OpenRaiser/Intern-Atlas huggingface dataset:
Representation Learning for Maximization of MI, Nonlinear ICA and Nonlinear Subspaces with Robust Density Ratio Estimation
Venue: Journal JMLR 2022
Year: 2022

Contrastive learning is a recent promising approach in unsupervised representation learning where a feature representation of data is learned by solving a pseudo classification problem from unlabelled data. However, it is not straightforward to understand what representation contrastive learning yields. In addition, contrastive learning is often based on the maximum likelihood estimation, which tends to be vulnerable to the contamination by outliers. To promote the understanding to contrastive learning, this paper first theoretically shows a connection to maximization of mutual information (MI). Our result indicates that density ratio estimation is necessary and sufficient for maximization of MI under some conditions. Thus, contrastive learning related to density ratio estimation as done in popular objective functions can be interpreted as maximizing MI. Next, with the density ratio, we establish new recovery conditions for the latent source components in nonlinear independent component analysis (ICA). In contrast with existing work, the established conditions include a novel insight for the dimensionality of data, which is clearly supported by numerical experiments. Furthermore, inspired by nonlinear ICA, we propose a novel framework to estimate a nonlinear subspace for lower-dimensional latent source components, and some theoretical conditions for the subspace estimation are established with the density ratio. Then, we propose a practical method through outlier-robust density ratio estimation, which can be seen as performing maximization of MI, nonlinear ICA or nonlinear subspace estimation. Moreover, a sample-efficient nonlinear ICA method is also proposed. We theoretically investigate outlier-robustness of the proposed methods. Finally, the usefulness of the proposed methods is numerically demonstrated in nonlinear ICA and through application to linear classification.
On the Efficiency of Entropic Regularized Algorithms for Optimal Transport
Venue: Journal JMLR 2022
Year: 2022

We present several new complexity results for the entropic regularized algorithms that approximately solve the optimal transport (OT) problem between two discrete probability measures with at most $n$ atoms. First, we improve the complexity bound of a greedy variant of Sinkhorn, known as \textit{Greenkhorn}, from $\widetilde{O}(n^2\varepsilon^{-3})$ to $\widetilde{O}(n^2\varepsilon^{-2})$. Notably, our result can match the best known complexity bound of Sinkhorn and help clarify why Greenkhorn significantly outperforms Sinkhorn in practice in terms of row/column updates as observed by~\citet{Altschuler-2017-Near}. Second, we propose a new algorithm, which we refer to as \textit{APDAMD} and which generalizes an adaptive primal-dual accelerated gradient descent (APDAGD) algorithm~\citep{Dvurechensky-2018-Computational} with a prespecified mirror mapping $ϕ$. We prove that APDAMD achieves the complexity bound of $\widetilde{O}(n^2\sqrtδ\varepsilon^{-1})$ in which $δ>0$ stands for the regularity of $ϕ$. In addition, we show by a counterexample that the complexity bound of $\widetilde{O}(\min\{n^{9/4}\varepsilon^{-1}, n^2\varepsilon^{-2}\})$ proved for APDAGD before is invalid and give a refined complexity bound of $\widetilde{O}(n^{5/2}\varepsilon^{-1})$. Further, we develop a \textit{deterministic} accelerated variant of Sinkhorn via appeal to estimated sequence and prove the complexity bound of $\widetilde{O}(n^{7/3}\varepsilon^{-4/3})$. As such, we see that accelerated variant of Sinkhorn outperforms Sinkhorn and Greenkhorn in terms of $1/\varepsilon$ and APDAGD and accelerated alternating minimization (AAM)~\citep{Guminov-2021-Combination} in terms of $n$. Finally, we conduct the experiments on synthetic and real data and the numerical results show the efficiency of Greenkhorn, APDAMD and accelerated Sinkhorn in practice.
No Weighted-Regret Learning in Adversarial Bandits with Delays
Venue: Journal JMLR 2022
Year: 2022

Consider a scenario where a player chooses an action in each round $t$ out of $T$ rounds and observes the incurred cost after a delay of $d_{t}$ rounds. The cost functions and the delay sequence are chosen by an adversary. We show that in a non-cooperative game, the expected weighted ergodic distribution of play converges to the set of coarse correlated equilibria if players use algorithms that have "no weighted-regret" in the above scenario, even if they have linear regret due to too large delays. For a two-player zero-sum game, we show that no weighted-regret is sufficient for the weighted ergodic average of play to converge to the set of Nash equilibria. We prove that the FKM algorithm with $n$ dimensions achieves an expected regret of $O\left(nT^{\frac{3}{4}}+\sqrt{n}T^{\frac{1}{3}}D^{\frac{1}{3}}\right)$ and the EXP3 algorithm with $K$ arms achieves an expected regret of $O\left(\sqrt{\log K\left(KT+D\right)}\right)$ even when $D=\sum_{t=1}^{T}d_{t}$ and $T$ are unknown. These bounds use a novel doubling trick that, under mild assumptions, provably retains the regret bound for when $D$ and $T$ are known. Using these bounds, we show that FKM and EXP3 have no weighted-regret even for $d_{t}=O\left(t\log t\right)$. Therefore, algorithms with no weighted-regret can be used to approximate a CCE of a finite or convex unknown game that can only be simulated with bandit feedback, even if the simulation involves significant delays.
Machine Learning on Graphs: A Model and Comprehensive Taxonomy
Venue: Journal JMLR 2022
Year: 2022

There has been a surge of recent interest in learning representations for graph-structured data. Graph representation learning methods have generally fallen into three main categories, based on the availability of labeled data. The first, network embedding (such as shallow graph embedding or graph auto-encoders), focuses on learning unsupervised representations of relational structure. The second, graph regularized neural networks, leverages graphs to augment neural network losses with a regularization objective for semi-supervised learning. The third, graph neural networks, aims to learn differentiable functions over discrete topologies with arbitrary structure. However, despite the popularity of these areas there has been surprisingly little work on unifying the three paradigms. Here, we aim to bridge the gap between graph neural networks, network embedding and graph regularization models. We propose a comprehensive taxonomy of representation learning methods for graph-structured data, aiming to unify several disparate bodies of work. Specifically, we propose a Graph Encoder Decoder Model (GRAPHEDM), which generalizes popular algorithms for semi-supervised learning on graphs (e.g. GraphSage, Graph Convolutional Networks, Graph Attention Networks), and unsupervised learning of graph representations (e.g. DeepWalk, node2vec, etc) into a single consistent approach. To illustrate the generality of this approach, we fit over thirty existing methods into this framework. We believe that this unifying view both provides a solid foundation for understanding the intuition behind these methods, and enables future research in the area.
Optimality and Stability in Non-Convex Smooth Games
Venue: Journal JMLR 2022
Year: 2022

Convergence to a saddle point for convex-concave functions has been studied for decades, while recent years has seen a surge of interest in non-convex (zero-sum) smooth games, motivated by their recent wide applications. It remains an intriguing research challenge how local optimal points are defined and which algorithm can converge to such points. An interesting concept is known as the local minimax point, which strongly correlates with the widely-known gradient descent ascent algorithm. This paper aims to provide a comprehensive analysis of local minimax points, such as their relation with other solution concepts and their optimality conditions. We find that local saddle points can be regarded as a special type of local minimax points, called uniformly local minimax points, under mild continuity assumptions. In (non-convex) quadratic games, we show that local minimax points are (in some sense) equivalent to global minimax points. Finally, we study the stability of gradient algorithms near local minimax points. Although gradient algorithms can converge to local/global minimax points in the non-degenerate case, they would often fail in general cases. This implies the necessity of either novel algorithms or concepts beyond saddle points and minimax points in non-convex smooth games.
A Perturbation-Based Kernel Approximation Framework
Venue: Journal JMLR 2022
Year: 2022

Kernel methods are powerful tools in various data analysis tasks. Yet, in many cases, their time and space complexity render them impractical for large datasets. Various kernel approximation methods were proposed to overcome this issue, with the most prominent method being the Nystr{ö}m method. In this paper, we derive a perturbation-based kernel approximation framework building upon results from classical perturbation theory. We provide an error analysis for this framework, and prove that in fact, it generalizes the Nystr{ö}m method and several of its variants. Furthermore, we show that our framework gives rise to new kernel approximation schemes, that can be tuned to take advantage of the structure of the approximated kernel matrix. We support our theoretical results numerically and demonstrate the advantages of our approximation framework on both synthetic and real-world data.
Simple and Optimal Stochastic Gradient Methods for Nonsmooth Nonconvex Optimization
Venue: Journal JMLR 2022
Year: 2022

We propose and analyze several stochastic gradient algorithms for finding stationary points or local minimum in nonconvex, possibly with nonsmooth regularizer, finite-sum and online optimization problems. First, we propose a simple proximal stochastic gradient algorithm based on variance reduction called ProxSVRG+. We provide a clean and tight analysis of ProxSVRG+, which shows that it outperforms the deterministic proximal gradient descent (ProxGD) for a wide range of minibatch sizes, hence solves an open problem proposed in Reddi et al. (2016b). Also, ProxSVRG+ uses much less proximal oracle calls than ProxSVRG (Reddi et al., 2016b) and extends to the online setting by avoiding full gradient computations. Then, we further propose an optimal algorithm, called SSRGD, based on SARAH (Nguyen et al., 2017) and show that SSRGD further improves the gradient complexity of ProxSVRG+ and achieves the optimal upper bound, matching the known lower bound of (Fang et al., 2018; Li et al., 2021). Moreover, we show that both ProxSVRG+ and SSRGD enjoy automatic adaptation with local structure of the objective function such as the Polyak-Łojasiewicz (PL) condition for nonconvex functions in the finite-sum case, i.e., we prove that both of them can automatically switch to faster global linear convergence without any restart performed in prior work ProxSVRG (Reddi et al., 2016b). Finally, we focus on the more challenging problem of finding an $(ε, δ)$-local minimum instead of just finding an $ε$-approximate (first-order) stationary point (which may be some bad unstable saddle points). We show that SSRGD can find an $(ε, δ)$-local minimum by simply adding some random perturbations. Our algorithm is almost as simple as its counterpart for finding stationary points, and achieves similar optimal rates.
Attraction-Repulsion Spectrum in Neighbor Embeddings
Venue: Journal JMLR 2022
Year: 2022

Neighbor embeddings are a family of methods for visualizing complex high-dimensional datasets using $k$NN graphs. To find the low-dimensional embedding, these algorithms combine an attractive force between neighboring pairs of points with a repulsive force between all points. One of the most popular examples of such algorithms is t-SNE. Here we empirically show that changing the balance between the attractive and the repulsive forces in t-SNE using the exaggeration parameter yields a spectrum of embeddings, which is characterized by a simple trade-off: stronger attraction can better represent continuous manifold structures, while stronger repulsion can better represent discrete cluster structures and yields higher $k$NN recall. We find that UMAP embeddings correspond to t-SNE with increased attraction; mathematical analysis shows that this is because the negative sampling optimisation strategy employed by UMAP strongly lowers the effective repulsion. Likewise, ForceAtlas2, commonly used for visualizing developmental single-cell transcriptomic data, yields embeddings corresponding to t-SNE with the attraction increased even more. At the extreme of this spectrum lie Laplacian Eigenmaps. Our results demonstrate that many prominent neighbor embedding algorithms can be placed onto the attraction-repulsion spectrum, and highlight the inherent trade-offs between them.
Weakly Supervised Disentangled Generative Causal Representation Learning
Venue: Journal JMLR 2022
Year: 2022

This paper proposes a Disentangled gEnerative cAusal Representation (DEAR) learning method under appropriate supervised information. Unlike existing disentanglement methods that enforce independence of the latent variables, we consider the general case where the underlying factors of interests can be causally related. We show that previous methods with independent priors fail to disentangle causally related factors even under supervision. Motivated by this finding, we propose a new disentangled learning method called DEAR that enables causal controllable generation and causal representation learning. The key ingredient of this new formulation is to use a structural causal model (SCM) as the prior distribution for a bidirectional generative model. The prior is then trained jointly with a generator and an encoder using a suitable GAN algorithm incorporated with supervised information on the ground-truth factors and their underlying causal structure. We provide theoretical justification on the identifiability and asymptotic convergence of the proposed method. We conduct extensive experiments on both synthesized and real data sets to demonstrate the effectiveness of DEAR in causal controllable generation, and the benefits of the learned representations for downstream tasks in terms of sample efficiency and distributional robustness.
PECOS: Prediction for Enormous and Correlated Output Spaces
Venue: Journal JMLR 2022
Year: 2022

Many large-scale applications amount to finding relevant results from an enormous output space of potential candidates. For example, finding the best matching product from a large catalog or suggesting related search phrases on a search engine. The size of the output space for these problems can range from millions to billions, and can even be infinite in some applications. Moreover, training data is often limited for the long-tail items in the output space. Fortunately, items in the output space are often correlated thereby presenting an opportunity to alleviate the data sparsity issue. In this paper, we propose the Prediction for Enormous and Correlated Output Spaces (PECOS) framework, a versatile and modular machine learning framework for solving prediction problems for very large output spaces, and apply it to the eXtreme Multilabel Ranking (XMR) problem: given an input instance, find and rank the most relevant items from an enormous but fixed and finite output space. We propose a three phase framework for PECOS: (i) in the first phase, PECOS organizes the output space using a semantic indexing scheme, (ii) in the second phase, PECOS uses the indexing to narrow down the output space by orders of magnitude using a machine learned matching scheme, and (iii) in the third phase, PECOS ranks the matched items using a final ranking scheme. The versatility and modularity of PECOS allows for easy plug-and-play of various choices for the indexing, matching, and ranking phases. We also develop very fast inference procedures which allow us to perform XMR predictions in real time; for example, inference takes less than 1 millisecond per input on the dataset with 2.8 million labels. The PECOS software is available at https://libpecos.org.
Mitigating the Effects of Non-Identifiability on Inference for Bayesian Neural Networks with Latent Variables
Venue: Journal JMLR 2022
Year: 2022

Bayesian Neural Networks with Latent Variables (BNN+LVs) capture predictive uncertainty by explicitly modeling model uncertainty (via priors on network weights) and environmental stochasticity (via a latent input noise variable). In this work, we first show that BNN+LV suffers from a serious form of non-identifiability: explanatory power can be transferred between the model parameters and latent variables while fitting the data equally well. We demonstrate that as a result, in the limit of infinite data, the posterior mode over the network weights and latent variables is asymptotically biased away from the ground-truth. Due to this asymptotic bias, traditional inference methods may in practice yield parameters that generalize poorly and misestimate uncertainty. Next, we develop a novel inference procedure that explicitly mitigates the effects of likelihood non-identifiability during training and yields high-quality predictions as well as uncertainty estimates. We demonstrate that our inference method improves upon benchmark methods across a range of synthetic and real data-sets.
A Primer for Neural Arithmetic Logic Modules
Venue: Journal JMLR 2022
Year: 2022

Neural Arithmetic Logic Modules have become a growing area of interest, though remain a niche field. These modules are neural networks which aim to achieve systematic generalisation in learning arithmetic and/or logic operations such as $\{+, -, \times, ÷, \leq, \textrm{AND}\}$ while also being interpretable. This paper is the first in discussing the current state of progress of this field, explaining key works, starting with the Neural Arithmetic Logic Unit (NALU). Focusing on the shortcomings of the NALU, we provide an in-depth analysis to reason about design choices of recent modules. A cross-comparison between modules is made on experiment setups and findings, where we highlight inconsistencies in a fundamental experiment causing the inability to directly compare across papers. To alleviate the existing inconsistencies, we create a benchmark which compares all existing arithmetic NALMs. We finish by providing a novel discussion of existing applications for NALU and research directions requiring further exploration.
Adversarial Classification: Necessary Conditions and Geometric Flows
Venue: Journal JMLR 2022
Year: 2022

We study a version of adversarial classification where an adversary is empowered to corrupt data inputs up to some distance $\varepsilon$, using tools from variational analysis. In particular, we describe necessary conditions associated with the optimal classifier subject to such an adversary. Using the necessary conditions, we derive a geometric evolution equation which can be used to track the change in classification boundaries as $\varepsilon$ varies. This evolution equation may be described as an uncoupled system of differential equations in one dimension, or as a mean curvature type equation in higher dimension. In one dimension, and under mild assumptions on the data distribution, we rigorously prove that one can use the initial value problem starting from $\varepsilon=0$, which is simply the Bayes classifier, in order to solve for the global minimizer of the adversarial problem for small values of $\varepsilon$. In higher dimensions we provide a similar result, albeit conditional to the existence of regular solutions of the initial value problem. In the process of proving our main results we obtain a result of independent interest connecting the original adversarial problem with an optimal transport problem under no assumptions on whether classes are balanced or not. Numerical examples illustrating these ideas are also presented.
Multivariate Boosted Trees and Applications to Forecasting and Control
Venue: Journal JMLR 2022
Year: 2022

Gradient boosted trees are competition-winning, general-purpose, non-parametric regressors, which exploit sequential model fitting and gradient descent to minimize a specific loss function. The most popular implementations are tailored to univariate regression and classification tasks, precluding the possibility of capturing multivariate target cross-correlations and applying structured penalties to the predictions. In this paper, we present a computationally efficient algorithm for fitting multivariate boosted trees. We show that multivariate trees can outperform their univariate counterpart when the predictions are correlated. Furthermore, the algorithm allows to arbitrarily regularize the predictions, so that properties like smoothness, consistency and functional relations can be enforced. We present applications and numerical results related to forecasting and control.
Active Structure Learning of Bayesian Networks in an Observational Setting
Venue: Journal JMLR 2022
Year: 2022

We study active structure learning of Bayesian networks in an observational setting, in which there are external limitations on the number of variable values that can be observed from the same sample. Random samples are drawn from the joint distribution of the network variables, and the algorithm iteratively selects which variables to observe in the next sample. We propose a new active learning algorithm for this setting, that finds with a high probability a structure with a score that is $ε$-close to the optimal score. We show that for a class of distributions that we term stable, a sample complexity reduction of up to a factor of $\widetildeΩ(d^3)$ can be obtained, where $d$ is the number of network variables. We further show that in the worst case, the sample complexity of the active algorithm is guaranteed to be almost the same as that of a naive baseline algorithm. To supplement the theoretical results, we report experiments that compare the performance of the new active algorithm to the naive baseline and demonstrate the sample complexity improvements. Code for the algorithm and for the experiments is provided at https://github.com/noabdavid/activeBNSL.
Greedification Operators for Policy Optimization: Investigating Forward and Reverse KL Divergences
Venue: Journal JMLR 2022
Year: 2022

Approximate Policy Iteration (API) algorithms alternate between (approximate) policy evaluation and (approximate) greedification. Many different approaches have been explored for approximate policy evaluation, but less is understood about approximate greedification and what choices guarantee policy improvement. In this work, we investigate approximate greedification when reducing the KL divergence between the parameterized policy and the Boltzmann distribution over action values. In particular, we investigate the difference between the forward and reverse KL divergences, with varying degrees of entropy regularization. We show that the reverse KL has stronger policy improvement guarantees, but that reducing the forward KL can result in a worse policy. We also demonstrate, however, that a large enough reduction of the forward KL can induce improvement under additional assumptions. Empirically, we show on simple continuous-action environments that the forward KL can induce more exploration, but at the cost of a more suboptimal policy. No significant differences were observed in the discrete-action setting or on a suite of benchmark problems. Throughout, we highlight that many policy gradient methods can be seen as an instance of API, with either the forward or reverse KL for the policy update, and discuss next steps for understanding and improving our policy optimization algorithms.