We introduce a new notion of generalization -- Distributional Generalization -- which roughly states that outputs of a classifier at train and test time are close *as distributions*, as opposed to close in just their average error. For example, if we mislabel 30% of dogs as cats in the train set of CIFAR-10, then a ResNet trained to interpolation will in fact mislabel roughly 30% of dogs as cats on the *test set* as well, while leaving other classes unaffected. This behavior is not captured by classical generalization, which would only consider the average error and not the distribution of errors over the input domain. Our formal conjectures, which are much more general than this example, characterize the form of distributional generalization that can be expected in terms of problem parameters: model architecture, training procedure, number of samples, and data distribution. We give empirical evidence for these conjectures across a variety of domains in machine learning, including neural networks, kernel machines, and decision trees. Our results thus advance our empirical understanding of interpolating classifiers.
Institute of Data Science, National University of Singapore（新加坡国立大学数据科学研究院）；Department of Industrial Systems Engineering and Management, National University of Singapore（新加坡国立大学工业系统工程与管理系）
Distributionally robust optimization (DRO) has attracted attention in machine learning due to its connections to regularization, generalization, and robustness. Existing work has considered uncertainty sets based on phi-divergences and Wasserstein distances, each of which have drawbacks. In this paper, we study DRO with uncertainty sets measured via maximum mean discrepancy (MMD). We show that MMD DRO is roughly equivalent to regularization by the Hilbert norm and, as a byproduct, reveal deep connections to classic results in statistical learning. In particular, we obtain an alternative proof …
Established approaches to obtain generalization bounds in data-driven optimization and machine learning mostly build on solutions from empirical risk minimization (ERM), which depend crucially on the functional complexity of the hypothesis class. In this paper, we present an alternate route to obtain these bounds on the solution from distributionally robust optimization (DRO), a recent data-driven optimization framework based on worst-case analysis and the notion of ambiguity set to capture statistical uncertainty. In contrast to the hypothesis class complexity in ERM, our DRO bounds depend on…
IBM Research AI（IBM研究院人工智能中心）；Colorado State University-Pueblo（科罗拉多州立大学普韦布洛分校）
Distributionally robust optimization (DRO) problems are increasingly seen as a viable method to train machine learning models for improved model generalization. These min-max formulations, however, are more difficult to solve. We therefore provide a new stochastic gradient descent algorithm to efficiently solve this DRO formulation. Our approach applies gradient descent to the outer minimization formulation and estimates the gradient of the inner maximization based on a sample average approximation. The latter uses a subset of the data in each iteration, progressively increasing the subset siz…
We propose a novel method for semi-supervised learning (SSL) based on data-driven distributionally robust optimization (DRO) using optimal transport metrics. Our proposed method enhances generalization error by using the unlabeled data to restrict the support of the worst case distribution in our DRO formulation. We enable the implementation of our DRO formulation by proposing a stochastic gradient descent algorithm which allows to easily implement the training procedure. We demonstrate that our Semi-supervised DRO method is able to improve the generalization error over natural supervised proc…
Predictive models that generalize well under distributional shift are often desirable and sometimes crucial to building robust and reliable machine learning applications. We focus on distributional shift that arises in causal inference from observational data and in unsupervised domain adaptation. We pose both of these problems as prediction under a shift in design. Popular methods for overcoming distributional shift make unrealistic assumptions such as having a well-specified model or knowing the policy that gave rise to the observed data. Other methods are hindered by their need for a pre-sp…
IBM Research AI（IBM研究院人工智能部门）；Thomas J. Watson Research Center（托马斯·J·沃森研究中心）
Seeking to improve model generalization, we consider a new approach based on distributionally robust learning (DRL) that applies stochastic gradient descent to the outer minimization problem. Our algorithm efficiently estimates the gradient of the inner maximization problem through multi-level Monte Carlo randomization. Leveraging theoretical results that shed light on why standard gradient estimators fail, we establish the optimal parameterization of the gradient estimators of our approach that balances a fundamental tradeoff between computation time and statistical variance. Numerical experi…
Applied AI Institute（应用人工智能研究所）；Deakin University（迪肯大学）；University of Oxford（牛津大学）
Offline policy learning (OPL) leverages existing data collected a priori for policy optimization without any active exploration. Despite the prevalence and recent interest in this problem, its theoretical and algorithmic foundations in function approximation settings remain under-developed. In this paper, we consider this problem on the axes of distributional shift, optimization, and generalization in offline contextual bandits with neural networks. In particular, we propose a provably efficient offline contextual bandit with neural network function approximation that does not require any func…
Reinforcement learning (RL) has achieved remarkable performance in numerous sequential decision making and control tasks. However, a common problem is that learned nearly optimal policy always overfits to the training environment and may not be extended to situations never encountered during training. For practical applications, the randomness of environment usually leads to some devastating events, which should be the focus of safety-critical systems such as autonomous driving. In this paper, we introduce the minimax formulation and distributional framework to improve the generalization abili…
EPFL（洛桑联邦理工学院）；UC San Diego（加州大学圣地亚哥分校）；UC Berkeley（加州大学伯克利分校）；Columbia University（哥伦比亚大学）
Having similar behavior at training time and test time $-$ what we call a "What You See Is What You Get" (WYSIWYG) property $-$ is desirable in machine learning. Models trained with standard stochastic gradient descent (SGD), however, do not necessarily have this property, as their complex behaviors such as robustness or subgroup performance can differ drastically between training and test time. In contrast, we show that Differentially-Private (DP) training provably ensures the high-level WYSIWYG property, which we quantify using a notion of distributional generalization. Applying this connect…
Tsinghua-Berkeley Shenzhen Institute, Tsinghua University（清华大学深圳国际研究生院（清华-伯克利深圳学院））；School of Industrial and Systems Engineering, Georgia Institute of Technology（佐治亚理工学院工业与系统工程系）
Given multiple source domains, domain generalization aims at learning a universal model that performs well on any unseen but related target domain. In this work, we focus on the domain generalization scenario where domain shifts occur among class-conditional distributions of different domains. Existing approaches are not sufficiently robust when the variation of conditional distributions given the same class is large. In this work, we extend the concept of distributional robust optimization to solve the class-conditional domain generalization problem. Our approach optimizes the worst-case perf…
On Generalization and Distributional Update for Mimicking Observations with Adequate Exploration
Shanghai Institute of Satellite Engineering（上海卫星工程研究所）；Department of Mathematics, College of Sciences, Shanghai University（上海大学数学系）；School of Computer Science, University of Bristol（布里斯托大学计算机科学学院）；Newtouch Center for Mathematics of Shanghai University（上海大学数学中心）
Institute of Information Engineering, Chinese Academy of Sciences（中国科学院信息工程研究所）；School of Cyber Security, University of Chinese Academy of Sciences（中国科学院大学网络空间安全学院）；Institute of Computing Technology, Chinese Academy of Sciences（中国科学院计算技术研究所）
In the absence of prior knowledge, ordinal embedding methods obtain new representation for items in a low-dimensional Euclidean space via a set of quadruple-wise comparisons. These ordinal comparisons often come from human annotators, and sufficient comparisons induce the success of classical approaches. However, collecting a large number of labeled data is known as a hard task, and most of the existing work pay little attention to the generalization ability with insufficient samples. Meanwhile, recent progress in large margin theory discloses that rather than just maximizing the minimum margi…
University of Waterloo（滑铁卢大学）；Waterloo Artificial Intelligence Institute（滑铁卢人工智能研究所）
In this study, we propose the Affine Variational Autoencoder (AVAE), a variant of Variational Autoencoder (VAE) designed to improve robustness by overcoming the inability of VAEs to generalize to distributional shifts in the form of affine perturbations. By optimizing an affine transform to maximize ELBO, the proposed AVAE transforms an input to the training distribution without the need to increase model complexity to model the full distribution of affine transforms. In addition, we introduce a training procedure to create an efficient model by learning a subset of the training distribution, …
Moderately Distributional Exploration for Domain Generalization
University of Science and Technology of China（中国科学技术大学）；Hong Kong Baptist University（香港浸会大学）；Australian Artificial Intelligence Institute（澳大利亚人工智能研究所）；University of Technology Sydney（悉尼科技大学）；Hefei Comprehensive National Science Center（合肥综合性国家科学中心）
Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）；Technion, Israel Institute of Technology（以色列理工学院）
We propose to analyse the conditional distributional treatment effect (CoDiTE), which, in contrast to the more common conditional average treatment effect (CATE), is designed to encode a treatment's distributional aspects beyond the mean. We first introduce a formal definition of the CoDiTE associated with a distance function between probability measures. Then we discuss the CoDiTE associated with the maximum mean discrepancy via kernel conditional mean embeddings, which, coupled with a hypothesis test, tells us whether there is any conditional distributional effect of the treatment. Finally, …
PDE+: Enhancing Generalization via PDE with Adaptive Distributional Diffusion
Institute of Computing Technology, Chinese Academy of Sciences（中国科学院计算技术研究所）；University of Chinese Academy of Sciences（中国科学院大学）；National University of Singapore（新加坡国立大学）
Understanding Generalization from Embedding Dimension and Distributional Convergence
Department of Biomedical Engineering, Southern University of Science and Technology（生物医学工程系，南方科技大学）；Department of Electronic and Electrical Engineering, Southern University of Science and Technology（电子与电气工程系，南方科技大学）
Submodular functions have applications throughout machine learning, but in many settings, we do not have direct access to the underlying function $f$. We focus on stochastic functions that are given as an expectation of functions over a distribution $P$. In practice, we often have only a limited set of samples $f_i$ from $P$. The standard approach indirectly optimizes $f$ by maximizing the sum of $f_i$. However, this ignores generalization to the true (unknown) distribution. In this paper, we achieve better performance on the actual underlying function $f$ by directly optimizing a combination …
Generalizing to Unseen Domains with Wasserstein Distributional Robustness under Limited Source Knowledge
Tsinghua-Berkeley Shenzhen Institute, Tsinghua University（清华大学-伯克利深圳学院，清华大学）；School of Data Science, The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳）数据科学学院）；School of Industrial and Systems and Engineering, Georgia Institute of Technology（佐治亚理工学院工业与系统工程学院）
École Normale Supérieure（巴黎高等师范学院）；CNRS（法国国家科学研究中心）；NTT Communication Science Laboratories（NTT通信科学实验室）
Building upon recent advances in entropy-regularized optimal transport, and upon Fenchel duality between measures and continuous functions , we propose a generalization of the logistic loss that incorporates a metric or cost between classes. Unlike previous attempts to use optimal transport distances for learning, our loss results in unconstrained convex objective functions, supports infinite (or very large) class spaces, and naturally defines a geometric generalization of the softmax operator. The geometric properties of this loss make it suitable for predicting sparse and singular distributi…
Nanyang Technological University（南洋理工大学）；Singapore Institute of Manufacturing Technology, A*STAR（新加坡科技研究局新加坡制造技术研究院）
Recent deep models for solving routing problems always assume a single distribution of nodes for training, which severely impairs their cross-distribution generalization ability. In this paper, we exploit group distributionally robust optimization (group DRO) to tackle this issue, where we jointly optimize the weights for different groups of distributions and the parameters for the deep model in an interleaved manner during training. We also design a module based on convolutional neural network, which allows the deep model to learn more informative latent pattern among the nodes. We evaluate t…
Mila（米拉研究所）；University of Montreal（蒙特利尔大学）；Vector（Vector研究所）；University of Toronto（多伦多大学）；McGill University（麦吉尔大学）；Facebook AI Research（脸书人工智能研究院）
Distributional shift is one of the major obstacles when transferring machine learning prediction systems from the lab to the real world. To tackle this problem, we assume that variation across training domains is representative of the variation we might encounter at test time, but also that shifts at test time may be more extreme in magnitude. In particular, we show that reducing differences in risk across training domains can reduce a model's sensitivity to a wide range of extreme distributional shifts, including the challenging setting where the input contains both causal and anti-causal ele…
VinAI Research（VinAI研究院）；Monash University（莫纳什大学）；Johns Hopkins University（约翰斯·霍普金斯大学）；University of Texas, Austin（德克萨斯大学奥斯汀分校）
Despite superior performance in many situations, deep neural networks are often vulnerable to adversarial examples and distribution shifts, limiting model generalization ability in real-world applications. To alleviate these problems, recent approaches leverage distributional robustness optimization (DRO) to find the most challenging distribution, and then minimize loss function over this most challenging distribution. Regardless of achieving some improvements, these DRO approaches have some obvious limitations. First, they purely focus on local regularization to strengthen model robustness, m…
University of California（加利福尼亚大学）；Northeastern University（东北大学）；Khoury College of Computer Science（Khoury计算机学院）；Computer Science and Engineering（计算机科学与工程学院）
Recent work has shown deep learning can accelerate the prediction of physical dynamics relative to numerical solvers. However, limited physical accuracy and an inability to generalize under distributional shift limit its applicability to the real world. We propose to improve accuracy and generalization by incorporating symmetries into convolutional neural networks. Specifically, we employ a variety of methods each tailored to enforce a different symmetry. Our models are both theoretically and experimentally robust to distributional shift by symmetry group transformations and enjoy favorable sa…
Adaptive Sample-Level Framework Motivated by Distributionally Robust Optimization with Variance-Based Radius Assignment for Enhanced Neural Network Generalization Under Distribution Shift
Bangladesh University of Engineering and Technology (BUET)（孟加拉国工程技术大学）
Distributional Matrix Completion via Nearest Neighbors in the Wasserstein Space
Industrial Engineering and Operations Research, Columbia University（哥伦比亚大学工业工程与运筹学系）；Statistics and Data Science, Cornell Tech, Cornell University（康奈尔大学数据科学与统计学系）；Operations Research and Information Engineering, Cornell Tech, Cornell University（康奈尔大学运筹学与信息工程系）
It is well-known that deep neural networks (DNNs) are susceptible to adversarial attacks, exposing a severe fragility of deep learning systems. As the result, adversarial training (AT) method, by incorporating adversarial examples during training, represents a natural and effective approach to strengthen the robustness of a DNN-based classifier. However, most AT-based methods, notably PGD-AT and TRADES, typically seek a pointwise adversary that generates the worst-case adversarial example by independently perturbing each data sample, as a way to "probe" the vulnerability of the classifier. Arg…
The use of Reinforcement Learning (RL) agents in practical applications requires the consideration of suboptimal outcomes, depending on the familiarity of the agent with its environment. This is especially important in safety-critical environments, where errors can lead to high costs or damage. In distributional RL, the risk-sensitivity can be controlled via different distortion measures of the estimated return distribution. However, these distortion functions require an estimate of the risk level, which is difficult to obtain and depends on the current state. In this work, we demonstrate the …
