Unraveling the Key Components of OOD Generalization via Diversification
Authors: Harold Luc Benoit
Liangze Jiang
Andrei Atanov
Oguzhan Fatih Kar
Mattia Rigotti
Amir Zamir
Venue: ICLR 2024 poster (2024)
Keywords: Algorithm Design
Diversity
OOD Generalization
Spurious Correlation
Understanding Neural Networks
Primary area: general machine learning (i.e., none of the above)
TL;DR: We distill the critical design factors of current state-of-the-art methods (multi-hypotheses/diversification methods) for spurious correlation situations.

Abstract: Supervised learning datasets may contain multiple cues that explain the training set equally well, i.e., learning any of them would lead to the correct predictions on the training data. However, many of them can be spurious, i.e., lose their predictive power under a distribution shift and consequently fail to generalize to out-of-distribution (OOD) data. Recently developed "diversification" methods (Lee et al., 2023; Pagliardini et al., 2023) approach this problem by finding multiple diverse hypotheses that rely on different features. This paper aims to study this class of methods and identify the key components contributing to their OOD generalization abilities.

We show that (1) diversification methods are highly sensitive to the distribution of the unlabeled data used for diversification and can underperform significantly when away from a method-specific sweet spot. (2) Diversification alone is insufficient for OOD generalization. The choice of the used learning algorithm, e.g., the model's architecture and pretraining, is crucial. In standard experiments (classification on Waterbirds and Office-Home datasets), using the second-best choice leads to an up to 20\% absolute drop in accuracy. (3) The optimal choice of learning algorithm depends on the unlabeled data and vice versa i.e. they are co-dependent. (4) Finally, we show that, in practice, the above pitfalls cannot be alleviated by increasing the number of diverse hypotheses, the major feature of diversification methods.

These findings provide a clearer understanding of the critical design factors influencing the OOD generalization abilities of diversification methods. They can guide practitioners in how to use the existing methods best and guide researchers in developing new, better ones.

Published as a conference paper at ICLR 2024
UNRAVELING THE KEY COMPONENTS OF OOD GENERALIZATION VIA DIVERSIFICATION
Harold Benoit".1.
Liangze Jiang
Andrei Atanov
Oguzhan Fatih Kar"
Mattia Rigotti²
Amir Zamir!
Swiss Federal Institute of Technology (EPFL)
IBM Research
ABSTRACT
Supervised learning datasets may contain multiple cues that explain the training set equally well, i.e., learning any of them would lead to the correct predictions on the training data. However, many of them can be spurious, i.e., lose their predictive power under a distribution shift and consequently fail to generalize to out-of- distribution (OOD) data. Recently developed "diversification" methods (Lee al.] 2023 Pagliardini et al.] (2023) approach this problem by finding multiple diverse hypotheses that rely on different features. This paper aims to study this class of methods and identify the key components contributing to their OOD) generalization abilities. We show that (1) diversification methods are highly sensitive to the distribution of the unlabeled data used for diversification and can underperform significantly when away from a method-specific sweet spot. (2) Diversification alone is insufficient for OOD generalization. The choice of the used learning algorithm, e.g., the model's architecture and pretraining, is crucial. In standard experiments (classification on Waterbirds and Office-Home datasets), using the second-best choice leads to an up to 20% absolute drop accuracy. (3) The optimal choice of learning algorithm depends on the unlabeled data and vice versa, i.e., they are co-dependent. (4) Finally, we show that, in practice, the above pitfalls cannot be alleviated by increasing the number of diverse hypotheses, the major feature of diversification methods. These findings provide a clearer understanding of the critical design factors influencing the OOD generalization abilities of diversification methods. They can guide practitioners in how to use the existing methods best and guide researchers in developing new, better ones.
INTRODUCTION
Achieving out-of-distribution (OOD) generalization is crucial milestone for the real-world de- ployment of machine learning models. A core obstacle in this direction is the presence of spurious features, i.c., features that are predictive of the true label on the training data distribution but fail under a distribution shift, They may appear due to, for example, a bias in the data acquisition process Lakden--Ranee et al. 2020)) or environmental cue closely related to the true predictive feature (Beery et 2018).
The presence of spurions correlation between spurious features and true underlying labels implies that there are multiple hypotheses (i.e., labeling functions) that all describe training data equally well, i.e., have a low training error, but only some generalize to the OOD test data. Previous works (Atanov jtt al. 2022] Battaglia et al.) 2018, Shah et al.) 2020)) have shown that in the presence of multiple predictive features, standard empirical risk minimization (Vapnik 199T) (ERM) using neural networks trained with stochastic gradient descent (SGD) converges to a hypothesis that is most aligned with the learning algorithm's inductive biases. When these inductive biases are not aligned well with the true underlying predictive feature, can cause ERM to choose a wrong (spurious) feature and, consequently, fail under distribution shift.
"Equal contribution. Corresponding author: wenald..hnnit@aaaumnen

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Published as a conference paper at ICLR 2024
Banss
Daps
(a) Original data
(b) Unlabeled data changes (c) Learning algo. changes
Figure 1: Diversification is a two-legged problem where unlabeled data and learning algorithm both matter and are co-dependent. and represent the training data points and their labels. represents unlabeled data. fERM represents the hypothesis found by empirical risk minimization (ERM), thus reflecting the inductive bias of the learning algorithm. f& represents second diverse hypothesis found by diversification method; it has low error on training data as Σрни does, but disagrees with on the unlabeled data. Compared (a) the original setting, we study how changing (b) unlabeled data and (c) the learning algorithm yield different solutions and, therefore, performance.
Recently, diversification methods (Lee et al., 2023 Pag ligddiidiii et al. 2023) have achieved sfate-of-the- art results classification settings in the presence of spurious correlations. Instead of training single model, these methods aim to find multiple plausible and diverse hypotheses that all describe the training data well, while relying on different predictive features, which is usually done by promoting different predictions on additional unlabeled data. The motivation is that among all the found features, there will be the true predictive one that causally linked to the label and, therefore, remains predictive under a distribution shift.
In this work, we identify and study the key factors that contribute the success of these diversification methods, adopting (Lee et al. (2033 Pagliardini et al. (023) as two recently proposed best-performing representative methods. Our contributions are as follows.
First, through theoretical and empirical analyses, show that diversification methods are sensi- tive to the distribution of the unlabeled data (Fig. [[a]]s vs. [bb). Specifically, each diversification method works best for different distributions of unlabeled data, and the performance drops significantly (up to 30% absolute accuracy) when diverging from the optimal distribution.
Second, we demonstrate that diversification alone cannot lead to OOD) generalization efficiently without additional biases. This similar to the in-distribution generalization with ERM, where "good"" learning algorithm's 's inductive biases are necessary for generalization (Vapnik & Cher vonenkis 2015). In particular, we show that these methods are sensitive to the choice of the architecture and pretraining method (Fig. [(a vs. [(c), and the deviation from best to second best model choice results in significant (up 20% absolute) accuracy drop (see Fig. Fig.
Further, we show that a co-dependence exists between unlabeled data and the learning algorithm, i.c., the optimal choice for one depends on the other. Specifically, for fixed training data, we can change unlabeled data in a targeted way to make one architecture (c.g., MLP) generalize and the other (e.g.. ResNet18) to have random guess test accuracy and vice versa.
Finally, we show that one of the expected advantages of diversification methods increasing the number of diverse hypotheses to improve OOD generalization does not hold up in practice and does not help alleviate the aforementioned pitfalls. Specifically, we do not observe any meaningful improvements using more than two hypotheses.
These findings provide a clearer understanding of the relevant design factors influencing the OOD generalization of diversification methods. They can guide practitioners in how to best use the existing methods and guide researchers in developing new, better ones. We provide guiding principles distilled from our study in each section and Sec. 6
2 RELATED WORK
Spurious correlation and underspecification. As special case of OOD generalization problem. spurious correlations can arise from the underspecified nature of the training data (D' D''Auuu et al. 2020 Koh al. 2021 Liang & Zou| 2022). In this setting, neural networks tend to learn simple (spurious) concepts rather than the true causal concepts, a phenomenon known as simplicity bias (Shah et al. 2020 Huh et al. 2023) shorteut learning (Geirhos et al. 2020 Scimeca et al.! 2022).
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Some works combat spurious correlation by improving worst-group performance (Sagawa al.
2020
Hu et al. 2018 Zhang et al. 202I), some require group annotation (Sagawa et al. [2020] Zhang
et al.
2022 Creager al. (0021) and others (Liu et al. [222] LaBonte et al. 2022; Sohoni et al.,
2020) aim at the no group information scenario. Diversification methods fit into the latter, as they only rely additional unlabeled data to promote diversity between multiple hypotheses.
Diversification methods. Recently proposed diversification methods (Lee et al., 2023 Pagliardini
al. 2023; Teney et al., 2022a b) find multiple diverse hypotheses during training handle spurious correlations. They introduce an additional diversification loss over multiple trained hypotheses, forcing them to rely on different features while still fitting training data well. (Teney et al. (2022a b) use input-spuce diversification that minimizes the alignment of input gradients over pairs of models at all training data points. DivDis (Lee et al. 2023) and D-BAT (Pagliardini et al., 2023) use output-space diversification, minimizing the agreement between models' predictions on additional unlabeled data. We focus on studying the latter, as these methods outperform the input-space ones by a large margin, achieving state-of-the- performance in the setting where true labels are close to or completely correlated with spurious attributes.
Inductive biases in learning algorithms. Different learning algorithms have different inductive biases (Shalev-Sh Shwlte & Ben-David 2014. Hüllermeier et al. 2013), which makes a given algorithm
prioritize a specific solution (Gunasekar et al. 2018: Ji & Telgarsky 2020). While being highly
overparameterized (Allen-Zhu et al.) (019) and able to fit even random labels (Zhang et al.
2017).
deep learning models were shown to benefit from architectural (Xu et al.) [022 Naseer et al.) 2021)
optimization (Kalimeris et al. 2019 Liu et al.) 2020) and pre-training (Immer et 20222 Lovering cct al., 2021) inductive biases. In this work, we study the influence of the choice of the learning algorithm and, hence, its inductive bias the performance of diversification methods. We show that diversification methods are sensitive to the choice of architecture and pretraining method.
3 LEARNING VIA DIVERSIFICATION
First, we formalize the problem of generalization under spurious correlation. Then, we present diversification framework along with the recent representative methods, DivDis (Lee et al. 2023) and D-BAT (Pagliardini et al. 2023 describing key differences between them: training strategies (sequential vs. simultaneous) and diversification losses (mutual information vs. agreement).
3.1 PROBLEM FORMULATION
For consistency, we follow a notation similar to that of D-BAT (Pagliardini et al. 2023). Let X' be the input space, y the output space. Both methods focus on classification, i.c. ) =!,..... (0,....g 9 1}. where q is the number of classes. We define a domain (D, h) as a distribution D over X and a hypothesis (labeling function) h: X' + V. The training data is drawn from the domain (D,, h²), and test data from a different domain (Dood, h)).. Given any domain (D, I(),,, a hypothesis ht and a loss function (e.g. cross-entropy loss) L: y x y + IR+, the expected loss is defined as: Ln h/) = EzZD [[(h()), (x))) Let H be the set of hypotheses expressed by given learning algorithm. We define Hi and H to be the optimal hypotheses set on the train and the OOD domains:
Hi == argmin CD, hh) Hood == argmin DDoa (h,""). hEH
(I)
hEW
Definition 1. (Spurious Ratio) Given spurious hypothesis h, the spurious ratio r with respect to a distribution D and its labeling function h² is defined as the proportion of data points where h and h agree, i.e., have the same prediction: T'D EE=D [h" (z) h(x)].
The spurious ratio describes how the spurious hypothesis h correlates with the true ht on data D. A spurious ratio of indicates that given data distribution has a complete spurious correlation. On the contrary, a spurious ratio of 0 indicates that the spurious hypothesis is always in opposition to the true labeling, namely inversely correlated. Finally, a spurious ratio of 0.5 means that the spurious hypothesis is not predictive of h², as there is no correlation between them. We will also refer to this setting as a "balanced" data distribution. We omit h" (and sometimes h) in the notation keep less cluttered, as they can be inferred from the context of a specific setting.
At the time of writing, these are the pest-pernogning diversification methods and the only existing output- space ones.
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Spurious correlation setting. In this setting, we assume that there exist one or more spurious hypotheses h E Hsp C Mi Hood, which generalize on Dr but not on Dood. Thus, the spurious ratio on training data is close to one: D. R 1. If there is a misalignment between the inductive bias of the learning algorithm and Hood the ERM hypothesis MERM may be closer to hypotheses from Hap than Hi n Hood i.c., have poor OOD generalization. The idea of diversification methods is to find multiple hypotheses from Hi with the aim to have one with good OOD generalization (see Sec.
3.2 DIVERSIFICATION FOR OOD GENERALIZATION
DivDis and D-BAT focus on the above spurious correlation setting. They assume access to additional unlabeled data D to find multiple diverse hypotheses that all fit the training data (De, h²) but disagree, i.e., make diverse predictions, on D,.. The motivation is to better cover the space H2 and, consequently, find a hypothesis from H2 n Hood that also generalizes to OOD data.
Optimization objective. Following DivDis and D-BAT, we define a diversification loss Ap. (h1 h2) that quantifies the agreement between two hypotheses on D,. Then, the case of finding K hypotheses, the training objective of diversification method is the sum of ERM loss and the diversification loss averaged over all pairs of hypotheses:
b1y lig argmin bi....ng ER
Σ Σ CD, h²)++ + K(K i=1 iØj
(hi,
(2)
Diversification loss. Let Ph., be the predictive distribution of hypothesis h; on given data D. Then, (Pr, The first term the mutual information, which is equal to 0 iff Ph, and Ph,, are independent. The second term is the KL-divergence between Ph, and prior distribution P (usually set the distribution of labels in Dz). It prevents hypotheses from collapsing to degenerate solutions.
DivDis [Lee et al. 2023): (11, h22 DEL (P .n2) l4h, @ Prrr )+ A
D-BAT (Pagliardini et al. 2023): y An (11,,,2
Ez--n- Ph., (ar; 0) Phg (2;; 1) + Pry (z; 1).
Phg (2;; Oj)], where Pr. [2; y) is the probability of class y predicted by h4-
In practice, they are computed and optimized on additional unlabeled data D,.- Note that it is usually
favorable to have the distribution of D, different from that of D,, i.e., r < rb, R 1, as this enables the diversification process to distinguish between spurious and semantic hypotheses (This is also confirmed by empirical results in Fig 2igght)ght) In Sec. [ we will show that both losses have their strengths, and the optimal choice depends on the spurious ratio of D,..
Sequential vs. simultancous optimization. In practice, when minimizing the diversification objective in Eq. 2 there are two choices: (i) optimize over all hypotheses simultaneously (ii) find hypotheses one by one. DivDis trains simultaneously and defines hypotheses as linear classifiers that share the same feature extractor. D-BAT, on the contrary, starts with h1 - hERM and finds new hypotheses, defined separate models sequentially. For consistency and comparability with D-BAT in Sec. D aanalysis, we also introduce DivDis-Seq. a version of DivDis using sequential optimization, allowing us concentrate only on the difference in diversification loss design.
The two-stage framework. After finding K hypotheses, one needs to be chosen to make the final prediction, leading to a two-stage approach (Lee et al., 2023): 1. Diversification: find K diverse hypotheses Hig C HG. 2. Disambiguation: select one hypothesis h H/sx given additional information (c.g., a few labeled test examples or human supervision). We identify the diversification stage as the most critical one. Indeed, if the desired hypothesis h² is not chosen (h* & Hx)) the second stage cannot make up for it as it is limited to only hypotheses from Hig We, therefore, focus on the first stage and assume an oracle that chooses the best available hypothesis in the second stage.
THE RELATIONSHIP BETWEEN UNLABELED DATA AND OOD GENERALIZATION VIA DIVERSIFICATION
In this section, we study how the different diversification losses of DivDis and D-BAT interact with the choice of unlabeled data. In an illustrative example and real- weal-world datasets, we identify that neither of the diversification losses is optimal in all scenarios and that their behavior and performance are highly dependent on the spurious ratio of the unlabeled OOD data D...
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80
Method
D-BAT
DivDis
64
Test (90
hypotheses
56
-
0.5
(lnversely Correlated)
(Balanced)
(Completely Spurious]
0.25
Class
Unatoksddtt
Spurious ratio on unlabeled data
Figure 2: Performance of diversification is highly dependent on unlabeled OOD data. Left: Top-left quadrant: The 2D binary classification task. Other quadrants: Show the second hypotheses (arrows are normal vectors) found by D-BAT (h2B) and DivDis-Seq (h22D) with varied spurious ratios of unlabeled OOD data rD.., {0, 0.25, 0.5) (from inversely correlated balanced). Right:
Best hypothesis test accuracy of D-BAT & DivDis(-Soc on MNIST-CIFAR (M/C) for varied spurious
ratios rD. and number of hypotheses K. The test accuracy measured on hold-out balanced data
Dood (i.e., DDodd 0.5, no spurious correlation).
4.1 THEORETICAL AND EMPIRICAL STUDY OF A SYNTHETIC EXAMPLE
Synthetic 2D binary classification task. In Fig. Left, we show 2D task with distribution
Dood spanning a 2D square, i.e., {x = (I1, T2) E =]] We define our hypothesis space H
to be all possible linear classifiers hu(:;; 3) where 3 is the radian of the classification plane w.r.t
horizontal axis T1- The ground truth labeling function is defined as h (x) = h(2;
5 [[2 > 0}
where Z is the indicator function, and the training distribution is defined as Dj = { [2) E
{[-1, x [0, 1}} U {[0, 1] [-1,0]] O]}}. We then define a spurious feature function hap (x)
(x:00)
I( xx= < 0} and assume that ERM converges to hap- This means that the first hypothesis
h (D-BAT) and h, (DivDis-Sev -ee)) of both methods converge to hap Finally, we define different distributions of unlabeled data Du to have different spurious ratios TD., from O to 0.5 (the construction described in Appendix(A) (A).
Proposition 1. (On Optimal Diversification Loss) the synthetic 2D binary task, let h hD and hD be the second hypotheses of D-BAT and DivDis-Seg, respectively. IfrD.. O, then hD h* and hDD h(:; 3). Otherwise, if TD. 0.5, then h h(x; πt) = 1 hap and h h". Increasing the spurious ratio DDo.foom 0 to 0.5 will lead to hD and hDD rotating counterclockwive.
See Appendix A for the full proof and Fig. Left for the visual demonstration. This proposition implies that D-BAT recovers h" when rp. = O (i.e., inversely correlated) and DivDis recovers h when r r... = 0.5 (i.e., balanced). For D-BAT, this happens because the optimal second hypothesis hB is the hypothesis that disagrees with hPs on all unlabeled data points i.e. hD {h H2 h(x) * 1 (x)) Vr Du). On the contrary, the optimal second hypothesis for DivDis- Seq is independent of the first one, i.e., disagrecing on half of the data points hDD {h H2 D. [h(x)= = P (x) = }.
In Fig. 2 Left, we empirically demonstrate this behavior by training linear classifiers with D-BAT S on such synthetic data, with 0.5k training 5k unlabeled OOD data points (following Lee et al., ) We observe that the behavior suggested in Proposition I is consistent with our experiments. This highlights that different diversification losses only recover the ground truth function in different specific spurious ratios.
and Div Sec
4.2 VERIFICATION ON REAL- WORLD IMAGE DATA
We further evaluate whether the suggested behavior holds with more complex classifiers and more complex datasets. Specifically, we use M/C (Shah al. 2020) and M/F (Pagliardini al. 2023).
II Appendix B we show additional results with more complex classifiers (i.c., MLP). For completeness, we also provide results with DivDis, which is deferred Appendix
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which are datasets that concatenate one image from MNIST with one image from either CIFAR- 10 Krizhevsky & Hinton 2009) or Fashion-MNIST (Xiao et al., 2017). We follow the setup of Lee et al. (2023); Paghardimi et al. (2023): we use Os and is from MNIST I and two classes from Fashion- MNNST (coats & dresses) and CIFAR- 10 (cars & trucks). The training data is designed to be completely spuriously correlated (e.g., g., Os always occur with cars and Is with trucks in M/C). We vary the spurious ratio TD. of the unlabeled data by changing the probability of Os occurring with cars/dresses. We use LeNet (Lecun et al. 1998) architecture for both D-BAT and DivDis(-Seq) methods.
Fig. Right shows that similar to Proposition D D-BAT is optimal when rD. = O whereas DivDis(- Seq) optimal setting is rD. 0.5. Both methods observe drastic decrease in performance away from their "sweet spot" (with up to 30% absolute accuracy drop). Note that
expected that both methods reach chance-level accuracy when rD. 1, it means that the spurious hypothesis becomes completely correlated to the true hypothesis on D .. and it is thus impossible to differentiate them by enforcing diversification on D,. In Appendix D Fig. [ shows that the same observation holds for the M/F dataset, and Tab. 4 also shows the results of different spurious ratios on a larger and more realistic dataset, CelebA-CC CC (Liu ct al. 2015 Lee et al., 2023).
Lee et al. )023); Pagliardini et al. (2023) note that both the number of hypotheses K and the diversification coefficient a (Eq. 2 are critical hyper-parameters, that may greatly influence the performance. However, controlling for these variables, in Fig. Right and Fig. [0 we find that
tuning and K is not sufficient to compensate the performance loss from the misalignment between unlabeled OOD data and the diversification loss.
Takeaway. Diversification methods' performance drops drastically when away from the spurious ratio (Def. TT "sweet spot", and neither diversification loss is optimal in all cases. Therefore, new methods should be designed to adapt to different unlabeled data distributions.
5 THE RELATIONSHIP BETWEEN LEARNING ALGORITHM AND OOD GENERALIZATION VIA DIVERSIFICATION
In this section, we study another key component of diversification methods the choice of the learning algorithm. First, we present theoretical result showing that diversification alone insufficient to achieve OOD generalization and requires additional biases (e.g.. the inductive biases of the learning algorithm). Then, we empirically demonstrate the high sensitivity of these methods to the choice of the learning al algorithm (architecture and pretraining method). Finally, empirically, we show that the optimal choices of the learning algorithm and unlabeled data are co-dependent.
5.1 DIVERSIFICATION ALONE IS INSUFFICIENT FOR OOD GENERALIZATINN
Diversification methods find hypotheses h,s that all minimize the training loss, i.e., h2 Hi, but disagree on the unlabeled data D.. The underlying idea is to cover the space H7 evenly and better approximate generalizable hypothesis from Hi n Hood (e.g., 2023)). However, if the original hypothesis space H. is expressive enough to include all possible labeling functions (c.g., neural networks (Homikk et al. 1989)), then H2 essentially only constrains its hypotheses' labeling on the training data D, while including all possible labelings over Dood. which implies gPoooal possible labelings, where g is the number of classes. Therefore, one might need to find exponentially many hypotheses before covering this space and approximating the desired hypothesis h" E Hi n Hand well enough.
Fig. 3 in (Pagliardini
al.,
Notably, we prove that having as many diverse hypotheses as the number of data points in Dood still insufficient to guarantee better than a random guess accuracy. Indeed, there always exists a set of hypotheses satisfying all the constraints of the diversification objective in Eq. 2 While having random accuracy w.r.t the true labeling h² on OOD data. The following Proposition 2 formalizes this intuition in the binary classification case. Please see its proof and extension to the multi-class case in Appendix E
Proposition 2. For K Dondl 1) and he the OOD labeling function, there exists set of diverse K hypotheses h1,... hK, i.e., A D (hs, h2) = 1fx & Dond h₂(2) = h/(x))//|Dooo Dood S 0.5 Vi,j (I,..., K), i # and it holds that maxweh dx Acc(h". ht)) < 0.5.

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WaC
Suparmaud
Office-tomme
37.35%
30.60%
(2.28%
Fan mokestoo
Now DEATE
Tarly
Figure 3: The performance of diversification methods is highly sensitive to the choice of archi- tecture and pretraining method. Left: DivDis and D-BAT best hypothesis (K = 2) performance with multiple pretraining strategies and architecture pairs Waterbirds- CC (Left) and Office-Home (Right). ResNet50 is used not specified. Right: Top-1 accuracy on ImageNet- Ik after fine-tuning.
Since in most cases, Lee al. (2023) Pagliardini et al. (2023) find 2 hypotheses to be sufficient to approximate ht and the size of the used datasets is larger than 2, these hypotheses should not only be diverse but also biased towards those that generalize under the considered distribution shift.
Takeaway. Diversification alone cannot lead to OOD generalization efficiently and requires additional biases to be brought by specific learning algorithm used in practice.
Properties of diverse hypotheses. In Appendix F:2 using the agreement score (AS) (Atanov et al.
2022. Hacohen et al.) 2020) as measure of the alignment of hypothesis with learning algorithm's 's inductive biases, study in what way D-BAT and DivDis diverse hypotheses are biased. We show that they find hypotheses that are not only diverse but aligned with the inductive bias of the used learning algorithm. According to the definition of AS, such alignment expected for a hypothesis found by empirical risk minimization (ERM). However, it is not generally expected from diverse hypotheses (as defined in Eq. 2) given that the additional diversification loss could destroy this alignment. This analysis sheds light on the process by which diverse hypotheses are found and emphasizes the choice of good learning algorithm, which is crucial, as shown in the next section.
5.2 LEARNING ALGORITHM SELECTION: A KEY TO EFFECTIVE DIVERSIFICATION
Sec.5.11 argues that the right learning algorithm's inductive biases (i.e.. those aligned well with the true causal hypothesis h²) are required for diversification to enable OOD generalization. In this section, we examine the "sensitivity" of this requirement by using DivDis and D-BAT with different choices of pretraining strategies and architectures several real-world datasets.
Experimental setup. We consider the following datasets. (1) A multi-class classification dataset Office-Home /Venkateswara et al. 2017) consists of images of 65 item categories across four domains: Art, Product, Clipart, and World. Following the experimental setting in Pagliardini et al. (2023) the Product and Clipart domains during training and the Real-Worle domain the out- distribution one. (2) A binary classification dataset: Waterbirds-CC/T. et al. (023);; Sagawa et al. (2020). modified version of Waterbirds where the background and bird features are completely spuriously correlated on the training data. We report worst-group accuracy for Waterbirds-CC.. i.e., the minimum accuracy among the four possible groups. We train both diversification methods using different architectures and pretraining methods, each resulting in a different learning algorithm with different inductive biases. Please full experimental details and results in Appendix G
Sensitivity to the model choice. Fig. shows that the performance of both diversification methods is highly sensitive the choice of the learning algorithm: 1) the gap between the best and second-best model is significant (10%-20%) 2) there no single model that performs the best over both datasets, and 3) there is 20% standard deviation of the performance over methods and datasets). Furthermore, similar to the findings of Wenzel et al. (2022), one cannot choose good model reliably based the ImageNet performance model, according to this proxy, ViT-MAE (He et al. 2021), underperforms significantly in all cases. Additionally, ViT-Dino (Caron et al. 2021). the third best ImageNet, completely fails for DivDis
the distribution of models (averaged
proxy. Indeed, the best
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Dataset
Method
K
K =
K
K =5
Waterbirds-CC
D-BAT (ViT-B/16 IN)
57.1
57.1
57.1
57.1
Div (MoCo-v2)
19.4
51.726.0
49.628.3
48.420.9
Office-Home
D-BAT (ViT-MAE)
61.9
62.64a.1
62.6+0.1
62.6+a.1
Dis (Resnet50 IN)
55.9
54.640.1
53.640.4
53. 53.1
Table 1: Increasing the number of hypotheses does not bridge the performance gap between different models. We increase the number of hypotheses found by diversification methods for the second-best model in Fig. [ and find that it is not enough to bridge the performance gap with the best-performing model. The best hypothesis accuracy is reported. Results are averaged over 3 seeds. Standard deviations for Waterbirds-CC are larger due the usage of the worst-group accuracy metrie.
on both datasets. Overall, these results emphasize the need for specific architecture and pretraining tailored for each dataset and method, which may require an expensive search.
Increasing K does not improve performance. Finally, we study whether the performance gap between the best and second-best models tested in Fig. can be closed by increasing the number of hypotheses K, as this is allegedly the major feature and motivation of diversification methods. Tab, shows that similar to the observation made Sec. 4.2 increasing K does not bring any improvements, suggesting that the choice of the model is more important for enabling OOD generalization. In Fig. 14 we further show that DivDis does not scale well larger K (e.g.. K 64) "out-of- -ftt--f---e--o".. and the performance drops as the number of hypotheses increases. Note that testing D-BAT in this regime would be prohibitively expensive.
Takeaway. Diversification methods are highly sensitive to the choice of the learning algorithm, e.g., architecture and pretraining method. The "built-in" mechanism of increasing the number of hypotheses K does not alleviate this issue and fails to improve performance.
5.3 ON THE CO-DEPENDENCE BETWEEN LEARNING ALGORITHM AND UNLABELED DATA
Sec. 4 and Sec. 5.2 show the sensitivity of the diversification methods to the distribution of the unlabeled data and the choice of learning algorithm, respectively. Here, further demonstrate that these choices are co-dependent, i.e., the optimal choice for one depends on the other. Specifically, we show that by only varying the distribution of unlabeled data, the optimal architecture can be changed.
Experimental setup. We consider two learning algorithms (architectures) AA E (MLP, ResNet18) (extension to other architectures is straightforward) and construct examples for D- B-BT where one architecture outperforms the other and vice versa. To do that we build on the idea of adversarial splits introduced in (Atanov et al. 2022), defined CIFAR-10 Krizhevssk & Hinton (0099 dataset D) Below, we briefly describe the construction and refer the reader to Appendix H for more details.
We start by considering two hypotheses with high agreement scores (Back et al. 2022) found by Atanov et al. (2022) for each architecture, such that the following holds:
ASMI.P((xM.P)) > ASMI.P(IRRN), ASRN > ASRx(hMmP).
where ASA stands for the agreement score measured with algorithm A. As shown in Atanov et al. (2022), the above inequalities suggest that each hypothesis A is more aligned with its corresponding learning algorithm i.e., ERM trained with MLP architecture will preferentially converge to /MLP over /RRn and vice-versa when training with Resnet18. Akin to adversarial splits (Atanov al. 2022), we then use these two high-AS hypotheses to construct a dataset to change, in a targeted way, what the first hypothesis of D-BAT h e hERM converges to, depending on the used learning algorithm. Different /1 in turn, lead to different h28 and, hence, different test performance.
As the true labeling A", we use a binary classification task constructed by splitting the original 10 classes into two sets of five. Then, Tab. Right illustrates, we construct training data D, to contain samples where all h" 1MLP, and haN agree, i.e., D, = {x E D h" (x) = hMLP(2) = (-)) Thus, by design, both MMLP and hRN completely spuriously conrelated with h Then, define unlabeled OOD data Da, either haar 0, haN 1)) (denoted as h* 1. hRN). or
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Untabeted (
Du
A
Test (cc.(%)
MLP
89.2+0.8
1 hXLP
ResNet18
56.7
MLP 55.4
10 MgN
ResNet18
77.0
Cosr
Duch 1 Luua
0g:A" A Prg
Table 2: The optimal choices of the (learning algorithm) architecture and unlabeled data are co-dependent. Left: Performance of the D-BAT method using two architectures and two created unlabeled data distributions. The test accuracy is on hold-out data Dood ~ Du. Right: Illustration of the two unlabeled data distributions used in the experiment. We keep the training data fixed and change Du s.t. h* is inversely correlated (denoted by 'L') to one of the "spurious" hypotheses hMLP, hRN- We show that the optimal choice of the architecture (A) depends on the unlabeled data distribution, and the best A is the one for which h" 1 hA on D..
DD (r = =. hRN = 0) (denoted as h' L hMLP). This means that h" is inversely correlated with
only one of hMLP or hRN, while not correlated to the other hypothesis.
Results. Keeping the training data fixed, we train D-BAT (K = 2) using different architecture and construct unlabeled data pairs (A, D.). Tab. Left shows that the performance of AA drops to almost random chance when 4 does not inversely correlate with h" on the unlabelled data (2 and @). This is consistent with Sec. 4 where we show that the setting with A/ DD, = is disadvantageous for D-BAT. In Appendix H Tab. T further shows similar observation for a different architecture pair (ViT & 18sNet18), and Fig. [1 extends the experiment with smooth interpolation from one unlabeled dataset setting to the other, showing linear transition where one architecture goes from optimal performance to random-chance accuracy, and vice-versa.
Takeaway. The optimal choices of the architecture and unlabeled data are co-dependent.
CONCLUSION AND LIMITATIONS
This paper aims to study diversification methods and identify key components enabling their OOD generalization: the diversification loss used, the distribution of the unlabeled data, and the choice of learning algorithm. Below, we distill some practical recommendations that follow from our analysis.
Unlabeled data and diversification loss. Sec. 4 shows that sub-optimal spurious ratio w.r.t to the chosen diversification loss may lead to significant performance drops. One possibility overcome this problem is to use mixture of diversification losses, determined by an estimate of the spurious ratio of unlabeled data. Another to try to collect unlabeled data with specific spurious ratio.
Choice of the learning algorithm. Sec. 5.2 demonstrates that the methods are highly sensitive to the choice of the learning algorithm inductive bias. Future methods should be made more resilient to this choice, e.g., by modeling each hypothesis with different architectures and pretraining methods or by implementing a mechanism to choose "good" model automatically.
Co-dependence. Sec. 5.3 suggests that practitioner should not expect the best learning algorithm (e.g., architecture or pretraining choice) found on one dataset to perform well on another one (as observed in Sec. 5.2). and an additional search might be needed to achieve good performance.
Then we discuss the limitations of our study:
Data characteristics. We characterize the influence of the OOD data distribution through its spurious ratio. The influence of other important properties of the OOD data may need be studied in future work. Furthermore, we mainly focused on image data to aid the comparison with Lee et al. (2023); Pagliardini al. (2023), but we expect our conclusions be mainly data-agnostic.
Co-dependence experiment only with D-BAT In Sec. [.3 the experiment is only performed with D-BAT. We expect DivDis to have similar co-dependence. However, its diversification loss (mutual information) and optimization strategy (simultaneous) make such a targeted experiment challenging to design. We leave an explicit demonstration for future work.

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REPRODUCIBILITY STATEMENT
In order to ensure that this work is reproducible, we have taken the following steps. Appendix A and F we provide proofs for each theoretical result (Proposition 1 and Proposition 2 For the experiments, in Appendix D G and H we provide a complete description of the datasets, used models, and hlpee-parameter settings. Additionally, all results from DivDis (Lee et al. 2023) and D-BAT (Pagliardini et al. 2023) are obtained using their respective published source code, ensuring faithful representation of their methods. Finally, provide the anonymized source code for the experiments performed the paper.
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Michael Zhang, Nimit S. Sohoni, Hongyang R. Zhang. Chelsea Finn, and Christopher Re. Correct-
N-Contrast: Contrastive Approach for Improving Robustness to Spurious Correlations. In Proceedings of the 39th International Conference Machine Learning, pp. (6444-265566 PMLR. June 2022. URL https: beattps:/pproeeeeet http:///pccccccc..o///o mlr. press /vsss/ zhang22z html ISSN: 2640-3498.
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Appendix
The appendix of this work is outlined as follows:
Appendix A proves Proposition of Sec. (synthetic 2D task), and shows that the optimal diversification loss depends on the spurious ratio of the unlabeled data.
Appendix extends the experiment done in Sec. 4.1 (synthetic 2D task) by training multilayer perceptron (MLP) instead a linear classifier, and shows empirically that PropositionTTextenee to more complex classifiers.
Appendix C provides additional experiments for Sec. 4.1 and shows empirically that Proposi-
extends to DivDis.
Appendix providees the implementation details of the experimental verification of Proposition T on real-world images (Sec. 4.2) We also provide additional results, using the M/F dataset (where MNIST and Fashion- MNIST (Xiao al. 2017) are concatenated), well the CelebA (Liu et al. 2015) dataset. We also show that tuning the diversification hyperparameter a is not sufficient compensate the performance loss from the misalignment between unlabeled data and diversification loss, i.c., the conclusion of Proposition [ still holds when tuning a.
Appendix proves Proposition 2 of Sec. 5.I proving the existence of a large number of pairwise diverse hypotheses which do not generalize. A proof for similar result in the multi-class classification case is also provided.
Appendix provides an overview of the important concepts from Task Discovery AAtanov al. 2022 used in this paper (agreement score, adversarial splits).
Appendix using agreement score, explains the experimental setup and results that demonstrate that D-BAT and DivDis find hypotheses that not only diverse but aligned with the inductive bias of the used learning algorithm.
Appendix G reports the experimental details and full results of Sec. 5.2
Appendix H provides detailed explanation of how to construct the training and unlabeled data of 5.3where we show that by only changing the distribution of unlabeled data, we can influence the optimal choice of the architecture. It also contains variant Tab. 2 with ReT&ResNet pair, as well as an extension of the experiment with a smooth interpolation from one unlabeled dataset setting the other, showing linear transition where one architecture goes from the optimal performance to random-chane chance accuracy, and vice-versa.
A PROOF AND DISCUSSION OF PROPOSITION
In Sec. 4 we make a proposition that, in the synthetic 2D example, the optimal choice diversification loss changes with the spurious ratio of unlabeled OOD data rD. Specifically, DivDis-Seq finds the ground truth hypothesis h* and only rp. = 0.5 (i.e., balanced or no spurious correlation), whereas D-BAT discovers ht if and only if rn. = 0 (i.e., inversely correlated). In this section, we provide the proof, method by method, and case by case.
We first restate the Proposition ass follows:
Synthetic 2D Binary Classification Task. We illustrate the setting in Fig. 4 and describe it below:
The data domain spans a 2D square, i.c., {i = (æ1, x2) E [-1, 1]3).
The training distribution is defined as D. = {a = x2) E {[-1,0ju[o, f}fl[ 1]]{-1,03).
i.c., contains data points the 1st and 4th quadrants.
Our hypothesis space H contains all possible linear classifiers h(x; B) where / is the radian of the classification plane w.r.t horizontal axis T1-
The ground truth hypothesis is ht h+(x)) ha;;; F) I(a 0}, where I is the indicator function.
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Class
Class
Unlabeled data
Aap
Figure 4: Synthetic 2D Binary Classification Task. and represent the training data points and their labels. represents unlabeled OOD data. In this setting, the unlabeled OOD data D, has spurious ratio rp. = 0 (i.e., inversely correlated).
The spurious hypothesis, i.c. the one that ERM converges to, is assumed to be hap(z) h(z;0) [æ < 0).
Thus, hup and htt agrec on the training data (Ist and 4th quadrants) and disagree on the 2nd and 3rd quadrants.
We vary the spurious ratio of the unlabeled OOD data distribution D, by varying the ratio of data points sampled from the 1st and 4th quadrants over the number of data points sampled from the 2nd and 3rd quadrants.
One possibility is to define D, = far (2),))) E {[R(r,.), 1] 1]} U ([1,,-R(rD.)] [-1,0]}}, and R(r) = - for 0 S S 0.5.
Let (z: y) the probability of class y predicted by hypothesis given sample z. The following proof assumes both the hypotheses hip and the second hypothesis hB or hiD discovered by D-BAT and DivDis-Seq have hard margin, i.e., Pa (2; y) E [0,1), Nonetheless, we also show empirically in Sec. 4.1 (Fig.) that when this hard margin condition does not hold, we get the same conclusion as Proposition
Proposition 1. (On Optimal Diversification Loss) In the synthetic 21) binary task, let hD and h2 be the second hypotheses of D-BAT and DivDis-Seq, respectively. TD, = 0, then h LB ht and h = h(:; 7). Otherwise, if TD. = 0.5, then h h(x; x) = hap and h h²
Proof.
In the following proof, -h denote the opposite hypothesis, i.e., -h(x) h(x).
D-BAT. Plugging in the unlabeled OOD data distribution D., the first hypothesis hap and the second hypothesis h. the diversification loss in D-BAT is:
An, (lap. AP = E Pr,,, (z; 0) (x) + Pr.,, (z; 1) Paoa
Let CERM be the ERM loss on fitting training data. D-BAT's objective is then L + a p., where is a hyperparameter. Bellow, we prove the proposition case by case:
When rp, = 0 (inversely correlated), the unlabeled OOD data spans {(0, 1] (]-,00)
-i.,),,iee the second and third quadrants. In this case, the diversification loss
Ap. E,- D. log (z; 0) Page (x; 1) Proy (2; 1) pzg=(x;0))
1
= Em log(1 Pngn (x; 1) + 0 P Pags (z; 0)) E (10, (0, 1J11
(2nd quadrant)
1 (3rd quadrant),
+ Bx-D. I 10000000000 Prpe (I; 1) + P (z; (z: 0)) = (|-1,0)0[-1,01)
(4)
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where we assume uniform distribution over Dood. The hypothesis h2? which minimizes the diversification loss in Eq. 4 should satisfy Page (z; = 1 and Pap= (z;0) = 1 for the data points in the second and third quadrants, respectively. Since the hypothesis space consists of linear classifiers, the two hypotheses that satisfy (i.e., with AD. 0) the above constraints are h* and Rap. where (z) = - -[ < 0} = [{2 > 0}. When considering the entire objective CBRM + a'p,,. only h* minimizes the objective to O, regardless of a. Therefore, in this case, the D-BAT's solution corresponds to the ground truth function ht.
When rD. = 0.5 (balanced), the unlabeled OOD data spans {[-1, [o i.e., all four quadrants. The diversification loss is, therefore:
AD. D. loogog (z; Phgs (z: 1) Prop (2; 1) Page (z: 0))]
1
= D, l Prps (z; 1) + 0- P (1
x E 10 1JH1
(2nd quadrant)
+ B. llog(1 Prpm (z; 1) + Pao (z; 0)) 00
([ U (0. 1}
(Ist quadrantt)
1
+ B, [-lo, llog(0 Prpe (x; 1) P (x;
x ( U[-10]
(3rd quadrant)
+ E Be- Pape (1; 1) + (1:=((;00) 0))
x E { 1| 11, 0|H (4th quadrant)
(5)
The hypothesis h2 which minimizes Eq. requires Pap=(1; 1) 1 for x in the 1st & 2nd quadrants, and Papa (z; 0) for x in the 3rd & 4th quadrants. The only hypothesis that satisfies these conditions is - -ssp- Although - -hpp doesn't 't minimize DD. we note that, given that the hypothesis function is mard-m any data point in the 1st and 4th quadrants drives the diversification loss to positive infinity if h (x) + (x)((x) This is because of the -log in the diversification loss. Therefore, in this case, D- D-BAT's s solution is - -hsp regardless of CERM and a. In practice (soft-margin regime), given that the a parameter is large enough to enforce diversification, this also holds as we empirically verified in Fig. L
The cases where 0 < rD. < 0.5 can be straightforwardly extended from the above two cases. Indeed, as said above, any unlabeled data point in the 1st and 4th quadrants drives the diver- sification loss to be positive infinity if h * -hap. and, thus, h2 = -hesp Note, that this "phase transition" arises in theory as consider all the points from Dood appearing in the 1st and 4th quadrants, i.e., {[R(DD.) ). U [o, I]} and {1o, R(TD, ) U [-,, ()] In practice, when Dood contains only some samples from these regions, the h B will rotate counterclockwise as we increase rD., from 0 to 0.5, starting at hGT and ending at -hap as seen in the empirical experiment shown in Fig. 2
Overall, when rD.. O, there are h² and -h. minimizing diversification loss with minimum 0. and only h² minimizes the whole loss (ERM + diversification loss). On the other hand, when rp., = 0.5. D-BAT finds hap that minimizes diversification loss but violates the ERM objective, regardless of the choice of a.
DivDis-Seq. The DivDis diversification loss is
AD. (Ktpp h DD) DKL, (ho) @ PAgo)) + DDRI IPD.).
The first term on the right-hand side is the mutual information between hap and h Minimizing mutual information on the unlabeled OOD data Dn yields a hypothesis h.2 that disagrees with hap Ny No data points while agreeing on the other 2 (where No is the size of unlabeled OOD data). In
on - finite sample size, this is equivalent the hypotheses being statistically independent.
The second term is the Kl--divergence between the class distribution of hi2 on unlabeled data D,, and the class distribution of D,, In this setting, an hypothesis h2 minimizing this metric simply needs to classify half of the samples in the first class and the other half in the second class. Therefore:
When rD_ = (inversely correlated), minimizing CERM (1DD)++App (hap: hDP) finds hDD
h(:; I ). Indeed, it is the only linear classifier that satisfies (minimizes to 0) both objectives.
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classifies all data points from D, correctly and "half" disagrees (i.e., statistically independent) 2nd and 3rd quadrants with hap and classifies half the unlabeled samples in each class.
When rD_ 0.5 (balanced), minimizing CD (h2 + a.Ap.. (hsp. h2 finds h2
h(æ; ) = ht as the only hypothesis satisfying both losses similar to the previous case.
In general, for 0 rD. 0.5, the classification boundary of h2p rotates counterclockwase (starting at (2;; 2) for rp 0) as the spurious ration increases i.e. h22 = h(2; ((r where 3(r) [0, 0.5] a 2 increasing function of r. More precisely, (3() J arctan( 1-3r). This is the solution that satisfies both losses, similar to the previous cases. The decision line lies the 2nd and 3rd quadrants, and, therefore classifies the labeled training data correctly. The angle be easily derived to satisfy the constraint that h2 and hep agree only half of the unlabeled OOD data. Since (r) is strictly increasing, is only when rp. = 0.5 the solution of Div DivDis-Seg coincides with the ground truth h" h(2; 2) ((0.))
In this setting, this means DivDis-Seq only finds h when the unlabeled OOD data balanced.
Conclusion. Overall, we see that the two methods find h in completely different conditions, which is consistent with the observation in Fig. and thus calls for attention on one of the key components spurious ratio of unlabeled OOD data TD.
DivDis. Since simultancous training introduces a more complex interaction between the two hy- potheses, we do not provide proof for DivDis. In Appendix C we give empirical results on the 2D example showing that DivDis only finds h* when rD. 0.5, as DivDis-Seq does, but the solutions different for other values of the spurious ratio.
B RESULTS FOR TRAINING MLPs ON 2D TASK
D-BAT
T0
D-BAT
Top
D-BAT To,
2D Binary Classiication
Pap
Seg To,
DivDls Seq
Top
DMDis Seq
To 0.5
Figure Performance of diversification is highly dependent on unlabeled OOD data (2D example + MLP). The unlabeled OOD data points are not shown in the plots. Left: the labeled training data Dr. the ground truth function he and the spurious function hap- Right: the second hypothesis for D-BAT and DivDis-Seq under different spurious ratios of unlabeled OOD data.
In this section, investigate whether the influence of the spurious ratio of unlabeled OOD data shown in Proposition [ still holds when the learning algorithm more flexible. Specifically, we the same 2D settings, described in Sec. And Appendix A but train a multilayer perceptron (MLP) instead of the linear classifier. The MLP consists of 3 fully connected layers (with a width of 40) and has ReL. as the activation function.
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Class
0
Agp
1o
1Du
2D Binary Task
Inverse Correlated
Partially Correlated
Balanced
Figure 6: (Complementary results for Fig. Left) Learning Dynamics of DivDis (Lee et al. 2023) on 2D binary task. Left: Top-left quadrant: The 2D binary classification task. Other quadrants: The two diverse hypotheses (arrows are normal vectors) found by DivDis with varied spurious ratios of unlabeled OOD data rD. {0, 0.25, 0.5) (from inversely correlated balanced).
As shown in Fig. 5 we observe that D-BAT (Pagliardini al. 2023) finds h" only in inversely correlated unlabeled OOD data, while DivDis-Seq, on the contrary. finds h under balanced unlabeled OOD data, which is consistent with the Proposition [ Indeed, for D-BAT, when rp, = 0.25 or rD. 0.5, the diversification loss contradicts the cross-entropy loss on the labeled training data, causing misclassification of the training data. On the contrary. DivDis-Seq's boundary rotates counterclockwise, and its diversification loss causes no contradiction with the cross-entropy training loss.
C RESULTS FOR DIVDIS ON 2D BINARY TASK
In Sec. 4.1 show how varying the spurious ratio influences the learning dynamics of DivDis-Seg and D-BAT. For completeness, we also provide results for DivDis (Fig.(6) The experimental setup is the same as in Sec. 4 with 0.5k 5k training and unlabeled OOD data (with varied spurious ratio). Similarly, the hypothesis space is restricted to linear classifiers. Because DivDis optimizes simultaneously (i.c., there is first/second model), do not fix the first classifier to hap contrary to what was done for D-BAT and DivDis-Seq.
As shown in Fig. 6 DivDis does not find the true hypothesis ht when the unlabeled OOD data inversely correlated. On the contrary, recovers /sр and h- when the unlabeled OOD data is balanced. Thus, DivDis and DivDis-Seg share similar learning dynamics when rD. = 0.5.
D MORE DETAILS, RESULTS AND DISCUSSION FOR SEC. 4.2
In Sec.
demonstrate real-world image data that of the key factors influencing the performance of diversification methods is the distribution of the unlabeled OOD data (more specifically, the spurious ratio TD..). Here provide
details for the experimental setup, and
results (in Fig. 9 for M/F (Pagliardini et al., 2023) dataset (where MNIST and
(Xiao)
et al. 2017) are concatenated). We then provide the results CelebA (Liu et al. 2015
Lee et al.
2023 as verification on a large-scale dataset, as shown in Tab.
Dataset & model details. We investigate two datasets: M/C and M/F, where:
In the training set (Fig. (7). the spurious dataset (MNIST) completely correlates with the "true" or semantic dataset (CIFAR-10 (Krizhevsky & Hinton 2009) Mushion-MNIST (Xiao al.) (017)).
Specifically, in M/C, MNIST Os and Is always concatenate with cars and trucks, respectively. In M/F, MNIST and Is always concatenate with coats and dresses, respectively.
For the unlabeled data D... where D-BAT & DivDis(-Seq)'s hypotheses make diverse predic- tions, the spurious ratio r',,, changes, exposing that the performance of diversification is highly dependent the unlabeled data.
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1
0
Figure 7: Training data samples from M/C (Top) and M/F (Bottom). Note that D, is completely spurious correlated (rD, = 1). Thus, MNIST Os/Is match with CIFAR-10 cars/trucks F-MNIST coats/dresses.
0
0
Figure 8: Test data samples from M/C (Top) and M/F (Bottom). Note that Dood balanced (r' = 0.5).
We can straightforwardly vary the rp, by changing the rules of concatenation in D... Specifically, take M/C as an example, we first take the samples of Os and Is from MNIST, well cars and trucks from CIFAR-10, 10, and we make sure they have the same size and are shuffled. Then, according spurious ratio rD,, we randomly select rp.. proportion of the samples from MNIST (Os / Is) and CIFAR- 10 (cars/trucks), and concatenate Os with cars and Is with trucks (so that the semantic feature correlated with the spurious feature in TD. of samples). We finally concatenate the remaining 1 rD. proportion of samples oppositely (i.e., Os with trucks and Is with cars).
TD. = means inversely correlated (all images are Os/trucks or Is/cars).
TD. 0.5 means balanced (half of the Os are concatenated with cars and the other half is concatenated with trucks, half of the is are concatenated with trucks and the other half is concatenated with cars).
TD, means completely spurious (all images are Os/cars 1s/trucks).
The test data is hold-out balanced OOD data Dood (Fig. i.e., I'Dood 0.5. in which there is no spurious correlation between the MNIST and target dataset (either CIFAR-10 10 or Fashion- MNIST), and the labels are assigned according to CIFAR-10 (in M/C) and Fashion-MNIST (in M/F).
We train LeNet (Lecun al. 1998). which contains 2 convolutional layers and 3 linear layers. Following Pagliardini et al. (2023) setup, depending on the dataset, we modify the number of channels and input output sizes of the linear layers. We summarize these parameters Tab.
Conv
Conv 2
Linear
Linear 2
Linear Pooling
M/C
3, 32, 5
32, 56, 5
2016, 512 960, 120
512, 256
256, 2
Average
M/F 6,
6, 16, 5
120, 84
84, 2
Max
Table 3: LeNet's parameters in Sec, 4.2 For Cony layers, the numbers represent the input channel, output channel and kernel size. For linear layers, the numbers are input and output sizes, respectively.
Results on M/F dataset. In the same manner of Fig 2 we show results on M/F dataset in Fig. Right. We similar trend as Fig 2
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Text Atc. balanced
80
en velenced
90
Nethed D-BAT DivOts
72
80
64
70
hypotheses
-66
60
0.0 (Inversily
5:25
0.5 (Bulanced)
1.0 (Complately Sportous)
6.0 Jevessely
0.5
Spurious (B-lanced)
(Complately Spurtous)
Spurious
Figure 9: Performance of diversification is highly dependent on unlabeled OOD data (M/C and M/F datasets). The test accuracy is measured on balanced data Dood (i.e. DDood 0.5, no spurious correlation). Left: Test accuracy D-BAT DivDis(-Seg) on MNIST/CIFAR-II for varied spurious ratios rD. Right: Test accuracy of D-BAT & DivDis(-Seq) on MNIST/Fashion-MNNSS for varied spurious ratios TD..
When rp. E [0, 0.5), (inversely correlated balanced), the results match our observations made in.
When rp, E [0.5, 1.0) (balanced to completely spurious), both on M/C and M/F, all methods have more and more difficulty diversify and use the semantic features. Indeed, the unlabeled OOD data distribution D., gets increasingly closer to the training distribution Dr, thus we cannot expect OOD generalization.
Overall the synthetic 2D binary task section, M/C, and M/F experiments suggest that, in practice, across different datasets, diversification methods' behavior and solutions are highly dependent on the spurious ratio of unlabeled OOD data.
Discussion on the a hyperparameter. the above experiments on both datasets, we use large coefficients for diversification losses (Ap, in Eq, 2 as 5/ 50/ / 50 for D-BAT DivDis DivDis-Seq, order to study the behavior these methods when the diversity objective is fully optimized.
In Fig. Left, we further show results for different values of a. We observe that tuning a is not sufficient to compensate the misalignment between the unlabeled OOD data and the diversification loss, and the performance for both methods has the same trend. Specifically, larger gives better test accuracy general, shown in Fig. Left. In Fig. [ight we select the best a for each scenario (i.e., each spurious ratio of unlabeled OOD data), and observe no meaningful difference in behavior (compared to Fig. 2 and Fig. 9). Therefore, a conclusion similar to Proposition still holds: even when tuning a for each unlabeled OOD data setting (i.e. spurious ratio), D-BAT performs best when the unlabeled data is inversely correlated, while DivDis performs best when the unlabeled data is balanced. This suggests that practitioner might not be able to compensate for a misalignment between unlabeled data and diversification loss by tuning the hyperparameter a.
Results on CelebA-CC dataset. In Tab. we further show results on large-scale real-world dataset, namely CelebA-CC (Liu al. 20155 Lee al.) (023). CelebA-CC is variant of CelebA, introduced by (Lee et al.) 2023), where the training data semantic attribute is completely correlated with the spurious attribute. Here, gender is used as the spurious attribute and hair color as the target. We take D-BAT and DivDis-Seq (for fair comparison on sequential training), and show their test accuracy on different degrees of spurious ratio of unlabeled OOD data (rD_ {0.0, 0.5, 1.03). Consistent with our previous observations, the results show that D-BAT performs the best when TD. = 0.0, and DivDis-Sed Seq performs the best when rD, = 0.5.
E PROOF OF PROPOSITION 2
We first remind our proposition:
Proposition 2. For K Doodl 1) and he the 0OD labeling function, there exists set of diverse K hypotheses hi,..., hi,, i.e., A A (hs, h2) = 1x & Dond h₂(2) = h/(x))//|Doom S
0.5 Vi,j (I,... ..... K), i + and it holds that maxweh dx Acc(h". ht) < 0.5.
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Teat Acc. with diferemt
DiO
0.005
81
0.05
500.0 0
Method BKT
72
0.1
(Inverscty Comrelated))
(Solonced)
Ilnversely Correisto))
(8olonced)
Spurious ratio on unlabeled data
Spurious rablo unlabeled data
Figure 10: Tuning a is not sufficient to compensate for the change in performance when the spurious ratio of unlabeled OOD data changes. The test aceuracy measured on balanced data Dood of MNIST/CIFAR-10, Left: Test accuracy of D-BAT and DivDis with varied spurious ratios of unlabeled OOD data, where various a are considered. Right: Test accuracy with the best for D-BAT and DivDis, where for each spurious ratio, the best is selected w.r.t the accuracy on hold-out balanced validation data.
VD. 0.0
TD. 0.5
VD. - 1.0
D-BAT
84.6_0.3
82.8_0.2
74.620.6
DivDis-Seq
84.8_0.2
86.1
73.2
Table Verification of the conclusion in Proposition CelebA-CC. Gender is used the spurious attribute and hair color as the target. We report average test accuracy. The trends of accuracy are consistent with what was shown in Fig(9] indicating that the conclusion that D-BAT reaches its optimal when rDD = 0.0 and DivDis(-Seq reaches its optimal when rp. = 0.5 can extend to much larger and more realistic dataset. DivDis-Seq used here for fair comparison.
This formulation covers methods of interest, D-BAT (Pagliardini al. 2023) and DivDis (Lee et al. 2023). Indeed, the maximum agreement is upper-bounded by 0.5. For DivDis, the optimal solution has maximum agreement of 0.5, seen in Appendix [A] For D-BAT, the optimal solution has the lowest agreement possible. Indeed, for = 2, the optimal solution has AD... (h1, h2) Thus, both methods optimal solutions covered when upper-boundeee the maximum agreement by 0.5 (as long as K < (2) Donal 1))-
We prove the existence of diverse set of K hypotheses, satisfying the condition of Proposition 2. using a classic construction from coding theory, called the Hadamard code (Bose & Shrikhande 1959).
Terminology. We first make explicit the equivalence between hypothesis space and coding theory terminology. In binary classification, a labeling function hypothesis h; Dood is binary codeword (vector) fixed length N, where N = |Dooal A set of hypotheses is now referred to code C' of size K. We define the Hamming distance between codewords h2 hy as d(hç, h,) >i Ih, (k) * hk)]]] where I is the indicator function and hk(k)) the hypothesis prediction the kth data point from Dood. The Hamming distance between two equal-length codewords symbols is the number of positions at which the corresponding symbols are different.
The agreement between two hypotheses h4 hy can now be rewritten using the Hamming distance as A ADoca (hs, h2) = + Σ (k), = h,(k)] = + (N - Σ (kh, + hk))) Similarly, the accuracy can also be rewritten Acc(h", " h²) Ap (h,, d(A
d(h,.h,)
N
Proof.
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We first use the fact that there exists binary code C with minimum distance d" min d( 2) = and [C] = 2N. This binary code is the Hadamard code (Bose & Shrikhande 1959] Rudra 2007p. also known as Walsh code. This binary code has 2N codewords of length N and has the miniaal distance of AV 2.
We show now that we can modify the Hadamard code to obtain another code C' with equivalent properties and h" E C For C², it then holds that max Acc(h", hh) I 0.5, as it was shown above that Aco(h", h!) construct such C".
ddh""h") S
N/2 N 0.5. Further, we show how to
Let hi be the first codeword of C. Let us now define function (or transformation) f(h) {0, 1) - {0, 1 such that f((1) = h", i. f transforms h1 into h², Since we are dealing with binary vectors, the function f(h) can be broken down into individual bit flips i.c.
h(()
if (1) - (1)
(()(1))
h(1) if (1) (()
Applying f to all codewords in code C' gives us a new code C' {h E C: f(h)) and h C². This operation maintains the minimum distance d since:
d(f(e) Σ + ((b)(())
Σ
[h.((k) h,)))]
(k))=))-()))
Σ
I1 hk( + 1 - h,(1))) (by definition of f)
[k: hi (K)
Σ
[h);(k) hy(1b)
(I: (x)
Σ
[hka(k + ky))
(k: (R)
N
Σ (k) + (b)] k=1
d(hs, hz)
Therefore, C' is the code satisfying all the conditions of Proposition 2.
As was shown before, a binary code C' is equivalent set of |C| hypotheses with the same properties. Thus, with = Doodl, the above construction gives us a set of (2) Dooal - 1) (2NN minus the true labeling h²' hypotheses satisfying the constraints of Proposition 2. This concludes proof.
Extension to multi-class classification
We used the mathematical framework coding theory and a classical result from it, the Hadamard code (Bose & Shrikhande 1959), to prove Proposition 2, specifically for binary hypotheses. However, coding theory, it has not been proven yet whether codes with similar "nice" properties, similar to Hadamard's s, exist for any q-ary codes i.e. for hypotheses with y possible classes. One exception is when y a prime number.
Proposition 3. Let q be the number of classes and a prime number. Let E N+ s.f. |Dood| = T. Then, for = (g. >Dooal 1) and hi the OOD labeling function, there exists a set of diverse K 4- ary hypotheses h1s ..,, hg, s.I., AD (hi h2) = xx E Dood 11(2)) = (x)) S 9 Vi.j E 1.... K # j, and it holds that max bg Aco(h" , M) <
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hp
hp
XS-0 06
X5-0
AS-2.06
Figure 11: Examples of high-AS hypotheses discovered in Atanov al. (2022). Each hypothesis is illustrated by exemplar images from each class as labeled be the corresponding discovered hypothesis hp. Neural networks can generalize by training the hn 's labeling.
2017 tells us that for any [Doal = 9th where m E N+, we can find code similar to Hadamard 's with minimum distance equal (()-1) and cardinality equal to q=* q |Dooal. By removing the semantic hypothesis from the count, we obtain that Proposition 3 holds for = (q |Dood]
Proof. Using similar argument from the proof of Proposition 2, (Stepanov 2006
F AGREEMENT SCORE AND IMPLICIT BIAS OF DIVERSE HYPOTHESES
F.1 AGREEMENT SCORE AND TASK DISCOVERY (ATANOV AL. 2022
In this section, we introduce more details on the background of Atanov et al. (022). as well as how we leverage the findings from
Agreement score as a measure of inductive bias alignment. We use the agreement score (AS) (Atanov et al. 2022: Hacohen al. 2020 Jiang al. 2022) measure the alignment between the found hypotheses and the inductive biases of learning algorithm. It is measured in the following way: given training dataset D, labeled with true hypothesis ht", unseen unlabeled data Dood, and a neural network learning algorithm A, train two networks from different initializations on the same training data, resulting in two hypotheses h2, he A(D,, h²), and measure the agreement between these two hypotheses on Dood:
(6) Baek et al. 2022) show that the AS correlates well with how well a learning algorithm A generalizes on a given training task represented by a hypothesis h. Indeed, high AS is necessary condition for generalization (Atanov al., 2022) (different outcomes of A have to at least converge to a similar solution). Finally, a learning algorithm will generalize on labeling the labeling is aligned with the learning algorithm's 's inductive biases, thus, we use AS as measure of how well a given hypothesis h is aligned with the inductive biases of AA
Ah; D., Dood) = E,, Aww.A(D,,.. Baw [hi = h2(x)]
Recent works Atanov et al. 2022
Task Discovery. Atanov et al. (2022) use bi-level optimization (also called meta-optimization) to optimize the agreement score (i.e., Eq. (6) and discover, on any dataset, high-AS hypotheses (tasks in the terminology of Task Discovery) that a given learning algorithm show that there are many diverse high-AS hypotheses different from semantic human annotations. In Fig. TT. show examples of the high-AS hypotheses discovered for the ResNet18 architecture CIFAR-10 10 (Krizhevsky & Hinton 2009).
generalize well on. They
Adversarial dataset splits (Fig. 12 Atanov et al. (022) also introduces the concept of adversarial dataset splits, which is a train-test dataset partitioning such that neural networks trained on the training set fail to generalize the test set. To do that, they induce spurious correlation between high-AS discovered hypothesis hp and the (target) semantic hypothesis h" on the training data, and the opposite correlation the test Specifically, they select data points training D,, such that a discovered high-AS hypothesis hn (specifically, AS(hp) > AS(h*)) )) completely spurious correlates with ht, i.e. {i E D, hn(x) = (x)). The test Drest is constructed such that the
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0,
Training sot
#=
0,4g
Test
At L,k = 0
Figure 12: Illustration of an adversarial split introduced in Ataanov et al., 2022). A high-AS discovered hypothesis hDp 'spuriously correlated' with semantic hypothesis h" on the training set, but inversely correlated with /t on test set. Training ERM on the training set and evaluating on test set give < 20% test accuracy (according to (Atanov et al., 2022)* 's Fig. 7). In the context of this work, the test set in given adversarial split is inversely correlated (r 0).
two hypotheses are inversely correlated, i.e., {i E Drest: hn(x) of 1e")())) Theoretically, a NN trained on such a training set D, should learn the hypotheses with a higher AS, i.e. hDs which would lead to a low accuracy when tested on Drest- This was indeed shown to hold in practice, where the test accuracy drops from 0.8 for a random split to 0.2 for an adversarial split.
Adversarial splits, therefore, show that neural networks favor learning the task with a higher AS (the background color the case of Fig. 12) when there are two hypotheses that can 'explain' the training data equally well. In this work, we refer to this ppreference' as an alignment between the neural network and hp. This creates controllable testbed for studying the effect of spurious correlations on NN training, which we also adopt in our study.
F.2 DIVERSIFICATION FINDS HYPOTHESES ALIGNED WITH INDUCTIVE BIASES
Disclaimer: For an introduction on agreement score, we refer to Appendix F.1
In this section, we study how the diversification process is biased in practice by the inductive biases of the chosen learning algorithm. Specifically, using agreement score, we demonstrate that D-BAT and DivDis find hypotheses that are not only diverse but aligned with the inductive bias of the learning algorithm.
Experimental setup
CIFAR-10. we build on top of the adversarial splits and construct a CIFAR-10 data split with complete spurious correlation the training data and balanced (no spurious correlation) unlabeled OOD data, as shown in Fig [3) This is a typical setting on which D-BAT (Pagliardini et al.) (202)) and DivDis (Lee al.) 2023) apply. More precisely, ht is semantic binary classification on CIFAR-10, 10, defined by choosing a 5 vs 5 split of the original 10 classes. We define the spuriously correlated CIFAR-IÓ data by using an arbitrary high AS binary labeling hD as the spurious hypothesis, similarly to "adversarial splits" introduced by (Atanov et al. 2022). There are two reasons for using this setting: one to easily control the data setup, and one for using (Atanov al. 2022) as reference of the AS for different hypotheses on CIFAR-10.
Measuring the AS of found hypotheses. For given dataset with training data Dr, unlabeled OOD data DV aod and test OOD data Dond, we train a diversification method (without pretraining) to find multiple diverse hypotheses .i,...,hu h and measure their agreement scores. More precisely, for each hypothesis hrs we measure (h,; D, U D Dood) where A is the same learning algorithm (e.g. ResNet18) 8) used to find the diverse hypotheses. This AS allows us to assess whether h; is labeling Dood randomly or in a way that aligns well with the inductive biases of the learning algorithms. We provide more details on the setting in Appendix F22 and illustrate the dataset creation in Fig. 13
Implicit bias of diverse hypotheses. Tab. 5 shows the agreement score of random hypotheses fRR (true labels D, but random labels Dona) and the diverse hypotheses found by both diversification methods. We observe a clear gap between the two, indicating that all diverse hypotheses label Dood in structured non-random manner.
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Training set (D₂)
h
0, hp
At 1,
airplane
sutomobile
bind
cat
deer
dag
frog
horse
ship
truck
Unlabeled data (D,,)
h, 0or1
ht 1, or 1
Illustration of the constructed CIFAR-10 data with spurious correlation. The semantic binary labeling h is defined by a vs. split of the original CIFAR-10 classes. Spurious hypothesis/feature hD (color) is predictive in the training set Dr, and non-predictive on unlabeled OOD data D..
Figure 13:
CIFAR-10
Waterbirds
Hypothesis
Test (%)
Acc.
Semantic (h")
0.83
100.0
0.84
100.0
Random (hg)
0.63
50.0
0.55
50.0
D-BAT
0.82+o.01
60. =0.2
0.89
25.3+3.0
DivDis
00.02
57.3
0.91 +9.04
28.9
Table 5: The diverse hypotheses found by diversification methods have high AS (i.e., aligned with the biases the learning algorithm). Nonetheless, without additional inductive biases (e.g.. the correct pretraining strategy, which is examined in 5.2 and Tab. 7). they do not generalize. The test accuracy CIFAR-I0 is measured on hold-out balanced data (Dood), and WG Acc. stands for Waterbirds worst-group test accuracy. Results are averaged over seeds.
MLP
VIT
Hypothesis
Test Acc.(%)
AS
Test Acc.(%)
Semantic (li))
0.76
100.0
D-BAT
0.8910.02
56. 19.4
0.90
59.2
DivDis
0.802
$8.3
0.87
57.7
Table 6: On CIFAR-10, D-BAT and DivDis also find high-AS hypotheses with MLP and ViT (Dosovitskiy al. 2021). Results averaged over 3 seeds.
Measuring the agreement expected AS value of hypothesis aligned with the inductive bias of A (otherwise, we wouldn's t expect A to be able learn lt*). We observe that the hypotheses found by DivDis and D-BAT have agreement scores similar to that of h², indicating good alignment with the inductive biases of A. Thus, optimizing Eq. 2 using neural networks as the learning algorithm, leads diverse hypotheses implicitly biased towards those favored by its inductive biases. According to the definition of AS, such alignment expected from hypothesis found through empirical risk minimization (ERM), however, it is not expected from diverse hypotheses (as defined in Eq. 2). given that the additional diversification loss could destroy this alignment. This analysis sheds light the process by which diverse hypotheses are found, and puts an emphasis on the choice of good learning algorithm, which crucial, as we show the subsequent Sec. 5.2
of the true semantic hypothesis h gives estimate of the
Similar to what shown with ResNet in Tab. in Tab. 6, we repeat the experiment with two different architectures, MLP and ViT (Dosovitskiy al. 2021), CIFAR-10. The diverse hypotheses found by D-BAT and DivDis have high AS. This demonstrates that our above conclusions also hold with different architectures.
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Hypothesis
AS
WG Ac).(%)
Semantic (h*)
0.84
100.0
Random (hR)
0.55
50.0
D-BAT (NP)
0.00
25.3=3.0
D-BAT (P)
0.86+0.02
59.1
DivDis (NP)
0.90
28.9
DivDis (P)
0.86
81.3
Table D-BAT and Dis find high-AS hypotheses on Waterbirds (Sagawa et al. 2020). The Agreement Score (AS) is measured on different hypotheses on Waterbirds. For D-BAT and DivDis. K = hypotheses are considered. 'WG Ace.' stands for worst-group accuracy. "NP" signifies non-pretrained, "P" signifies ImageNet pretrained. Two ResNet-50 models are trained from scratch when measuring the AS. The best model performance is shown for DivDis. For D-BAT, we always show the performance of the second model. Results are averaged over 3 seeds.
Diverse hypotheses cannot generalize without the correct pretraining
As seen above, D-BAT and DivDis produce diverse hypotheses implicitly biased towards those favored by the inductive biases of its learning algorithm. Nonetheless, this implicit bias may not lead OOD generalization, i.e., ht & H as the test accuracies in Tab. 5 are found to be near the chance level.
In Tab. 7 Waterbirds, repeat the same experiment in Tab. 5 with additional variable. The ResNet50 model is either trained from scratch or starting from ImageNet- I supervised pretraining weights. We can that pretraining does not affect whether DivDis and D-BAT find high AS hypotheses, however it greatly influences the generalization capability the found hypotheses. These results corroborate with Sec. 5.2 that the correct choice of inductive bias is crucial unlock OOD generalization.
G RESULTS AND IMPLEMENTATION DETAILS OF SEC. 5.2
G.1 EXPERIMENTAL DETAILS
Remarks on D-BAT and DivDis.
All experiments were run using DivDis and D-BAT respective codebases to ensure closest reproducibility to their presented methods and results.
DivDis' default setting is to augment the data while training. This option was disabled to ensure a fair comparison to D-BAT.
For their results, DivDis rebuilt the Waterbirds(Sagawa al. 2020 dataset from scratch. On contrary, D-BAT used the one provided by the WILDS(Koh |l.] 202 library. To ensure fair comparison, both methods were run using the latter version of the dataset.
If not precised, all train, validation and test splits are taken as provided from Pagliardini et al. (2023); ; Lee et al. (2023) or WILDS.
The best models are selected according to validation accuracy.
Computational resources. Each experiment can be run on a single A100 40GB GPU.
Models. If not precised, the model used in most experiments is ResNet50 (He et al. 2015). Otherwise, when using a Vision Transformer (ViT), we use ViT-B/16" (Dosovitskiy et al., 202 The last exception is for DivDis Camelyon17 (DenseNet121 (Huang al. (017)).
DivDis parameters. For Waterbirds variants, the optimizer SGD, the number of epochs 100, the learning rate is 0.001, the weight decay is 0.0001. The a parameter (referred as À in DivDis) was tuned over {0.1, 10). For Office-HHome, the optimizer SGD, the number of epochs is 50, the learning rate is 0.001, the weight decay is 0.0001, and the a parameter was tuned over (0.1, 1, 10). For Camelyon17, the original berf-rrrn setting from DivDis was used.
http:///wtttiiiiioooommo///eeeor 16.html
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Selessgeriseed
Supervised
Resnet50
WT
SUMC1RR22
MoCreet
WI-MAR
VIT-R/16
OfficeHoune
58,2139
79.2.044
DivDts
55.9...
70.7.1.
FIRM (D-RAT Ax)
13.5
70zns
9.ass
35.
11.9
90.n
90
Waterhirds CC
D BAT by
15.0
8.6
22,
15.4
36.1asn
55.5-xx
47.6.a0
49.7
07.7as3
$7.2
Divits (best)
25.0 25.0ga
7.8
20.4..0
22.
494.mm3
38.9.9
1122
47.
70.5
227 227
Table 8: The performance of diversification methods highly sensitive to the choice of architec- ture and pretraining method.. Average accuracy on Office- and worst-group accuracy on Waterbirds-CC for different pretraining methods. All methods are pretrained on ImageNet-Ik, not specified, the used model is ResNet50. Results are averaged over seeds. Bold and underline stand for the best and second best, respectively.
Dataset
Method
==
K =
K
=
Waterbirds
D-BAT (VIT-B/16 IN)
$7.1
45.0+1.1
45.5
44.5
Div (MoCO-v2)
49.4
51.746.0
49.648.3
48.4
Office-Home
D-BAT (ViT-MAE)
61.9
62.640.1
60.8
61.7.0.7
DivDis (Resnet50 IN)
55.9
54.610.1
53.610.4
53.1
Table Increasing the number of hypotheses, while using the second-best inductive bias, does not bridge the performance gap with the best inductive bias. Best hypothesis performance for DivDis and corresponding hypothesis performance D-BAT are reported. The second-best inductive bias was chosen according to Fig. Results are averaged over 3 seeds.
D-BAT parameters. For Waterbirds variants and Office-Home, the optimizer is SGD, the learning rate is 0.001, the weight decay is 0.0001. Given that D-BAT optimizes sequentially, the number of epochs is an important parameter to tune. We tuned over epochs E (30, 100} and (0.0001,0.1). For Camelyon17, the original D-BAT best-performing setting was used.
G.2 COMPLETE RESULTS OF SEC.
We provide the full results for Fig. 3 (ERM baseline included) in Tab. g We also provide the accuracy of each new head of D-BAT for Tab. 1 in Tab. Finally, in Fig. 14 we further show that DivDis does not scale well to larger (e.g. 64) "out-of-the-box" and the performance drops as the number of hypotheses increases. Note that testing D-BAT in this regime would be prohibitively expensive.
G.3 PRETRAINING STRATEGY AND ARCHITECTURE DETAILS.
In Fig. we vary the pretraining method and architecture, and measure the effects on performance. We provide additional details here. If not precised, the methods use He et al. 2015)
model. All 8 variations are pretrained on the ImageNet- MeeNNNeeekkRuuskkoooe kK(ussakkvsky et al. 2015) dataset:
Self-
SWAV (Caron et al. 2020)
SimCLRy2 (Chen et al. 2020a)
MoCo-v2 (Chen et al.| 2020b)
ViT-B/16 MAE (He et al./ 2021
ViT-B/16 Dino (Caron et al.) 2021
Supervised
-Adversarially robust classifiers (Salman et al. 2020).
Resset50 IN (He al. 2015). supervised pretraining ImageNet-Ik, 1k. This is the pretraining method used by (Lee et al. 2023 Pagliardini et al. 2023) in their papers.
ViT--/116 IN (Dosovitskiy al. 2021). supervised pretraining ImageNet-1k.
Experimental details For the adversarially robust classifier, the L2-Robust ImageNet ResNet-50 0.05) model was chosen, following the advice of (Salman al., 2020), as it is hypothesized that smaller values of e tend work better on datasets where leveraging finer-grained features are necessary (i.e., where there is less norm-separation between classes in the input space), such as
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Best hypothesis worst-group accuracy
Pretrained
Worst-group accuracy
From Scratch
0.6
ERM pretrained
0.4
Worst-grouw
0.2
0.0
64
Number of hypotheses K
Figure 14: Naively increasing the number of hypotheses in DivDis is detrimental to performance. y-axis shows the worst-group accuracy of the best hypothesis on Waterbirds-CC. We use Resnet50 as backbone. Pretraining is supervised pretraining on ImageNet-1k, 1k. "ERM pretrained" baseline stands for standard empirical risk minimization (ERM) without the usage of diversification. Results are averaged over 3 seeds. Preliminary investigation of this behavior suggests that the diversification loss not optimized well when K grows large (e.g. K 8) due to the averaging over all pairs of hypotheses, and that another aggregation function, c.g., maximum, could work better.
Waterbirds-CC CC or procedure described in G.T
Each variation hyperparameters were tuned following the same
H DETAILED EXPERIMENTAL SETUP AND ADDITIONAL RESULTS FOR SEC. 5.3
In 5.3 we demonstrate that using different inductive biases can drastically and predictably influence diversification method. Here we provide more details on how we construct such examples. Additionally, in Tab. T we also provide results where the examples are constructed using ViT- ResNet pair (instead of MLP- R-ReeNetet pair). Finally, we provide an extension Tab. 2 showing how an inductive bias gradually gets favorable and vice versa through the transition spurious ratios, in Fig. 15
Construction
Prerequisite: We consider semantic (5-vs-5) binary classification task h* CIFAR-10 (Krizhevsky & Hinton 2009) (i.e., airplane, automobile, bird, cat, deer original classes as class and dog, frog. horse, ship, truck original classes as class 0).
Step 1 (Selecting hypotheses aligned with learning algorithms from (Atanov et al., 2022)
We take two high-AS hypotheses (see Fig. for examples of such hypotheses) discovered in (Atanov al. 2022) for MLP and ResNet18 (He et 2015). where the hypotheses (haLP and
(RR)) satisfy Eq.
(hMLP()
(SSML((RRN) ASRN(hRN) > ASRN(hMLP)) (hMLP),
For example, this means the MLP hypothesis hMLP has a high AS when training MLPs but lower AS when training ResNet18s. Also, we ensure these two hypotheses have higher AS than the true hypothesis h² to make sure that they are able act as a spurious hypothesis.
Step 2 (Constructing training data where h², hMLP and haN completely correlates): As presented in Tab. TO in Dr row, we select the data points as training set D, such that h², hSLP and JRN agree. As shown in (Atanov et al. 2022) by adversarial splits, when two hypotheses correlate with each other (i.e., their labels are the same on training data), a neural network tends converge to the hypothesis with higher AS. Thus, combining with the conditions in step (i.e., Eq. 3) training MIP on Dj with ERM should converge to /MLP and training ResNet on D, with ERM to hRN, which is illustrated in Tab. Right.
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ht
PMLP
ARN
ht
hMLP
HRN
Dt
D
0
D2
Du
0
Table 10: The rules for constructing the data in Sec. 5.3 covering the two cases in Tab. Left: h 1 MMLP on Pu. Right: Ji* hRN on Du- '1. means inversely correlated on the unlabeled OOD data D..
Step 3 (Further improving the alignment between the two hypotheses and their correspond- ing architectural inductive biases): This step is not necessary in general, but it allows us to find hypotheses that are better aligned with the inductive biases of the network. This is because the Task Discovery framework from (Atanov et al.] (022) might not provide globally optimal hypotheses. The improvement goes follows: their AS for a better alignment with the corresponding learning algorithm (i.e., MLP and hMLP, ResNet18 and hRN). Specifically, predictions on CIFAR-I data except for the training data i.e. \. D,- We replace the old labels of /MLP and İRN D \ D, by the new labels predicted by MLP and ResNet18. This step gives us higher AS hypotheses (thus more preferred by the given architecture) that satisfy Eq. (equation also shown in Step 1).
update /MLP and hRN further increase
train with ERM an MLP and R8sNet18
D, and make
Step 4 (Constructing unlabeled OOD data such that ResNet18 or MLP fails): As shown in Tab. TO D, row, we select data points with specific hypothesis labels as the unlabeled OOD data. By design, in Tab. To Loft, h' is inversely correlated to /MIL and is not correlated (i.e. balanced) to ARN- We know training an MI.P with ERM on D, will choose /MLP- Therefore, D-BAT will perform well (i.., find htt) by minimizing its diversification loss. On the contrary, training a ResNet with ERM on D, will choose hRN. Therefore, as shown in Sec. 4 D-BAT cannot perform well by minimizing its diversification loss. The opposite conclusion holds for Tab. 10 Right.
Step 5: we take D, and D., (which are around 12k and 24k images, respectively) and run D-BAT (Pagliardini et al. 2023) (the labels on D, are inaccessible), and measure the test accuracy on hold-out D. aod D.. which is shown in Tab. Left.
The construction process is thus white-box process (or attack), similar adversarial attacks, but the architectural inductive bias aspect. We reiterate that the purpose of this example to illustrate that the choice of architectural inductive bias have very drastic influence on the behavior of diversification methods and this choice is co-dependent on the properties of the unlabeled OOD data.
Additionally, can demonstrate this co-dependence in "fine- grained manner, as shown in Fig. 15 Here we still keep the training data the same (according to D, in Tab. [0], and construct the unlabeled OOD data D,, such that the spurious ratios of /MLP and ARN gradually switch from low to high high to low. Hence, the Tab. Left and the Tab. Right correspond to the left and right extremitics of the x-axis of Fig. 15] and in between are interpolations between the two distributions (spurious ratios). This gives better view of how one inductive bias gets favorable through the transitions of spurious ratios, and vice versa.
Da,
A
Test (Cc
VIT
76.4
h 1 livrr
ResNet18
58. 58.140.4
ViT
68. 40.3
1r LAR
ResNet18
79.040.6
Table 11: Tathhwwith with another architecture pair. VIT far exceeds ResNet18 when h² hver and vice versa. Results are averaged over seeds.
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MLP ResNet18
Test Accuracy (%)
70
60
(0. 1/2) h, 1 hNL
(1/8, 3/8)
(1/4, 1/4)
(3/8, 1/8)
(1/2, 0) h. hAN
Transition of spurious ratio (rt m
Figure 15: The favorable inductive bias changes over transitions of spurious ratios. As the unlabeled OOD data D changes, the optimal inductive bias switches from MLP to Resnet18, indicating their co-dependence. Results are averaged over seeds.
32


Reviews:
[{"review_id": "T3cpMbGWwJ", "reviewer": "Reviewer_shsk", "summary": "This paper examines two recently proposed algorithms for improving out-of-domain generalization through diverse hypothesis selection with respect to both a labeled and unlabeled dataset. The algorithms try to find hypotheses that agree with the labels on the labeled data but disagree with one another on the unlabeled data.\n\nThe authors essentially find that there is \"no free lunch\" for improving out of domain generalization in that the set of diverse hypotheses selected will depend on which unlabeled data was used to find the diverse hypotheses, which underlying model class was used, and which metric is used to quantify the diverseness of a set of hypotheses.\n\nThe diverse hypothesis selection algorithms examined in the paper are:\n1. diversify and disambiguate aka \"DivDis\" (Lee et al. 2023)\n2. diverisity by disagreement training aka \"DBAT\" (Pagliardini et al., 2023)\n\nDivDis and DBAT use different metrics to assess the diversity of a set of hypotheses.\n\nThe main contributions of the paper are:\n1. Proposition #1 which states that DivDis and DBAT will select different second hypotheses w.r.t. an unlabeled dataset depending on the extent to which the first hypothesis selected (i.e., the ERM) agrees with the true hypothesis on the unlabeled data.\n2. Proposition #2 which states that the number of diverse hypotheses generated by an algorithm like DivDis or DBAT needs to be super-linear in the number of unlabeled datapoints in order to ensure that at least one has greater than 50% accuracy with respect to the true hypothesis.\n3. Experiments on synthetic and real datasets to support propositions #1 and #2, and which also show that the success of a diverse hypothesis generation algorithm jointly depends on the underlying model class and the unlabeled set of data selected.", "questions": "Regarding figure 2, h* is defined in the text such that instances with x1 > 0 should be labeled as positive examples. I assume that x1 is the horizontal axis such that points to the right are positive. But the plot and its legend shows points to the left are positive (Class 1). Is my interpretation correct here?\n\nAssuming it is correct, hyperplanes are typically defined by their normal, in which case is should be that h*(x) = h(x; 0), not h*(x) = h(x; pi/4), since theta=0 radians points in the positive horizontal direction to the right, whereas theta=pi/4 radians points up to the top of the page.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "This paper really dives into the intricacies of the diverse hypothesis generation problem and does a wonderful job illustrating how complex the problem truly it is; that is, success simultaneously depends on all variables. In my opinion, this message should be communicated more frequently in conference proceedings.\n\nIn particular, proposition #1 is quite illuminating in that it shows how DivDis and DBAT select different diverse hypotheses from one another, and there are different regimes defined in terms of the agreement with the true hypothesis on the unlabeled data where each is superior. And here, it was nice to see how the experiments on the synthetically constructed datasets (e.g., MNIST/CIFAR) supported the theoretical findings.", "weaknesses": "This paper has a number of weaknesses. I found the presentation more confusing and dense than it could be:\n\n  1. In particular, there is some terminology and notation that can be improved for greater understanding. The \"spurious ratio\" index is a poorly named quantity because it's literally the accuracy of the selected hypothesis with respect to the true hypothesis h*. Namely, h* has the maximum spurious ratio value of 1.0, but it's definitely NOT spurious as it's the true hypotheis. Another name, like the \"agreement ratio\" would be much clearer.\n\n  2. Similarly, the plots on the left side of figure 2 do not seem to agree with the description of the synthetic problem in the first paragraph of section 4.1, which made it very hard to understand what was being communicated (I elaborate on my confusion below).\n\nAnd though a good message to repeat, the no-free-lunch findings discussed in the paper are known in the supervised learning setting, so they definitely need to hold for this harder setting with labeled + unlabeled data. Could this research direction become even more constructive by making certain statistical assumptions? In the paper, D_u can be any \"out of distrubtion\" distribution; and the capacity of the learning algorithm is assumed to be infinite in proposition 2, even though in other parts of the paper, inductive biases of different model classes are highlighted (which suggests that there the model class has less than infinite capacity in practice).\n\nThe results for DivDis in table 1, section 5.2 for increasing K are not convincing because alpha needs to increase as the square of the number of hypotheses in the set (i.e., O(K^2)). Otherwise the regularization becomes too strong and dwarfs the empirical risk. But the text makes it seem like a fixed alpha was used, which would needlessly harm DivDis' performance as K gets larger.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "7ZE4TiweU4", "reviewer": "Reviewer_xY3D", "summary": "The paper critiques recent trend of building diverse models and then selecting one at test time, as a way of enhancing OOD generalization. It presents empirical and theoretical results on why the methods may not always work.", "questions": "See the weaknesses above. \nIn particular, can the authors setup a model selection expt where the spurious ratio of the unseen test set is known and the test set is simulated. Would it always lead to the correct model? What else do we need to know about the test set? Can that be summarized, or is that not possible (and hence there is no way to know the best model apriori)?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "I really like this paper. Given so many methods being proposed for OOD generalization, it is important to take a step back and analyze which ones are likely to work and under what conditions. \n\nThis paper finds that the literature on diversification of hypotheses may not be conceptually well-motivated. The key result is that the success of this technique depends on inductive bias of the model architecture and the same architecture may not work well for different kinds of test set. \n\nIn hindsight, many of the observations seem obvious. for example, given that the methods use 2-3 different hypotheses, they obviously are relying on the training procedure's inductive bias (otherwise how can 2-3 samples explore the full space of \"good\" hypotheses?). But still, the authors do a good job of articulating multiple such issues in a single paper.", "weaknesses": "While the analysis is compelling, I'm wondering whether these limitations matter in practice. What if we do model selection over multiple architectures and multiple diversity algorithms? Is the risk that the results we get on a cross-validation set may not generalize to the test set?\n\nIf so, what are the summary statistics of the test set that the above procedure would need to know? For example, if the spurious ratio of the (unseen) test set is known, can that be used to simulate a pseudo-test set and then do model selection over it?\n\nOverall, I'm unclear of main takeaway of this paper. Should we not use diversification algorithms? What is the better alternative?\nPersonally, I feel a better message from the paper can be that it identifies the axes on which model selection should be done, if some summary statistics of the test set can be provided in advance.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "j2xyF3QlfC", "reviewer": "Reviewer_MHwE", "summary": "The authors study 2 recently proposed supervised learning \"diversification\" methods (DivDis and D-Bat) which aim to achieve high accuracy out-of-distribution (OOD) by generating a diversity of hypotheses each of which fit the training data but which disagree on additional unlabelled data from a different distribution  and then picking one final hypothesis from the list . The process is intended to reduce the odds of yielding a model which relies on spurious correlations which would not persist under distributional shift.  Through a mix of theory , toy experiments and real-world-data experiments, the authors arrive at a number of findings which warn that neither DivDis nor D-Bat (nor any particular diversification method ) is likely to work well in all situations. The performance of diversification methods is shown to be highly sensitive to the distribution of the unlabelled data on which diversity is measured.  There is an interaction between the learning algorithm and the unlabelled data distribution such that each affects the optimality of the other. The appropriateness of the inductive bias of the learning algorithm is shown to be critical to the success of diversification methods. Increasing the number of hypotheses generated by the diversification methods does not necessarily fix these concerns...the interplay between learning algorithm and unlabelled distribution and the importance of inductive bias persist.", "questions": "Can you add some more background for how OOD concerns can arise in industry? Even though the experiments use real world data in some sense, the tweaking of the real world data to introduce spurious correlations makes the experiments a bit less compelling in terms of real-world evidence. \n\nCan you explain a bit about why I can assume that stage 2 (disambiguation ) probably is doable? I need to be reasonably confident that stage 2 is also doable in order to care about the success of stage 1.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "The work is original to the best of my knowledge.  \n\nThe paper is well written and clear. I have a few suggested typo-style edits in the weaknesses section, but the writing is certainly strong. \n\nI did not notice any errors or incorrect conclusions in the findings. Although I didn't have time to go through every result in extreme detail, I have a reasonable amount of confidence in the correctness of the results, generally.\n\nGeneralization which is robust to out-of-distribution shifts is certainly a worthy topic.", "weaknesses": "My main concern with the paper is whether its results are significant enough to merit acceptance at ICLR. I'm open to being persuaded that the paper is significant enough, but that's not clear to me, for a few different reasons.  I offer these concerns with only moderate confidence, since I am not an expert specifically on distributional shift literature.\n\nThe paper focuses largely on the pitfalls and limitations of 2 papers from ICLR 2023, the Lee DivDis and the Pagliardini D-Bat approaches. I don't doubt that these papers are high-quality and significant, but they simply haven't been around long enough to know for sure how significant it is to critique them and scrutinize their flaws. I'm not saying that critiquing them is unworthy- it's just hard to tell whether it's a highly significant contribution. \n\nAlso, although the authors do use real-world data to an extent, I find the applications to still be a bit contrived and not illustrative of a clear real-world situation where distributional shift needs to be handled and cannot be avoided.  The MNIST-CIFAR concatenations and Waterbirds-CC are both contrived, i.e., constructed to have spurious correlations, rather than spurious correlations arising naturally.  Office-Home with Art, Product, Clipart, Real-World is somewhat better, but nonetheless, the authors of that dataset did manage to assemble all four sources and so the sensible thing to do would seem to be to train on a dataset with all 4 sources mixed together. I can see how you could have a real-life situation where e.g. only Art, Product, Clipart are available and then you have to design a system which peforms well on Real-World, but the paper would be stronger if a problem domain was studied where that was a real constraint that made distributional shift unavoidable. The most obvious scenario would be distribution shift over time, e.g. , we had to collect images during sunny weather in the summer and then had to deploy the system in darker weather when it was raining...something along those lines. \n\nMy other significance concern is the findings (while common-sense) seem a bit unsurprising to me. We've  known since the 1990s from the Wolpert No Free Lunch theorems, the bias-variance tradeoff, etc  that the right inductive bias is crucial to supervised learning success, in general. While it's worthwhile to illustrate a version of this principle for these recently-developed diversification methods, the finding that the principle holds seems more like common sense to me than like a surprising finding that advances the field in a major way. Maybe I'm wrong, though, and maybe I\"m expecting too much from a paper worthy of ICLR acceptance.\n\nAnother significance concern I have comes from a reaction I had to a sentence on page 4: \"We, therefore, focus on studying the first stage and assume access to the oracle that chooses the best available hypothesis in the second stage\". While I can see that it's true that the first stage is crucial to the success of the whole diversification approach, as someone who is not that familiar with diversification methods or OOD-robust algorithms, it's not that clear to me that we can assume that the second stage will succeed if the first stage succeeds. I guess the idea is that if K is small, then we only need a modest amount of supervised holdout data to choose from among the K diverse hypotheses generated, e.g. the in-sample bias of picking from among a low-K list using validation data is modest? I suppose that could be true, but I would appreciate more background on that topic. \n\nGiven what I wrote in the previous paragraphs and given the focus on 2 papers just published in 2023, the whole paper leans a little bit too far in the direction of assuming that what the authors are studying is very important rather than making the case for the importance of the topic to someone who isn't already deeply involved in OOD and diversification.\n\nTypos:\n\npage 5 function. And the training distribution -> function, and the training distribution (don't start a new sentence here)\npage 9 \"This is in consistence with\" -> This is consistent with", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "C7e5KXjYyK", "reviewer": "Reviewer_WG9i", "summary": "This paper studies the class of  “diversification” methods that address OOD problem, and identify the key components contributing to their performance. Some findings are provided through research，that can help guide the development of approaches. Some experiments are conducted to support their results.", "questions": "In addition to the methods used in the paper, does it need to consider more models to support the conclusions of the paper?", "limitations": "", "rating": 5, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "1) This paper investigates the “diversification” methods from theoretically from different aspects. Some key results are provided.\n2) Valid experiments have proved the rationality of their views. \n3) Some showcase are provided to validate the meaning of the findings.", "weaknesses": "1) What are the unique advantages of “diversification” methods to solve OOD problems？Does it contribute much to OOD community to study the components of “diversification”? More discussion of motivation and related works are needed.\n2) Despite the limitation talked in paper, I think experimental setup of the paper is still simple. More complicated diversification loss, approaches, need to be included.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (poster)