50 \| 40.72 ± 1.55 \| \| MixHop \| 87.61 ± 0.85 \| 76.26 ± 1.33 \| 85.31 ± 0.61 \| 32.22 ± 2.34 \| 60.50 ± 2.53 \| 43.80 ± 1.48 \| \| H2GCN \| 87.87 ± 1.20 \| 77.11 ± 1.57 \| 89.49 ± 0.38 \| 35.70 ± 1.00 \| 60.11 ± 2.15 \| 36.48 ± 1.86 \| \| LINKX \| 84.64 ± 1.13 \| 73.19 ± 0.99 \| 87.86 ± 0.77 \| 36.10 ± 1.55 \| 68.42 ± 1.38 \| 61.81 ± 1.80 \| \| WRGAT \| 88.20 ± 2.26 \| 76.81 ± 1.89 \| 88.52 ± 0.92 \| 36.53 ± 0.77 \| 65.24 ± 0.87 \| 48.85 ± 0.78 \| \| ACM-GCN \| 87.91 ± 0.95 \| 77.32 ± 1.70 \| 90.00 ± 0.52 \| 36.28 ± 1.09 \| 66.93 ± 1.85 \| 54.40 ± 1.88 \| \| GloGNN++ \| 88.33 ± 1.09 \| 77.22 ± 1.78 \| 89.24 ± 0.39 \| 37.70 ± 1.40 \| 71.21 ± 1.84 \| 57.88 ± 1.76 \| \| UniFilter \| 89.12 ± 0.87 \| 80.28 ± 1.31 \| 90.19 ± 0.41 \| 37.79 ± 1.11 \| 73.66 ± 2.44 \| 64.26 ± 1.46 \| to graph sizes and propagation hops. The superior performance of UniFilter strongly confirms the superb effectiveness and generality of UniBasis. 5.2 Spectrum distribution of datasets The superior performance of UniFilter explicitly implies the superb capability of UniBasis to capture the spectral characteristics of graphs. For better demonstration, we first calculate the spectral signal frequency of each basis vector for all $d$-dimensions, resulting in $d$ spectrum of length $K + 1$. We then average the spectrum and associate it with the learned weights $w$ accordingly, where weights $w \in \mathbb{R}^{K+1}$ of UniBasis are trained for each dataset. The spectrum distributions of trained UniBasis for the 6 datasets are plotted in Figure 2. Recall that signals in specific frequencies with weights in large absolute are enhanced while signals with small weights are suppressed. As displayed, the majority of signals of the three homophily datasets lie within the relatively low-frequency intervals as expected, e.g., [0.3, 0.5]. We also observe some minor high-frequency information which also provides insightful information for node classification (Klicpera et al., 2019b; Chen et al., 2019; Balcilar et al., 2020). On the contrary, UniBasis of the three heterophily datasets intends to remove low-frequency signals with negative weights and preserve high-frequency information. The distinct spectrum distributions of UniBasis disclose the unique spectral characteristics of each dataset. Those results manifest the capability of UniBasis as a new method to analyze graphs with varying heterophily degrees in the spectral domain with enriched interpretability. 5.3 Ablation Study Universality of UniBasis. We compare UniFilter with three of its variants using distinct polynomial bases in order to verify the effectiveness of UniBasis. To this end, we alter UniFilter by changing UniBasis into 1) a filter simply using the heterophily basis (setting $\tau = 0$) denoted as HetFilter, 2) a filter simply using the homophily basis (setting $\tau = 1$) denoted as HomFilter, and 3) a filter using the orthonormal basis (adopting $\{v_0, v_1, \cdots, v_k\}$) denoted as OrtFilter. For easy control, we generate a synthetic dataset $G_s$ by adopting the graph structure and label set of Cora. W.l.o.g., we generate a random one-hot feature vector in 100 dimensions for each node in $G_s$. To vary homophily ratios of $G_s$, we permute nodes in a random sequence and randomly reassign node labels progressively, resulting in homophily ratios in $\{0.13, 0.20, 0.30, 0.40, 0.50, 0.60, 0.70, 0.81\}$ accordingly. The performance advantage gaps of UniFilter over the three variants are presented in Figure 3. We omit the results of HetFilter beyond $h \geq 0.3$ since the corresponding performance gaps become significantly larger, which is as expected since the heterophily basis is incapable of tackling homophily graphs. In particular, the performance advantage of UniFilter over HomFilter gradually decreases as $h$ grows larger. Contrarily, performance gaps of OrtFilter from UniFilter peak at $h = 0.3$ with a notable shortfall and then erratically decrease, ending with an accuracy gap of 0.71% at $h = 0.81$. The fluctuation of OrtFilter states the inferiority of orthonormal basis over UniBasis. Sensitivity of $\tau$. To explore the sensitivity of UniFilter towards the hyperparameter $\tau$, we vary $\tau$ in $\{0, 0.1, \cdots, 0.9, 1\}$ and test UniFilter on the strong homophily dataset Cora and the strong heterophily dataset Squirrel. Figure 4 plots the performance development along varying $\tau$. As displayed, UniFilter prefers the homophily basis on Cora, and the performance peaks at $\tau = 0.9$. On the contrary, the performance of UniFilter slightly fluctuates when $\tau \leq 0.7$ and then moderately --- 2Note that this is the smallest homophily ratio we can possibly acquire by random reassignments. decreases along the increase of $\tau$ on Squirrel. When $\tau = 1.0$, the accuracy score drops sharply since only the homophily basis is utilized in this scenario. 6 RELATED WORK Polynomial filters. As the seminal work, ChebNet (Defferrard et al., 2016) utilizes a $K$-order truncated Chebyshev polynomials (Mason & Handscomb, 2002; Hammond et al., 2011) and provides $K$-hop localized filtering capability. GPR-GNN (Chien et al., 2021) simply adopts monomials instead and applies the generalized PageRank (Li et al., 2019) scores as the coefficients to measure node proximity. In comparison, SGC (Wu et al., 2019) simplifies the propagation by keeping only the $K$-th order polynomial and removing nonlinearity. ASGC (Chapnuriya & Musco, 2022) simplifies the graph convolution operation by calculating a trainable Krylov matrix so as to adapt various heterophily graphs, which, however, is suboptimal as demonstrated in our experiments. To enhance controllability and interpretability, BernNet (He et al., 2021) employs nonnegative Bernstein polynomials as the basis. Later, Wang & Zhang (2022) examine the expressive power of existing polynomials and propose JacobiConv by leveraging Jacobi polynomial (Askey, 1974), achieving better adaptability to underlying graphs. Subsequently, He et al. (2022) revisit ChebNet and pinpoint the over-fitting issue in Chebyshev approximation. To address the issue, they turn to Chebyshev interpolation and propose ChebNetII. Recently, polynomial filter OptBasisGNN (Guo & Wei, 2023) orthogonalizes the polynomial basis in order to maximize the convergence speed. Instead of using fixed-order polynomials, Specformer (Bo et al., 2023) resorts to Transformer (Vaswani et al., 2017) to derive learnable bases for each feature dimension. While Specformer demonstrates promising performance, it requires conducting eigendecomposition with the cost of $O(n^3)$, rendering it impractical for large social graphs. Contrarily, the time complexity of UniFilter is linear to both graph sizes and propagation hops. Nonetheless, none of the above polynomial filters take the varying heterophily degrees of graphs into consideration when utilizing polynomials, which leads to suboptimal empirical performance, as verified in our experiments. Model-optimized GNNs. One commonly adopted technique in model design is to combine both low-pass and high-pass filters. GNN-LF/HF (Zhu et al., 2021) devises variants of the Laplacian matrix to construct a low-pass and high-pass filter respectively. HOG-GNN (Wang et al., 2022) designs a new propagation mechanism and considers the heterophily degrees between node pairs during neighbor aggregation, which is optimized from a spatial perspective. DualGR (Ling et al., 2023) focuses on the multi-view graph clustering problem and proposes dual label-guided graph refinement to handle heterophily graphs, which is a graph-level classification task. ACM-GCN (Luan et al., 2022) trains both low-pass and high-pass filters in each layer and then adopts the embeddings from each filter adaptively. Another applied model design aims to extract homophily from both local and global graph structures. H$_2$GCN (Zhu et al., 2020) incorporates ego and neighbor embeddings, and high-order neighborhood and intermediate representations. Similarly, GloGNN++ (Li et al., 2022) trains a coefficient matrix in each layer to measure the correlations between nodes so as to aggregate homophilous nodes globally. To explicitly capture the relations between distant nodes, WRGAT (Suresh et al., 2021) leverages the graph rewiring (Topping et al., 2022; Karhadkar et al., 2023) technique by constructing new edges with weights to measure node proximity. Additionally, there are GNNs handling heterophily graphs from other aspects. LINKX (Lim et al., 2021) learns embeddings from node features and graph structure in parallel. Then the two embeddings are concatenated and fed into MLP for node predictions. Ordered GNN (Song et al., 2023) establishes the hierarchy structure of neighbors and then constrains the neighbor nodes within specific hops into the specific blocks of neurons, avoiding feature mixing within hops. 7 CONCLUSION In this paper, we propose a universal polynomial basis UniBasis by incorporating the graph heterophily degrees in the premise of thorough theoretical analysis for spectral graph neural networks. Upon UniBasis, we devise a general graph filter UniFilter. By a comprehensive evaluation of UniFilter on both real-world and synthetic datasets against a wide range of baselines, the remarkably superior performance of UniFilter significantly supports the effectiveness and generality of UniBasis for graphs with varying heterophily. In addition, UniBasis is exhibited as a promising new method for graph analysis by capturing the spectral characteristics of graphs and enriching the interpretability. 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2XkTz7gdpc	The equation after eq.1 combines two independent distributions together, which uses less information to predict v_l, and can lead to error. This is because that edges being refined can be very important for identifying which node is expected to expand.	EFFICIENT AND SCALABLE GRAPH GENERATION THROUGH ITERATIVE LOCAL EXPANSION Andreas Bergmeister* Karolis Martinkus† Nathanaël Perraudin† Roger Wattenhofer ETH Zürich Prescient Design SDSC, ETH Zürich DISCO, ETH Zürich ABSTRACT In the realm of generative models for graphs, extensive research has been conducted. However, most existing methods struggle with large graphs due to the complexity of representing the entire joint distribution across all node pairs and capturing both global and local graph structures simultaneously. To overcome these issues, we introduce a method that generates a graph by progressively expanding a single node to a target graph. In each step, nodes and edges are added in a localized manner through denoising diffusion, building first the global structure, and then refining the local details. The local generation avoids modeling the entire joint distribution over all node pairs, achieving substantial computational savings with subquadratic runtime relative to node count while maintaining high expressivity through multiscale generation. Our experiments show that our model achieves state-of-the-art performance on well-established benchmark datasets while successfully scaling to graphs with at least 5000 nodes. Our method is also the first to successfully extrapolate to graphs outside of the training distribution, showcasing a much better generalization capability over existing methods. 1 INTRODUCTION Graphs are mathematical structures representing relational data. They comprise a set of nodes and a set of edges, denoting pairwise relations between them. This abstraction is ubiquitous in modeling discrete data across domains like social networking (Fan et al., 2019), program synthesis (Nguyen et al., 2012) Bieber et al., 2020), or even origami design (Geiger et al., 2023). A crucial task is the generation of new graphs that possess characteristics similar to those observed. For example, in drug discovery, this involves creating graphs that encode the structure of a desired type of protein (Ingraham et al., 2022; Martinkus et al., 2023) or molecule (Jin et al., 2018; Vignac et al., 2023b). Traditional graph generation methods (Albert & Barabási, 2002) estimate parameters of known models like Stochastic Block Model (SBM) (Holland et al., 1983) or Erdos-Renyi (Erdős et al., 1960), but often fail to capture the complexity of real-world data. Deep learning offers a promising alternative, with approaches falling into two categories depending on the factorization of the data-generating distribution. Autoregressive techniques build graphs incrementally, predicting edges for each new node (You et al., 2018; Liao et al., 2020; Dai et al., 2020). One-shot methods generate the entire graph at once using techniques such as variational autoencoders (Simonovsky & Komodakis, 2018), generative adversarial networks (Cao & Kipf, 2022; Martinkus et al., 2022), normalizing flows (Liu et al., 2019), score-based and denoising diffusion models (Niu et al., 2020; Haefeli et al., 2023). Despite the success of these methods in generating graphs comprising several hundred nodes, scaling beyond this range poses challenges. The computational cost of predicting edges between all node pairs scales at least quadratically with the number of nodes, which is inefficient for sparse graphs typical of real-world data. Sample fidelity is also an issue, as autoregressive methods struggle with node-permutation-invariant training due to the factorial increase in node orderings, and one-shot methods often fail to capture both global and local graph structure simultaneously. Also, in contrast to algorithmic approaches (Babiac et al., 2023), neither have been shown to generalize to larger unseen graph sizes. Finally, the neural architectures employed either exhibit limited expressiveness, Correspondence to: andreas.bergmeister@inf.ethz.ch †Equal contribution. Figure 1: Example of a 4-level coarsening sequence. Colors indicate the node contraction sets $\mathcal{V}^{(p)}$. Our generation process aims at reversing with expansions and refinements the $T$ steps of this sequence from $G_T$ to $G_0$. The details of a single step are provided in Figure 2. although with linear complexity in the number of edges for message passing neural networks (Xu et al., 2019), or are computationally expensive with quadratic or even higher scaling factors for more expressive architectures (Dwivedi & Bresson, 2021; Maron et al., 2020). We present a novel approach to graph generation through iterative local expansion. In each step, we expand some nodes into small subgraphs and use a diffusion model to recover the appropriate local structure. The model is trained to reverse a graph coarsening process, as depicted in Figure 1, applied to the dataset graphs (Loukas, 2018; Loukas & Vandergheynst, 2018; Hermsdorff & Gunderson, 2019; Jin et al., 2020b; Kumar et al., 2022; 2023). We argue that this is inherently suitable for generating graphs, as it allows for the generation of an approximate global structure initially, followed by the addition of local details. This generative process effectively represents a particular kind of network growth, which we find to be much more robust to changes in generated graph sizes than existing approaches. Moreover, our method enables modeling the distribution of edges without the need to represent the entire joint distribution over all node pairs, enhancing scalability for larger graphs. Our theoretical analysis shows that, under mild conditions, our method exhibits sub-quadratic sampling complexity relative to the number of nodes for sparse graphs. We also introduce a more efficient local version of the Provably Powerful Graph Network (PPGN) (Maron et al., 2020), termed Local PPGN. This variant is especially well suited for our iterative local expansion approach, maintaining the high expressive power in the local subgraphs that we process while providing better computational efficiency. To demonstrate the effectiveness of our approach, we conducted experiments with widely used benchmark datasets. First, in the standard graph distribution modeling task from (Martinkus et al., 2022; Vignac et al., 2023a), our model achieves state-of-the-art performance with the highest Validity-Uniqueness-Novelty score on the planar and tree datasets. Additionally, it generated graphs most closely matching the test set’s structural statistics for protein and point cloud datasets. Second, we evaluate our method’s ability to generalize beyond the training distribution, by generating graphs with an unseen number of nodes and verifying if they retain the defining characteristics of the training data. In this setting, our method is the only one capable of preserving these characteristics across the considered datasets. Third, we show that for sparse graphs our model exhibits subquadratic sampling complexity relative to the number of nodes, and validate this empirically by generating planar graphs of increasing size. Our implementation is available at \url{https://github.com/AndreasBergmeister/graph-generation}. 2 RELATED WORK The seminal work by You et al. (2018) pioneered graph generation using recurrent neural networks, creating the adjacency matrix sequentially. Liao et al. (2020) improved this approach by simultaneously sampling edges for newly added nodes from a mixture of independent Bernoulli distributions with parameters obtained from a message passing graph neural network. Kong et al. (2023) conceptualized this method as the inverse of an absorbing state diffusion process (Austin et al., 2021) and proposed reinforcement learning to optimize node addition sequences. Lately, diffusion models have come to dominate alternative approaches in terms of sample quality and diversity. Although initially only effective for graphs with tens of nodes (Niu et al., 2020), subsequent improvements using discrete diffusion (Vignac et al., 2023a,b; Haefeli et al., 2023), refining the diffusion process with a destination-predicting diffusion mixture (Jo et al., 2023), or dropping... Figure 2: Single step schematic representation of the proposed methodology. The upper row delineates two sequential coarsening steps, using color differentiation to denote the contraction sets $V^{(p)}$. Commencing from the right in the lower row, the expansion of $G_{l+1}$ into $\tilde{G}_l = \tilde{G}(G_{l+1}, v_{l+1})$ is shown, assuming a known cluster size vector $v_{l+1}$. Colors distinguish membership within expansion sets while dashed lines indicate edges to be removed as per the edge selection vector $e_l$. The resultant refined graph $G_l = G(\tilde{G}_l, e_l)$ is shown in the central box, where node features correspond to the cluster size vector $v_l$, used in expanding $G_l$ into $\tilde{G}_{l-1}$ (illustrated in the leftmost box). Permutation equivariance (Yan et al., 2023) allowed for successful generation of graphs with a few hundred nodes. Nevertheless, scalability and computational complexity remain challenges for these models. As a countermeasure, Diamant et al. (2023) suggest limiting the maximal bandwidth of generated graphs. They leverage the observation that real-world graph nodes can often be ordered to confine non-zero adjacency matrix entries within a narrow diagonal band (Cuthill & McKee, 1969). Within this band, generation can be achieved using models such as GraphRNN (You et al., 2018), variational autoencoders (Grover et al., 2019), or score-based generative models (Niu et al., 2020). Alternatively, Chen et al. (2023b) introduce degree-guided diffusion, which begins with an RNN-generated degree sequence to condition the graph diffusion model. During each step, the model only considers edge connections between nodes predicted to require degree increases. This non-local process requires a simple, non-expressive, message passing graph neural network for efficient execution. However, it does offer an increase in empirical computational efficiency. Goyal et al. (2020) propose a different approach by generating a canonical string representation of the graph using a long short-term memory network. Although the length of the string is linear in the number of graph edges, generating the strings for model training has worst-case factorial complexity, which limits the practicality of this approach for general large-scale graph generation tasks. An orthogonal line of research leverages hierarchical constructions for more efficient graph generation. Dai et al. (2020) improve the original RNN-based adjacency generation by You et al. (2018) using binary tree-structured conditioning on rows and columns of the matrix, cutting the complexity from $O(n^2)$ to $O((n + m) \log n)$, with $n$ representing nodes and $m$ edges. Shirzad et al. (2022) suggest a two-stage process starting with tree-based cluster representation, followed by incremental subgraph construction. Another two-level approach to generation is proposed by Davies et al. (2023) using DiGress (Vignac et al., 2023a) to create cluster graphs, followed by independent generation of cluster subgraphs and intra-cluster edges. In a related vein, Karami (2023) present a methodology that extends to multiple levels of hierarchy, with autoregressive generation of cluster subgraphs. Limmios et al. (2023) propose another method to enhance DiGress’s scalability, which involves a divide-and-conquer strategy for sampling subgraph coverings. Although the independence assumptions of these hierarchical methods improve scalability, they may compromise sample accuracy, contrasting with our approach that avoids such assumptions. Both Davies et al. (2023) and Karami (2023) utilize the Louvain algorithm (Blondel et al., 2008) for pre-generating clusterings for training, unlike our method, which employs random sampling of coarsening sequences during training. Additionally, Guo et al. (2023) introduce a graph expansion layer for inclusion in the generator of a generative adversarial network or the decoder of a variational autoencoder, with parameter training carried out through reinforcement learning. Hierarchical approaches have also been developed for molecular generation (Jin et al., 2018, 2020a; Kuznetsov & Polykovskiy, 2021), with the aim of improving efficiency and performance by integrating domain knowledge. However, these methods are not optimized for general graph generation tasks. 3 METHOD This section presents our proposed method for graph generation through iterative local graph expansion. A graph is a tuple $ G = (V, E) $, where $ V $ is a set of $ n = \|V\| $ vertices and $ E $ a set of $ m = \|E\| $ undirected edges. Assuming an arbitrary indexing of the nodes from 1 to $ n $, we use $ v^{(i)} $ to denote the $ i $-th node in $ V $ and $ e^{(i,j)} = \{v^{(i)}, v^{(j)}\} \in E $ to denote the undirected edge connecting the nodes $ v^{(i)} $ and $ v^{(j)} $. Although the generated graphs are unattributed, the proposed method internally generates node and edge features denoted by $ v $ and $ e $ respectively. Their $ i $-th component, denoted by $ v[i] $ and $ e[i] $, corresponds to the feature of the $ i $-th node or edge in the graph. $ W \in \mathbb{R}^{n \times n} $ is a symmetric adjacency matrix with non-zero entries $ W[i,j] = W[j,i] $ assigning positive (unary for the dataset graphs) weight to edges $ e^{(i,j)} \in E $. Consequently, the combinatorial Laplacian matrix is defined as $ L = D - W $, where $ D $ is the diagonal degree matrix with $ D[i,i] = \sum_{j=1}^{n} W[i,j] $. All graphs are assumed to be connected. 3.1 Graph Expansion Starting from a singleton graph $ G_L = (\{v\}, \emptyset) $, we construct a sequence of graphs with increasing size in an auto-regressive fashion as \[ G_l \xrightarrow{\text{expand}} \tilde{G}_{l-1} \xrightarrow{\text{refine}} G_{l-1}, \] with $ G_0 $ being the graph to be generated. In every step, we expand each node in $ G_l $ into a cluster of nodes, connecting nodes within the same cluster and between neighboring clusters, resulting in a graph $ \tilde{G}_{l-1} $ with $ n_{l-1} $ nodes. Subsequently, we refine $ \tilde{G}_{l-1} $ into $ G_{l-1} $ by selectively eliminating certain edges present in $ \tilde{G}_{l-1} $. Figure 2 illustrates this process. Let us now formalize the definitions of the expansion and refinement steps. Definition 1 (Graph Expansion) Given a graph $ G = (V,E) $ with $ \|V\| = n $ nodes and a cluster size vector $ v \in \mathbb{N}^n $ denoting the expansion size of each node, let $ \tilde{G}(G,v) = (\tilde{V},\tilde{E}) $ denote the expansion of $ G $. It contains $ v[p] $ nodes, $ v^{(p_1)}, \ldots, v^{(p_{v[p]})} $, for each node $ v^{(p)} \in V $ in the initial graph. As such, the expanded node set is given by $ \tilde{V} = V^{(1)} \cup \cdots \cup V^{(n)} $, where $ V^{(p)} = \{v^{(p_i)} \| 1 \leq i \leq v[p]\} $ for $ 1 \leq p \leq n $. The edge set $ \tilde{E} $ includes all intraclass edges, $ \{e^{(p_i,p_j)} \| 1 \leq p \leq n, 1 \leq i < j \leq v[p]\} $, as well as the cluster interconnecting edges, $ \{e^{(p,q)} \| e^{(p,q)} \in E, v^{(p_i)} \in V^{(p)}, v^{(q_j)} \in V^{(q)}\} $. Definition 2 (Graph Refinement) Given a graph $ \tilde{G} = (\tilde{V},\tilde{E}) $ with $ \tilde{m} = \|\tilde{E}\| $ edges and an edge selection vector $ e \in \{0,1\}^{\tilde{m}} $, let $ G(\tilde{G},e) = (V,E) $ denote the refinement of $ \tilde{G} $, with $ V = \tilde{V} $ and $ E \subseteq \tilde{E} $ such that the $ i $-th edge $ e^{(i)} \in E $ if and only if $ e[i] = 1 $. Probabilistic Model Starting from a given dataset $ \{G^{(1)}, \ldots, G^{(N)}\} $ of i.i.d. graph samples, we aim to fit a distribution $ p(G) $ that matches the unknown true generative process as closely as possible. We model the marginal likelihood of a graph $ G $ as the sum of likelihoods over expansion sequences \[ p(G) = \sum_{\varpi \in \Pi(G)} p(\varpi). \] Here, $ \Pi(G) $ denotes the set of all possible expansion sequences $ (G_L = (\{v\}, \emptyset), G_{L-1}, \ldots, G_0 = G) $ of a single node into the target graph $ G $, with each $ G_{l-1} $ being a refined expansion of its predecessor, that is, $ \tilde{G}_{l-1} = \tilde{G}(G_l,v_l) $ is the expansion of $ G_l $ according to Definition 1 with the cluster size vector $ v_l $, and $ G_{l-1} = G(\tilde{G}_{l-1},e_{l-1}) $ is the refinement of $ \tilde{G}_{l-1} $ according to Definition 2 and the edge selection vector $ e_{l-1} $. Factorization We factorize the likelihood of a fixed expansion sequence $ \varpi = (G_L, \ldots, G_0) $ into a product of conditional likelihoods of single expansion and refinement steps, assuming a Markovian structure, as \[ p(\varpi) = p(G_L) \cdot \prod_{l=L}^{1} p(G_{l-1} \| G_l) = \prod_{l=L}^{1} p(e_{l-1} \| \tilde{G}_{l-1}) p(v_l \| G_l). \] To avoid modeling two separate distributions $ p(e_t \mid \tilde{G}_t) $ and $ p(v_t \mid G_t) $, we rearrange terms as \[ p(\varpi) = \frac{p(v_L \mid G_L)}{p(v_L)} \cdot \prod_{l=L-1}^{1} p(v_l \mid G_l)p(e_l \mid \tilde{G}_l) \cdot p(e_0 \mid \tilde{G}_0), \] (1) and model $ v_t $ to be conditionally independent of $ \tilde{G}_t $ given $ G_t $, i.e. $ p(v_t \mid G_t, \tilde{G}_t) = p(v_t \mid G_t) $, allowing us to write \[ p(v_t \mid G_t)p(e_t \mid \tilde{G}_t) = p(v_t, e_t \mid \tilde{G}_t). \] We represent the expansion and refinement vectors as node and edge features of the expanded graph, respectively. This enables us to model a single joint distribution over these features for each refinement and consecutive expansion step. ### 3.2 Learning to Invert Graph Coarsening We now describe how we construct expansion sequences $ \varpi \in \Pi(G) $ for a given graph $ G $ and use them to train a model for conditional distributions $ p(v_t, e_t \mid \tilde{G}_t) $. For this, we introduce the notion of graph coarsening as the inverse operation of graph expansion. Intuitively, we obtain a coarsening of a graph by partitioning its nodes into nonoverlapping, connected sets and contracting the induced subgraph of each set into a single node. Definition 3 (Graph Coarsening)* Let $ G = (V, E) $ be an arbitrary graph and $ P = \{V^{(1)}, \ldots, V^{(n)}\} $ be a partitioning of the node set $ V $, such that each partition $ V^{(p)} \in P $ induces a connected subgraph in $ G $. We construct a coarsening $ \tilde{G}(G, P) = (\tilde{V}, \tilde{E}) $ of $ G $ by representing each partition $ V^{(p)} \in P $ as a single node $ v^{(p)} \in \tilde{V} $. We add an edge $ e^{(p,q)} \in \tilde{E} $ between distinct nodes $ v^{(p)} \neq v^{(q)} \in \tilde{V} $ in the coarsened graph if and only if there exists an edge $ e^{\{i,j\}} \in E $ between the corresponding disjoint clusters in the original graph, i.e. $ v^{(i)} \in V^{(p)} $ and $ v^{(j)} \in V^{(q)} $. An important property of this coarsening operation is that it can be inverted through an appropriate expansion and subsequent refinement step, as elaborated in Appendix A. Based on this premise, it can be deduced through an inductive argument that for any given coarsening sequence $ (G = G_0, G_1, \ldots, G_L = (\{v\}, \emptyset)) $ that transforms a graph $ G $ into a single node, there exists a corresponding expansion sequence $ \varpi \in \Pi(G) $ with the same elements in reverse order, i.e. $ \varpi = (G_L, \ldots, G_0) $. Note that successive coarsening steps always result in a single-node graph, as long as the original graph is connected, and every coarsening step contains at least one non-trivial contraction set, i.e. a set of nodes with more than one node. We define the distribution $ p(\pi) $ over coarsening sequences symmetrically to $ p(\varpi) $ in Equation 1 and use $ \Pi(G) $ to denote the set of all possible coarsening sequences of a graph $ G $. With this, it holds that \[ p(G) = \sum_{\varpi \in \Pi(G)} p(\varpi) \geq \sum_{\pi \in \Pi(G)} p(\pi). \] (2) Note that this inequality is strict, as there exist expansion sequences that are not the reverse of any coarsening sequence.$^2$ As we can easily generate samples from $ \Pi(G) $, this is a suitable lower bound on the marginal likelihood of $ G $ that we can aim to maximize during training. Contraction Families Without further restrictions on the allowed partitioning of the node set in Definition 3, for an arbitrary graph $ G $ there can potentially be exponentially many coarsenings of it, rendering the computation of the sum $ \sum_{\pi \in \Pi(G)} p(\pi) $ intractable. Therefore, we further restrict the possible contraction sets in graph coarsening to belong to a given contraction family $ F(G) $. We use $ \Pi_F(G) $ to denote the set of all possible coarsening sequences of $ G $ that only use contraction sets from $ F(G) $ in each step. $ \Pi_F(G) $ is a subset of $ \Pi(G) $, and hence Equation 2 with $ \Pi(G) $ replaced by $ \Pi_F(G) $ still holds. Following Loukas [2018], we experiment with edge contraction $ F(G) = E $ and neighborhood contraction $ F(G) = \{ \{v^{(j)} \mid e^{\{i,j\}} \in E \} \mid v^{(i)} \in V \} $. $^2$For example, the refinement step might split the graph into two connected components, which cannot, from Definition 3, be coarsened back into a single connected graph. Variational Interpretation Given a distribution $ q(\pi \mid G) $ over coarsening sequences $ \Pi_F(G) $ for a graph $ G $, it holds that \[ p(G) \geq \sum_{\pi \in \Pi_F(G)} p(\pi) \geq \mathbb{E}_{\pi \sim q(\pi \mid G)} \left[ \frac{p(\pi)}{q(\pi \mid G)} \right], \] and one can derive the evidence lower bound on the log-likelihood under the given model as \[ \log p(G) \geq \mathbb{E}_{\pi \sim q(\pi \mid G)} \left[ \log p(v_L \mid G_L) + \sum_{l=L-1}^{1} \log p(v_l, e_l \mid \tilde{G}_l) + \log p(e_0 \mid \tilde{G}_0) \right] + H(q(\pi \mid G)), \] leading to a variational interpretation of the model. Spectral Guided Generation The above formulation is agnostic to the distribution $ q(\pi \mid G) $ over coarsening sequences $ \Pi_F(G) $, giving us the flexibility to choose a distribution that facilitates the learning process and improves the generative performance of the model. While the uniform distribution over all possible coarsening sequences $ \Pi_F(G) $ gives the tightest bound in Equation 3, as the entropy term vanishes, arbitrary coarsening sequences could destroy important structural properties of the original graph $ G $, making it difficult for the model to learn to invert them. Therefore, we propose a distribution $ q $ that prioritizes coarsening sequences preserving the spectrum of the graph Laplacian, which is known to capture important structural properties of a graph. Note that the distribution $ q $ does not need to be explicitly defined. Instead, for training the model, we only need a sampling procedure from this distribution. In Appendix D, we propose a sampling procedure for coarsening sequences which is parametric in a cost function. It iteratively evaluates the cost function across all contraction sets and subsequently selects a cost-minimizing partition of the contraction sets in a greedy and stochastic fashion. When instantiating the cost function with the Local Variation Cost [7] proposed by Loukas (2018), we obtain a Laplacian spectrum-preserving distribution over coarsening sequences. In Appendix C, we summarize the work of Loukas (2018) and show how our generic sampling procedure can be instantiated with the Local Variation Cost. In Section D.1, we empirically validate the effectiveness of this approach by comparing the generative performance of the model with and without spectrum-preserving sampling. While numerous graph coarsening techniques exist (Loukas, 2018; Hermisdorff & Gunderson, 2019; Jin et al., 2020b; Kumar et al., 2022, 2023), our chosen method stands out for two key reasons. It adheres to our coarsening definition with an efficient local cost function guiding contraction set selection. Additionally, it’s a multilevel scheme that maintains the original graph’s Laplacian spectrum at each level, essential for our goals. 3.3 Modeling and Training We now turn to the modeling of conditional distributions $ p(v_l, e_l \mid \tilde{G}_l) $ within our marginal likelihood factorization for $ p(G) $. Let $ p_\theta(v_l, e_l \mid \tilde{G}_l) $ denote the parameterized distribution, with $ \theta $ as the parameters. We use the same model for all $ 1 \leq l < L $ conditional distributions $ p_\theta(v_l, e_l \mid \tilde{G}_l) $ as well as $ p_\theta(e_0 \mid \tilde{G}_0) $ and $ p_\theta(v_L \mid G_L) = p_\theta(v_L) $, with the parameters $ \theta $ being shared between all distributions. For the latter two distributions, we disregard the edge and node features, respectively, but maintain the same modeling approach as for the other distributions. In the following, we describe the modeling of $ p_\theta(v_l, e_l \mid \tilde{G}_l) $ for arbitrary but fixed level $ 1 \leq l < L $. Modeling with Denoising Diffusion Models An effective method should be capable of representing complex distributions and provide a stable, node permutation-invariant training loss. Denoising diffusion models meet these criteria. This method entails training a denoising model to restore the original samples—in our setting, node and edge features $ v_l $ and $ e_l $—from their corrupted counterparts. Inference proceeds iteratively, refining predictions from an initial noise state. Although this requires multiple model queries per graph expansion, it does not affect the algorithm’s asymptotic complexity nor impose restrictive assumptions on the distribution, unlike simpler models such as mixtures of independent categorical distributions. We adopt the formulation proposed by Song et al. (2021), enhanced by contributions from Karras et al. (2022). This method represents the forefront in image synthesis, and preliminary experiments indicate its superior performance for our application. For a comprehensive description of the framework and its adaptation to our context, see Appendix E. 3.4 LOCAL PPGN A key component of our proposed methodology is the specialized architecture designed to parameterize the conditional distributions $ p_\theta(v_t, e_t \mid G_t) $, or equivalently, the denoising model. Our design incorporates a novel edge-wise message passing layer, termed Local PPGN. When designing this layer, we drew inspiration from the PPGN model (Maron et al., 2020), which is provably more expressive than graph message passing networks at the expense of increased computational complexity (cubic in the number of nodes). Recognizing that our suggested methodology only locally alternates graphs at every expansion step and that these graphs possess a locally dense structure, as a result of the expansion process (Definition 1), we designed a layer that is locally expressive, resembles the PPGN layer on a dense (sub)graph, but retains efficiency on sparse graphs, with linear runtime relative to the number of edges. An elaborate explanation of this layer and its placement within existing graph neural network models can be found in Appendix F. In-depth architectural details of the overall model are presented in Appendix F.2. 3.5 SPECTRAL CONDITIONING Martinkus et al. (2022) found that using the principal Laplacian eigenvalues and eigenvectors of a target graph as conditional information improves graph generative models. A salient aspect of our generative methodology is that it generates a graph $ G_t $ from its coarser version, $ G_{t+1} $. Given the preservation of the spectrum during coarsening, the Laplacian spectrum of $ G_t $ is approximated by that of $ G_{t+1} $. The availability of $ G_{t+1} $ during the generation of $ G_t $ allows computing its principal Laplacian spectrum and subsequently conditioning the generation of $ G_t $ on it. Specifically, we accomplish this by computing the smallest $ k $ non-zero eigenvalues and their respective eigenvectors of the Laplacian matrix $ L_{t+1} $ of $ G_{t+1} $. We then employ SignNet (Lim et al., 2022) to obtain node embeddings for nodes in $ G_{t+1} $, which are then replicated across nodes in the same expansion set to initialize $ G_t $'s embeddings. This shared embedding feature also aids the model in cluster identification. Our Local PPGN model, while inherently capturing global graph structures, can benefit from explicit conditioning on spectral information. We adjust the number of eigenvalues $ k $ as a tunable hyperparameter; when $ k = 0 $, node embeddings are drawn from an isotropic normal distribution. 3.6 PERTURBED EXPANSION As noted, the given Definitions 1 and 2 are sufficient to reverse a contraction step with an appropriate expansion and subsequent refinement step. However, we have observed that introducing an additional source of randomness in the expansion is beneficial for the generative performance of the model, particularly in the context of datasets with limited samples where overfitting is a concern. Therefore, we introduce the concept of perturbed expansion, where in addition to the edges in $ \tilde{E} $, we add edges between nodes whose distance in $ G $ is bounded by an augmented radius independently with a given probability. A formal definition and an illustrative explanation of this concept can be found in Appendix B. 3.7 DETERMINISTIC EXPANSION SIZE Our graph expansion method iteratively samples a cluster size vector $ v $ to incrementally enlarge the graph. The process halts when $ v $ is entirely composed of ones, indicating no further node expansion is necessary. However, this stochastic approach may not reliably produce graphs of a predetermined size. To remedy this, we propose a deterministic expansion strategy, primarily applicable in cases of edge contraction where the maximum expansion size is two. In this strategy, $ v $ is treated as binary. We set the target size for the expanded graph at each expansion step and, instead of sampling $ v $, we select the required number of nodes with the highest probabilities for expansion to reach the predefined size. Additionally, we introduce the reduction fraction, calculated as one minus the ratio of node counts between the original and expanded graphs, as an additional input to the model during training and inference. More details are discussed in Appendix C. ### Table 1: Sample quality on synthetic graphs. \| Model \| Deg. ↓ \| Clus. ↓ \| Orbit ↓ \| Spec. ↓ \| Wavelet ↓ \| Ratio ↓ \| Valid ↑ \| Unique ↑ \| Novel ↑ \| V.U.N. ↑ \| \|------------------------\|--------\|---------\|---------\|---------\|-----------\|---------\|---------\|----------\|---------\|----------\| \| Training set \| \| \| \| \| \| \| \| \| \| \| \| GraphRNN [You et al., 2018] \| 0.00049 \| 0.2779 \| 1.2543 \| 0.0459 \| 0.1034 \| 490.2 \| 0.0 \| 100 \| 100 \| 0.0 \| \| GRAN [Liao et al., 2020] \| 0.00046 \| 0.0426 \| 0.0009 \| 0.0075 \| 0.0019 \| 2.0 \| 97.5 \| 85.0 \| 2.5 \| 0.0 \| \| SPECTRE [Matinfarik et al., 2022] \| 0.0005 \| 0.0012 \| 0.0003 \| 0.0003 \| 0.0019 \| 9.0 \| 99.0 \| 25.0 \| 100 \| 25.0 \| \| DiGress [Vignac et al., 2023a] \| 0.0007 \| 0.0780 \| 0.0079 \| 0.0098 \| 0.0031 \| 3.1 \| 77.5 \| 100 \| 100 \| 77.5 \| \| EDGE [Chen et al., 2023b] \| 0.0761 \| 0.3229 \| 0.7737 \| 0.0957 \| 0.3627 \| 43.1 \| 0.0 \| 100 \| 100 \| 0.0 \| \| BwR (HPF-GNN) [Dranjant et al., 2023] \| 0.0231 \| 0.2596 \| 0.5473 \| 0.0444 \| 0.1314 \| 251.9 \| 0.0 \| 100 \| 100 \| 0.0 \| \| BiGG [Dai et al., 2020] \| 0.0007 \| 0.0570 \| 0.0367 \| 0.0105 \| 0.0052 \| 16.0 \| 62.5 \| 85.0 \| 42.5 \| 5.0 \| \| GraphGen [Goyal et al., 2020] \| 0.0328 \| 0.2106 \| 0.4236 \| 0.0430 \| 0.0989 \| 210.3 \| 7.5 \| 100 \| 100 \| 100 \| \| Ours (one-shot) \| 0.0003 \| 0.0245 \| 0.0006 \| 0.0104 \| 0.0030 \| 1.7 \| 67.5 \| 100 \| 100 \| 67.5 \| \| Ours \| 0.0005 \| 0.0626 \| 0.0017 \| 0.0075 \| 0.0013 \| 2.1 \| 95.0 \| 100 \| 100 \| 95.0 \| \| Model \| Deg. ↓ \| Clus. ↓ \| Orbit ↓ \| Spec. ↓ \| Wavelet ↓ \| Ratio ↓ \| Valid ↑ \| Unique ↑ \| Novel ↑ \| V.U.N. ↑ \| \|------------------------\|--------\|---------\|---------\|---------\|-----------\|---------\|---------\|----------\|---------\|----------\| \| Stochastic Block Model (n_{max} = 187, n_{avg} = 104) \| \| \| \| \| \| \| \| \| \| \| \| Training set \| \| \| \| \| \| \| \| \| \| \| \| GraphRNN [You et al., 2018] \| 0.00055 \| 0.0584 \| 0.0785 \| 0.0065 \| 0.0431 \| 14.7 \| 5.0 \| 100 \| 100 \| 5.0 \| \| GRAN [Liao et al., 2020] \| 0.0113 \| 0.0553 \| 0.0540 \| 0.0054 \| 0.0212 \| 9.7 \| 25.0 \| 100 \| 100 \| 25.0 \| \| SPECTRE [Matinfarik et al., 2022] \| 0.0015 \| 0.0521 \| 0.0412 \| 0.0056 \| 0.0028 \| 2.2 \| 52.5 \| 100 \| 100 \| 52.5 \| \| DiGress [Vignac et al., 2023a] \| 0.0018 \| 0.0485 \| 0.0415 \| 0.0045 \| 0.0014 \| 1.7 \| 60.0 \| 100 \| 100 \| 60.0 \| \| EDGE [Chen et al., 2023b] \| 0.0279 \| 0.1113 \| 0.0854 \| 0.0251 \| 0.1500 \| 51.4 \| 0.0 \| 100 \| 100 \| 0.0 \| \| BwR (HPF-GNN) [Dranjant et al., 2023] \| 0.0112 \| 0.0604 \| 0.0667 \| 0.0059 \| 0.0370 \| 38.0 \| 7.5 \| 100 \| 100 \| 7.5 \| \| BiGG [Dai et al., 2020] \| 0.0112 \| 0.0604 \| 0.0667 \| 0.0059 \| 0.0370 \| 11.9 \| 10.0 \| 100 \| 100 \| 10.0 \| \| GraphGen [Goyal et al., 2020] \| 0.0550 \| 0.0623 \| 0.1189 \| 0.0182 \| 0.1193 \| 48.8 \| 5.0 \| 100 \| 100 \| 5.0 \| \| Ours (one-shot) \| 0.0141 \| 0.0528 \| 0.0809 \| 0.0071 \| 0.0205 \| 10.5 \| 75.0 \| 100 \| 100 \| 75.0 \| \| Ours \| 0.0119 \| 0.0517 \| 0.0669 \| 0.0067 \| 0.0219 \| 10.2 \| 45.0 \| 100 \| 100 \| 45.0 \| \| Model \| Deg. ↓ \| Clus. ↓ \| Orbit ↓ \| Spec. ↓ \| Wavelet ↓ \| Ratio ↓ \| Valid ↑ \| Unique ↑ \| Novel ↑ \| V.U.N. ↑ \| \|------------------------\|--------\|---------\|---------\|---------\|-----------\|---------\|---------\|----------\|---------\|----------\| \| Tree Graphs (n_{max} = 64, n_{avg} = 64) \| \| \| \| \| \| \| \| \| \| \| \| Training set \| \| \| \| \| \| \| \| \| \| \| \| GRAN [Liao et al., 2020] \| 0.1884 \| 0.0080 \| 0.0199 \| 0.2751 \| 0.3274 \| 607.0 \| 0.0 \| 100 \| 100 \| 0.0 \| \| DiGress [Vignac et al., 2023a] \| 0.0002 \| 0.0000 \| 0.0000 \| 0.0113 \| 0.0043 \| 1.6 \| 90.0 \| 100 \| 100 \| 90.0 \| \| EDGE [Chen et al., 2023b] \| 0.2674 \| 0.0000 \| 0.0000 \| 0.0000 \| 0.0000 \| 850.7 \| 0.0 \| 7.5 \| 100 \| 0.0 \| \| BwR (HPF-GNN) [Dranjant et al., 2023] \| 0.0016 \| 0.1239 \| 0.0003 \| 0.0480 \| 0.0388 \| 11.0 \| 0.0 \| 100 \| 100 \| 0.0 \| \| BiGG [Dai et al., 2020] \| 0.0014 \| 0.0000 \| 0.0000 \| 0.0119 \| 0.0058 \| 5.2 \| 100 \| 87.5 \| 50.0 \| 75.0 \| \| GraphGen [Goyal et al., 2020] \| 0.0105 \| 0.0000 \| 0.0000 \| 0.0153 \| 0.0122 \| 33.2 \| 95.0 \| 100 \| 100 \| 95.0 \| \| Ours (one-shot) \| 0.0004 \| 0.0000 \| 0.0000 \| 0.0080 \| 0.0055 \| 2.1 \| 82.5 \| 100 \| 100 \| 82.5 \| \| Ours \| 0.0001 \| 0.0000 \| 0.0000 \| 0.0117 \| 0.0047 \| 4.0 \| 100 \| 100 \| 100 \| 100 \| ### Table 2: Sample quality on real-world graphs. All models achieve perfect uniqueness and novelty. Several models fail on the point cloud dataset due to memory limitations (OOM), and GraphGen is unable to generate the canonical string representations within a reasonable timeframe (OOT). ### 4 EXPERIMENTS Our experiments evaluate three main aspects of our model: (1) its ability to generate graphs with structural properties similar to the training data on common synthetic graph generation datasets (planar, SBM, tree); (2) its ability to scale to much larger real-world graphs (proteins and point clouds); (3) extrapolation to out-of-distribution graph sizes. We rely on the standard metrics, datasets and evaluation procedures introduced by [Martinkus et al., 2022]. Details on this and the hyperparameters we used are covered in Appendix I. #### Simple Graph Generation. In Table 1, the most critical metric is the percentage of valid, unique, and novel graphs (V.U.N.) in the generated set. Validity for synthetic graphs indicates the adherence to the defined properties, e.g. planarity or acyclicity. Uniqueness and novelty metrics report the diversity of the output, serving as an indicator for non-overfitting. Our method demonstrates strong performance, surpassing our baseline, which operates without iterative expansion, but directly generates the full graph using the diffusion model (Ours (one-shot)). The exception is the SBM dataset, where the inherent randomness of the graphs and the absence structure aside from large clusters, likely affects the results. Nevertheless, our model still attains a satisfactory V.U.N. score. The first five columns of the table show the maximum mean discrepancy (MMD) between the generated and test graphs for the degree distribution, clustering coefficient, orbit counts, spectrum, and wavelet coefficients. We summarize these metrics by the average ratio between the generated and training MMDs. Although DiGress is the overall best performer with respect to this metric, our method achieves competitive results and is the best for planar graphs. The benefits of our approach become clearer with larger, complex real-world graphs. In Table 2 we show model performance on protein graphs with up to 500 nodes and point cloud graphs with up to 5037 nodes (see Table 6). In both cases, our method outperforms competitors by a large margin in structural similarity to the test set. Note that several methods are unable to scale to 5037 nodes. Appendix J offers a runtime comparison, affirming our method’s subquadratic scaling when generating sparse graphs of increasing size. This section also includes a theoretical analysis of the model’s complexity. Sample graphs generated by our model can be found in Appendix J. Extrapolation and Interpolation. We assess our model’s capability to generate graphs with node counts beyond the training distribution through extrapolation (creating larger graphs) and interpolation (varying sizes within observed ranges). We use a planar and a tree dataset, each comprising 128 training graphs with sizes uniformly sampled from [32, 64] for extrapolation and from [32, 64] ∪ [128, 160] for interpolation. Our evaluation involves generating graphs with 48 to 144 nodes, producing 32 graphs per size for validation and 40 for testing. We report the validity and uniqueness rates of generated graphs. Figure 3 demonstrates that our method is uniquely capable of reliably extrapolating and interpolating to out-of-distribution graph sizes across both datasets. We note that GRAN, DiGress and Ours (one shot) fail, in general, to generate larger graphs in contrast to their performance on smaller versions of the datasets (see Table 1). Therefore, our experiment does not fully determine whether these methods fail because they cannot interpolate/extrapolate or because they are unable to generate larger graphs. 5 CONCLUSION In this work, we present the first graph generative method based on iterative local expansion, where generation is performed by a single model that iteratively expands a single node into the full graph. We made our method efficient (with sub-quadratic complexity) by introducing the Local PPGN layer that retains high expressiveness while performing only local computation. We performed tests on traditional graph generation benchmarks, where our method achieved state-of-the-art results. Furthermore, to the best of our knowledge, our method is the only one able to generate graphs outside of the training distribution (with different numbers of nodes) while retaining the main graph characteristics across different datasets. 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F76bwRSLeK	In section 6 about limitations, I would venture that two reasons for the poor reconstruction could be that (a) the proposed linear encoder does not have enough expressive power or (b) there exists no sparse linear basis to explain the LLM layers or (c) there is no compact natural language description for the sparse features.	Sparse Autoencoders Find Highly Interpretable Features in Language Models Hoagy Cunningham∗12, Aidan Ewart∗13, Logan Riggs∗1, Robert Huben, Lee Sharkey4 1EleutherAI, 2MATS, 3University of Bristol, 4Apollo Research {hoagycunningham, aidanprattewart, logansmith5}@gmail.com Abstract One of the roadblocks to a better understanding of neural networks’ internals is polysemanticity, where neurons appear to activate in multiple, semantically distinct contexts. Polysemanticity prevents us from identifying concise, human-understandable explanations for what neural networks are doing internally. One hypothesised cause of polysemanticity is superposition, where neural networks represent more features than they have neurons by assigning features to an over-complete set of directions in activation space, rather than to individual neurons. Here, we attempt to identify those directions, using sparse autoencoders to reconstruct the internal activations of a language model. These autoencoders learn sets of sparsely activating features that are more interpretable and monosemantic than directions identified by alternative approaches, where interpretability is measured by automated methods. Moreover, we show that with our learned set of features, we can pinpoint the features that are causally responsible for counterfactual behaviour on the indirect object identification task (Wang et al., 2022) to a finer degree than previous decompositions. This work indicates that it is possible to resolve superposition in language models using a scalable, unsupervised method. Our method may serve as a foundation for future mechanistic interpretability work, which we hope will enable greater model transparency and steerability. 1 Introduction Advances in artificial intelligence (AI) have resulted in the development of highly capable AI systems that make decisions for reasons we do not understand. This has caused concern that AI systems that we cannot trust are being widely deployed in the economy and in our lives, introducing a number of novel risks (Hendrycks et al., 2023), including potential future risks that AIs might deceive humans in order to accomplish undesirable goals (Ngo et al., 2022). Mechanistic interpretability seeks to mitigate such risks through understanding how neural networks calculate their outputs, allowing us to reverse engineer parts of their internal processes and make targeted changes to them (Cammarata et al., 2021; Wang et al., 2022; Elhage et al., 2021). To reverse engineer a neural network, it is necessary to break it down into smaller units (features) that can be analysed in isolation. Using individual neurons as these units has had some success (Olah et al., 2020; Bills et al., 2023), but a key challenge has been that neurons are often polysemantic, activating for several unrelated types of feature (Olah et al., 2020). Also, for some types of network activations, such as the residual stream of a transformer, there is little reason to expect features to align with the neuron basis (Elhage et al., 2023). Elhage et al. (2022b) investigate why polysemanticity might arise and hypothesise that it may result from models learning more distinct features than there are dimensions in the layer. They call this phenomenon superposition. Since a vector space can only have as many orthogonal vectors as it has dimensions, this means the network would learn an overcomplete basis of non-orthogonal features. Features must be sufficiently sparsely activating for superposition to arise because, without ∗Equal contribution Code to replicate experiments can be found at https://github.com/HoagyC/sparse_coding high sparsity, interference between non-orthogonal features prevents any performance gain from superposition. This suggests that we may be able to recover the network’s features by finding a set of directions in activation space such that each activation vector can be reconstructed from a sparse linear combinations of these directions. This is equivalent to the well-known problem of sparse dictionary learning (Olshausen & Field, 1997). Building on Sharkey et al. (2023), we train sparse autoencoders to learn these sets of directions. Our approach is also similar to Yun et al. (2021), who apply sparse dictionary learning to all residual stream layers in a language model simultaneously. Our method is summarised in Figure 1 and described in Section 2. We then use several techniques to verify that our learned features represent a semantically meaningful decomposition of the activation space. First, we show that our features are on average more interpretable than neurons and other matrix decomposition techniques, as measured by autointerpretability scores (Section 3) (Bills et al., 2023). Next, we show that we are able to pinpoint the features used for a set task more precisely than other methods (Section 4). Finally, we run case studies on a small number of features, showing that they are not only monosemantic but also have predictable effects on the model outputs, and can be used for fine-grained circuit detection. (Section 5). 2 Taking Features out of Superposition with Sparse Dictionary Learning To take network features out of superposition, we employ techniques from sparse dictionary learning (Olshausen & Field, 1997; Lee et al., 2006). Suppose that each of a given set of vectors $ \{x_i\}_{i=1}^{n_{\text{vec}}} \subset \mathbb{R}^d $ is composed of a sparse linear combination of unknown vectors $ \{g_j\}_{j=1}^{n_{\text{feat}}} \subset \mathbb{R}^d $, i.e. $ x_i = \sum_j a_{ij} g_j $ where $ a_i $ is a sparse vector. In our case, the data vectors $ \{x_i\}_{i=1}^{n_{\text{vec}}} $ are internal activations of a language model, such as Pythia-70M (Biderman et al., 2023), and $ \{g_j\}_{j=1}^{n_{\text{feat}}} $ are unknown, ground truth network features. We would like to learn a dictionary of vectors, called dictionary features, $ \{f_k\}_{k=1}^{n_{\text{dict}}} \subset \mathbb{R}^d $ where for any network feature $ g_j $ there exists a dictionary feature $ f_k $ such that $ g_j \approx f_k $. To learn the dictionary, we train an autoencoder with a sparsity penalty term on its hidden activations. The autoencoder is a neural network with a single hidden layer of size $ d_{\text{hid}} = Rd_{\text{in}} $, where $ d_{\text{in}} $ is the dimension of the language model internal activation vectors$^1$ and $ R $ is a hyperparameter that controls the ratio of the feature dictionary size to the model dimension. We use the ReLU activation function in the hidden layer (Fukushima, 1975). We also use tied weights for our neural network, meaning the weight matrices of the encoder and decoder are transposes of each other$^2$. Thus, on --- 1 We mainly study residual streams in Pythia-70M and Pythia 410-M, for which the residual streams are of size $ d_{\text{in}} = 512 $ and $ d_{\text{in}} = 1024 $, respectively (Biderman et al., 2023). 2 We use tied weights because (a) they encode our expectation that the directions which detect and define the feature should be the same or highly similar, (b) they halve the memory cost of the model, and (c) they remove input vector $ x \in \{x_i\} $, our network produces the output $ \hat{x} $, given by \[ c = \text{ReLU}(Mx + b) \tag{1} \] \[ \hat{x} = M^T c \tag{2} \] \[ = \sum_{i=0}^{d_{\text{hid}}-1} c_i f_i \tag{3} \] where $ M \in \mathbb{R}^{d_{\text{hid}} \times d_{\text{in}}} $ and $ b \in \mathbb{R}^{d_{\text{hid}}} $ are our learned parameters, and $ M $ is normalised row-wise.$^3$ Our parameter matrix $ M $ is our feature dictionary, consisting of $ d_{\text{hid}} $ rows of dictionary features $ f_i $. The output $ \hat{x} $ is meant to be a reconstruction of the original vector $ x $, and the hidden layer $ c $ consists of the coefficients we use in our reconstruction of $ x $. Our autoencoder is trained to minimise the loss function \[ L(x) = \frac{\\|x - \hat{x}\\|^2_2}{\dim(x)} + \alpha \\|c\\|_1 \tag{4} \] where $ \alpha $ is a hyperparameter controlling the sparsity of the reconstruction, $ l_m $ is the width of the original activation. The $ \ell^1 $ loss term on $ c $ encourages our reconstruction to be a sparse linear combination of the dictionary features. It can be shown empirically (Sharkey et al., 2023) and theoretically (Wright & Ma, 2022) that reconstruction with an $ \ell^1 $ penalty can recover the ground-truth features that generated the data. Figure 2: The tradeoff between the average number of active features and the fraction of the variance that is unexplained, as the $ \ell^1 $ coefficient $ \alpha $ is varied. Model is Pythia70M. Black dot represents the $ R = 2, \alpha = 0.00086 $ point used for autointerpretation. ambiguity about whether the learned direction should be interpreted as the encoder or decoder direction. They do not reduce performance when training on residual stream data but we have observed some reductions in performance when using MLP data. $^3$Normalisation of the rows (dictionary features) prevents the model from reducing the sparsity loss term $ \\|c\\|_1 $ by increasing the size of the feature vectors in $ M $. \| Feature \| Description (Generated by GPT-4) \| Interpretability Score \| \|---------\|----------------------------------\|------------------------\| \| I-0000 \| parts of individual names, especially last names. \| 0.33 \| \| I-0001 \| actions performed by a subject or object. \| -0.11 \| \| I-0002 \| instances of the letter ‘W’ and words beginning with ‘w’. \| 0.55 \| \| I-0003 \| the number ‘5’ and also records moderate to low activation for personal names and some nouns. \| 0.57 \| \| I-0004 \| legal terms and court case references. \| 0.19 \| Table 1: Results of autointerpretation on the first five features found in the layer 1 residual stream, with $R = 2$, $\alpha = 0.00086$ on Pythia70m. Autointerpretation produces a description of what the feature means and a score for how well that description predicts other activations. ## 3 INTERPRETING DICTIONARY FEATURES ### 3.1 INTERPRETABILITY AT SCALE Having learned a set of dictionary features, we want to understand whether our learned features display reduced polysemy, and are therefore more interpretable. To do this in a scalable manner, we require a metric to measure how interpretable a dictionary feature is. We use the automated approach introduced in Bills et al. (2023) because it scales well to measuring interpretability on the thousands of dictionary features our autoencoders learn. In summary, the autointerpretability procedure takes samples of text where the dictionary feature activates, asks a language model to write a human-readable interpretation of the dictionary feature, and then prompts the language model to use this description to predict the dictionary feature’s activation on other samples of text. The correlation between the model’s predicted activations and the actual activations is that feature’s interpretability score. See Appendix A and Bills et al. (2023) for further details. We show descriptions and top-and-random scores for five dictionary features from the layer 1 residual stream in Table 1. The features shown are the first five under the (arbitrary) ordering in the dictionary. ### 3.2 SPARSE DICTIONARY FEATURES ARE MORE INTERPRETABLE THAN BASELINES We assess our interpretability scores against a variety of alternative methods for finding dictionaries of features in language models. In particular, we compare interpretability scores on our dictionary features to those produced by a) the default basis, b) random directions, c) Principal Component Analysis (PCA), and d) Independent Component Analysis (ICA). For the random directions and for the default basis in the residual stream, we replace negative activations with zeros so that all feature activations are nonnegative. Figure 3 shows that our dictionary features are far more interpretable by this measure than dictionary features found by comparable techniques. We find that the strength of this effect declines as we move through the model, being comparable to ICA in layer 4 and showing minimal improvement in the final layer. This could be a result of our use of a consistent $\alpha = 0.00086$, $R = 2$ in our automatic interpretation results, which as seen in Figure 2 led to a higher number of average active features in the later layers. However, it may also indicate that sparse autoencoders work less well in later layers but also may be connected to the difficulties of automatic interpretation, both because by building on earlier layers, later features may be more complex, and because they are often best explained by their effect on the output. Bills et al. (2023) showed that GPT-4 is able to generate explanations that are very close to the average quality of the human-generated explanations given similar data. However, they also showed that current LLMs are limited in the kinds of patterns that they can find, sometimes struggling to find patterns that center around next or previous tokens rather than the current token, and in the current protocol are unable to verify outputs by looking at changes in output or other data. --- 4For PCA we use an online estimation approach and run the decomposition on the same quantity of data we used for training the autoencoders. For ICA, due to the slower convergence times, we run on only 2GB of data, approximately 4 million activations for the residual stream and 1m activations for the MLPs. Figure 3: Average top-and-random autointerpretability score of our learned directions in the residual stream, compared to a number of baselines, using 150 features each. Error bars show 95% confidence intervals around means. The feature dictionaries used here were trained for 10 epochs using $\alpha = .00086$ and $R = 2$ on Pythia 70M. We do show, in Section 5, a method to see a feature’s causal effect on the output logits by hand, but we currently do not send this information to the language model for hypothesis generation. The case studies section also demonstrates a closing parenthesis dictionary feature, showing that these final layer features can give insight into the model’s workings. See Appendix C for a fuller exploration of different learned dictionaries through the lens of automatic interpretability, looking at both the MLPs and the residual stream. 4 IDENTIFYING CAUSALLY-IMPORTANT DICTIONARY FEATURES FOR INDIRECT OBJECT IDENTIFICATION In this section, we quantify whether our learned dictionary features localise a specific model behaviour more tightly than the PCA decomposition of the model’s activations. We do this via activation patching, a form of causal mediation analysis (Vig et al., 2020), through which we edit the model’s internal activations along the directions indicated by our dictionary features and measure the changes to the model’s outputs. We find that our dictionary features require fewer patches to reach a given level of KL divergence on the task studied than comparable decompositions (Figure 4). Specifically, we study model behaviour on the Indirect Object Identification (IOI) task (Wang et al., 2022), in which the model completes sentences like “Then, Alice and Bob went to the store. Alice gave a snack to ____.” This task was chosen because it captures a simple, previously-studied model behaviour, which in particular has been widely explored through causal mediation analysis (Wang et al., 2022) (Conmy et al., 2023) and it captures a simple model behaviour. Recall that the training of our feature dictionaries does not emphasize any particular task. 4.1 ADAPTING ACTIVATION PATCHING TO DICTIONARY FEATURES In our experiment, we run the model on a counterfactual target sentence, which is a variant of the base IOI sentence with the indirect object changed (e.g., with “Bob” replaced by “Vanessa”); save the encoded activations of our dictionary features; and use the saved activations to edit the model’s residual stream when run on the base sentence. In particular, we perform the following procedure. Fix a layer of the model to intervene on. Run the model on the target sentence, saving the model output logits $ y $ and the encoded features $ \tilde{c}_1, \ldots, \tilde{c}_k $ of that layer at each of the $ k $ tokens. Then, run the model on the base sentence up through the intervention layer, compute the encoded features $ c_1, \ldots, c_k $ at each token, and at each position replace the residual stream vector $ x_i $ with the patched vector \[ x'_i = x_i + \sum_{j \in F} (\tilde{c}_{i,j} - c_{i,j}) f_j \] where $ F $ is the subset of the features which we intervene on (we describe the selection process for $ F $ later in this section). Let $ z $ denote the output logits of the model when you finish applying it to the patched residual stream $ x'_1, \ldots, x'_k $. Finally, compute the KL divergence $ D_{KL}(z\|y) $, which measures how close the patched model’s predictions are to the target’s. We compare these interventions to equivalent interventions using principal components found as in Section 3.2. To select the feature subset $ F $, we use the Automated Circuit Discovery (ACDC) algorithm of Conny et al. (2023). In particular, we use their Algorithm 4.1 on our features, treating them as a flat computational graph in which every feature contributes an independent change to the $ D_{KL} $ output metric, as described above and averaged over a test set of 50 IOI data points. The result is an ordering on the features so that patching the next feature usually results in a smaller $ D_{KL} $ loss than each previous feature. Then our feature subsets $ F $ are the first $ k $ features under this ordering. We applied ACDC separately on each decomposition. ### 4.2 Precise Localisation of IOI Dictionary Features We show in Figure 4 that our sparse feature dictionaries allow the same amount of model editing, as measured by KL divergence from the target, in fewer patches (Left) and with smaller edit magnitude (Right) than the PCA decomposition. We also show that this does not happen if we train a non-sparse dictionary $ (\alpha = 0) $. However, dictionaries with a larger sparsity coefficient $ \alpha $ have lower overall reconstruction accuracy which appears in Figure 4 as a larger minimum KL divergence. In Figure 4, we consider interventions on layer 11 of the residual stream, and we plot interventions on other layers in Appendix F. ![Figure 4](image) Figure 4: (Left) Number of features patched vs KL divergence from target, using various residual stream decompositions. We find that patching a relatively small number of dictionary features is more effective than patching PCA components and features from the non-sparse $ \alpha = 0 $ dictionary. (Right) Mean edit magnitude vs KL divergence from target as we increase the number of patched features. We find that our sparse dictionaries improve the Pareto frontier of edit magnitude vs thoroughness of editing. In both figures, the feature dictionaries were trained on the first 10,000 elements of the Pile (Gao et al., 2020) (approximately 7 million activations) using the indicated $ \alpha $ values and $ R = 4 $, on layer 11 of Pythia-410M (see Appendix F for results on other layers). 5 CASE STUDIES In this section, we investigate individual dictionary features, highlighting several that appear to correspond to a single human-understandable explanation (i.e., that are monosemantic). We perform three analyses of our dictionary features to determine their semantic meanings: (1) Input: We identify which tokens activate the dictionary feature and in which contexts, (2) Output: We determine how ablating the feature changes the output logits of the model, and (3) Intermediate features: We identify the dictionary features in previous layers that cause the analysed feature to activate. 5.1 INPUT: DICTIONARY FEATURES ARE HIGHLY MONOSEMANTIC We first analyse our dictionary directions by checking what text causes them to activate. An idealised monosemantic dictionary feature will only activate on text corresponding to a single human-understandable concept, whereas a polysemantic dictionary feature might activate in unrelated contexts. Figure 5: Histogram of token counts for dictionary feature 556 in layer 4 of Pythia-70M-deduped. (Left) For all datapoints that activate the feature, we show the count of each token in each activation range. The majority of activations are apostrophes, particularly for higher activations. Notably the lower activating tokens are conceptually similar to apostrophes, such as other punctuation. (Right) We show which token predictions are suppressed by ablating the feature, as measured by the difference in logits between the ablated and unabladed model. We find that the token whose prediction decreases the most is the “s” token. Note that there are 12k logits negatively effected, but we set a threshold of 0.1 for visual clarity. The autoencoder hyperparameters used were $R = 4$, $\alpha = 0.0014$. To better illustrate the monosemanticity of certain dictionary features, we plot the histogram of activations across token activations. This technique only works for dictionary features that activate for a small set of tokens. We find dictionary features that only activate on apostrophes (Figure 5): periods; the token “the”; and newline characters. The apostrophe feature in Figure 5 stands in contrast to the default basis for the residual stream, where the dimension that most represents an apostrophe is displayed in Figure 10 in Appendix D.1; this dimension is polysemantic since it represents different information at different activation ranges. Although the dictionary feature discussed in the previous section activates only for apostrophes, it does not activate on all apostrophes. This can be seen in Figures 13 and 14 in Appendix D.2 showing two other apostrophe-activating dictionary features, but for different contexts (such as “[I/We/They]’ll” and “[don/won/wouldn’t]”). Details for how we searched and selected for dictionary features can be found in Appendix D.3. 5.2 OUTPUT: DICTIONARY FEATURES HAVE INTUITIVE EFFECTS ON THE LOGITS In addition to looking at which tokens activate the dictionary feature, we investigate how dictionary features affect the model’s output predictions for the next token by ablating the feature from the residual stream$^5$. If our dictionary feature is interpretable, subtracting its value from the residual $^5$Specifically we use less-than-rank-one ablation, where we lower the activation vector in the direction of the feature only up to the point where the feature is no longer active. stream should have a logical effect on the predictions of the next token. We see in Figure 5 (Right) that the effect of removing the apostrophe feature mainly reduces the logit for the following “s”. This matches what one would expect from a dictionary feature that detects apostrophes and is used by the model to predict the “s” token that would appear immediately after the apostrophe in possessives and contractions like “let’s”. 5.3 Intermediate Features: Dictionary Features Allow Automatic Circuit Detection We can also understand dictionary features in relation to the upstream and downstream dictionary features: given a dictionary feature, which dictionary features in previous layers cause it to activate, and which dictionary features in later layers does it cause to activate? To automatically detect the relevant dictionary features, we choose a target dictionary feature such as layer 5’s feature for tokens in parentheses which predicts a closing parentheses (Figure 6). For this target dictionary feature, we find its maximum activation $M$ across our dataset, then sample 20 contexts that cause the target feature to activate in the range $[M/2, M]$. For each dictionary feature in the previous layer, we rerun the model while ablating this feature and sort the previous-layer features by how much their ablation decreased the target feature. If desired, we can then recursively apply this technique to the dictionary features in the previous layer with a large impact. The results of this process form a causal tree, such as Figure 6. Being the last layer, layer 5’s role is to output directions that directly correspond to tokens in the unembedding matrix. In fact, when we unembed feature 59027, the top-tokens are all closing parentheses variations. Intuitively, previous layers will detect all situations that precede closing parentheses, such as dates, acronyms, and phrases. ![Figure 6: Circuit for the closing parenthesis dictionary feature, with human interpretations of each feature shown. Edge thickness indicates the strength of the causal effect between dictionary features in successive residual stream layers, as measured by ablations. Many dictionary features across layers have similar interpretations and often point in similar directions in activation space, as measured by cosine similarity. Model used was Pythia-70M-deduped, with the autoencoder hyperparameters $R = 4$, $\alpha = 0.0014$.] 6 Discussion 6.1 Related Work Several previous works have attempted to decompose language representations into sparsely-activating features, varying both the representation studied and the technique used. Our approach, training a neural network with a sparsity term in the loss function, is similar to the approaches in Faruqui et al. (2015); Subramanian et al. (2018); Sharkey et al. (2023). In other works, such as Yun et al. (2021); Zhang et al. (2019), the decomposition is found via the FISTA algorithm, and Murphy et al. (2012) uses the Non-Negative Sparse Embeddings method. Of these works, Faruqui et al. (2015); Subramanian et al. (2018); Zhang et al. (2019); Murphy et al. (2012) applied these techniques to word embeddings, while only Sharkey et al. (2023); Yun et al. (2021) found sparse decompositions of the activations of a language model. Many of these works, including Murphy et al. (2012); Subramanian et al. (2018); Yun et al. (2021) also find improved interpretability of their features, as measured by techniques such as crowd-sourced judgements, the word intrusion detection test, and word-level polysemy disambiguation, respectively. The works most similar to ours are Sharkey et al. (2023), which inspired this work, and Subramanian et al. (2018). The latter use sparse autoencoders to learn their decomposition of word embeddings, though for their main results they use losses which train the learned features to approximate a sparse binary unit, finding in preliminary experiments that this outperformed the use of an $\ell^1$ penalty. Other previous works have tried to encourage sparsity in neural networks via changes to the architecture or training process. These approaches include altering the attention mechanism (Correia et al., 2019), adding $\ell^1$ penalties to neuron activations (Kasioumis et al., 2021; Georgiadis, 2019), pruning neurons (Frankle & Carbin, 2018), and using the softmax function as the non-linearity in the MLP layers (Elhage et al., 2022a). However, training a state-of-the-art foundation model with these additional constraints is difficult (Elhage et al., 2022a), and improvements to interpretability are not always realized (Meister et al., 2021). 6.2 Limitations and Future Work The approach we present in this paper found interpretable directions, but depending on the choice of hyperparameters leaves a significant fraction of the model’s variance unexplained (Figure 2). Future work could seek to improve the Pareto frontier of sparsity and reconstruction accuracy by exploring alternative architectures for the autoencoder or incorporating information about the weights of the model or dictionary features found in adjacent layers into the training process. This approach could also be applied to other components of a transformer, such as the output of the MLP or attention sublayers, as our attempt to find sparse directions in the MLP layer met only mixed success (see Appendix C). In Section 4, we show that for the IOI task, behaviour is dependent on a relatively small number of features. We expect that, because our dictionary is trained in a task-agnostic way, these result will generalize to similar tasks and behaviours, but more work is needed to confirm this suspicion. If this property generalizes, we would have a set of features which allow for understanding many model behaviours using just a few features per behaviour. We would also like to trace the causal dependencies between features in different layers, with the overarching goal of providing a lens for viewing language models under which causal dependencies are sparse. This would hopefully be a step towards the eventual goal of building an end-to-end understanding of how a model computes its outputs. 6.3 Conclusion Sparse autoencoders are a scalable, unsupervised approach to disentangling language model network features from superposition. Our approach requires only unlabelled model activations and uses orders of magnitude less compute than the training of the original models. We have demonstrated that the dictionary features we learn are more interpretable by autointerpretation, letting us pinpoint the features responsible for a given behaviour more finely, and are more monosemantic than comparable methods. This approach could facilitate the mapping of model circuits, targeted model editing, and a better understanding of model representations. An ambitious dream in the field of interpretability is enumerative safety (Elhage et al., 2022b): producing a human-understandable explanation of a model’s computations in terms of a complete list of the model’s features and thereby providing a guarantee that the model will not perform dangerous behaviours such as deception. We hope that the techniques we presented in this paper also provide a step towards achieving this ambition. ACKNOWLEDGMENTS We would like to thank the OpenAI Researcher Access Program for their grant of model credits for the autointerpretation and CoreWeave for providing EleutherAI with the computing resources for this project. We also thank Nora Belrose, Arthur Conmy, Jake Mendel, and the OpenAI Automated Interpretability Team (Jeff Wu, William Saunders, Steven Bills, Henk Tillman, and Daniel Mossing) for valuable discussions regarding the design of various experiments. We thank Wes Gurnee, Adam Jermyn, Stella Biderman, Leo Gao, Curtis Huebner, Scott Emmons, and William Saunders for their feedback on earlier versions of this paper. Thanks to Delta Hessler for proofreading. AE and LR are supported by the Long Term Future Fund. RH is supported by an Open Philanthropy grant. HC was greatly helped by the MATS program, funded by AI Safety Support. REFERENCES Stella Biderman, Hailey Schoelkopf, Quentin Gregory Anthony, Herbie Bradley, Kyle O’Brien, Eric Hallahan, Mohammad Aflah Khan, Shivanshu Purohit, USVSN Sai Prashanth, Edward Raff, et al. Pythia: A suite for analyzing large language models across training and scaling. In International Conference on Machine Learning, pp. 2397–2430. PMLR, 2023. Steven Bills, Nick Cammarata, Dan Mossing, Henk Tillman, Leo Gao, Gabriel Goh, Ilya Sutskever, Jan Leike, Jeff Wu, and William Saunders. Language models can explain neurons in language models. 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9TG42oozQP	Since the authors have modeled the potential causal graph (Figure 1(c)), why not directly use the causal relationships on the graph to model variables C and M using VAE, instead of modeling variable Z first and then using causal discovery algorithms to distinguish them?	Causal Effect Estimation with Mixed Latent Confounders and Post-treatment Variables Anonymous authors Paper under double-blind review Abstract Causal inference from observational data has attracted considerable attention in recent years. One main obstacle is the handling of confounders. As the direct measure of confounders may not always be feasible, recent methods seek to address the confounding bias with proxy variables, which are covariates researchers postulate to be conducive to the inference of latent confounders. However, observed covariates may scramble both latent confounders and latent post-treatment variables in observational study, where existing methods risk biasing the estimation by unintentionally controlling for variables affected by the treatment. In this paper, we systematically investigate the bias due to latent post-treatment variables, i.e., latent post-treatment bias, in causal effect estimation. We first derive the bias of existing methods when selected proxies scramble both latent confounders and latent post-treatment variables, which we demonstrate can be arbitrarily bad. We then propose a novel Confounder-identifiable VAE (CiVAE) to address the bias. CiVAE is built upon the assumption that the prior of the latent variables belongs to a general exponential family with at least one invertible sufficient statistic in the factorized part. Based on this, we show that latent confounders and latent post-treatment variables can be individually identified up to simple bijective transformations. Finally, we prove that the true causal effects can be unbiasedly estimated with the transformed confounders inferred by CiVAE. Experiments on both simulated and real-world datasets demonstrate that CiVAE is significantly more robust to latent post-treatment bias than existing methods for causal effects estimation. 1 Introduction Causal inference, which seeks to draw conclusions about cause-and-effect relationships among variables of interest, has gained increasing prominence in various fields, such as social science, economics, and public health (Glass et al., 2013; Johansson et al., 2016; Prosperi et al., 2020). Traditional methods rely on randomized control trials (RCT) to draw valid causal conclusions from experimentation (Cook et al., 2002). Recently, more attention has been dedicated toward causal inference from observational datasets, which contain samples with passively observed past treatment, the associated outcome, and possibly features, and in which researchers have no control over the treatment assignment mechanism (Shalit et al., 2017; Shi et al., 2019; Wager & Athey, 2018). One main obstacle to inferring causal relations from observational data is confounding bias, which occurs when past treatments were determined by variables that causally influence the outcome, i.e., confounders. In such cases, the difference in the average outcome between the treatment group and the non-treatment group cannot be attributed solely to the treatment, but may also be due to the systematic difference of samples in the two groups (Mickey & Greenland, 1989). If the confounders can be observed, a simple strategy to address such a bias is to control them via methods such as covariate adjustment (Pocock et al., 2002) or propensity score re-weighting (Li et al., 2018). However, confounders are not always measurable (Kuroki & Pearl, 2014). Therefore, recent methods seek to adjust for the influence of confounders based on their noisy proxies, which are generally covariates researchers postulated to be conducive for the inference of confounders (Miao et al., 2018; Yao et al., 2018; Madras et al., 2019). One exemplar work from this strain is the causal effect variational auto-encoder (CEVAE) (Louizos et al., 2017) (Fig. 1(a)), which has demonstrated that confounding bias can be mitigated by controlling latent variables inferred from proxies of confounders. Although proxy-of-confounder-based methods have achieved substantial progress, we argue that these algorithms may risk controlling latent post-treatment variables (i.e., variables causally affected by the treatment) scrambled in the proxy variables, where post-treatment bias may be unintentionally introduced in the estimated treatment effect. Here, we note that the negative effects of controlling post-treatment variables have been investigated in prior research [Acharya et al., 2016; Elwert & Winship, 2014; King & Zeng, 2006]. For example, Montgomery et al. (2018) found that more than 50% of the papers published in top journals of politics inadvertently control post-treatment variables in the experimental setting, although researchers have complete control over the treatment assignment mechanism and the covariates to control for. On this basis, we postulate that post-treatment bias could be even worse for proxy-based methods in the setting of observational study, as when treatments are passively recorded, it is difficult to determine which variables causally influence the treatment and which variables are influenced by it (as both confounders and post-treatment variables are correlated with the treatment and the outcome). In addition, the post-treatment variables can be latent, which may be scrambled into the observed covariates together with the latent confounders. Consider the following real-world example that researchers from the Company$^1$ have encountered when estimating the average causal effects of switching a job from onsite to online mode to the statistics of the applicants (e.g., average age, gender/geographical diversity, etc.). In this case, the Company collected a dataset of two groups of online (i.e., the treatment group) and onsite jobs (i.e., the non-treatment group), where for each job, the statistics of the applicants (i.e., the average age) are calculated as the outcome. Clearly, the seniority of the job is a confounder between the treatment and the outcome, as less senior jobs (e.g., internships) are more likely to permit online work, and applicants for these jobs tend to be younger on average. The seniority of a job can be difficult to measure. Therefore, the required skills of the job, which the recruiter must provide when publishing a job Ad in the Company, can be used as the proxy of the confounder "seniority". However, a caveat is that, switching to an online working mode may also alter the required skills of a job, thereby affecting the qualification of the applicants (where these altered skills are mediators). Consequently, directly using the required skills as the proxy of the confounder "seniority" could unintentionally control latent mediators, which introduces post-treatment bias in the causal effect estimation results. Addressing the latent post-treatment bias faces multi-faceted challenges. First, there lacks a theoretical formulation of the bias when the selected proxies scramble latent post-treatment variables for proxy-of-confounder-based methods; the trade-off between deconfounding and introducing new post-treatment bias is not clear. In addition, it is difficult to distinguish confounders and post-treatment variables in the latent space. Existing covariate disentanglement-based methods, e.g., TEDVAE [Zhang et al., 2021], mainly focus on an easier task of disentangling latent confounders with latent adjusters and instrumental variables. This can be achieved by using their different predictive abilities w.r.t. the treatment and outcome (see Fig. 1(b)). However, since latent confounders and post-treatment variables correlate with both the treatment and outcome, the two cannot be disentangled by these methods. One solution is to assume the proxy of latent post-treatment variables can be observed, from which post-treatment variables can be inferred and disentangled from the latent confounders. However, this assumption is too strong, as in the previous online/onsite job case, we can never know which skills are causally influenced by the work mode. Finally, even if latent confounders can be distinguished, since general latent variable models have no identifiability guarantee [Khemakhem et al., 2020], it is unclear whether controlling the inferred latent variables, which may be arbitrary transformations of the true confounders, can provide unbiased estimations of the causal effects. To address the aforementioned challenges, we provide a systematic investigation of the latent post-treatment bias in causal inference. We first analyze the behavior of existing proxy-based causal inference methods when the selected proxies scramble both latent confounders and post-treatment variables, where we demonstrate that the estimated average causal effects can be arbitrarily biased. --- $^1$ Anonymized due to double-blind review policy. We then propose the Confounder-identifiable VAE (CiVAE) to address such biases. Specifically, we show that based on a mild assumption that the prior distribution of latent variables (i.e., the latent confounders and post-treatment variables) belongs to a general exponential family with at least one invertible sufficient statistic in the factorized part, latent confounders and latent post-treatment variables can be individually identified up to simple bijective transformations. In addition, based on the causal relations among confounders, mediators, and treatment, we further demonstrate that the inferred confounders (which are actually transformed proxies of the true confounders) could be properly distinguished from the inferred latent post-treatment variables with pair-wise conditional independence tests. Finally, we prove that the true causal effects can be unbiasedly estimated based on transformed confounders inferred by CiVAE. Experiments on both simulated and real-world datasets demonstrate that CiVAE shows more robustness to latent post-treatment bias than existing methods. 2 Problem Formulation and Analysis 2.1 Problem Formulation Throughout this paper, we assume the causal model in Fig. 1-(c), where the dashed lines denote indeterminate causal mechanisms that might vary in different cases. We use a binary random variable $T$ to denote the treatment, a random vector $\mathbf{X} \in \mathbb{R}^{K_X}$ to denote the observed covariates, and a random scalar $Y \in \mathbb{R}$ to denote the outcome. Furthermore, observed covariates $X$ are assumed to be generated from $K_C$ independent latent confounders $C \triangleq [C_1, C_2, ..., C_{K_C}]$ and $K_M$ latent post-treatment variables $M \triangleq [M_1, M_2, ..., M_{K_M}]$ under the causal influence of treatment $T$. We use the random vector $Z \triangleq [C \|\| M] \in \mathbb{R}^{K_Z = K_C + K_M}$ to denote all latent factors. Our aim is to estimate the average causal effects of treatment $T$ on outcome $Y$ with auxiliary confounder information in $X$, where the estimation should be devoid of both confounding bias and post-treatment bias. 2.2 Analysis of Latent Post-Treatment Bias 2.2.1 Preliminaries and Assumptions To achieve such a purpose, we first formally define the (conditional) average treatment effects (C/ATE) when covariates $X$ scramble both latent confounders $C$ and post-treatment variables $M$. We then define the post-treatment bias when covariates $X$ are used directly as the proxy of confounders. To facilitate the analysis, we make the following assumption regarding the causal generative process. Assumption 1. (Noisy-Injectivity). We assume $X = f(C, M) + \epsilon$, where $f$ is a deterministic function that combines latent confounders $C$ and latent post-treatment variables $M$ into observations $X$ and $\epsilon$ is random noise. In addition, we assume that the function $f$ is injective; beyond injectivity, $f$ can be arbitrarily nonlinear. We use $f^\dagger : X \rightarrow [C \|\| M]$ to denote its left inverse. We use $f^\dagger_C : X \rightarrow C$ and $f^\dagger_M : X \rightarrow M$ to denote the mapping from $X$ to $C$, $M$, respectively. Noisy-Injectivity is a common assumption made either explicitly or implicitly in most existing proxy-of-confounder-based causal inference algorithms. For example, if both $X$ and $C$ are categorical, Pearl (2012) assumes that $X$ has at least the same number of categories as $C$, whereas the effect restoration algorithm (Rothman et al., 2008) assumes that the matrix of $p(C, X)$ to be full-rank. Although CEVAE (Louizos et al., 2017) makes no explicit injectivity assumption between $C$ and $X$, it requires that the joint distribution $p(C, X, T, Y)$ can be fully recovered from the observations $(X, T, Y)$. The literature shows that some of the possible identification criteria are 1) multiple independent views of $C$ in $X$ (Edwards et al., 2015), and 2) $C$ is categorical and $X$ is a mixture of Gaussian components determined by $C$ (that is, $X$ is generated by bijective mapping of $C$ to the mean of the corresponding component with added Gaussian noise) (Anandkumar et al., 2014). In the following part of this section, we omit the noise $\epsilon$ to gain better intuition of latent post-treatment bias (but all the exact conclusions will still hold in the posterior sense). In Section 3, we assume that noise exists and demonstrate that our method can still adequately identify latent confounders. 2.2.2 Causal Estimand and the True ATE Based on Assumption 1, we are ready to define the estimated average treatment effect (ATE) by controlling the covariates $X$, as well as the true (conditional) average treatment effects. Definition 1. We define the Difference in Conditional Expected Values (DCEV) as \[ DCEV(x) = \mathbb{E}[Y\|T = 1, X = x] - \mathbb{E}[Y\|T = 0, X = x], \] which is the difference of the expected value of the outcome $ Y $ for units with variable $ X = x $ in the treatment group and the non-treatment group. Based on $ DCEV(x) $, we define the Difference in Expected Value (DEV), i.e., $ DEV(X) = \mathbb{E}_p(x)[DCEV(X)] $ as the expected value DCEV. $ DEV(X) $ denotes the ATE estimand by controlling covariates $ X $. If $ X = \emptyset $, $ DEV(\emptyset) $ represents the naive estimator that directly calculates the expected difference of $ Y $ between the treatment group and the non-treatment group. With the causal estimand $ DEV(X) $ introduced, we then define the true causal effects (i.e., CATE) when covariates $ X $ scramble both latent confounders and post-treatment variables according to the generative process described in Assumption 1. The main issue that hinders a direct definition of CATE with $ DCEV(x) $ and $ DEV(X) $ is that, since $ X $ contains latent post-treatment variables $ M $, conditional on $ X $, the strong ignorability assumption (Imbens & Rubin, 2015) widely used for the identification of causal effects does not hold. Accordingly, we have: Definition 2. Under Assumption 1, we define the Conditional Average Treatment Effect (CATE) for individuals with observed covariates $ X = x $ as follows: \[ CATE(x) = \mathbb{E}[Y\|T = 1, C = f_C^\dagger(x)] - \mathbb{E}[Y\|T = 0, C = f_C^\dagger(x)], \] with the Average Treatment Effect (ATE) of treatment $ T $ defined as \[ ATE = \mathbb{E}[Y\|do(T = 1)] - \mathbb{E}[Y\|do(T = 0)] = \mathbb{E}_p(C)[\mathbb{E}[Y\|T = 1, C] - \mathbb{E}[Y\|T = 0, C]]. \] In Definition 2, we only consider the latent confounder component of $ X $ for CATE in Eq. (2), as the causal relationship between the post-treatment variables $ M $ and the outcome $ Y $ is indeterminate (see Fig. 1(c)). However, if the specific relationship between $ M $ and $ Y $ can be further established by the researcher (e.g., all elements of $ M $ are latent mediators), more precise forms of CATE can be derived with path-specific counterfactual analysis (Imai et al., 2010; Cheng et al., 2022). 2.2.3 LATENT POST-TREATMENT BIAS With $ DEV(X) $ (the ATE estimator that controls the covariates $ X $), CATE, and ATE defined in Section 2.2.2, in this section, we analyze the latent post-treatment bias of existing proxy-of-confounder-based causal inference methods, such as CEVAE (Louizos et al., 2017), that control latent variables inferred from the covariates $ X $ to estimate the ATE of $ T $ on $ Y $, when $ X $ scrambles both latent confounders and post-treatment variables. In our analysis, Lemma 2.1 will be frequently used. Lemma 2.1. For an injective function $ g $, $ \mathbb{E}[Y\|X = x] = \mathbb{E}[Y\|g(X) = g(x)] $ holds. The proof when $ g $ is differentiable a.e. can be referred to in Appendix A.1. Since the latent variable models used in existing methods (such as VAE with factorized Gaussian prior in CEVAE) lack identifiability guarantee (i.e., the recovery of the exact latent variables), we assume that these models can recover the true latent space $ Z = [C, M] $ up to invertible transformations $ \tilde{f} $, where the inference process can be represented as $ \tilde{Z} = \tilde{f}(X) = \tilde{f} \circ f^\dagger(X) $. With such an assumption, we have the following theorem regarding the latent post-treatment bias when $ X $ mixes post-treatment variables. Theorem 2.2. If the observed covariates $ X $ are generated from latent confounders $ C $ and latent post-treatment variables $ M $ according to Assumption 1, the latent post-treatment bias of a proxy-of-confounder-based causal inference algorithm that controls latent variables $ \tilde{Z} $ inferred from $ X $ via $ \tilde{f} = \tilde{f} \circ f^\dagger : \mathbb{R}^{K_X} \rightarrow \mathbb{R}^{K_C + K_M} $ to estimate the ATE can be formulated as follows: \[ Bias(\tilde{X}) = ATE - DEV(\tilde{f}(X)) = ATE - \mathbb{E}[\mathbb{E}[Y\|T = 1, \tilde{f}(X)] - \mathbb{E}[Y\|T = 0, \tilde{f}(X)]] \] \[ = ATE - \mathbb{E}[\mathbb{E}[Y\|T = 1, \tilde{f} \circ f^\dagger(f(C, M))] - \mathbb{E}[Y\|T = 0, \tilde{f} \circ f^\dagger(f(C, M))]] \] \[ = \mathbb{E}[\mathbb{E}[Y\|T = 1, C] - \mathbb{E}[Y\|T = 0, C]] - \mathbb{E}[\mathbb{E}[Y\|T = 1, C, M] - \mathbb{E}[Y\|T = 0, C, M]], \] which can be arbitrarily bad. Therefore, the estimator of existing proxy-of-confounder-based methods, i.e., $ DEV(\tilde{f}(X)) $, is an arbitrarily biased estimator of the ATE, when the selected proxy of confounders $ X $ accidentally mixes in latent post-treatment variables $ M $. Equivalently, we could say that given covariates $ X $, the backdoor criteria between $ T $ and $ Y $ does not hold, which requires the conditional set of variables contains no descendants of the treatment $ T $ (Glymour et al., 2016). The final step of Eq. (4) can be proved since $ f $ is injective and $ \bar{f} $ bijective, the composite $ \bar{f} \circ f^\dagger \circ f : [C, M] \rightarrow \hat{Z} $ is bijective, so we can use Lemma 2.1 to remove $ \bar{f} \circ f^\dagger \circ f $ in the condition. ### 2.2.4 Examples in the Linear Cases Generally, the latent post-treatment bias defined in Eq. (4) cannot be simplified because 1) the causal relationship between $ M $ and $ Y $ is indeterminate, and 2) the causal influence of $ C, M, $ and $ T $ on $ Y $ can be arbitrary. However, for linear structural causal models with causal relationships determined between $ M $ and $ Y $ (e.g., $ M $ are mediators, which are post-treatment variables that have causal influences on the outcomes), stronger conclusions can be drawn as follows: Corollary 2.3. (MixedMediator) For the linear Structural Causal Model (SCM) defined as: \[ T \leftarrow 1(\alpha_T + \sum \beta_i \cdot C_i > a) \\ M_j \leftarrow \alpha_M + \gamma_j \cdot T \\ X \leftarrow \alpha_X + A[M\|\|C] \\ Y \leftarrow \alpha_Y + \tau \cdot T + \sum \theta_j \cdot M_j + \sum \kappa_i \cdot C_i, \] where the mixture function $ f = A \in \mathbb{R}^{K_X \times (K_C + K_M)} $ is a full column-rank matrix, the CATE, ATE, and the bias of proxy-of-confounder-based causal inference model that controls the latent variables $ \hat{Z} $ inferred via $ \hat{Z} = \tilde{f}(X) = B^T X $ can be formulated as follows: \[ ATE = CATE = \tau + \sum \gamma_j \cdot \theta_j \\ DEV(\hat{Z}) = \mathbb{E}[DCEV(\hat{Z})] = DCEV(\hat{Z}) = \tau \\ Bias(\hat{Z}) = ATE - DEV(\hat{Z}) = \sum \gamma_j \cdot \theta_j, \] where $ B \in \mathbb{R}^{K_X \times (K_C + K_M)} $ is another full column-rank matrix. Since $ \sum \gamma_j \cdot \theta_j $ is arbitrary, the estimator $ DEV(\hat{Z}) = \mathbb{E}[DCEV(B^T X)] $ is arbitrarily biased for ATE estimation. The proof of Eq. (6) is provided in Appendix A.2. In addition, we show that the post-treatment variables $ M $ DO NOT necessarily need to have direct causal effects on the outcome $ Y $ to incur arbitrary bias in ATE estimation. In Appendix A.3, we provide another example (i.e., MixedCorrelator) in the linear case where $ M $ is correlated with $ Y $ through unobserved confounders $ U $ in Corollary A.1. ### 3 METHODOLOGY In this section, we introduce the proposed Confounder-identifiable Variational Auto-Encoder (CiVAE) to address latent post-treatment bias. Specifically, we first prove that if the prior distribution of the true latent variables $ Z = [C, M] $ satisfies certain weak assumptions, identifiability criterion holds, and each dimension of the inferred latent variables $ \hat{Z} $, i.e., $ \hat{Z}_i $, corresponds to the invertible transformation of either a true confounder $ C_j $ or a true post-treatment variable $ M_k $. Then, utilizing the causal relations between $ C, M, $ and $ T $, we novelly transform the challenging confounder-identifiability problem into a tractable pair-wise conditional independence test problem, which can be effectively solved with kernel-based methods. Finally, we demonstrate that controlling the transformed confounders inferred by CiVAE can yield an unbiased estimation of the true ATE. #### 3.1 Generative Process The fundamental work of deep variational inference with identifiability guarantee, i.e., the identifiable VAE (iVAE) (Khemakhem et al., 2020), makes a strict assumption that the prior of true latent variables $ Z $ (i.e., $ [C, M] $ in our case) is conditionally factorized given the available covariates (i.e., the treatment $ T $ and the outcome $ Y $ in our case). However, since both latent confounders $ C $ and latent post-treatment variables $ M $ form fork structures with the outcome $ Y $ (see Fig. 1(c)) (Koller & Friedman, 2009), $ C_i, C_j, M_i, $ and $ M_j $ are not independent given $ Y $. Recently, Non-Factorized iVAE (NF-iVAE) (Lu et al., 2021) was proposed that allows arbitrary dependence among the true latent variables $ Z $ in the conditional priors, where $ Z $ can be identified up to arbitrary non-linear transformations. However, the transformation are not necessarily invertible, which is risky for causal inference, as multiple values of the confounders may collapse, leading to bias when estimating the ATE by averaging the $ DCEV $ calculated in each stratum of the inferred confounders. The proposed NF-iVAE guarantees the identifiability of $ Z $ by putting a general exponential family distribution with at least one invertible sufficient statistic in the factorized part as its prior when conditioning on treatment $ T $ and outcome $ Y $, which can be formulated as follows. Assumption 2. Let $ Z = [C\|\|M] $ be the random vector for latent variables that causally generate the observed covariates $ X $ according to Assumption 7. We assume that the conditional prior of $ Z $ given the outcome $ Y $ and the treatment $ T $ belongs to a general exponential family with parameter vector $ \lambda(Y,T) $ and sufficient statistics $ S(Z) = [S_f(Z)^T, S_{nf}(Z)^T]^T $. Specifically, $ S(Z) $ is composed of (i) the sufficient statistics of a factorized exponential family, i.e., $ S_f(Z) = [S_1(Z_1)^T, \ldots, S_K(Z_K)^T]^T $, where all components $ S_i(Z_i) $ have dimension larger than or equal to 2 and each $ S_i $ has at least one invertible dimension, and (ii) $ S_{nf}(Z) $, where $ S_{nf} $ is a neural network with ReLU activation. The density of the conditional prior can be formulated as: \[ p_{S,\lambda}(Z\|Y,T) = Q(Z)/C(Y,T) \exp[S(Z)^T \lambda(Y,T)], \] where $ Q(Z) $ is the base measure and $ C(Y,T) $ not dependent on $ Z $ is the normalizing constant. We justify that assumption 2 is weak and practical as follows. 1) Neural networks with ReLU activation have universal approximation ability of distributions (Lu & Lu, 2020). Therefore, Eq. (7) can model arbitrary dependence between true latent confounders $ C $ and true post-treatment variables $ M $ conditional on $ T $ and $ Y $. 2) Although CiVAE makes an extra assumption that $ \forall i $, at least one dimension of $ S_i $ is invertible, this can be easily satisfied as most commonly used exponential family distributions, such as Gaussian, Bernoulli, etc., has at least one invertible sufficient statistic. The reason why we use ReLU as the activation is that, the identifiability of iVAE relies on the condition that the sufficient statistics $ S $ have zero second-order cross-derivative. The factorized part, i.e., $ S_f $, satisfies it trivially since all cross-derivatives of $ S_f $ are zero. In addition, since the ReLU neural networks are linear a.e., all second-order derivatives of $ S_{nf} $ are zero. Therefore, identifiability holds after adding $ S_{nf} $ in the prior that allows the capturing of arbitrary dependence among $ Z $. ### 3.2 Optimization Objective Combining Assumptions 1 and 2, the generative process of CiVAE can be formulated as follows: \[ p_\theta(X,Z\|Y,T) = p_f(X\|Z)p_{S,\lambda}(Z\|Y,T), \] \[ p_f(X\|Z) = p_e(X-f(Z)). \] where $ \theta = (f, \lambda, S) \in \Theta $ are the parameters of the generative distribution. Since the generative process of CiVAE is parameterized by deep neural networks, the posterior distribution of $ Z $, i.e., $ p_\theta(Z\|X,Y,T) $, is intractable. Therefore, we resort to variational inference (Blei et al., 2017), where we introduce approximate posterior $ q_\phi(Z\|X,Y,T) $ parameterized by deep neural network with trainable parameter $ \phi $, and in $ q_\phi(Z\|\cdot) $ measures the one closes to $ p_\theta(Z\|\cdot) $ measured by KL divergence. Minimization of the KL is equivalent to maximization of the evidence lower bound (ELBO) as: \[ L(\theta, \phi) := \mathbb{E}_{q_\phi(Z\|X,Y,T)} \left[ \log p_f(X\|Z) + \log p_{S,\lambda}(Z\|Y,T) - \log q_\phi(Z\|X,Y,T) \right]. \] Since the normalization constant $ C $ in Eq. (7) is generally intractable, it is infeasible to directly learn $ S, \lambda $ by optimizing Eq. (10). Therefore, we substitute the KL term in Eq. (10) with the widely-used score matching (Hyvärinen & Dayan, 2005) to learn unnormalized densities instead as follows: \[ L(S, \lambda, \phi) := \mathbb{E}_{q_\phi(Z\|X,Y,T)} \left[ \\| \nabla_Z \log q_\phi(Z\|X,Y,T) - \nabla_Z \log p_{S,\lambda}(Z\|Y,T) \\|^2 \right] + \text{const.} \] --- There are a few exponential family with no invertible sufficient statistics, e.g., Weibull distribution when shape parameter $ k $ is even. Note that although $ f $ is a function, we include it in the parameter set to be consistent with the iVAE paper. 3.3 Identifiability of CiVAE With the generative process and optimization objective of CiVAE introduced in the previous subsections, we are ready to introduce the final assumption of CiVAE, which, combined with Assumptions 1 and 2 leads to the main theorem of this paper, which states the identifiability of CiVAE. Assumption 3. Assume the following: (i) The set $ \{ X \in \mathcal{X} : \phi(X) = 0 \} $ has measure zero, where $ \phi $ is the characteristic function of the density $ p_f $ in Eq. (9). (ii) The sufficient statistics, $ S_i $ in $ S_f $ are all twice differentiable. (iii) The mixture function $ f $ in Eq. (9) has all second-order cross derivatives. (iv) There exist $ k + 1 $ distinct points $(Y, T)_0, \ldots, (Y, T)_k$ such that the matrix $ L = [\lambda((Y, T)_1) - \lambda((Y, T)_0), \ldots, \lambda((Y, T)_k) - \lambda((Y, T)_0)] $ of size $ k \times k $ is invertible, where $ k = \text{Dim}(S) $. (i) - (iii) are trivial for neural networks. (iv) denotes that independent samples of $(Y, T)$ are required to identify $ C $ and $ M $. The identifiability theorem of CiVAE can be formulated as follows. Theorem 3.1. If Assumptions 1, 2, and 3 hold, and if $ \theta, \tilde{\theta} \in \Theta \rightarrow p_{\theta}(X\|Y, T) = p_{\tilde{\theta}}(X\|Y, T) $, the true latent variables $ Z $ are identifiable up to permutation and element-wise bijective transformation. Furthermore, in the case of variational inference, if we denote the true parameter that generates the data as $ \theta^* $, if (i) the distribution family $ q_{\phi}(Z\|X, Y, T) $ contains the posterior $ p_{\theta}(Z\|X, Y, T) $, and $ q_{\phi}(Z\|X, Y, T) > 0 $, (ii) we optimize Eq. (4) w.r.t. both $ \theta, \phi $, then in the limit of infinite data, true parameters $ \theta^* $ can be learned up to a permutation and bijective transformation of $ Z $. The proof of Theorem 3.1 is based on the NF-iVAE paper (Lu et al., 2021), with the new assumption introduced in CiVAE that each $ S_i $ has at least one invertible dimension incorporated to ensure that the transformation of each $ Z_i $ is bijective. The detailed proof is provided in Appendix A.4. 3.4 Identification of Latent Confounders Theorem 3.1 ensures that latent variables $ \hat{Z} $ inferred by CiVAE cannot 1) mix confounders and post-treatment variables in each dimension, or 2) collapse different values of the latent confounders into the same value. To further determine the dimensions of confounder and post-treatment variable in $ \hat{Z} $, we rely on the causal relations between latent variables $ Z = [C, M] $ and treatment $ T $ and the associated marginal/conditional dependence properties. These are discussed as follows. - Case 1. Intra-Confounders. Latent confounders $ C_i, C_j $ and the treatment $ T $ form the V-structure $ C_i \rightarrow T \leftarrow C_j $. Therefore, $ C_i $ and $ C_j $ are marginally independent, whereas they become dependent when conditioning on the assigned treatment $ T $. - Case 2. Intra-Post Treatment Variables. Latent post-treatment variables $ M_i, M_j $ and the treatment $ T $ form a fork-structure $ M_i \leftarrow T \rightarrow M_j $, where $ M_i, M_j $ are marginally dependent, but they become independent after conditioning on the assigned treatment $ T $. - Case 3. Cross-Confounder and Post-Treatment Variables. Latent confounder $ C_i $, latent post-treatment variable $ M_j $, and the treatment $ T $ forms a chain structure $ C_i \rightarrow T \rightarrow M_j $, where $ C_i, M_j $ are marginally dependent, and they become independent after conditioning on $ T $. From the above analysis we can find that, the dependence between two latent variables $ Z_i $ and $ Z_j $ increases after conditioning on the treatment $ T $ ONLY in the case of intra-confounders. Therefore, if more than one latent confounders exist, which is highly probable when covariates $ X $ are high-dimensional, we can conduct independence test $ \text{Ind}(\hat{Z}_i, \hat{Z}_j) $ and $ \text{CInd}(\hat{Z}_i, \hat{Z}_j\|T) $ for all pairs of inferred latent variables, which can be implemented via kernel-based methods as (Zhang et al., 2012), and select the pairs where p-value of $ \text{CInd} $ is larger than that of $ \text{Ind} $ as latent confounders. Here, we note that the kernel-based (conditional) independence test incurs $ N^2 \times K_Z^2 $ complexity in the training phase. However, once the dimensions of the confounders in $ \hat{Z} $ are determined, CiVAE has the same complexity as CEVAE for the estimation of CATE and ATE in the test phase. Therefore, we argue that the additional complexity of model training is worthy due to the substantially increased robustness toward latent post-treatment bias (which will be demonstrated in Section 4). 3.5 ATE Estimator with Transformed Confounders Finally, we show that controlling transformed confounders $ \hat{C} $ inferred by CiVAE provides an unbiased estimation of ATE. Although assumptions weaker than Assumption 2 e.g., inferred confounders have the same propensity score as the true confounders (i.e., $ \hat{C} $ does not have to be a bijective transformation of $ C $), could lead to the same unbiasedness results (Imbens & Rubin, 2015), since our main purpose is to analyze the latent post-treatment bias and propose a viable solution accordingly, this introduces unnecessary complexity, which could be explored as a direction for future study. Theorem 3.2. Controlling bijective of confounders is equivalent to controlling true confounders in ATE estimation, i.e., $ \text{DEV}(\hat{C}) = \text{DEV}(g(C)) = \text{ATE} $, if transformation function $ g $ is bijective. The proof of Theorem 3.2 for discrete $ C $ is trivial (where $ \hat{C} = g(C) $ represents a simple relabeling of the stratum that we calculate the DCEV and take the expectation). The proof in the continuous case where $ g $ is differentiable is provided in Appendix A.5. With Theorem 3.2 we can control the identified latent confounders as true confounders, providing an unbiased estimate of ATE. ### 4 Empirical Study #### 4.1 Datasets We establish two simulated datasets, i.e., MixedMediator and MixedCorrelator, that consider two types of post-treatment variables, i.e., 1) mediators and 2) variables that are correlated with the outcome $ Y $ via latent confounders $ U $. The generative process of the two datasets can be referred to in Corollary 2.3 and Corollary A.1 respectively, where the latent confounders $ C $ are generated from Gaussian as $ C \sim \text{Gaussian}(0, I_{KC}) $. For MixedMediator, $ \gamma $ is set as $[-1, -1, -1]$, $ \theta $ is set as $[1, 1, 1]$, and $ \tau $ is set as 2, which results in $ \text{ATE} = -1 $. For MixedMediator, we set the same $ \gamma $ and $ \theta $ as MixedMediator, where parameters $ \phi = 1 $ and $ \tau = 1 $, which results in $ \text{ATE} = 1 $. In addition, we build a real-world dataset based on the job Ads data from the Company, aiming to estimate the ATE of switching a job from onsite to online working mode to the statistics of the applicants (here we choose the average age as the outcome). In the dataset, treatment $ T $ represents the working mode of the job, where $ T = 1 $ represents the job is online, whereas $ T = 0 $ represents the job is onsite, $ Y $ is the standardized age, and $ X \in \{0, 1\}^{K_X} $ indicates the required skills of the job. We select 3,228 jobs from Bay Area, where a primary study shows that $ \text{DEV}(\emptyset) \approx -2 $ years (i.e., online job applicants are two years younger than onsite job applicants). To simulate the latent confounder $ C $ and post-treatment variables $ M $, we first learn a generative model as follows: \[ Z \sim \text{Gaussian}(0, I_{K_Z}), \quad X \sim \text{Multi}(NN_f(Z)), \quad Y \sim \text{Gaussian}(w \odot Z, 1) \] where Multi represents multinomial distribution, $ NN_f $ is a neural network with softmax activation, $ Z, w \in \mathbb{R}^{K_Z}, K_Z = 6 $, and $ \odot $ represents the element-wise product operator, respectively. We then treat the first $ K_C = 3 $ dimensions of $ Z $ as the latent confounders $ C $ and the remaining $ K_M = K_Z - K_C $ dimensions as the latent mediators $ M $. After learning $ NN_f $ and $ w $ according to Eq. [12], we draw latent confounders $ C \in \text{Gaussian}(0, I) $, latent mediators $ M = T \cdot \gamma $, and set the outcome $ Y = w \odot [C\|M] + \tau \cdot T $, where the true ATE can be calculated as $ \text{sum}(\gamma \odot w_{-K_M}) + \tau $. #### 4.2 Comparisons with the State-of-the-Art The baselines we include for comparisons can be categorized into three classes. 1) Unawareness, where no information in $ X $ is used for ATE estimation. We implement the naive LR0 estimator, which regresses $ Y $ on $ T $ and uses the coefficient to estimate the ATE (Imbens & Rubin, 2015) (LR0 equals to $ \text{DEV}(\emptyset) $, i.e., the difference of average outcome between the treatment and non-treatment group). 2) Control-$ X $, which directly controls the covariates $ X $. In this class, LR1 regresses $ Y $ on $ T $ and $ X $, whereas TarNet uses a two-branch neural network to estimate the $ \text{DEV}(X) $. 3) Control-$ Z $, which controls latent variables $ Z $ learned from the covariates $ X $. Methods from this class include the CEVAE (Louizos et al., 2017) and covariate disentanglement methods (see Fig. 1(b)), such as DR-CFR (Hassanpour & Greiner, 2020) and TEDVAE (Zhang et al., 2021). The comparisons are summarized in Table 1. From Table 1 we can empirically verify the correctness of Theorem 2.2 that post-treatment bias indeed poses a serious issue for proxy-of-confounder-based methods, because for the MixedMediator and MixedConfounder datasets, CEVAE is worse than the naive LR0 estimator that directly calculates the difference of mean outcome between the --- 5 which leads to -0.178 after standardization. Code demo see https://anonymous.4open.science/r/CiVAE-demo-54B9 (a) Case 1: Intra-Confounder (b) Case 2: Intra-Mediator (c) Case 3: Confounder-Mediator Figure 2: Visualization of p-value of independence test before and after conditioning on treatment $T$. Table 1: Comparison of CiVAE with baselines on ATE estimation with latent post-treatment bias. \| Dataset \| MixedMediator \| MixedCorrelator \| Company \| \|---------\|---------------\|-----------------\|---------\| \| \| ATE. Err. \| ATE. Err. \| ATE. Err. \| \| LR0 \| 0.975 ± 0.032 \| 1.975 \| 2.977 ± 0.032 \| 1.977 \| 0.131 ± 0.015 \| 0.399 \| \| LR1 \| 1.457 ± 0.167 \| 2.457 \| 3.400 ± 0.130 \| 2.400 \| 0.093 ± 0.071 \| 0.361 \| \| TarNet \| 1.461 ± 0.172 \| 2.461 \| 3.414 ± 0.146 \| 2.414 \| 0.112 ± 0.085 \| 0.380 \| \| CEVAE \| 1.550 ± 0.292 \| 2.550 \| 3.323 ± 0.167 \| 2.323 \| 0.106 ± 0.078 \| 0.374 \| \| DR-CFR \| 1.239 ± 0.324 \| 2.239 \| 3.185 ± 0.319 \| 2.185 \| 0.094 ± 0.089 \| 0.362 \| \| TEDVAE \| 1.042 ± 0.315 \| 2.042 \| 3.138 ± 0.281 \| 2.138 \| 0.097 ± 0.093 \| 0.365 \| \| CiVAE \| -0.822 ± 0.753 \| 0.178 \| 1.199 ± 0.765 \| 0.199 \| -0.140 ± 0.137 \| -0.128 \| \| True ATE\| -1.000 ± 0.000 \| 0.000 \| 1.000 ± 0.000 \| 0.000 \| -0.268 ± 0.000 \| 0.000 \| treatment and non-treatment groups. In addition, for MixedMediator and Company datasets, all methods except the proposed CiVAE fail to predict the negativity of the ATE. Covariates disentanglement-based methods, i.e., DR-CFR and TEDVAE, achieve similar performance as CEVAE. The reason is that, these methods disentangle latent confounders $C$ from latent instrumental variables $I$ and latent adjusters $A$ by utilizing their causal relations with $T$ and $Y$, i.e., $I$ is predictive only for $T$, $A$ is predictive only for $Y$, whereas $C$ is predictive for both $T$ and $Y$. For example, TEDVAE includes three encoders to infer three sets of latent variables $\hat{I}$, $\hat{A}$, $\hat{C}$ from $X$ and adds classification losses $p(T\|\hat{I}, \hat{C})$ and $p(Y\|T, \hat{C}, \hat{A})$ on the CEVAE loss. However, when latent post-treatment bias exists, since both latent confounders $C$ and latent post-treatment variables $M$ are correlated with both $T$ and $Y$, $\hat{C}$ inferred by TEDVAE still cannot disentangle $C$ from $M$. CiVAE achieves significantly better results compared to CEVAE and TEDVAE, which demonstrates its effectiveness in identifying and distinguishing latent confounders from post-treatment variables in proxies. However, we also notice that a downside of CiVAE is the comparatively large variance across ten dataset splits, as misidentifying latent mediators as confounders may result in severe performance degradation when the mediation effects are strong or the number of latent confounders is small. 4.3 Disentangling of Latent Confounders and Post-treatment Variables We show the p-value of the pairwise independence test of the true latent variables before and after conditioning on the assigned treatment $T$. From Fig. 2, we can find that the difference between the three cases discussed in Subsection 3.4 is significant. Here, we should note that the distinction of the intra-confounder case from other cases relies on the assumption that latent confounders are independent. If the latent confounders are correlated, we can first use causal discovery techniques such as the PC algorithm [Spirites et al., 2000] to find direct parents of $T$, and use our algorithm as the refinement to determine the true confounders $C$ from the misidentified post-treatment variables. 5 Conclusions In this paper, we systematically investigated the latent post-treatment bias in causal inference from observational data. We first prove that unresolved latent post-treatment variables scrambled in the proxy of confounders can arbitrarily bias the ATE estimation. To address the bias, we proposed the Confounder-identifiable VAE (CiVAE), which, utilizing a mild assumption regarding the prior of latent factors, guarantees the identifiability of latent confounders up to bijective transformations. Finally, we show that controlling the latent confounders inferred by CiVAE can provide an unbiased estimation of the ATE. Experiments on both simulated and real-world datasets demonstrated that CiVAE has superior robustness to latent post-treatment bias compared with state-of-the-art methods. REFERENCES Avidit Acharya, Matthew Blackwell, and Maya Sen. Explaining causal findings without bias: Detecting and assessing direct effects. American Political Science Review, 110(3):512–529, 2016. Animashree Anandkumar, Rong Ge, Daniel Hsu, Sham M Kakade, and Matus Telgarsky. Tensor decompositions for learning latent variable models. Journal of Machine Learning Research, 15: 2773–2832, 2014. David M Blei, Alp Kucukelbir, and Jon D McAuliffe. Variational inference: A review for statisticians. Journal of the American Statistical Association, 112(518):859–877, 2017. Lu Cheng, Ruocheng Guo, and Huan Liu. Causal mediation analysis with hidden confounders. In Proceedings of the Fifteenth ACM International Conference on Web Search and Data Mining, pp. 113–122, 2022. 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EhmEwfavOW	I'm concerned about the datasets used in the node classification task. Results in Table 1 show that the performance of MagNet is far way from FaberNet. I guess that this is because MagNet operates as a low-pass filter and therefore cannot perform well in the heterophilic datasets.	HoloNets: Spectral Convolutions Do Extend to Directed Graphs Christian Koke & Daniel Cremers Technical University Munich and Munich Center for Machine Learning {christian.koke,cremers}@tum.de Abstract Within the graph learning community, conventional wisdom dictates that spectral convolutional networks may only be deployed on undirected graphs: Only there could the existence of a well-defined graph Fourier transform be guaranteed, so that information may be translated between spatial- and spectral domains. Here we show this traditional reliance on the graph Fourier transform to be superfluous and – making use of certain advanced tools from complex analysis and spectral theory – extend spectral convolutions to directed graphs. We provide a frequency-response interpretation of newly developed filters, investigate the influence of the basis used to express filters and discuss the interplay with characteristic operators on which networks are based. In order to thoroughly test the developed theory, we conduct experiments in real world settings, showcasing that directed spectral convolutional networks provide new state of the art results for heterophilic node classification on many datasets and – as opposed to baselines – may be rendered stable to resolution-scale varying topological perturbations. Our code is available at https://github.com/ChristianKoke/HoloNets. 1 Introduction A particularly prominent line of research for graph neural networks is that of spectral convolutional architectures. These are among the theoretically best understood graph learning methods (Levie et al., 2019a; Ruiz et al., 2021a; Koke, 2023) and continue to set the state of the art on a diverse selection of tasks (Bianchi et al., 2019; He et al., 2021; 2022a; Wang & Zhang, 2022b). Furthermore, spectral interpretations allow to better analyse expressivity (Balciar et al., 2021), shed light on shortcomings of established models (NT & Maehara, 2019) and guide the design of novel methods (Bo et al., 2023). Traditionally, spectral convolutional filters are defined making use of the notion of a graph Fourier transform: Fixing a self-adjoint operator on an undirected $N$-node graph – traditionally a suitably normalized graph Laplacian $L = U^\top \Lambda U$ with eigenvalues $\Lambda = \text{diag}(\lambda_1, ..., \lambda_N)$ – a notion of Fourier transform is defined by projecting a given signal $x$ onto the eigenvectors of $L$ via $x \mapsto U x$. Since $L$ is self-adjoint, the eigenvectors form a complete basis and no information is lost in the process. In analogy with the Euclidean convolution theorem, early spectral networks then defined convolution as multiplication in the "graph-Fourier domain" via $x \mapsto U^\top \cdot \text{diag}(\theta_1, ..., \theta_N) \cdot U x$, with learnable parameters $\{\theta_1, ..., \theta_N\}$ (Bruna et al., 2014). To avoid calculating an expensive explicit eigendecomposition $U$, Defferrard et al. (2016) proposed to instead parametrize graph convolutions via $x \mapsto U^\top g_\theta(\Lambda) U x$, with $g_\theta$ a learnable scalar function applied to the eigenvalues $\Lambda$ as $g_\theta(\Lambda) = \text{diag}(g_\theta(\lambda_1), ..., g_\theta(\lambda_N))$. This precisely reproduces the mathematical definition of applying a scalar function $g_\theta$ to a self-adjoint operator $L$, so that choosing $g_\theta$ to be a (learnable) polynomial allowed to implement filters computationally much more economically as $g_\theta(L) = \sum_{k=1}^{K} \theta_k L^k$. Follow up works then considered the influence of the basis in which filters $\{g_\theta\}$ are learned (He et al., 2021; Levie et al., 2019b; Wang & Zhang, 2022a) and established that such filters provide networks with the ability to generalize to unseen graphs (Levie et al., 2019a; Ruiz et al., 2021b; Koke, 2023). Common among all these works, is the need for the underlying graph to be undirected: Only then are the corresponding operators symmetric, so that a complete set of orthogonal eigenvectors exists and the graph Fourier transform $U$ (used to define the filter $g_\theta(L)$ via $x \mapsto U g_\theta(\Lambda) U^\top x$) is well-defined.\footnote{Strictly speaking it is not symmetry ($L = L^\top$) but normality ($LL^\top = L^\top L$) of $L$ that ensures this.} Currently however, the graph learning community is endeavouring to finally also account for the previously neglected directionality of edges, when designing new methods (Zhang et al., 2021; Rossi et al., 2023; Beami et al., 2021; Geisler et al., 2023; He et al., 2022b). Since characteristic operators on digraphs are generically not self-adjoint, traditional spectral approaches so far remained inaccessible in this undertaking. Instead, works such as Zhang et al. (2021); He et al. (2022b) resorted to limiting themselves to certain specialized operators able to preserve self-adjointness in this directed setting. While this approach is not without merit, the traditional adherence to the graph Fourier transform remains a severely limiting factor when attempting to extend spectral networks to directed graphs. Contributions: In this paper we argue to completely dispose with this reliance on the graph Fourier transform and instead take the concept of learnable functions applied to characteristic operators as fundamental. This conceptual shift allows us to consistently define spectral convolutional filters on directed graphs. We provide a corresponding frequency perspective, analyze the interplay with chosen characteristic operators and discuss the importance of the basis used to express these novel filters. The developed theory is thoroughly tested on real world data: It is found that directed spectral convolutional networks provide new state of the art results for heterophilic node classification and – as opposed to baselines – may be rendered stable to resolution-scale varying topological perturbations. 2 SIGNAL PROCESSING ON DIRECTED GRAPHS Our work is mathematically rooted in the field of graph signal processing; briefly reviewed here: Weighted directed graphs: A directed graph $ G := (G, E) $ is a collection of nodes $ G $ and edges $ E \subseteq G \times G $ for which $(i, j) \in E$ does not necessarily imply $(j, i) \in E$. We allow nodes $ i \in G $ to have individual node-weights $ \mu_i > 0 $ and generically assume edge-weights $ w_{ij} \geq 0 $ not necessarily equal to unity or zero. In a social network, a node weight $ \mu_i = 1 $ might signify that a node represents a single user, while a weight $ \mu_j > 1 $ would indicate that node $ j $ represents a group of users. Similarly, edge weights $ \{w_{ij}\} $ could be used to encode how many messages have been exchanged between nodes $ i $ and $ j $. Importantly, since we consider directed graphs, we generically have $ w_{ij} \neq w_{ji} $. Edge weights also determine the so called reaches of a graph, which generalize the concept of connected components of undirected graphs (Veerman & Lyons, 2020): A subgraph $ R \subseteq G $ is called reach, if for any two vertices $ a, b \in R $ there is a directed path in $ R $ along which the (directed) edge weights do not vanish, and $ R $ simultaneously possesses no outgoing connections (i.e. for any $ c \in G $ with $ c \notin R $: $ w_{ca} = 0 $). For us, this concept will be important in generalizing the notion of scale insensitive networks (Koke et al., 2023) to directed graphs in Section 3.3 below. Feature spaces: Given $ F $-dimensional node features on a graph with $ N = \|G\| $ nodes, we may collect individual node-feature vectors into a feature matrix $ X $ of dimension $ N \times F $. Taking into account our node weights, we equip the space of such signals with an inner-product according to $ \langle X, Y \rangle = \text{Tr}(X^* M Y) = \sum_{i=1}^{N} \sum_{j=1}^{F} \overline{X}_{ij} Y_{ij} \mu_i $ with $ M = \text{diag}(\{\mu_i\}) $ the diagonal matrix of node-weights. Here $ X^* $ denotes the (hermitian) adjoint of $ X $ (c.f. Appendix B for a brief recapitulation). Associated to this inner product is the standard 2-norm $ \\|X\\|_2^2 = \sum_{i=1}^{N} \sum_{j=1}^{F} \|X_{ij}\|^2 \mu_i $. Characteristic Operators: Information about the geometry of a graph is encapsulated into the set of edge weights, collected into the weight matrix $ W $. From this, the diagonal in-degree and out-degree matrices $ D_{ii}^{\text{in}} = \sum_j W_{ij}, D_{jj}^{\text{out}} = \sum_i W_{ij} $ may be derived. Together with the node-weight matrix $ M $ defined above, various characteristic operators capturing the underlying geometry of the graph may then be constructed. Relevant to us – apart from the weight matrix $ W $ – will especially be the (in-degree) Laplacian $ L^{\text{in}} := M^{-1}(D^{\text{in}} - W) $, which is intimately related to consensus and diffusion on directed graphs (Veerman & Kummel, 2019). Importantly, such characteristic operators $ T $ are generically not self-adjoint. Hence they do not admit a complete set of orthogonal eigenvectors and their spectrum $ \sigma(T) $ contains complex eigenvalues $ \lambda \in \mathbb{C} $. Appendix B contains additional details on such operators, their canonical (Jordan) decomposition and associated generalized eigenvectors. 3 Spectral Convolutions on Directed Graphs Since characteristic operators on directed graphs generically do not admit a complete set of orthogonal eigenvectors, we cannot make use of the notion of a graph Fourier transform to consistently define filters of the form $ g_\theta(T) $. While this might initially seem to constitute an insurmountable obstacle, the task of defining operators of the form $ g(T) $ for a given operator $ T $ and appropriate classes of scalar-valued functions $ \{g\} $ – such that relations between the functions $ \{g\} $ translate into according relation of the operators $ \{g(T)\} $ – is in fact a well studied problem [Haase, 2006; Colombo et al., 2011]. Corresponding techniques typically bear the name "functional calculus" and importantly are also definable if the underlying operator $ T $ is not self-adjoint [Cowling et al., 1996]. 3.1 The Holomorphic Functional Calculus In the undirected setting, it was possible to essentially apply arbitrary functions $ \{g\} $ to the characteristic operator $ T = U^\top \Lambda U $ by making use of the complete eigendecomposition as $ g(T) := U^\top g_\theta(\Lambda) U $. However, a different approach to consistently defining the matrix $ g(T) $ – not contingent on such a decomposition – is available if (and only if) one restricts $ g $ to be a holomorphic function: For a given subset $ U $ of the complex plane, these are the complex valued functions $ g : U \to \mathbb{C} $ for which the complex derivative $ dg(z)/dz $ exists everywhere on the domain $ U $ (c.f. Appendix D for more details). The property of holomorphic functions that allows to consistently define the matrix $ g(T) $ is the fact that any function value $ g(\lambda) $ can be reproduced by calculating an integral of the function $ g $ along a path $ \Gamma $ encircling $ \lambda $ (c.f. also Fig. 2) as \[ g(\lambda) = -\frac{1}{2\pi i} \oint_{\Gamma} g(z) \cdot (\lambda - z)^{-1} dz. \tag{1} \] Here "$dz"$ denotes the complex line-integration-measure in $ \mathbb{C} $. In order to define the matrix $ g(T) $, the formal replacement $ \lambda \mapsto T $ is then made on both sides of (1), with the path $ \Gamma $ now not only encircling a single value $ \lambda $ but all eigenvalues $ \lambda \in \sigma(T) $ (c.f. also Fig. 3): \[ g(T) := -\frac{1}{2\pi i} \oint_{\Gamma} g(z) \cdot (T - z \cdot Id)^{-1} dz \tag{2} \] Note that $ (T - z \cdot Id)^{-1} $ – and hence the integral in (2) – is indeed well-defined: All eigenvalues of $ T $ are assumed to lie inside the path $ \Gamma $. For any choice of integration variable $ z $ on this path $ \Gamma $, the matrix $ (T - z \cdot Id) $ is thus indeed invertible, since $ z $ is never an eigenvalue. The integral in (2) defines what is called the holomorphic functional calculus [Gindler, 1966; Kato, 1976]. Importantly (c.f. Appendix E), the definition of $ g(T) $ in (2) agrees with algebraic relations: Theorem 3.1. Applying a polynomial $ g(\lambda) := \sum_{k=0}^{K} a_k \lambda^k $ to $ T $ yields $ g(T) = \sum_{k=0}^{K} a_k T^k $. Similarly applying the function $ g(\lambda) = 1/\lambda $ yields $ g(T) = T^{-1} $ provided $ T $ is invertible. 3.2 Spectral Convolutional Filters on Directed Graphs Since the holomorphic functional calculus is evidently no longer contingent on $ T $ being self-adjoint, it indeed provides an avenue to consistently define spectral convolutional filters on directed graphs. Parametrized Spectral Convolutional Filters: In practice it is of course prohibitively expensive to continuously compute the integral (2) as the learnable function $ g $ is updated during training. Instead, we propose to represent a generic holomorphic function $ g $ via a set of basis functions $ \{\Psi_i\}_{i \in I} $ as $ g_\theta(z) := \sum_{i \in I} \theta_i \cdot \Psi_i(z) $ with learnable coefficients $ \{\theta_i\}_{i \in I} $ parametrizing the filter $ g_\theta $. For the 'simpler' basis functions $ \{\Psi_i\}_{i \in I} $, we either precompute the integral (2), or perform it analytically (c.f. Section 3.3 below). During training and inference the matrices $ \Psi_i(T) := -\frac{1}{2\pi i} \oint_{\Gamma} \Psi_i(z) \cdot (T - z \cdot Id)^{-1} dz $ are then already computed and learnable filters are given as \[ g_\theta(T) := \sum_{i \in I} \theta_i \cdot \Psi_i(T). \tag{3} \] Generically, each coefficient $ \theta_i $ may be chosen as a complex number; equivalent to two real parameters. If the functions $ \{ \Psi_i \}_{i \in I} $ are chosen such that each matrix $ \Psi_i(T) $ contains only real entries (e.g., for $ \Psi $ a polynomial with real coefficients), it is possible to restrict convolutional filters to being purely real: In this setting, choosing the parameters $ \{ \theta_i \} $ to be purely real as well, leads to $ g_\theta(T) = \sum_{i \in I} \theta_i \cdot \Psi_i(T) $ itself being a matrix that contains only real entries. In this way, complex numbers need never to appear within our network, if this is not desired. In Theorem 4.1 of Section 4 below, we discuss how, under mild and reasonable assumptions, such a complexity-reduction to using only real parameters can be performed without decreasing the expressive power of corresponding networks. Irrespective of whether real or complex weights are employed, the utilized filter bank $ \{ \Psi_i \}_{i \in I} $ determines the space of learnable functions $ g_\theta \in \text{span}(\{ \Psi_i \}_{i \in I}) $ and thus contributes significantly to the inductive bias present in the network. It should thus be adjusted to the respective task at hand. The Action of Filters in the Spectral Domain: In order to determine which basis functions are adapted to which tasks, a "frequency-response" interpretation of spectral filters is expedient: In the undirected setting this proceeded by decomposing any characteristic operator $ T $ into a sum $ T = \sum_{\lambda \in \sigma(T)} \lambda \cdot P_\lambda $ over its distinct eigenvalues. The spectral action of any function $ g $ was then given by $ g(T) = \sum_{\lambda \in \sigma(T)} g(\lambda) \cdot P_\lambda $. Here the spectral projections $ P_\lambda $ project each vector to the space spanned by the eigenvectors $ \{ v_i \} $ corresponding to the eigenvalue $ \lambda $ (i.e., satisfying $ (T - \lambda \cdot \text{Id})v_i = 0 $). In the directed setting, there only exists a basis of generalized eigenvectors $ \{ w_i \}_{i=1}^{N-1} $; each satisfying $ (T - \lambda \cdot \text{Id})^m w_i = 0 $ for some $ \lambda \in \sigma(T) $ and $ m \in \mathbb{N} $ (c.f. Appendix B). Denoting by $ P_\lambda $ the matrix projecting onto the space spanned by these generalized eigenvectors associated to the eigenvalue $ \lambda \in \sigma(T) $, any operator $ T $ may be written as $ T = \sum_{\lambda \in \sigma(T)} \lambda \cdot P_\lambda + \sum_{\lambda \in \sigma(T)} (T - \lambda \cdot \text{Id}) \cdot P_\lambda $. It can then be shown (Kato [1976]), that the spectral action of a given function $ g $ is given as \[ g(T) = \sum_{\lambda \in \sigma(T)} g(\lambda) P_\lambda + \sum_{\lambda \in \sigma(T)} \left[ \sum_{n=1}^{m_\lambda} \frac{g^{(n)}(\lambda)}{n!} (T - \lambda \cdot \text{Id})^n \right] P_\lambda. \tag{4} \] Here the number $ m_\lambda $ is the algebraic multiplicity of the eigenvalue $ \lambda $; i.e., the dimension of the associated generalized eigenspace. The notation $ g^{(n)} $ denotes the $ n $-th complex derivative of $ g $. The appearance of such derivative terms in (4) is again evidence, that we indeed needed to restrict from generic- to differentiable functions, in order to sensibly define directed spectral convolutional filters. It is instructive to gain some intuition about the second sum on the right-hand-side of the frequency response (4), as it is not familiar from undirected graphs (since it vanishes if $ T $ is self-adjoint): As an example consider the un-weighted directed path graph on three nodes depicted in Fig. 4 and choose as characteristic operator $ T $ the adjacency matrix (i.e., $ T = W $). It is not hard to see (c.f. Appendix C for an explicit calculation) that the only eigenvalue of $ W $ is given by $ \lambda = 0 $ with algebraic multiplicity $ m_\lambda = 3 $. Since spectral projections always satisfy $ \sum_{\lambda \in \sigma(T)} P_\lambda = \text{Id} $ (c.f. Appendix B), and here $ \sigma(W) = \{ 0 \} $ we thus have $ P_{\lambda=0} = \text{Id} $ in this case. Suppose now we are tasked with finding a (non-trivial) holomorphic filter $ g(\lambda) $ such that $ g(T) = 0 $. The right-hand sum in (4) implies, that beyond $ g(0) = 0 $, also the first and second derivative of $ g(\lambda) $ needs to vanish at $ \lambda = 0 $ to achieve this. Hence the zero of $ g(\lambda) $ at $ \lambda = 0 $ must be at least of order three; or equivalently for $ \lambda \to 0 $ we need $ g(\lambda) = o(\lambda^3) $. This behaviour is of course exactly mirrored in the spatial domain: As applying $ W $ simply moves information at a given node along the path, applying $ W $ once or twice still leaves information present. After two applications, only node 3 still contains information and thus applying $ W^k $ precisely removes all information if and only if $ k \geq 3 $. Without the assumption of acyclicity, the spectrum of characteristic operators of course generically does not consist only of the eigenvalue $ \lambda = 0 $. Thus generically $ P_{\lambda=0} \neq \text{Id} $ and the role played by the operator $ T = W $ in the considerations above is instead played by its restriction $ (T \cdot P_{\lambda=0}) $ to the generalized eigenspace corresponding to the eigenvalue $ \lambda = 0 $. For us, the spectral response (4) provides guidance when considering scale-insensitive convolutional filters on directed graphs in Sections 3.3 and 4 below. The spectral response (4) is however never used to implement filters: As discussed above, this is achieved much more economically via (3). --- 2 Additional details on this so called Jordan Chevalley decomposition are provided in Appendix B. 3 N.B.: A once-complex-differentiable function is automatically infinitely often differentiable (Ahlfors [1966]). 3.3 Explicit Filter Banks Having laid the theoretical foundations, we consider examples of task-adapted filter banks $ \{ \Psi_i \}_{i \in I} $. 3.3.1 Bounded Spectral Domain: Faber Polynomials First, let us consider spectral networks on a single graph with a fixed characteristic operator $ T $. From the holomorphic functional calculus [2], we infer that convolutional filters $ \{ g(T) \} $ are in principle provided by all holomorphic functions $ \{ g \} $ defined on a domain $ U $ which contains all eigenvalues $ \lambda \in \sigma(T) $ of $ T $. As noted above, implementing an arbitrary holomorphic $ g $ is however too costly, and we instead approximate $ g $ via a collection of simpler basis functions $ \{ \Psi_i \}_{i \in I} $ as $ g(\lambda) \approx \sum_{i \in I} \theta_i \Psi_i(\lambda) $. In order to choose the filter bank $ \{ \Psi_i \}_{i \in I} $, we thus need to answer the question of how to optimally approximate arbitrary holomorphic functions on a given fixed domain $ U $. The solution to this problem is given in the guise of Faber polynomials [Ellacott 1983; Coleman & Smith 1987] which generalize the familiar Chebyshev polynomials utilized in [Defferrard et al. 2016] to subsets $ U $ of the complex plane [Elliott 1978]. Faber polynomials provide near mini-max polynomial approximation to any holomorphic function defined on a domain $ U $ satisfying some minimal conditions (c.f. Elliott [1978] for exact details). What is more, they have already successfully been employed in numerically approximating matrices of the form $ g(T) $ for $ T $ not necessarily symmetric [Moret & Novati 2001]. While for a generic domain $ U $ Faber polynomials are impossible to compute analytically, this poses no limitations to us in practice: Short of a costly explicit calculation of the spectrum $ \sigma(T) $, the only information that is generically available, is that eigenvalues may be located anywhere within a circle of radius $ \\|T\\| $. This circle must thus be contained in any valid domain $ U $. Making the minimal choice by taking $ U $ to be exactly this circle, the $ n $-th Faber polynomial may be analytically calculated [He 1995]: Up to normalization (absorbed into the learnable parameters) it is given by the monomial $ \lambda^n $. We thus take our $ n $-th basis element $ \Psi_n(\lambda) $ to be given precisely by this monomial: $ \Psi_n(\lambda) = \lambda^n $. Thus Faber polynomials evaluate to $ \Psi_k(T) = T^k $ on our characteristic operator $ T $ (c.f. Thoerem 3.1). In a setting where more detailed information on $ \sigma(T) $ is available, the domain $ U $ may of course be adapted to reflect this. Corresponding Faber polynomials might then be pre-computed numerically. 3.3.2 Unbounded Spectral Domain: Functions Decaying at Complex Infinity In the multi-graph setting – e.g. during graph classification – we are confronted with the possibility that distinct graphs may describe the same underlying object [Levie et al. 2019a; Maskey et al. 2021; Koke 2023]. This might for example occur if two distinct graphs discretize the same underlying continuous space; e.g. at different resolution scales. In this setting – instead of precise placements of nodes – what is actually important is the overall structure and geometry of the respective graphs. Un-normalized Laplacians provide a convenient multi-scale descriptions of such graphs, as they encode information corresponding to coarse geometry into small (in modulus) eigenvalues, while finer graph structures correspond to larger eigenvalues [Chung 1997; Ng et al. 2001]. When designing networks whose outputs are not overly sensitive to fine print articulations of graphs, the spectral response [4] then provides guidance on determining which holomorphic filters $ g $ are able to suppress this superfluous high-lying spectral information: It is sufficient that $ g^{(n)}(\lambda)/n! \approx 0 $ for $ \|\lambda\| \gg 1 $. It can be shown that no holomorphic function with such large-$\|\lambda\|$ asymptotics defined on all of $ C $ exists.$^5$ We thus make the minimal necessary change and assume $ g $ to be defined on a punctured domain $ U = C \setminus \{ y \} $ instead. The choice of $ y \in C $ is treated as a hyperparameter, which may be adjusted to the task at hand. Any such $ g $ may then be expanded as $ g(\lambda) = \sum_{j=1}^{\infty} \theta_j (\lambda - y)^{-j} $ for some coefficients $ \{ \theta_j \}_{j=1}^{\infty} $ [Bak & Newman 2017]. Evaluating the defining integral [2] for the Laplacian $ L^{in} $ on the atoms $ \Psi_j(\lambda) = (\lambda - y)^{-j} $ yields $ \Psi_j(L^{in}) = ([L^{in} - y \cdot Id])^{-1}y $; as proved in Appendix E. Hence corresponding filters are polynomials in the resolvent $ R_y(L^{in}) := [L^{in} - y \cdot Id]^{-1} $ of $ L^{in} $. Such resolvents are traditionally used as tools to compare operators with potentially divergent norms [Teschl 2014]. Recently [Koke et al. 2023] utilized them in the undirected setting to construct networks provably assigning similar feature-vectors to weighted graphs describing the same underlying object at different resolution-scales. Our approach extends these networks to the directed setting: $^4$I.e. minimizing the maximal approximation error on the domain of definition $ U $. $^5$This is an immediate consequence of Liouville’s theorem in complex analysis [Ahlfors 1966]. Effective directed Limit Graphs: From a diffusion perspective, information in a graph equalizes faster along edges with large weights. In the limit where the edge-weights within certain sub-graphs tend to infinity, information within these clusters equalizes immediately and such sub-graphs should thus effectively behave as single nodes. Extending undirected-graph results (Koke & Kutyniok, 2022; Koke et al., 2023), we here establish rigorously that this is indeed also true in the directed setting. Mathematically, we make our arguments precise by considering a graph $G$ with a weight matrix $W$ admitting a (disjoint) two-scale decomposition as $W = W_{\text{regular}} + c \cdot W_{\text{high}}$ (c.f. Fig. 5). As the larger weight scale $c \gg 1$ tends to infinity, we then establish that the resolvent $R_y(L^{\text{in}})$ on $G$ converges to the resolvent $R_y(L^{\text{in}})$ of the Laplacian $L^{\text{in}}$ on a coarse-grained limit graph $\mathcal{G}$. This limit $\mathcal{G}$ arises by collapsing the reaches $R$ of the graph $G_{\text{high}} = (G, W_{\text{high}})$ (c.f. Fig. 5(c)) into single nodes. For technical reasons, we here assume equal in- and out-degrees within $G_{\text{high}}$ (i.e. $\sum_i W_{ij}^{\text{high}} = \sum_i W_{ji}^{\text{high}}$). Appendix G contains proofs corresponding to the results below. ![Figure 5](image) When defining $\mathcal{G}$, directed reaches now replace the undirected components of Koke et al. (2023): Definition 3.2. The node set $\mathcal{G}$ of $\mathcal{G}$ is constituted by the set of all reaches in $G_{\text{high}}$. Edges $\mathcal{E}$ of $\mathcal{G}$ are given by those elements $(R, P) \in \mathcal{G} \times \mathcal{G}$ with non-zero agglomerated edge weight $W_{RP} = \sum_{r \in R} \sum_{p \in P} W_{rp}$. Node weights in $\mathcal{G}$ are defined similarly by aggregating as $\mu_R = \sum_{r \in R} \mu_r$. To map signals between these graphs, translation operators $J^\downarrow$, $J^\uparrow$ are needed. Let $x$ be a scalar graph signal and let $1_R$ be the vector that has 1 as entry for nodes $r \in R$ and zero otherwise. Denote by $u_R$ the entry of $u$ at node $R \in \mathcal{G}$. The projection operator $J^\downarrow$ is then defined component-wise by evaluation at node $R \in \mathcal{G}$ as $(J^\downarrow x)_R = \langle 1_R; x \rangle / \mu_R$. Interpolation is defined as $J^\uparrow u = \sum_{R \in \mathcal{G}} u_R \cdot 1_R$. The maps $J^\downarrow$, $J^\uparrow$ are then extended from single features $\{x\}$ to feature matrices $\{X\}$ via linearity. With these preparations, we can now rigorously establish the suspected effective behaviour of clusters: Theorem 3.3. In the above setting, we have $\\|R_y(L^{\text{in}}) \cdot X - J^\uparrow R_y(L^{\text{in}}) J^\downarrow \cdot X \\| \longrightarrow 0$ as $c \rightarrow \infty$. For $c \gg 1$, applying the resolvent $R_y(L^{\text{in}})$ on $G$ is thus essentially the same as first projecting to the coarse-grained graph $\mathcal{G}$ (where all strongly connected clusters are collapsed), applying the corresponding resolvent there and then interpolating back up to $G$. The geometric information within $R_y(L^{\text{in}})$ is thus essentially reduced to that of the coarse grained geometry within $\mathcal{G}$. Large weights within a graph typically correspond to fine-structure articulations of its geometry: For graph-discretisations of continuous spaces, edge weights e.g. typically correspond to inverse discretization lengths ($w_{ij} \sim 1/d_{ij}$) and strongly connected clusters describe closely co-located nodes. In social-networks, edge weights might encode a closeness-measure, and coarse-graining would correspond to considering interactions between (tightly connected) communities as opposed to individual users. In either case, fine print articulations are discarded when applying resolvents. Stability of Filters: This reduction to a limit description on $\mathcal{G}$ is respected by our filters $\{g_\theta\}$: Theorem 3.4. In the above setting, we have $\\|g_\theta(L^{\text{in}}) \cdot X - J^\uparrow g_\theta(L^{\text{in}}) J^\downarrow \cdot X \\| \longrightarrow 0$ as $c \rightarrow \infty$. If the weight-scale $c$ is very large, applying the learned filter $g_\theta(\lambda) = \sum_{i=1}^K \theta_i (\lambda - y)^{-i}$ to a signal $X$ on $G$ as $X \mapsto g_\theta(L^{\text{in}}) \cdot X$ thus is essentially the same as first discarding fine-structure information by projecting $X$ to $\mathcal{G}$, applying the spectral filter $g_\theta$ there and subsequently interpolating back to $G$. Information about the precise articulation of a given graph $G$ is thus suppressed in this propagation scheme; it is purely determined by the graph structure of the coarse-grained description $\mathcal{G}$. Theorem 4.2 below establishes that this behaviour persists for entire (directed) spectral convolutional networks. This is known as Kirchhoff’s assumption (Balti, 2018): reproducing the eponymous law of electrical circuits. 4 SPECTRAL NETWORKS ON DIRECTED GRAPHS: HOLONETS We now collect holomorphic filters into corresponding networks; termed HoloNets. In doing so, we need to account for the possibility that given edge directionalities might limit the information-flow facilitated by filters $ \{ g_\theta(T) \} $: In the path-graph setting of Fig. 4, for example, a polynomial filter in the adjacency matrix would only transport information along the graph; features of earlier nodes would never be augmented with information about later nodes. To circumvent this, we allow for two sets of filters $ \{ g_\theta^{\text{fwd}}(T) \} $ and $ \{ g_\theta^{\text{bwd}}(T^) \} $ based on the characteristic operator $ T $ and its adjoint $ T^ $. Allowing these forward- and backward-filters to be learned in different bases $ \{ \Psi_i^{\text{fwd/bwd}} \}_{i \in I^{\text{fwd/bwd}}} $, we may write \[ g_\theta^{\text{fwd/bwd}}(\lambda) = \sum_{i \in I^{\text{fwd/bwd}}} \theta_i^{\text{fwd/bwd}} \Psi_i^{\text{fwd/bwd}}(\lambda). \] With bias matrices $ B^{\ell+1} $ of size $ N \times F_{\ell+1} $ and weight matrices $ W_k^{\text{fwd/bwd}, \ell+1} $ of dimension $ F_\ell \times F_{\ell+1} $, our update rule is then efficiently implemented as \[ X^\ell = \rho \left( \alpha \sum_{i \in I^{\text{fwd}}} \Psi_i^{\text{fwd}}(T) \cdot X^{\ell-1} \cdot W_i^{\text{fwd}, \ell} + (1 - \alpha) \sum_{i \in I^{\text{bwd}}} \Psi_i^{\text{bwd}}(T^) \cdot X^{\ell-1} \cdot W_i^{\text{bwd}, \ell} + B^\ell \right). \] Here $ \rho $ is a point-wise non-linearity, and the parameter $ \alpha \in [0, 1] $ – learnable or tunable – is introduced following Rossi et al. (2023) to allow for a prejudiced weighting of the forward or backward direction. We additionally provide a pseudocode description of the corresponding models in Appendix H. The generically complex weights & biases may often be restricted to $ \mathbb{R} $ without losing expressivity: Theorem 4.1.* Suppose for filter banks $ \{ \Psi_i^{\text{fwd/fwd}} \}_{i \in I^{\text{fwd}}} $ that the matrices $ \Psi_i^{\text{fwd}}(T), \Psi_i^{\text{bwd}}(T^) $ contain only real entries. Then any HoloNet with layer-widths $ \{ F_\ell \} $ with complex weights & biases and a non-linearity that acts on complex numbers componentwise as $ \rho(a + ib) = \tilde{\rho}(a) + i\tilde{\rho}(b) $ can be exactly represented by a HoloNet of widths $ \{ 2 \cdot F_\ell \} $ utilizing $ \tilde{\rho} $ and employing only real weights & biases. This result (proved in Appendix H) establishes that for the same number of real parameters, real HoloNets theoretically have greater expressive power than complex ones. In our experiments in Section 5 below, we empirically find that complex weights do provide advantages on some graphs. Thus we propose to treat the choice of complex vs. real parameters as a binary hyperparameter. FaberNet:* The first specific instantiation of HoloNets we consider, employs the Faber Polynomials of Section 3.3.1 for both the forward and backward filter banks. Since Rossi et al. (2023) established that considering edge directionality is especially beneficial on heterophilic graphs, this is also our envisioned target for the corresponding networks. We thus use as characteristic operator a matrix that avoids direct comparison of feature vectors of a node with those of immediate neighbours: We choose \[ T = (D^{\text{in}})^{-\frac{1}{2}} \cdot W \cdot (D^{\text{out}})^{-\frac{1}{2}} \] since it has a zero-diagonal and its normalization performed well empirically. For the same reason of heterophily, we also consider the choice of whether to include the Faber polynomial $ \Psi_0(\lambda) = 1 $ in our basis as a hyperparameter. As non-linearity, we choose either $ \rho(a + ib) = \text{ReLU}(a) + i\text{ReLU}(b) $ or $ \rho(a + ib) = \|a\| + i\|b\| $. Appendix I contains additional details. Dir-ResolvNet: In order to build networks that are insensitive to the fine-print articulation of directed graphs, we take as filter bank the functions $ \{ \Psi_i(\lambda) = (\lambda - y)^{-1} \}_{i > 0} $ evaluated on the Laplacian $ L^m $ for both the forward and backward direction. To account for individual node-weights when building up graph-level features, we use an aggregation $ \Omega $ that aggregates $ N \times F $-dimensional node-feature matrices as $ \Omega(X)_j = \sum_{i=1}^{N} \|X_{ij}\| \cdot \mu_i $ to a graph-feature $ \Omega(X) \in \mathbb{R}^F $. Graph-level stability under varying resolution scales is then captured by our next result: Theorem 4.2. Let $ \Phi $ and $ \Phi $ be the feature maps associated to Dir-ResolvNets with the same weights and biases deployed on graphs $ G $ and $ G $ as defined in Section 3.3.2. With $ \Omega $ the aggregation method specified above and $ W = W^{\text{regular}} + c \cdot W^{\text{high}} $ as in Theorem 3.4, we have for $ c \to \infty $: \[ \\|\Omega(\Phi(X)) - \Omega(\Phi(J^\dagger X))\\| \to 0 \] Appendix C contains proofs of this and additional stability results. From Theorem 4.2 we conclude that graph-level features generated by a Dir-ResolvNet are indeed insensitive to fine print articulations of weighted digraphs: As discussed in Section 3.3.2, geometric information corresponding to such fine details is typically encoded into strongly connected sub-graphs; with the connection strength $ c $ corresponding to the level of detail. However, information about the structure of these sub-graphs is precisely what is discarded when moving to $ G $ via $ J^\dagger $. Thus the greater the level of detail within $ G $, the more similar are generated feature-vectors to those of a (relatively) coarse-grained description $ G $. 5 EXPERIMENTS We present experiments on real-world data to evaluate the capabilities of our HoloNets numerically. 5.1 FaberNet: Node Classification We first evaluate on the task of node-classification in the presence of heterophily. We consider multiple heterophilic graph-datasets on which we compare the performance of our FaberNet instantiation of the HoloNet framework against a representative array of baselines: As simple baselines we consider MLP and GCN (Kipf & Welling [2017]), H₂GCN (Zhu et al. [2020]), GPR-GNN (Chien et al. [2021]), LINKX (Lim et al. [2021]), FSGNN (Maurya et al. [2021]), ACM-GCN (Luan et al. [2022]), GloGNN (Li et al. [2022]) and Gradient Gating (Rusch et al. [2023]) constitute heterophilic state-of-the-art models. Finally state-of-the-art models for directed graphs are given by DiGCN (Tong et al. [2020]), MagNet (Zhang et al. [2021]) and Dir-GNN (Rossi et al. [2023]). Appendix I contains dataset statistics as well as additional details on baselines, experimental setup and hyperparameters. Table 1: Results on real-world directed heterophilic datasets. OOM indicates out of memory. \| Homophily \| Squirrel \| Chameleon \| Arxiv-year \| Snap-patents \| Roman-Empire \| \|-----------\|----------\|-----------\|------------\|--------------\|--------------\| \| MLP \| 28.77 ± 1.56 \| 46.21 ± 2.99 \| 36.70 ± 0.21 \| 31.34 ± 0.05 \| 64.94 ± 0.62 \| \| GCN \| 53.43 ± 2.01 \| 64.82 ± 2.24 \| 46.02 ± 0.26 \| 51.02 ± 0.06 \| 73.69 ± 0.74 \| \| H₂GCN \| 37.90 ± 2.02 \| 59.39 ± 1.98 \| 49.09 ± 0.10 \| OOM \| 60.11 ± 0.52 \| \| GPR-GNN \| 54.35 ± 0.87 \| 62.85 ± 2.90 \| 45.07 ± 0.21 \| 40.19 ± 0.03 \| 64.85 ± 0.27 \| \| LINKX \| 61.81 ± 1.80 \| 68.42 ± 1.38 \| 56.00 ± 0.17 \| 61.95 ± 0.12 \| 37.55 ± 0.36 \| \| FSGNN \| 74.10 ± 1.89 \| 78.27 ± 1.28 \| 50.47 ± 0.21 \| 65.07 ± 0.03 \| 79.92 ± 0.56 \| \| ACM-GCN \| 67.40 ± 2.21 \| 74.76 ± 2.20 \| 47.37 ± 0.59 \| 55.14 ± 0.16 \| 69.66 ± 0.62 \| \| GloGNN \| 57.88 ± 1.76 \| 71.21 ± 1.84 \| 54.79 ± 0.25 \| 62.09 ± 0.27 \| 59.63 ± 0.69 \| \| Grad. Gating \| 64.26 ± 2.38 \| 71.40 ± 2.38 \| 63.30 ± 1.84 \| 69.50 ± 0.39 \| 82.16 ± 0.78 \| \| DiGCN \| 37.74 ± 1.54 \| 52.24 ± 3.65 \| OOM \| OOM \| 52.71 ± 0.32 \| \| MagNet \| 39.01 ± 1.93 \| 58.22 ± 2.87 \| 60.29 ± 0.27 \| OOM \| 88.07 ± 0.27 \| \| DirGNN \| 75.13 ± 1.95 \| 79.74 ± 1.40 \| 63.97 ± 0.30 \| 73.95 ± 0.05 \| 91.3 ± 0.46 \| \| FaberNet \| 76.71 ± 1.92 \| 80.33 ± 1.19 \| 64.43 ± 0.28 \| 75.10 ± 0.03 \| 92.24 ± 0.43 \| As can be inferred from Table 1, FaberNet sets new state of the art results on all five heterophilic graph datasets above; out-performing intricate undirected methods specifically designed for the setting of heterophily. What is more, it also outperforms directed spatial methods such as Dir-GNN, whose results can be considered as reporting a best-of-three performance over multiple directed spatial methods (c.f. Appendix I or Rossi et al. [2023] for details). FaberNet also significantly out-performs MagNet. This method is a spectral model, which relies on the graph Fourier transform associated to a certain operator that is able to remain self-adjoint in the directed setting. We thus might take this gap in performance as further evidence of the utility of transcending the classical graph Fourier transform: Utilizing the holomorphic functional calculus – as opposed to the traditional graph Fourier transform – allows to base filters on (non-self-adjoint) operators more adapted to the respective task at hand. On Squirrel and Chameleon, our method performed best when using complex parameters (c.f. Table 7 in Appendix I). With MagNet being the only other method utilizing complex parameters, its performance gap to Dir-ResolvNet also implies that it is indeed the interplay of complex weights with judiciously chosen filter banks and characteristic operators that provides state-of-the-art performance; not the use of complex parameters alone. 5.2 Dir-ResolvNet: DiGraph Regression and Scale-Insensitivity We test the properties of our Dir-ResolvNet HoloNet via graph regression experiments. Weighted-directed datasets containing both node-features and graph-level targets are currently still scarce. Hence we follow Koke et al. [2023] and evaluate on the task of molecular property prediction. While neither our Dir-ResolvNet nor baselines of Table 1 are designed for this task, such molecular data still allows for fair comparisons of expressive power and stability properties of (non-specialized) graph learning methods (Hu et al. [2020a]). We utilize the QM7 dataset (Rupp et al. [2012]), containing graphs of 7165 organic molecules; each containing hydrogen and up to seven types of heavy atoms. Prediction target is the molecular atomization energy. While each molecule is originally represented by its Coulomb matrix $W_{ij} = Z_i \cdot Z_j / \|\vec{x}_i - \vec{x}_j\|$, we modify these edge-weights: Between each heavy atom and all atoms outside its respective immediate hydrogen cloud we set $W_{ij} = Z_i^{\text{outside}} \cdot (Z_j^{\text{heavy}} - 1) / \|\vec{x}_i - \vec{x}_j\|$. While the sole reason for this change is to make the underlying graphs directed (enabling comparisons of directed methods), we might heuristically interpret it as arising from a (partial) shielding of heavy atoms by surrounding electrons (Heinz & Suter [2004]). Digraph-Regression: With $W$ as directed weight-matrix and setting $y = -1$, we evaluate Dir-ResolvNet against all other directed methods of Table 1. Atomic charges are used as node weights ($\mu_i = Z_i$) where applicable and one-hot encodings of atomic charges $Z_i$ provide node-wise input features. As evident from Table 2, our method produces significantly lower mean-absolute-errors (MAEs) than corresponding directed baselines: Competitors are out-performed by a factor of two and more. We attribute this to Dir-ResolvNets ability to discard superfluous information; thus better representing overall molecular geometries. Scale-Insensitivity: To numerically investigate the stability properties of Dir-ResolvNet that were mathematically established in Theorems 3.4 and 4.2, we translate Koke et al. [2023]'s undirected setup to the directed setting: We modify (all) molecular graphs on QM7 by deflecting hydrogen atoms (H) out of equilibrium towards the respective nearest heavy atom. This introduces a two-scale setting as in Section 3.3.2. Edge weights between heavy atoms remain the same, while weights between H-atoms and closest heavy atoms increasingly diverge. Given an original molecular graph $G$, the corresponding limit $\hat{G}$ corresponds to a coarse grained description, with heavy atoms and surrounding H-atoms aggregated into super-nodes. Feature vectors of aggregated nodes are now normalized bag-of-word vectors whose individual entries encode how much total charge of a given super-node is contributed by individual atom-types. Appendix I provides additional details. In this setting, we compare feature vectors of collapsed graphs with feature vectors of molecules where hydrogen atoms have been deflected but have not yet arrived at the positions of nearest heavy atoms. Feature vectors are generated with the previously trained networks of Table 2. As evident from Fig. 6, Dir-ResolvNet’s feature-vectors converge as the scale $c \sim \|\vec{x}_H - \vec{x}_{\text{heavy}}\|^{-1}$ increases; thus numerically verifying the scale-invariance Theorem 4.2. Feature vectors of baselines do not converge: These models are sensitive to changes in resolutions when generating graph-level features. This difference in sensitivity is also apparent in our final experiment, where collapsed molecular graphs $\{\hat{G}\}$ are treated as a model for data obtained from a resolution-limited observation process unable to resolve individual H-atoms. Given models trained on directed higher resolution digraphs $\{G\}$, atomization energies are then to be predicted solely using coarse grained molecular digraphs. While Dir-ResolvNet’s prediction accuracy remains high, performance of baselines decreases significantly if the resolution scale is reduced during inference: While Dir-ResolvNet out-performed baselines by a factor of two and higher before, this lead increases to a factor of up to 240 if resolutions vary (c.f. Table 3). ### Table 2: QM7 [kcal/mol] \| Method \| MAE \| \|------------\|--------\| \| DirGNN \| 59.01±2.54 \| \| MagNet \| 45.31±4.24 \| \| DiGCN \| 39.95±6.23 \| \| Dir-ResolvNet \| 17.12±0.63 \| ### Table 3: QM7 coarse [kcal/mol] \| Method \| MAE \| \|------------\|--------\| \| DirGNN \| 195.64±2.20 \| \| MagNet \| 663.63±190.358 \| \| DiGCN \| 6672.71±2243.61 \| \| Dir-ResolvNet \| 27.34±7.55 \| ### Conclusion We introduced the HoloNet framework, which allows to extend spectral networks to directed graphs. Key building blocks of these novel networks are newly introduced holomorphic filters, no longer reliant on the graph Fourier transform. We provided a corresponding frequency perspective, investigated optimal filter-banks and discussed the interplay of filters with characteristic operators in shaping inductive biases. Experiments on real world data considered two particular HoloNet instantiations: FaberNet provided new state-of-the-art results for node classification under heterophily while Dir-ResolvNet generated feature vectors stable to resolution-scale-varying topological perturbations. 7 ACKNOWLEDGEMENTS The authors thank Mariia Gladkova and Tarun Yenamandra for helpful discussions. This work was supported by the ERC Advanced Grant SIMULACRON. 8 REPRODUCIBILITY We are taking great care in ensuring reproducibility of our work: • We give complete mathematical definitions of all utilized concepts. Proofs of all Theorems are provided in the appendix (c.f. e.g. Appendices G and H). • We exactly detail our newly introduced (HoloNet) framework and its two particular instances (FaberNet, Dir-ResolvNet) in the main body of our paper (c.f. Section 4). • The experimental setups for all our experiments are described exactly in Appendix I. • All hyperparameter settings of our methods for all experiments are detailed in the appendix (c.f. Table 6, Table 7 and Section 12). • Our code is available at https://github.com/ChristianKoke/HoloNets. REFERENCES L. V. Ahlfors. Complex Analysis. McGraw-Hill Book Company, 2 edition, 1966. Joseph Bak and Donald J. Newman. Complex analysis. Springer, 2017. Muhammet Balcilar, Guillaume Renton, Pierre Héroux, Benoit Gaüzère, Sébastien Adam, and Paul Honeine. Analyzing the expressive power of graph neural networks in a spectral perspective. In 9th International Conference on Learning Representations, ICLR 2021, Virtual Event, Austria, May 3-7, 2021. 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Specformer: Spectral graph neural networks meet transformers. In The Eleventh International Conference on Learning Representations, ICLR 2023, Kigali, Rwanda, May 1-5, 2023. OpenReview.net, 2023. URL https://openreview.net/pdf?id=0pdSt3oyJa1. Joan Bruna, Wojciech Zaremba, Arthur Szlam, and Yann Lecun. Spectral networks and locally connected networks on graphs. In International Conference on Learning Representations (ICLR2014), CBLS, April 2014, 2014. Eli Chien, Jianhao Peng, Pan Li, and Olgica Milenkovic. Adaptive universal generalized pagerank graph neural network. In 9th International Conference on Learning Representations, ICLR 2021, Virtual Event, Austria, May 3-7, 2021. OpenReview.net, 2021. URL https://openreview.net/forum?id=n6j17fLxrP.
XyB4VvF01X	Premise selection is one important step for theorem proving, which is not discussed in this work. How to guarantee that all relevant theorems/lemmas are properly included when constructing the graph representation?	GRAPH2TAC: LEARNING HIERARCHICAL REPRESENTATIONS OF MATH CONCEPTS IN THEOREM PROVING Anonymous authors Paper under double-blind review ABSTRACT Concepts abound in mathematics and its applications. They vary greatly between subject areas, and new ones are introduced in each mathematical paper or application. A formal theory builds a hierarchy of definitions, theorems and proofs that reference each other. When an AI agent is proving a new theorem, most of the mathematical concepts and lemmas relevant to that theorem may have never been seen during training. This is especially true in the Coq proof assistant, which has a diverse library of Coq projects, each with its own definitions, lemmas, and even custom tactic procedures used to prove those lemmas. It is essential for agents to incorporate such new information into their knowledge base on the fly. We work towards this goal by utilizing a new, large-scale, graph-based dataset for machine learning in Coq. We leverage a faithful graph-representation of Coq terms that induces a directed graph of dependencies between definitions to create a novel graph neural network, Graph2Tac (G2T), that takes into account not only the current goal, but also the entire hierarchy of definitions that led to the current goal. G2T is an online model that is deeply integrated into the users’ workflow and can adapt in real time to new Coq projects and their definitions. It complements well with other online models that learn in real time from new proof scripts. Our novel definition embedding task, which is trained to compute representations of mathematical concepts not seen during training, boosts the performance of the neural network to rival state-of-the-art k-nearest neighbor predictors. 1 INTRODUCTION Interactive theorem provers (ITPs) are special programming languages which assist users in writing formal proofs. They not only check the correctness of a proof, but give the user detailed feedback along the way. Coq is one such ITP based on the Calculus of Inductive Constructions [Paulin-Mohring (1993)]. It has been used to verify the four color theorem [Gonthier (2008)] and create a certifiably correct C compiler [Blazy & Leroy (2005)]. The need for provably secure hardware and software is increasingly urgent, but formal theorem proving remains laborious. In Coq, users input proofs using tactics which advance the state of the proof. At every proof step, Coq provides the user with a list of proof states for which they must supply a proof. They do so one step at a time by entering tactic commands for which Coq returns a new set of proof states to solve. A number of works have explored neural and machine learning guidance to Coq and other ITPs in order to assist users in writing proofs. However, one particular challenge is that, like all programming languages, a model trained on one particular set of projects may not be able to adapt to a new project with its own set of definitions. We need online models, as illustrated in Figure 1, which can take into account new information in a file or project without needing to be retrained. This topic has been previously explored in the Tactician framework [Blaauwbroek et al. (2020b)], which contains several types of online models [Zhang et al. (2021)] such as the powerful $k$-nearest neighbor ($k$-NN) model. These models learn on-the-fly from tactic scripts written by users and, as a result, are able to make highly relevant proof suggestions for new theorems. However, despite their strength, these models are somewhat simplistic. They have no knowledge of the current state of the global context within Coq, and are unable to adapt predicted tactics to the environment. We present a graph neural network (GNN) model, Graph2Tac (G2T), that in addition to training on existing works, can adapt to new definitions and theorems (which in Coq are special cases of Figure 1: Overview of the online learning setting where a model is amended with unseen math concepts and proofs. Figure 2: A graph-based representation of the example given in Figure 1. Definitions) in the environment in real time. Unlike the $k$-NN model, it cannot learn to use new tactics, but it rather learns to understand and work with new definitions while synthesizing tactic scripts. This ability provides a strong complement to the $k$-NN model. Like the $k$-NN, Graph2Tac interfaces with Coq through the Tactician framework. Tactician provides both a Suggest command which lets our model suggest tactics for a given proof state (as shown in Figure 1) and a synth tactic to use our model’s suggestions to search for a complete proof. Due to the integration into the Tactician framework, G2T is the first practical neural network-based solver for Coq that a user can run on a consumer-grade computer. It will automatically adapt to the users’ Coq project and can assist the user in writing proofs from the comfort of their editor. This paper presents a novel neural definition embedding task presented in Figure 3 which uses an embedding table to learn representations for definitions seen during training. The embedding of a definition is subsequently used to build representations of proof states that reference the definition. Additionally, it can be used to predict probable arguments for tactics from the global context. To dynamically update the definition embedding table for new definitions, we train a definition model which during training aligns its output to the definition embedding table. During inference, when a new definition is encountered, we use the definition model to compute an embedding of that definition. When proof states or other definitions that depend on that definition are encountered, we use our newly calculated embedding. Hence our embeddings take into account not only the form of the definition, but the hierarchy of definitions leading up to it. Contributions Our contributions are the following: (1) We showcase a new definition training task and show it improves theorem proving in Coq from 17.4% to 26.1% of theorems solved in the challenging setting of new Coq packages not seen during training using consumer hardware. (2) We show the usefulness of graphs which represent the entire environment of definitions as a mono-graph. (3) To our knowledge, we give the first comprehensive comparison of many symbolic and machine learning solvers in Coq (or any ITP for that matter) including Graph2Tac, a transformer model, Tactician’s $k$-NN, and CoqHammer. (4) We demonstrate that the $k$-NN solver, despite its simplicity, is one of the most powerful available solvers, solving 25.8% of test theorems, out-performing both CoqHammer and our transformer baseline. (5) Appendix B shows that our G2T and $k$-NN solvers are state of the art among existing Coq solvers, including ASTactic, TacTok, and ProverBot9001. (6) We show that G2T and the $k$-NN are complementary online solvers, together proving 33.2% of test theorems. (7) G2T is one of the first neural solvers conveniently available to end-users. Background and related work In recent years, there have been machine learning approaches for various interactive theorem proving systems. In the Coq ecosystem specifically, a series of articles (First et al., 2020; First & Brun, 2022; Sanchez-Stern et al., 2023) was based on the dataset provided by CoqGym (Yang & Deng, 2019), which contains many Coq packages to benchmark machine learning systems. Other projects in similar directions were GamePad (Huang et al., 2019), an interaction system based on the coqtop REPL, and ProverBot9001 (Sanchez-Stern et al., 2020). Another line of work is based on the Tactician system (Blaauwbroek et al., 2020b), for which implementations of k-nearest neighbours and random forest algorithms were built (Zhang et al., 2021). Early systems for Coq were Proof General (Komendantskaya et al., 2013) and SEPIA (Gransden et al., 2015) and the experiments by (Kaliszyk et al., 2014). A system that is often used for comparisons is CoqHammer (Czajka & Kaliszyk, 2018). For other interactive theorem proving systems, there has also been a lot of work done in recent years. Machine learning guidance was used to prove problems from the Mizar ITP system (Urban, 2003; Kaliszyk & Urban, 2015). TacticToe (Gauthier et al., 2017) is an ITP machine learning system for HOL4. For HOLight, there is the HOList (Bansal et al., 2019) system. In Isabelle, the Sledgehammer system was also extended with machine learning (Kühlwein et al., 2013; Blanchette et al., 2016a). For the recently developed Lean system, there are, for example, LeanDojo by (Yang et al., 2023), the integrated decision trees by (Piotrowski et al., 2023) and work using Language Models (LMs) (Han et al., 2021; Lample et al., 2022). There have also been LM-based approaches for other systems, for example Magnushammer by (Mikuta et al., 2023) for Isabelle and the GPT-f system (Polu & Sutskever, 2020) for MetaMath. We touch upon some more related work in Section C of the Appendix. Our work is novel compared to the previous related work: we use a graph-based dataset extracted with Coq kernel knowledge which allows us to develop a graph neural network that learns the meaning of definitions by exploiting the hierarchical structure of the data. This architecture tackles the problem of learning online, making use of new definitions in new projects. 2 GRAPH-BASED INTERACTION WITH THE COQ PROOF ASSISTANT There are many formats in the literature for representing the proof state and definitions in the environment, both for training a model and for communication between the model and Coq. In this work, we use a new graph-based format where all Coq definitions and proof states are stored in one large interconnected graph. Figure 2 shows a subset of this large graph representing the proof state and environment at the start of the Suggest command in Figure 1. This provides the following advantages: (1) The graph is created from the terms coming from the Coq kernel, faithfully encoding all objects in the Coq environment. (2) References to global definitions, for example, $+$ in the example, are explicitly referenced by edges avoiding any ambiguity with name resolution. (3) Local variables are connected to their binders, e.g., $\forall, \lambda$, via edges, eliminating the need for local variable names. (4) Equal terms are shared across parts of the graph leading to more compact representations. This large mono-graph (containing millions of nodes) is too large for a graph neural network. See Appendix E for an illustration. Instead, we extract smaller proof state and definition graphs for input into our models. Each proof state and definition has a root node in the large graph. To obtain a smaller graph, we calculate the forward closure from the root node, stopping when a definition node is encountered. Mutually dependent definitions, such as inductive definitions, have multiple mutually dependent root nodes. Theorems are special cases of definitions where the theorem node points to both the theorem statement and its proof term. We omit the proof term from the theorem’s definition graph to reduce the graph size. The subgraphs extracted from the mono-graph are topologically ordered according to their dependencies, so that they can be processed in an appropriate order by the neural network. In Figure 2, the subgraphs are highlighted for definitions $T_1$, $=, f, +,$ and $N$, as well as the current proof state of theorem $T_2$, which is still under construction. Notice $T_1$ and the proof state share the subterm associated with $2^$ (not drawn in full detail). In the dataset of proofs, each proof state has an associated tactic, forming the training data for the prediction model. A tactic such as rewrite plus_comm in $H$ is decomposed into a base tactic rewrite _ in _, and arguments plus_comm and $H$. While arguments may be arbitrary Coq terms and even other kinds of objects, our prediction model only predicts local hypotheses or global definitions. The dataset we utilize is extracted from 120 different Coq packages from the Opam package manager. These packages were selected by a SAT solver as the largest mutually consistent set of packages available for Coq v8.11. Their topics vary wildly, including analysis, compiler and programming language formalization, separation logic, homotopy type theory, and much more. The graph extracted from these formalizations consists of over 250 million nodes, which encode 520k definitions, of which 266k are theorems, and 4.6M proof state transformations. We divide packages into training and test where no test package depends on a training package. To do so, we induced a random topological order on the Coq packages, with regard to the dependency graph. The resulting list was then split such that the average percentage of theorems and of proof states in the training split is close to 90% (in our case, it is 91.3% of all theorems and 88.8% of all proof states). The Tactician synth tactic and Suggest command can communicate via this graph format with a Python client running a machine learning model. Tactician sends the entire mono-graph of global definitions along with the current proof state. The client returns a list of suggested tactics and scores. This integration makes our solver usable in practice, and allows us to perform a massively parallel benchmark of our model on any Coq Opam package. 3 PROOF AUTOMATION METHODS Here, we describe all solvers that will be compared in this paper. Section 3.1 describes the architecture of Graph2Tac, while Section 3.2 summarizes other systems for comparison. The transformer was developed in conjunction with Graph2Tac. All other systems were developed elsewhere. Note that comparisons with highly relevant solvers, such as Proverb9001 [Sanchez-Stern et al. (2020)], --- 1This is justified by the principle of proof irrelevance: To use a theorem one does not need to know its proof. 2Roughly half of the definitions are derived from each other through Coq’s module and section mechanism. are missing because they do not provide proof search through a Coq tactic. This makes direct comparison challenging (nonetheless, see Appendix B for an informal comparison). Other symbolic solvers, such as SMTCoq [Armand et al., 2011] and Itauto Besson [2021] are excluded because they are not general purpose solvers. 3.1 Graph Neural Network and Definition Task Graph2Tac primarily consists of two parts, a definition task and a prediction task, shown in Figure 4. The input to each of these is a directed graph with labeled nodes and edges representing either a definition for the definition task, or a proof-state for the prediction task. We additionally associate metadata with each graph. Definition graphs have root nodes (node $ f $ in the figure), one for each definition defined in that graph. Inductive data types define multiple mutually dependent concepts in the same graph, e.g., `bool`, `true`, `false`, and may therefore have multiple roots. A proof state graph comes with local context nodes, one for each hypothesis. Both types of graphs contain definition nodes, which are leaf nodes that represent referenced definitions from the global context. With each definition node, we store an index to the definition embedding table described in the next paragraph. For the prediction task, in addition to the proof state graph, the input includes the indices of all global definitions and all tactics currently available in Coq’s global context. There are four learned embedding tables: edge labels and node labels (not shown in Figure 4), base tactics that occur in the training dataset, and definitions that occur in the training dataset. All embeddings have dimension equal to the model’s hidden dimension, $ h_{\text{dim}} = 128 $. The definition and node embeddings are constrained to be unit-normalized. During inference, the definition table will be dynamically updated with new definitions as discussed later. We transform the input graph, be it a proof state or definition graph, using the following approach (Figure 4 step A). Each graph is pruned to 1024 nodes. For effective message passing, we duplicate all edges to go in both directions and also add self-edges. The edges are assigned edge embeddings from the corresponding embeddings table. This table has $ 2E + 1 $ entries, accounting for the $ E $ original edge labels, the $E$ reverse edge labels, and the self-edge label. Non-definition nodes are assigned embeddings from the node embeddings table based on their node label. Definition nodes are instead assigned embeddings from the definition embeddings table, except for the root nodes in the definition task. Those are masked with null embeddings, as the goal of the definition task is to predict the corresponding definition embeddings. The transformed graphs are put into a message-passing GNN (Figure 4 step B). The GNN consists of 8 hops where the $t$th hop transforms a node embedding $x^t_n$ according to the following two steps: $$\hat{x}^{t+1}_n = \text{ReLU}\left(\frac{1}{\deg(n)} \sum_{m:e:m \rightarrow n} \text{Dense}_{\theta_t}(e, x^t_m)\right)$$ $$x^{t+1}_n = \text{LayerNorm}\left(x_t + \text{Dropout}_{0.1}(\text{MLP}_{\psi_t}(\hat{x}^{t+1}_n))\right)$$ The first step is a graph convolution layer where each node embedding is updated according to incoming edges from neighboring nodes. Here, $\deg(n)$ is the number of incoming edges $m \rightarrow n$ with target $n$ and edge embedding $e$. The dense layer has output dimension $h_{\text{dim}}$. The next step consists of a 2-layer MLP (Dense, ReLU, Dense) with an inner hidden dimension of $2h_{\text{dim}}$, then Dropout, Residual, and LayerNorm. The weights for each hop are separate, but the definition and prediction tasks both use the same GNN backbone, sharing the same weights. The output of the GNN is a graph with the same structure as the input, but with updated node embeddings. Both the definition and prediction tasks use mean pooling (Figure 4 step C) to obtain a single vector embedding for the graph. For the definition task, the pooled embedding is concatenated with each of the embeddings of the root nodes for the definition, then fed into a two-layer MLP and finally unit normalized (step D). Optionally, along with each root node embedding, we additionally concatenate a name embedding for the definition, using a bidirectional LSTM (not-shown) to embed the fully qualified Coq identifier string, e.g. “Coq.Init.Logic.and”. For the prediction task, the pooled embedding is fed into a two-layer MLP (step E). The output is multiplied by the tactic embedding table using inner product and put through a softmax layer to obtain the base tactic probabilities. To predict the arguments for a tactic, we use a simple two-layer RNN (Figure 4 step F). The initial hidden state of the RNN is the embedding for the chosen tactic in the tactic embedding table. (During training the ground truth tactic is provided as input. The decoding method used during inference is described below.) The number of steps of the RNN corresponds to the number of arguments required by the given tactic. Each argument is predicted as follows (Figure 4 step G). The RNN output is fed into a pair of two-layer MLPs (Dense, ReLU, Dense), resulting in a pair of query embeddings, one for global argument prediction and one for local argument prediction. The global argument prediction query is unit normalized. For global arguments, we take the inner product of the global argument query with each definition in the embedding table resulting in a logit for each definition. Since the inner product is bounded (as both vectors in the inner product are unit normalized), we scale the global logits by a learned temperature parameter. For each local argument, we use the GNN output embedding for that local node in the graph and take the inner product with the local argument query. Concatenating the local and global logits and performing softmax we get a distribution over all possible arguments. Since many of the calculations used in our model take place on variable-size lists of objects, e.g. local arguments, our implementations rely heavily on ragged tensors in TensorFlow [Abadi et al., 2015] and graph tensors in TF-GNN [Ferludin et al., 2022]. We train on batches of 512 definitions and 512 proof states. The loss for the definition task is cosine similarity. The loss for the training task is the cross-entropy for the full tactic sequence. For example, for the tactic apply T1 the loss is $-\log P(\text{apply } _) - \log P(T1 \| \text{apply } _)$ using the probabilities coming from the model. The combined loss of the two tasks is $L = 1000 L_{\text{def}} + L_{\text{tactic}}$. During inference, the tactic embedding table is masked for tactics which both occur in the training data and are available in the Coq’s current state. Similarly, the definition embedding table is masked 3If a definition graph contains multiple definitions, the loss is divided by $\sqrt{n}$ where $n$ is the number of entry points. for all definitions available in the current global context. However, for new definitions not seen during training, we first calculate an embedding using the definition task. If there are multiple new definitions, we compute embeddings from each definition graph individually, updating the embeddings in a topologically sorted order so that those for dependencies are computed before those for latter definitions which depend on those dependencies. At inference time, the output of the tactic task is a list of tactic suggestions, where each sequence starts with a base tactic, e.g., `apply _` and then contains the arguments for that tactic, if they are required. We use beam search decoding with a beam width of 256 to generate 256 tactic suggestions for each proof state. We train three models: G2T-Named uses the name embedding in step D, whereas G2T-Anon does not. G2T-NoDef is trained without a definition task. Each of these is run with three configurations on the definition model during inference: The Recalc configuration calculates embeddings for all definitions, Update only calculates embeddings for new definitions—i.e., those not seen during training, and Frozen uses random unit normalized embeddings in place of the definition model. G2T-NoDef is only used as G2T-NoDef-Frozen. G2T-Anon-Update is the primary model. 3.2 Non-Graph Approaches firstorder auto with ∗* As a baseline, we use Coq’s built-in `firstorder` reasoning tactic combined with the programmable proof search tactic `auto`, with all available hint databases. CoqHammer Czajka & Kaliszyk (2018) translates theories expressed in Coq to a first-order language understood by the external automated theorem provers Vampire, CVC4, Z3 and Eprover. Once an external proof is found, the premises required for the proof are extracted, and the proof is reconstructed inside of Coq through the `sauto` family of higher-order solving tactics Czajka (2020). k-Nearest Neighbor The fastest practical solver currently available in the Tactician framework is a fast k-nearest neighbor (k-NN) model Plaauwbroek et al. (2020c). It builds a database of proof states and associated tactics and extracts hand-crafted features from those proof states Zhang et al. (2021). When the model is queried, it looks up proof states with the most similar features to the current proof state and returns the corresponding tactics, ordered by similarity. It does not require any training. Despite its simplicity, the k-NN solver is the strongest baseline because, like Graph2Tac, it is able to adapt in real-time to the changing Coq environment. It is highly complementary because instead of learning from new definitions it rather learns from recent proof scripts. Proof State Text-based Transformer We implement a decoder-only transformer baseline that operates on the textual representations of the proof states, and predicts a textual representation of the tactic. We use the GPT2 implementation available via the Transformers Python library Wolf et al. (2020). The embedding size is set to 768 and it has 12 layers, as in one of the models described in that paper. Our approach is similar to that of Han et al. (2021), Jiang et al. (2022), Yang et al. (2023), except our transformer models are trained from scratch only on Coq proof data. 4 Evaluation Experimental setup To evaluate the performance of the solvers described above, we randomly choose 2000 theorems from the Coq packages in our testing set. The solvers are given a time limit of 10 minutes per theorem to synthesize a proof. To look at package-specific results, we sample up to 500 test theorems per package with a time limit of 5 minutes. The search procedure utilized by the solvers based on the Tactician framework is a modification of Dijkstra’s algorithm that performs iterative deepening in order to relieve memory pressure and has an optimization that avoids backtracking between independent subgoals. For more information, see Appendix D. During evaluation, each solver is limited by the operating system to one CPU with two hyperthreads. All processes, including Coq and any external processes such as neural networks and ATP’s, must share this CPU. An exception is made for the Transformer-GPU solver, which is permitted to perform model inference on a dedicated GPU instead of a CPU. We explore two artificially aggregated solvers, that simulate running $ n $ solvers concurrently while still adhering to the one-CPU computational budget. The number of theorems “solved” by the aggregate solvers in time $ t $ is the number solved by any of the components in time $ t/n $. “G2T-Anon-Update + k-NN” is intended to simulate a solver capable of learning both from new definitions in the global context and new tactic proof scripts. “CoqHammer combined” is an aggregate of all four ATP backends and the reconstruction tactic called \texttt{best}. This simulates the intent of CoqHammer’s authors to run many backends in parallel, while maintaining a fair comparison with other solvers. See Appendix I for an in-depth analysis of our CoqHammer experiments. Results The left plot of Figure 5 shows the fraction of test theorems solved over time for various solvers. One can see both the progress over time, but an indication of the startup cost of each solver. The right plot of Figure 5 replaces time with the number of calls to the underlying model, giving an indication of the usefulness of that model’s predictions in search irrespective of model speed.$^4$ This and other plots remove the results from the \texttt{hoti} because HoTT replaces Coq’s built-in standard library, upon which CoqHammer depends. The \texttt{tlc} package is also removed because G2T-Anon-Update was able to exploit an inconsistency as explained in Appendix K. CoqHammer combined fairs better than the transform solver, even the variant using a GPU for model inference. (See Appendix II for a detailed breakdown of the CoqHammer results.) Among the G2T solvers shown, the two with additional definition task outperforms the G2T-Frozen-Def baseline (26.1% vs 17.4%) demonstrating that the definition task helps to improve results. Note, G2T-Frozen-Def performs similarly to both transformer variants in terms of model calls. This suggests that the advantage of the graph model over the transformer is due to model speed and not prediction quality. The addition of names in G2T-Named-Update fares slightly worse than the main G2T solver G2T-Anon-Update. The $ k $-NN solver outperforms the G2T-Anon-Update model at smaller time limits, but for later time steps the latter starts to overtake. We see a similar picture relative to model calls. The ability of the $ k $-NN to repeat the tactics of recent proofs may be especially powerful, and indeed, in Appendix B we --- $^4$Although the number of tactics returned by a model can also impact the pass rate, especially if a particular branch of the search runs out of tactics to apply. The faster models happen to also return more results. suggest the $k$-NN is at least as powerful as existing neural Coq solvers in the literature. Nonetheless, if larger amounts of search are required, there appears to be value in the more sophisticated G2T-Anon-Update model. Both solvers are quite complementary as we see in the results of combination solver G2T-Anon-Update + $k$-NN as well as in the Venn Diagram of solved theorems in Figure[6]. Both models show the success of online methods in this setting and both use different online information (Figure[1]). Figure[7] breaks down the performance for the largest 15 packages. We see that neither the G2T-Anon-Update nor the $k$-NN always performs better. In many cases G2T-Anon-Update either overtakes the $k$-NN or appears as if it will overtake the $k$-NN if given enough time. The tlc package is special in that G2T-Anon-Update was able to find a significant number of theorems using an inconsistent axiom in the environment, which is why it was removed from the results above. ![Cumulative solving curves](image) Figure 7: Package-specific cumulative solving curves. We show the behaviors of $k$-NN, GNN, and CoqHammer (except on HoTT which is incompatible with CoqHammer). ## 5 Discussion and Future Work Our definition task improved a neural theorem prover from 17.4% to 26.1% in the difficult setting of proving theorems in never-before-seen packages. This, in addition to the success of the $k$-NN approach, shows the importance of online learning in this setting. We leave as future work how to unify the G2T and $k$-NN approaches shown in Figure[1]. Ideally, such a model should also account for new tactics, as well as learn from how new definitions and tactics are used in new theorems. One avenue is exploring if our model can be fine-tuned in real-time. To improve the definition task, given that G2T-Anon-Update outperformed G2T-Named-Update, we wonder if adding the names makes the definition task too easy for the model. There may also be alternatives to our definition task, using ideas in self-supervised or contrastive learning, or using social-network-size graph models to process the entire graph of interconnected definitions at once. Theorem proving and programming share a lot of similarities and concerns, so it is useful to explore how this work relates to code generation, and we leave open how our methodology could apply to text-based models. Retrieval augmented transformers are a possible approach [Yang et al. (2023)], but may not scale to the full definition hierarchy. 6 REPRODUCIBILITY This paper carefully describes our methods for building, training, and testing our models and solvers. The full code for each stage of the pipeline is available in the following open-source repositories: [REDACTED]. This includes the code for training the models, running them as part of a solver, interfacing with Coq, and benchmarking. We use an open-source dataset (which the dataset authors will release soon at [REDACTED]), and all of our models are trained from scratch so there is no use of proprietary pre-training data. Our solvers are intended for use by both researchers and Coq end users, and usable on a typical Linux machine with no GPU. The model can also be hosted on an external server (for example, with a GPU), but used on a local Coq instance via a TCP connection. Instructions for setup are included in [REDACTED]. We hope to share our trained models, but users can also train their own with the code we provided and the open dataset. While we trained the graph models for three weeks from scratch on two A100s, we noticed that a model trained in two days showed similar results. Users may also train models on a different set of Coq packages, including new or private Coq developments, via the tools provided with the dataset and with our training code. Our benchmarking framework allows testing on most Coq Opam packages compatible with Coq 8.11, leaving the possibility of testing our solvers (or solvers trained with our code) on future Coq benchmarks. We plan to share the full test results at [REDACTED], including which theorems were tested on, which were proved, and how long the solver took in terms of seconds, model calls, and tactic executions. This is all in hopes of facilitating future comparison and collaboration.
9rzEPbs4Wg	According to the impact of masking probability alpha on generalization and safety metrics, it shows a direct correlation between alpha and corruption accuracy. However, why such a correlation is held? Will this correlation be held for more datasets or applications?	IMPROVING GENERALIZATION AND SAFETY OF DEEP NEURAL NETWORKS WITH MASKED ANCHORING Anonymous authors Paper under double-blind review ABSTRACT Anchoring is a recent architecture and task-agnostic technique that can produce state-of-the-art epistemic uncertainty estimates, and improve extrapolation capabilities. However, the differences between anchored models and non-anchored variants is not well studied – as there is little insight into the kinds of functions anchoring induces and how they behave under distribution shifts. In this paper, we analyze and improve anchoring as a training protocol for deep neural networks, evaluating them on important tasks of out of distribution generalization, task adaptation, anomaly rejection and calibration. We pinpoint the impact of anchoring on generalization as being inversely related to the sensitivity of the model to the distribution of residuals. We further improve this sensitivity using a new technique called Random Anchor Masking (RAM) that significantly improves the quality of anchored models. We build evidence for the superiority of RAM-training using a range of benchmarks of varying size, using neural networks of varying complexity and scale. 1 INTRODUCTION Anchoring is a simple, architecture-agnostic protocol for training neural networks; it has enabled several capabilities ranging from state-of-the-art uncertainty estimates and calibration (Thiagarajan et al., 2022), outlier rejection (Anirudh & Thiagarajan, 2022), and extrapolation (Netanyahu et al., 2023). At a high level, anchoring replaces the input to the network using a tuple comprising an “anchor” randomly chosen from the training dataset, and the residual between the input and the anchor. This is done such that, the prediction on the input should be consistent regardless of the anchor choice. By reposing the prediction task into the joint space of anchors and residuals, this trivial transformation has been shown to provide significant gains in performance over standard deep models. However, the behaviour of anchored training is not sufficiently clear from the existing literature. For example, in (Thiagarajan et al., 2022), the fact that anchoring leads to meaningful uncertainties is justified via studying anchoring as perturbations to the neural tangent kernel (NTK) (Jacot et al., 2018); while this explains why meaningful uncertainties arise, it does not shed light into the quality of functions anchoring can produce. Moreover, in (Netanyahu et al., 2023), the anchored model is essentially considered equivalent to the non-anchored model in terms of the function it approximates. As a result, there is little clarity on anchoring as a training mechanism on its own, and its impact on generalization and safety characteristics. This is essential as models are being increasingly adopted in a variety of applications across different domains such as healthcare (Davenport & Kalakota, 2019) and autonomous driving (Bogdoll et al., 2022), where it is prudent to holistically evaluate and understand the model behavior under challenging distribution or task shifts. In that context, generalization to data beyond the training distribution (Yang et al., 2021) as well as the ability to accurately detect changes in the input data are critical to promote safe model adoption (Hendrycks et al., 2021b). While generalization includes producing accurate predictions on covariate shifts i.e., images of a particular modality collected from different sensors or adapting to new task shifts (Andreassen et al., 2021), sensitivity to input data variations include producing well-calibrated confidences (Guo et al., 2017) under data shifts and the ability to accurately detect anomalous samples (Hendrycks & Gimpel, 2017b) of disparate semantic characteristics with respect to the training data. Existing research has demonstrated significant performance improvements in each task considered independently (Hendrycks & Gimpel, 2017a; Lee et al., 2018; Hendrycks et al., 2018; Sehwag et al., Since a model can non-trivially trade-off between the different safety objectives, in practice, it is challenging to effectively train and assess models (Hendrycks et al., 2022). This motivates the need to devise novel training protocols that are not specific to architectures and do not require sophisticated priors (e.g., PixMix Hendrycks et al. (2022), but simultaneously improve generalization and safety of the trained models. In this paper, we study the viability of anchoring as a training protocol for large-scale datasets and sophisticated model architectures. Using a variety of architectures of varying complexity and size, we study different aspects of generalization such as prediction under severe corruptions, calibration, anomaly rejection, adaptation under task/covariate shifts and robustness under label noise to establish the benefits of anchoring over standard network training. We summarize our key findings below: - Anchoring is able to boost generalization performance without sacrificing on safety metrics such as calibration or anomaly rejection; - We pinpoint the improved generalization of anchored models as being linked to the diversity of residuals exposed to the model during training; - Building upon this insight, we propose Random Anchor Masking (RAM), an efficient and effective regularization for improving diversity, which shows significantly improved generalization over both standard anchoring and non-anchored models. - We observe significant improvements in generalization and safety across datasets of varying size and complexity (Imagenet-1K/CIFAR100.10) and architectures of varying scales (RegNet/WRN/ResNet/WRN/DEIT-T/DEIT-S/Swinv2-T/SWINv2-S/SWINv2-B/ViT-B). 2 BACKGROUND AND RELATED WORK Notations: Let $ F_\theta $ be a multi-class classifier parameterized by $ \theta $ that is trained on a dataset $ D = \{(x_i, y_i)\}_{i=1}^M $ with $ M $ samples. The classifier works on an input image $ x \in X \subseteq \mathbb{R}^{C \times H \times W} $. The objective of this classifier is to predict an output label $ \hat{y} \in Y $ where $ Y $ is defined as the set $ Y = \{1, 2, \ldots, K\} $, where $ K $ represents the total number of distinct classes. Here, $ x $ is an RGB image from the space of inputs $ X $ with $ C $ channels, height $ H $, and width $ W $. Anchoring in Deep Models: The principle of anchoring introduced in (Thiagarajan et al., 2022) involves the reparameterization of an input $ x $ into a tuple comprising an anchor $ r $, drawn at random from an anchor distribution $ P(R) $, and the residual $ \Delta x $ denoted by $ [r, \Delta x] = [r, x - r] $. For image data, the tuple is constructed by concatenating the anchor and residual along the channel axis, resulting in a 6-channel tensor for every 3-channel RGB image $ x $. Apart from this architectural change, the optimization objective of the deep network is left unchanged (Thiagarajan et al., 2022). The simple re-parameterization of the input leads to a joint distribution that depends not only on $ P(R) $, but also on the distribution of the residuals $ P(\Delta) $. Formally, the training objective can be written as: \[ \theta^* = \arg \min_{\theta} L(y, F_\theta([r, x - r])), \forall r \in P(R) \] where $ L(.) $ is a loss function. Effectively, anchoring enforces that for every input sample $ x $, \[ F_\theta([r_1, x - r_1]) = F_\theta([r_2, x - r_2]) = \cdots = F_\theta([r_k, x - r_k]), \] where $ F_\theta $ is the anchored model that operates on the tuple $[r_k, x - r_k]$ to predict $ y $. During both training and testing, the anchors need to be drawn from $ P(R) $, which is set to the training distribution $ P(X) $ itself in our implementation. However, given that equation 1 explicitly marginalizes the choice of anchor, any random training sample (or a small number of them) can be used to obtain predictions at inference time. While the idea of enforcing prediction consistency across different anchor choices might appear similar to data augmentation methods, we want to clarify that anchoring does not impose any invariance to data characteristics, but only expands the space of (anchor, residual) pairs with each additional anchor. This general principle can be used with any model architecture or task, and several recent works have demonstrated the utility of anchoring in design optimization (Thiagarajan et al., 2022), reinforcement learning (Netanyahu et al., 2023), generalization gap prediction (Narayanaswamy et al., 2022) and graph neural network calibration (Trivedi et al., 2023). Why does anchoring improve generalization? In order to understand this, we directly refer to the following two key results from existing literature: (a) In (Thiagarajan et al., 2022), it was shown that centering a dataset using different constant inputs (or anchors) will lead to different solutions, due to inherent lack of shift invariance in NTKs induced by commonly adopted neural networks. Building on this principle, anchored training uses different anchors for the same sample across different epochs with the goal of marginalizing out the effect of anchor choice at inference time. But in this process, it implicitly helps explore a large class of hypotheses, thus resulting in a more generalizable solution; (b) In (Netanyahu et al., 2023), it was theoretically showed that an anchored model can better extrapolate to unseen data regimes where (independently) the anchor $ r \in P(R) $ and the residual for the unseen sample $ \Delta x_t \in P(\Delta) $. Furthermore, it was argued that the problem of generalizing to “out of support” (OOS) samples (i.e., no evidence of observing such a sample in the training data) can be made more tractable by carefully choosing anchors $ \tilde{r} \sim P(R) $ at inference time, such that $ x_t - \tilde{r} = \Delta x_t \sim P(\Delta) $, even if the specific combination of $ [\tilde{r}, x_t - \tilde{r}] $ was not observed during training. While such an out-of-combination (OOC) setting can still be challenging to handle, the hope is that the predictions might be better calibrated, i.e., low confidence for OOC tuples. Building upon this insight, we argue that, by exposing the model to more diverse combinations of (anchor, residual) pairs during training, generalization can be systematically improved. To this end, we explore a novel regularization strategy to effectively increase the diversity of $ P(\Delta) $. Furthermore, existing works have not rigorously studied the viability of anchoring as a training protocol for large-scale datasets, modern architectures or even practical distribution shifts. Hence, for the first time, we empirically benchmark anchored training across dataset sizes (CIFAR-10 to Imagenet), architectures (ResNet to ViT) and network sizes (5M to 88M parameters), using important safety metrics including OOD generalization calibration, anomaly rejection and adaptation (both ID and OOD evaluation). 3 Proposed Approach In anchoring, since $ r $ is always drawn from $ P(X) $ by design, handling novelty to $ \Delta x_t $ becomes key to improving the OOD generalization. Intuitively, with wide anchor distributions (e.g., $ P(R) = P(X) $ in ImageNet training), the residual distribution $ P(\Delta) $ obtained through the extensive space of (anchor, residual) pairs is expressive enough to support a wide variety of OOD scenarios, when compared to conventional deep models. A direct implication of this statement is that an anchored model can behaves unreliably when presented with novel residuals. While this might not be of concern when test data comes from the in-distribution (ID), i.e., $ \Delta x \in P(\Delta) $, its effect is more pronounced when handling out of distribution (OOD) data in practical tasks such as generalizing under corruptions or distribution shifts (Shen et al., 2021), anomaly rejection (Hendrycks et al., 2019; Hendrycks & Dietterich, 2019) or adaptation under task shifts (Trivedi et al., 2023). Consequently, without ensuring sufficient generalization and safety properties, anchoring becomes a less attractive choice in practice, particularly with large-scale models. A naïve way to improve the diversity of $ P(\Delta) $ (or equivalently $ P(r, \Delta) $) is to consider a much wider anchor distribution (i.e., $ P(R) \supseteq P(X) $). However, it is non-trivial to characterize the anchor distribution (e.g., for a large dataset such as ImageNet) and to arbitrarily find additional data to improve the diversity. Even in cases where we can find such additional data, it can lead to increased computational complexity for anchored training. For example, when training on ImageNet (regardless of the architecture), we find that anchored training requires 20 additional epochs to converge to the same level of validation loss as a vanilla model. On the other hand, with CIFAR-10 or CIFAR-100, anchored models converge effectively with the standard training recipe. To circumvent these challenges, we introduce RAM (Random Anchor Masking), a simple and efficient regularization strategy that leads to improved generalization, without impacting the complexity of training. 3.1 RAM: Improving Generalization via Noisy Residuals In this paper, we adopt an alternative approach to increasing diversity of $ P(\Delta) $ by making the residuals noisy, and propose a simple implementation in the form of RAM. Given the inherent challenge in defining a suitable residual noise distribution (and inferring its hyper-parameters), we define the noise distribution to be same as the anchor distribution itself, i.e., $ P(R) = P(N) = P(X) $, and implement it efficiently using the RAM regularizer. Formally, for a given tuple $[r, x - r]$, anchor masking zeroes out the anchor while keeping the residual fixed, i.e., $[0, x - r]$, while making the prediction for the sample $x$. In general, the tuple for making a prediction for $x$ with a zero anchor (note: zero vector is a valid anchor in our anchor distribution) should have been written as $[0, x - 0]$. In this context, anchor masking can be re-interpreted as making a prediction using the zero anchor, but with a noised residual $x + \epsilon$ where $\epsilon = -r$ and $\epsilon \in P(N)$. Using noisy residuals during training naturally improves the diversity of $P(\Delta)$ and more interestingly, avoids the over-reliance of OOD test samples only on the anchors, which can lead to highly mis-calibrated predictions. Figure 1: We examine how two anchored ResNet-18 (He et al., 2016) models trained on CIFAR10 (Krizhevsky, 2009) respond to input data corruption. We measure the entropy (left) and accuracy (right) as a function of perturbation strength. Note, the anchor is fixed to be the same in both cases. A well-calibrated model is expected to produce higher entropy predictions (less confident) as severity of perturbation increases, while the accuracy should correspondingly drop. We average these metrics across 20 perturbation strengths in $[0.0, 1.0]$, in 10 random directions applied to 100 OOD examples (CIFAR − 10C/Gaussian noise/severity 5). Standard deviation is measured across the 10 trials. In addition to improving the training process, RAM controls the sensitivity of the model predictions under data noise, thus leading to significantly improved calibration in practice. Figure 1 highlights the distinction between the prediction calibration (measured in terms of the entropy) of two anchored models trained with and without RAM regularization respectively. With increasing severity of the data corruption (Gaussian noise in this case) for fixed anchor (a random sample from $P(R)$), we witness improved calibration and superior generalization behavior. This hypotheses holds true even with more challenging distribution shifts and corruptions as we demonstrate in our empirical studies. Furthermore, we find in our experiments that, RAM helps improve other safety metrics such as adaptation under task/covariate shifts and anomaly rejection. Next, we describe the implementation of anchored training with RAM regularization. 3.2 Implementation Details For all models trained in this study, we followed the training recipes from the torchvision library (https://pytorch.org/vision/stable/models.html) and directly adopted the same hyper-parameter configurations even for anchored training. The hyper-parameter $\alpha \in [0, 1]$ directly controls the schedule of residual corruption during training. Note that, with the schedule $\alpha$ ($\alpha = 0.2$ corresponds to every 5th batch), we perform anchor masking for an entire batch and obtain gradient updates with noisy residuals. However, this is only an implementation choice and one can consider alternative approaches; for example, the residual corruption can be applied to $\alpha$ fraction of samples from each batch or this can be included as an additional loss objective. While using a high $\alpha$ value can help improve generalization, it can also adversely affect the training convergence and eventually the ID performance itself. We chose $\alpha$ such that the validation loss was low on ID test data and it needs the same number of epochs as standard anchoring. Based on our experiments with CIFAR-10, CIFAR-100 and ImageNet across multiple architectures (RegNet, ResNet, WRN40-2, DEIT, ViTb and SWINv2), we recommend the use of $\alpha = 0.2$. In terms of memory overheads and inferencing efficiency, we find that anchored training is very similar to conventional neural network training. protocols. The only difference we notice is that, with larger datasets like ImageNet, anchored training requires 20 additional epochs to converge to the same level of validation loss. 4 EXPERIMENTS AND RESULTS We extensively assess the performance of anchored training (w/ and w/o RAM) across diverse benchmarks, architectures and generalization tasks. Our key goal is to examine how RAM behaves across a spectrum of datasets with varying complexities and sizes, as well as different choices of model architectures. Following this, we broadly evaluate the behavior of models in three tasks: (i) Generalization to out-of-distribution (OOD) data, (ii) Anomaly rejection, and (iii) Adaptation of representations under task (and distribution) shifts. 4.1 SETUP Training Protocol: All models unless specified are trained on ImageNet-1K (Russakovsky et al., 2015), a large-scale vision benchmark comprising of 1.3 million training images across 1000 diverse categories. Following standard practice, we adopt the optimization and pre-processing settings provided in the torchvision library to train all models. As mentioned earlier, we train all anchored models on ImageNet for an additional 20 epochs beyond the one provided in the recipe to closely match the top-1 accuracy of a non-anchored model and to effectively leverage the diversity in $P(\Delta)$. Architectures: We consider a family of model architectures with different levels of structural and parameter complexity to rigorously assess the viability of anchoring as a standard training protocol. Specifically, we consider RegNetY-800-MF (6.4M) (Radosavovic et al., 2020), DEIT-T (5M), DEIT-S (22M) (Touvron et al., 2021), SWINv2-T (28.4M), SWINv2-S (49.7M), SWINv2-B (87.8M) (Liu et al., 2022) and ViT-B-16 (86.6M) (Dosovitskiy et al., 2021) models. Note that for training anchored models, we modify the first convolution layer or the transformer layer in each architecture to handle the reparameterized 6 channel images. Baselines and Evaluation Metrics: We compare anchoring with RAM against the non-anchored counterparts as well as the standard anchored variants across the three tasks. We report the top-1 accuracy to evaluate model performance for OOD generalization and adaptation tasks. For assessing anomaly rejection performance, we resort to the AUROC between the ID and OOD energy scores (Liu et al., 2020). In addition, we report the calibration error on datasets considered for OOD generalization using the recently proposed Smoothed ECE metric (Blasiok & Nakkiran, 2023) to assess the quality of model confidences under different test-time conditions. Finally, we use the standard accuracy metric for evaluating adaptation fidelity. 4.2 MAIN FINDINGS AND DISCUSSION OOD Generalization: An important assessment of model safety is the ability to generalize to distributional shifts from the training data. As such, we expect the models to encode non-trivial semantic concepts from the training data and use the same to generalize even when the test-time distributions change. To that end, we conduct a zero-shot evaluation of the pre-trained models on (i) ImageNet-C (Hendrycks & Dietterich, 2019) with 19 natural image corruptions across 5 severity levels, (ii) ImageNet-C (Mintun et al., 2021) with 10 noise corruptions across 5 severity levels; (iii) ImageNet-R (Hendrycks et al., 2021a) containing different renditions of 200 classes from ImageNet; (iv) ImageNet-S (Wang et al., 2019) comprising black and white sketch images from each class of ImageNet; and (v) ImageNet-V2 (Recht et al., 2019) containing three new test datasets for ImageNet models in addition to the standard evaluation set. In Table 1, we report the OOD generalization performance across the different benchmarks. It can be observed from the results that anchored training and in particular, the variant with RAM consistently yields improvements over their non-anchored architectures for OOD generalization across all architectures and different distribution shifts. A striking observation is that, network capacity plays a significant role in effectively leveraging the increased diversity produced by RAM. For example, with ImageNet, as we move from RegNet (5M) to SWINv2-B (88M), we witness larger performance improvements over both anchoring w/o RAM as well as standard training. On the contrary with the DEIT-T model with only 5M parameters, the benefits of incorporating RAM are somewhat limited. Finally, following our observation in Table 1: Out-of-distribution (OOD) generalization performance (corruptions and distribution shifts) on models trained with Imagenet-1K. We report the Top1 accuracy in each case and highlight the best performing model in each with pink. \| Architecture \| Anchoring? \| RAM? \| ImageNet-v2 \| ImageNet-R \| ImageNet-S \| ImageNet-C \| ImageNet-C \| \|--------------\|------------\|------\|-------------\|------------\|------------\|------------\|------------\| \| \| \| \| \| \| \| Sec. 1 \| Sec. 2 \| Sec. 3 \| Sec. 4 \| Sec. 5 \| \| DEiT-T (58) \| ✗ \| ✗ \| 67.9 \| 32.7 \| 19.89 \| 60.33 \| 53.01 \| 46.07 \| 36.56 \| 26.02 \| \| \| ✗ \| ✗ \| 68.04 \| 33.01 \| 20.44 \| 61.01 \| 53.95 \| 46.89 \| 37.17 \| 26.51 \| \| \| ✗ \| ✗ \| 68.83 \| 32.57 \| 19.61 \| 60.0 \| 53.09 \| 46.41 \| 36.8 \| 26.26 \| \| RegNet (6.4M)\| ✗ \| ✗ \| 71.85 \| 33.03 \| 22.28 \| 61.31 \| 51.34 \| 42.8 \| 31.63 \| 21.02 \| \| \| ✗ \| ✗ \| 71.39 \| 32.13 \| 22.41 \| 61.12 \| 51.15 \| 42.42 \| 30.79 \| 20.12 \| \| \| ✗ \| ✗ \| 71.43 \| 32.45 \| 21.51 \| 62.24 \| 53.48 \| 45.48 \| 33.89 \| 22.9 \| \| DEiT-S (22M)\| ✗ \| ✗ \| 75.19 \| 41.88 \| 29.12 \| 70.62 \| 64.74 \| 59.0 \| 49.94 \| 38.39 \| \| \| ✗ \| ✗ \| 75.88 \| 43.11 \| 30.01 \| 71.5 \| 65.81 \| 60.36 \| 52.05 \| 41.2 \| \| \| ✗ \| ✗ \| 75.46 \| 42.14 \| 29.19 \| 70.66 \| 65.04 \| 59.71 \| 51.52 \| 40.72 \| \| SWINv2-T (29.4M)\| ✗ \| ✗ \| 77.21 \| 40.84 \| 27.08 \| 71.63 \| 64.89 \| 57.77 \| 47.77 \| 35.60 \| \| \| ✗ \| ✗ \| 77.34 \| 40.36 \| 27.56 \| 72.32 \| 65.85 \| 58.95 \| 49.51 \| 37.41 \| \| \| ✗ \| ✗ \| 77.16 \| 41.17 \| 27.68 \| 72.13 \| 65.71 \| 59.21 \| 50.01 \| 38.58 \| \| SWINv2-S (49.7M)\| ✗ \| ✗ \| 79.21 \| 45.17 \| 32.25 \| 74.48 \| 68.8 \| 62.84 \| 54.32 \| 42.85 \| \| \| ✗ \| ✗ \| 79.3 \| 45.95 \| 32.08 \| 74.75 \| 68.87 \| 63.12 \| 54.7 \| 43.11 \| \| \| ✗ \| ✗ \| 78.93 \| 46.63 \| 33.3 \| 74.7 \| 69.12 \| 63.65 \| 55.5 \| 44.33 \| \| VITB16 (86.6M)\| ✗ \| ✗ \| 76.31 \| 44.06 \| 29.4 \| 72.37 \| 66.57 \| 61.6 \| 52.88 \| 41.09 \| \| \| ✗ \| ✗ \| 76.17 \| 45.56 \| 32.32 \| 72.64 \| 67.14 \| 62.33 \| 54.46 \| 43.48 \| \| \| ✗ \| ✗ \| 76.28 \| 46.39 \| 33.0 \| 72.52 \| 67.38 \| 62.87 \| 55.13 \| 44.52 \| \| SWINv2-B (87.8M)\| ✗ \| ✗ \| 79.39 \| 45.7 \| 31.91 \| 74.45 \| 68.55 \| 62.34 \| 53.66 \| 41.87 \| \| \| ✗ \| ✗ \| 79.35 \| 47.6 \| 33.42 \| 74.95 \| 69.28 \| 63.43 \| 55.08 \| 43.8 \| \| \| ✗ \| ✗ \| 79.76 \| 48.16 \| 33.34 \| 75.24 \| 69.63 \| 64.05 \| 56.08 \| 45.19 \| Table 2: Measuring calibration under distribution shifts and anomaly rejection performance of models trained on Imagenet-1K. We report the Smoothed ECE (.) and the AUROC (↑) scores to assess calibration and anomaly rejection performance respectively. For smoothed ECE, we report the mean and standard deviation across all ImageNet OOD datasets. \| Architecture \| Anchoring? \| RAM? \| Calibration (ECE) \| LSUN (C) \| LSUN (R) \| ISUN \| Textures \| Places365 \| NINCO \| \|--------------\|------------\|------\|------------------\|----------\|----------\|------\|----------\|-----------\|-------\| \| DEiT-T \| ✗ \| ✗ \| 0.116 ± 0.016 \| 94.75 \| 85.03 \| 84.02 \| 85.55 \| 67.55 \| 74.42 \| \| \| ✗ \| ✗ \| 0.116 ± 0.015 \| 96.12 \| 85.73 \| 85.95 \| 85.96 \| 70.08 \| 75.78 \| \| \| ✗ \| ✗ \| 0.112 ± 0.015 \| 94.54 \| 85.88 \| 85.26 \| 86.25 \| 70.36 \| 76.31 \| \| RegNet \| ✗ \| ✗ \| 0.154 ± 0.064 \| 98.79 \| 97.61 \| 97.77 \| 88.37 \| 83.03 \| 80.18 \| \| \| ✗ \| ✗ \| 0.164 ± 0.068 \| 98.77 \| 98.04 \| 98.01 \| 87.6 \| 83.39 \| 80.44 \| \| \| ✗ \| ✗ \| 0.144 ± 0.071 \| 98.58 \| 95.82 \| 96.42 \| 90.05 \| 83.24 \| 82.18 \| \| DEiT-S \| ✗ \| ✗ \| 0.111 ± 0.029 \| 93.6 \| 88.68 \| 87.86 \| 80.51 \| 60.66 \| 70.1 \| \| \| ✗ \| ✗ \| 0.112 ± 0.027 \| 95.04 \| 90.72 \| 90.64 \| 81.82 \| 66.23 \| 72.41 \| \| \| ✗ \| ✗ \| 0.113 ± 0.027 \| 94.93 \| 89.82 \| 89.58 \| 82.76 \| 67.33 \| 72.64 \| \| SWINv2-T \| ✗ \| ✗ \| 0.121 ± 0.034 \| 91.73 \| 78.93 \| 80.25 \| 78.83 \| 72.53 \| 77.46 \| \| \| ✗ \| ✗ \| 0.121 ± 0.032 \| 89.53 \| 78.73 \| 78.68 \| 76.64 \| 74.75 \| 76.49 \| \| \| ✗ \| ✗ \| 0.117 ± 0.027 \| 90.25 \| 78.15 \| 77.69 \| 78.09 \| 77.16 \| 78.49 \| \| SWINv2-S \| ✗ \| ✗ \| 0.126 ± 0.039 \| 94.54 \| 82.21 \| 82.89 \| 77.87 \| 70.63 \| 74.73 \| \| \| ✗ \| ✗ \| 0.122 ± 0.045 \| 95.35 \| 87.46 \| 87.73 \| 80.83 \| 76.67 \| 77.79 \| \| \| ✗ \| ✗ \| 0.119 ± 0.041 \| 94.71 \| 83.43 \| 84.18 \| 79.66 \| 74.81 \| 79.47 \| \| VITB16 \| ✗ \| ✗ \| 0.109 ± 0.037 \| 91.59 \| 87.34 \| 86.92 \| 79.24 \| 65.72 \| 65.98 \| \| \| ✗ \| ✗ \| 0.106 ± 0.035 \| 90.87 \| 85.81 \| 85.17 \| 76.88 \| 66.16 \| 68.49 \| \| \| ✗ \| ✗ \| 0.105 ± 0.028 \| 89.88 \| 85.5 \| 84.55 \| 78.91 \| 67.18 \| 70.32 \| \| SWINv2-B \| ✗ \| ✗ \| 0.132 ± 0.055 \| 95.05 \| 85.32 \| 85.32 \| 76.35 \| 65.99 \| 72.13 \| \| \| ✗ \| ✗ \| 0.129 ± 0.058 \| 95.82 \| 85.4 \| 85.98 \| 77.88 \| 70.75 \| 72.63 \| \| \| ✗ \| ✗ \| 0.124 ± 0.051 \| 95.84 \| 86.5 \| 87.34 \| 75.74 \| 73.66 \| 74.53 \| Figure 1, anchoring w/ RAM withstands high noise severity better than the other models, achieving improvements of 2% – 7% at severity 5. Calibration and Anomaly Rejection: While generalization to distribution shifts is key to improve model utility, it must be ensured that the models produce well-calibrated prediction probabilities that match the likelihood of correctness. Hence, calibration is a vital test to understand how tempered the model predictions are under shifts and ensure that they do not provide over-confident predictions to OOD inputs. On the other hand, when the inputs are semantically disparate and do not share the same label space as the training data, we require the models to appropriately flag them as anomalies. Table 3: LP-based adaptation for models trained on Imagenet-1K on target datasets. We measure the accuracy (†) of the adapted model using the validation split of the target dataset. \| Architecture \| Anchoring? \| RAM? \| DTD \| UCF101 \| Flowers102 \| Food101 \| OxfordPets \| StanfordCars \| CIFAR-10 \| CIFAR-100 \| Caltech \| Aircraft \| Average \| \|------------------\|------------\|------\|-----\|--------\|------------\|---------\|-----------\|-------------\|----------\|-----------\|---------\|----------\|---------\| \| DEIT (tiny) \| X \| X \| 63.65 \| 65.50 \| 89.96 \| 60.16 \| 90.40 \| 37.11 \| 88.51 \| 69.98 \| 91.54 \| 41.49 \| 69.78 \| \| \| \| \| 63.42 \| 65.0 \| 89.24 \| 60.44 \| 89.89 \| 36.28 \| 89.15 \| 70.82 \| 91.95 \| 41.19 \| 69.71 \| \| \| \| \| 64.3 \| 66.56 \| 90.58 \| 60.17 \| 89.1 \| 37.33 \| 89.19 \| 70.12 \| 91.97 \| 42.27 \| 70.16 \| \| RegNet \| X \| X \| 68.09 \| 70.34 \| 94.03 \| 66.33 \| 90.43 \| 53.55 \| 91.88 \| 73.21 \| 94.0 \| 54.67 \| 75.58 \| \| \| \| \| 66.67 \| 72.14 \| 93.91 \| 65.89 \| 90.9 \| 50.21 \| 90.21 \| 73.12 \| 93.6 \| 51.97 \| 74.86 \| \| \| \| \| 68.03 \| 71.21 \| 94.29 \| 66.24 \| 90.76 \| 52.87 \| 92.03 \| 74.74 \| 93.38 \| 54.64 \| 75.92 \| \| SWIN-v2 (tiny) \| X \| X \| 72.1 \| 74.28 \| 96.37 \| 72.5 \| 92.53 \| 55.76 \| 91.82 \| 74.89 \| 94.53 \| 57.49 \| 78.13 \| \| \| \| \| 71.87 \| 74.91 \| 95.41 \| 73.16 \| 93.51 \| 57.23 \| 91.78 \| 75.88 \| 93.96 \| 57.64 \| 78.54 \| \| \| \| \| 71.34 \| 74.97 \| 94.88 \| 72.68 \| 92.34 \| 54.40 \| 92.12 \| 75.87 \| 94.58 \| 58.15 \| 78.14 \| \| DEIT (small) \| X \| X \| 68.56 \| 73.57 \| 93.5 \| 67.17 \| 91.71 \| 49.61 \| 92.46 \| 75.89 \| 94.05 \| 48.96 \| 75.55 \| \| \| \| \| 68.32 \| 73.91 \| 93.5 \| 68.41 \| 92.15 \| 50.62 \| 93.63 \| 77.81 \| 94.69 \| 48.63 \| 76.17 \| \| \| \| \| 68.5 \| 74.94 \| 93.46 \| 68.68 \| 92.37 \| 50.0 \| 93.41 \| 77.34 \| 94.81 \| 49.29 \| 76.34 \| \| VITb16 \| X \| X \| 69.21 \| 76.13 \| 94.97 \| 71.22 \| 92.59 \| 59.96 \| 95.58 \| 81.82 \| 95.22 \| 56.44 \| 79.31 \| \| \| \| \| 68.97 \| 75.42 \| 94.32 \| 70.94 \| 91.74 \| 61.56 \| 95.35 \| 81.91 \| 95.17 \| 58.6 \| 79.4 \| \| \| \| \| 68.79 \| 76.47 \| 94.44 \| 71.25 \| 92.4 \| 61.95 \| 98.1 \| 82.07 \| 94.94 \| 58.54 \| 79.7 \| To that end, we conduct an extensive evaluation of model calibration under distribution shifts using the ImageNet-C/C/R/S/V2 variants, and for anomaly rejection, we consider the benchmarks: (i) LSUN (C) (Yu et al., 2015), (ii) LSUN (R) (Yu et al., 2015), (iii) iSUN, (iv) Textures (Cimpoi et al., 2014b), and (v) Places365 (Zhou et al., 2017) and (vi) NINCO (Bitterwolf et al., 2023), a recent OOD benchmark which comprises of images with semantic overlap with ImageNet but with no class overlap. We report the calibration and the anomaly rejection performance of all models in Table 2. It can be observed that, incorporating RAM leads to significantly improved model calibration irrespective of the choice of architecture, thus demonstrating the importance of increasing $P(\Delta)$ diversity. Similar to any existing regularization strategy (e.g., Mixup) adopted for improving generalization, RAM with high $\alpha$ might have the risk of producing models with reduced sensitivity towards anomalous data (i.e., compared to anchoring w/o RAM). However, we find that with $\alpha = 0.2$, the anomaly rejection trade-off is minimal and interestingly, it can even lead to higher rejection AUROC scores on challenging cases (e.g., NINCO) using networks with higher capacity. Downstream Adaptation: To investigate the effectiveness of the features obtained through the proposed training strategies, we employ two evaluation protocols: Adaptation (ID Eval.) and Adaptation (OOD Eval.). While both protocols involve keeping the feature extractor $F_\theta$ (pre-trained on ImageNet) frozen and training a linear probe on the acquired features, the protocols primarily differ in terms of the distribution of the test set. In Adaptation (ID Eval.), we assume that the distribution of the dataset used for linear probing is the same as that of the test set, for example, performing linear probing on UCF-101 train set and evaluating on UCF-101 test samples. This protocol enables us to explore the transferability of our features under various task shifts. Through Adaptation (OOD Eval.), we aim to study the transferability and generalizability of the features under both task and distribution shifts. To this end, we first train the linear probe with a dataset that introduces a task shift compared to ImageNet. We then evaluate the linear probe with data drawn from a different distribution, characterized by covariate shifts, compared to the probing dataset. Note, that we only consider covariate shifts for our evaluation. Adaptation (ID Eval): We consider the following suite of target datasets representing varying levels of task shifts. (i) UCF101 (Soomro et al., 2012); (ii) Food101 (Bossard et al., 2014); (iii) Flowers102 (Nilsback & Zisserman, 2008); (iv) OxfordPets (Parkhi et al., 2012); (v) StanfordCars (Krause et al., 2013); (vi) DTD (Cimpoi et al., 2014a); (vii) Caltech101 (Fei-Fei et al., 2004); (viii) FGVC-Aircraft (Maji et al., 2013); (ix) CIFAR-10 (Krizhevsky et al., 2009); and (x) CIFAR-100 (Krizhevsky et al., 2014) datasets. We employ LogisticRegression from Scikit-Learn to derive our linear probes. The regularization parameter $C$ is determined through k-fold cross-validation. From Table 3, we can see that, the proposed masked training of anchored models consistently yields adaptation gains on multiple datasets across different architectures. Notably, we observe an upward trend compared to non-anchored models with an increase in architecture complexity. These results Table 4: LP-based adaptation for models trained on DomainNet on domains Real and Sketch respectively. Contrasting Table 3, we measure the zero-shot accuracy (†) of the adapted model on the remaining domain shifted datasets in DomainNet. \| Architecture \| Anchoring? \| RAM? \| Train: Real \| Train: Sketch \| \|--------------\|------------\|------\|-------------\|--------------\| \| \| \| \| Real \| Sketch \| Clipart \| Painting \| Average \| Real \| Sketch \| Clipart \| Painting \| Average \| \| DEiT-T \| ✗ \| ✗ \| 75.03 \| 19.78 \| 28.9 \| 39.52 \| 40.81 \| 36.35 \| 42.79 \| 27.23 \| 26.54 \| 33.23 \| \| \| ✗ \| ✔ \| 75.1 \| 20.77 \| 29.45 \| 39.68 \| 41.25 \| 36.48 \| 43.58 \| 28.21 \| 27.17 \| 33.86 \| \| \| ✔ \| ✗ \| 74.72 \| 20.07 \| 28.1 \| 38.94 \| 40.86 \| 35.68 \| 42.77 \| 27.23 \| 26.32 \| 33.0 \| \| \| ✔ \| ✔ \| 79.0 \| 20.78 \| 32.18 \| 41.4 \| 43.34 \| 38.73 \| 47.49 \| 29.87 \| 27.78 \| 35.97 \| \| RegNet \| ✗ \| ✗ \| 78.61 \| 20.99 \| 32.37 \| 40.34 \| 43.08 \| 38.39 \| 47.41 \| 30.03 \| 26.86 \| 35.67 \| \| \| ✗ \| ✔ \| 78.52 \| 20.49 \| 32.04 \| 41.34 \| 43.1 \| 38.22 \| 47.48 \| 29.9 \| 27.97 \| 35.89 \| \| \| ✔ \| ✗ \| 80.88 \| 22.69 \| 34.91 \| 44.09 \| 45.64 \| 44.22 \| 50.3 \| 33.78 \| 31.91 \| 40.05 \| \| SWINv2-T \| ✗ \| ✗ \| 81.0 \| 22.89 \| 34.83 \| 44.11 \| 45.71 \| 42.0 \| 49.66 \| 32.5 \| 30.8 \| 38.74 \| \| \| ✗ \| ✔ \| 80.77 \| 23.09 \| 35.42 \| 44.3 \| 45.9 \| 42.23 \| 49.35 \| 32.61 \| 30.54 \| 38.68 \| \| \| ✔ \| ✗ \| 79.47 \| 25.2 \| 35.31 \| 45.01 \| 46.25 \| 42.71 \| 50.67 \| 34.02 \| 31.23 \| 39.66 \| \| DEiT-S \| ✗ \| ✗ \| 79.78 \| 25.76 \| 36.75 \| 45.22 \| 46.88 \| 42.3 \| 51.72 \| 35.28 \| 31.94 \| 40.31 \| \| \| ✗ \| ✔ \| 79.4 \| 25.92 \| 36.04 \| 45.54 \| 46.72 \| 43.18 \| 51.75 \| 34.96 \| 32.04 \| 40.48 \| \| \| ✔ \| ✗ \| 80.7 \| 25.83 \| 37.38 \| 46.3 \| 47.56 \| 41.35 \| 53.61 \| 35.4 \| 31.42 \| 40.44 \| \| VITb16 \| ✗ \| ✗ \| 80.41 \| 27.67 \| 39.48 \| 46.66 \| 48.56 \| 44.55 \| 55.02 \| 38.24 \| 33.39 \| 42.8 \| \| \| ✗ \| ✔ \| 80.48 \| 28.02 \| 38.98 \| 46.97 \| 48.61 \| 44.81 \| 55.3 \| 37.76 \| 32.7 \| 42.64 \| Figure 2: Evaluating OOD generalization for DeIT(S) models on Imagenet, comparing non-anchored and RAM variants trained with varying levels of label noise. Each sub-plot depicts the top-1 accuracy across 3 severities from Imagenet-C/C. We observe that RAM offers increased robustness to label noise compared to the non-anchored counterparts providing an accuracy improvement of ≈2%. Thus indicate that anchored training (w/ and w/o RAM) provide feature representations that are transferable even with sophisticated architectures. Adaptation (OOD Eval): For this task, we use DomainNet (Peng et al., 2019), a large-scale benchmark with images from multiple domains, with images from 345 categories. More specifically, we pick four domains, namely real, sketch, clipart, and painting, to evaluate generalization of adapted probes under challenging domain shifts. We conduct two sets of out-of-domain (OOD) evaluations – one involves training the linear probe with images from the real domain, and the other with images from the sketch domain. We then directly test the linear probes on the held-out domains. From Table 4, we make similar findings as above, that anchored models produce richer representation than their non-anchored counterparts, thus leading to improved transferability. RAM regularization continues to provide improvements, and especially under complex shifts (e.g., sketch → painting), with large architectures (e.g., VITb16), we notice non-trivial performance improvements. 4.3 ANALYSIS Impact of Training Label Noise: In the previous section, we evaluated the effectiveness of different models on generalization to OOD corruptions. While we found RAM to be particularly effective, here, we seek to further evaluate its efficacy in the realistic, but more challenging setting where the training data may be compromised for e.g., by label noise (Chen et al., 2023). Introducing label noise during training adds confusion to the system, challenging the model to maintain robustness to noise while preserving generalization. To that end, we randomly flip the labels of 1% of the data during the training of a DEIT(S) model on ImageNet, and repeat this experiment with varying levels of label noise $ l = \{5, 20, 30\} $. Subsequently, we evaluate OOD generalization on ImageNet C/C. Figure 2 illustrates that, even with higher levels of label noise and increasing data corruption complexities, anchored training w/ RAM demonstrates robustness and superior generalization compared to the non-anchored model. Table 5: Evaluating OOD Generalization, Calibration, Anomaly Rejection, and ID Adaptation Performance for CIFAR-10/100 models trained with and without RAM. In most cases, anchored models, particularly with RAM, consistently outperform non-anchored and standard anchoring baselines in generalization, calibration, and adaptation across different architectures. \| Dataset \| Architecture \| Anchoring? \| RAM? \| Corruption Accuracy (%) \| Calibration (±) \| Ano. Rej. (%) \| Adaptation-ID (%) \| \|---------\|--------------\|------------\|------\|------------------------\|----------------\|---------------\|------------------\| \| CIFAR-10 \| ResNet-18 \| ✗ \| ✗ \| 89.44, 84.47, 77.91, 70.74, 58.72 \| 0.15 ± 0.07 \| 92.46 \| 19.05 \| \| \| \| ✗ \| ✔ \| 88.99, 84.28, 79.16, 72.09, 59.82 \| 0.14 ± 0.07 \| 91.38 \| 22.15 \| \| \| \| ✔ \| ✔ \| 90.98, 87.15, 83.17, 77.81, 67.26 \| 0.09 ± 0.05 \| 94.19 \| 22.61 \| \| CIFAR-100 \| ResNet-18 \| ✗ \| ✗ \| 61.2, 53.6, 48.8, 42.6, 33.3 \| 0.12 ± 0.04 \| 75.96 \| 31.61 \| \| \| \| ✗ \| ✔ \| 64.5, 55.5, 49.9, 43.3, 33.1 \| 0.13 ± 0.04 \| 83.58 \| 32.45 \| \| \| \| ✔ \| ✔ \| 67.78, 59.63, 54.34, 48.03, 37.87 \| 0.13 ± 0.04 \| 87.48 \| 34.1 \| \| WRN 40-2 \| ResNet-18 \| ✗ \| ✗ \| 62.26, 52.82, 46.85, 40.12, 30.05 \| 0.26 ± 0.06 \| 83.76 \| 33.91 \| \| \| \| ✗ \| ✔ \| 64.55, 55.47, 49.43, 42.84, 32.75 \| 0.24 ± 0.06 \| 84.79 \| 33.64 \| \| \| \| ✔ \| ✔ \| 66.0, 57.77, 52.33, 45.64, 35.52 \| 0.19 ± 0.06 \| 79.42 \| 37.08 \| Impact on Dataset Size: To better understand the behavior of anchored training with smaller datasets, we conducted extensive experiments involving the training of ResNet-18 models on CIFAR-10 and CIFAR-100 datasets, along with WideResNet40-2 (WRN-40-2) on CIFAR-100, following hyper-parameters and training configurations as in (Thiagarajan et al., 2022). Similar to our ImageNet experiments, we considered OOD generalization, calibration, anomaly rejection and adaptation performance evaluations. Remarkably, our results, detailed in Table 5, highlight that the incorporation of RAM regularization during anchored training significantly enhances robustness to corruptions across all severity. The improvement in accuracy under corruptions surpasses 5.2% compared to the non-anchored model and even outperforms standard anchoring by more than 4%. Beyond generalization, anchoring improves calibration errors and anomaly rejection fidelity. Notably, the anchoring variant with RAM achieves the lowest calibration error on average, showcasing its effectiveness. In anomaly rejection, RAM outperforms standard anchoring by almost 3% and non-anchored training by about 2%. Finally, linear probing on CIFAR-10 trained models provides consistent gains, with anchored models showing an average improvement of over 3.5%. We extended our study to WRN-40-2, on CIFAR-100 and observed the persistence of benefits in OOD generalization, calibration and adaptation. Interestingly, improvements are evident even at lower severity levels on CIFAR-100 compared to CIFAR-10. The adaptation results in Table 5 further underscore the clear advantages of RAM across both architectures. While anomaly rejection gains were substantial for ResNet-18, WRN-40-2 exhibited a trade-off between OOD generalization and anomaly rejection (discussed in Section A of the appendix). 5 CONCLUSION Through this work, we find that, across varying dataset sizes (CIFAR-10 to ImageNet), model architectures (ResNet to ViT) and network sizes (5M to 88M parameters), anchored training can provide significant gains in OOD generalization, anomaly rejection and adaptation, compared to conventional training. In particular, when the train recipe includes high-capacity architectures or advanced mechanisms (e.g., Mixup, EMA, label-smoothing, cutmix), anchored training tends to provide bigger performance gains over the base models. However, we realize that state-of-the-art results in OOD generalization are often obtained using model souping (Wortsman et al., 2022) or by fine-tuning large scale pre-trained models (Goyal et al., 2023). Hence, we believe an important future direction of work is to integrate anchoring (w/ RAM) into these approaches. While we have not theoretically characterized the behavior of RAM regularization, our hypothesis is rooted in existing theory and our empirical results provide evidence for the hypothesis. However, building upon the empirical success of anchoring, carrying out a theoretical study on generalization in anchored models is crucial. REFERENCES Anders Andreassen, Yasaman Bahri, Behnam Neyshabur, and Rebecca Roelofs. 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In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pp. 19338–19347, June 2023. Chuan Guo, Geoff Pleiss, Yu Sun, and Kilian Q Weinberger. On calibration of modern neural networks. In International Conference on Machine Learning, pp. 1321–1330. PMLR, 2017. Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 770–778, 2016. Dan Hendrycks and Thomas Dietterich. Benchmarking neural network robustness to common corruptions and perturbations. In International Conference on Learning Representations, 2019. URL https://openreview.net/forum?id=HJz6tiCqYm.

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