We study a model for adversarial classification based on distributionally robust chance constraints. We show that under Wasserstein ambiguity, the model aims to minimize the conditional value-at-risk of the distance to misclassification, and we explore links to adversarial classification models proposed earlier and to maximum-margin classifiers. We also provide a reformulation of the distributionally robust model for linear classification, and show it is equivalent to minimizing a regularized ramp loss objective. Numerical experiments show that, despite the nonconvexity of this formulation, standard descent methods appear to converge to the global minimizer for this problem. Inspired by this observation, we show that, for a certain class of distributions, the only stationary point of the regularized ramp loss minimization problem is the global minimizer.
In many applications in statistics and machine learning, the availability of data samples from multiple possibly heterogeneous sources has become increasingly prevalent. On the other hand, in distributionally robust optimization, we seek data-driven decisions which perform well under the most adverse distribution from a nominal distribution constructed from data samples within a certain discrepancy of probability distributions. However, it remains unclear how to achieve such distributional robustness in model learning and estimation when data samples from multiple sources are available. In thi…
École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）；TU Delft（代尔夫特理工大学）
We study a distributionally robust mean square error estimation problem over a nonconvex Wasserstein ambiguity set containing only normal distributions. We show that the optimal estimator and the least favorable distribution form a Nash equilibrium. Despite the non-convex nature of the ambiguity set, we prove that the estimation problem is equivalent to a tractable convex program. We further devise a Frank-Wolfe algorithm for this convex program whose direction-searching subproblem can be solved in a quasi-closed form. Using these ingredients, we introduce a distributionally robust Kalman filt…
Construction of ambiguity set in robust optimization relies on the choice of divergences between probability distributions. In distribution learning, choosing appropriate probability distributions based on observed data is critical for approximating the true distribution. To improve the performance of machine learning models, there has recently been interest in designing objective functions based on Lp-Wasserstein distance rather than the classical Kullback-Leibler (KL) divergence. In this paper, we derive concentration and asymptotic results using Bregman divergence. We propose a novel asymme…
University of Texas at Austin（德克萨斯大学奥斯汀分校）；New York University（纽约大学）；Georgia Institute of Technology（佐治亚理工学院）
Wasserstein distributionally robust optimization (DRO) has recently achieved empirical success for various applications in operations research and machine learning, owing partly to its regularization effect. Although connection between Wasserstein DRO and regularization has been established in several settings, existing results often require restrictive assumptions, such as smoothness or convexity, that are not satisfied for many problems. In this paper, we develop a general theory on the variation regularization effect of the Wasserstein DRO - a new form of regularization that generalizes tot…
We propose to align distributional data from the perspective of Wasserstein means. We raise the problem of regularizing Wasserstein means and propose several terms tailored to tackle different problems. Our formulation is based on the variational transportation to distribute a sparse discrete measure into the target domain. The resulting sparse representation well captures the desired property of the domain while reducing the mapping cost. We demonstrate the scalability and robustness of our method with examples in domain adaptation, point set registration, and skeleton layout.
Inverse multiobjective optimization provides a general framework for the unsupervised learning task of inferring parameters of a multiobjective decision making problem (DMP), based on a set of observed decisions from the human expert. However, the performance of this framework relies critically on the availability of an accurate DMP, sufficient decisions of high quality, and a parameter space that contains enough information about the DMP. To hedge against the uncertainties in the hypothetical DMP, the data, and the parameter space, we investigate in this paper the distributionally robust appr…
The Chinese University of Hong Kong（香港中文大学）；Nanjing University（南京大学）
Wasserstein \textbf{D}istributionally \textbf{R}obust \textbf{O}ptimization (DRO) is concerned with finding decisions that perform well on data that are drawn from the worst-case probability distribution within a Wasserstein ball centered at a certain nominal distribution. In recent years, it has been shown that various DRO formulations of learning models admit tractable convex reformulations. However, most existing works propose to solve these convex reformulations by general-purpose solvers, which are not well-suited for tackling large-scale problems. In this paper, we focus on a family of W…
Calibrating Ambiguity Set via Diagnostic Transport for Distributionally Robust Optimization
Heinz College of Information Systems and Public Policy, Carnegie Mellon University（卡内基梅隆大学海因茨信息系统与公共政策学院）；Machine Learning Department, Carnegie Mellon University（卡内基梅隆大学机器学习系）；Department of Statistics and Data Science, Carnegie Mellon University（卡内基梅隆大学统计与数据科学系）
Wasserstein distance-based distributionally robust optimization (DRO) has received much attention lately due to its ability to provide a robustness interpretation of various learning models. Moreover, many of the DRO problems that arise in the learning context admits exact convex reformulations and hence can be tackled by off-the-shelf solvers. Nevertheless, the use of such solvers severely limits the applicability of DRO in large-scale learning problems, as they often rely on general purpose interior-point algorithms. On the other hand, there are very few works that attempt to develop fast it…
Enhancing Distributional Robustness in Principal Component Analysis by Wasserstein Distances
The Hong Kong Polytechnic University（香港理工大学）；Chinese Academy of Sciences（中国科学院）；Academy of Mathematics and Systems Science（数学系统科学研究院）；University of Chinese Academy of Sciences（中国科学院大学）
Tsinghua University（清华大学）；University of Toronto（多伦多大学）
In prescriptive analytics, the decision-maker observes historical samples of $(X, Y)$, where $Y$ is the uncertain problem parameter and $X$ is the concurrent covariate, without knowing the joint distribution. Given an additional covariate observation $x$, the goal is to choose a decision $z$ conditional on this observation to minimize the cost $\mathbb{E}[c(z,Y)|X=x]$. This paper proposes a new distributionally robust approach under Wasserstein ambiguity sets, in which the nominal distribution of $Y|X=x$ is constructed based on the Nadaraya-Watson kernel estimator concerning the historical dat…
We develop Distributionally Robust Optimization (DRO) formulations for Multivariate Linear Regression (MLR) and Multiclass Logistic Regression (MLG) when both the covariates and responses/labels may be contaminated by outliers. The DRO framework uses a probabilistic ambiguity set defined as a ball of distributions that are close to the empirical distribution of the training set in the sense of the Wasserstein metric. We relax the DRO formulation into a regularized learning problem whose regularizer is a norm of the coefficient matrix. We establish out-of-sample performance guarantees for the s…
University of Texas at Austin（德克萨斯大学奥斯汀分校）；Stanford University（斯坦福大学）
We propose a distributionally robust classification model with a fairness constraint that encourages the classifier to be fair in view of the equality of opportunity criterion. We use a type-$\infty$ Wasserstein ambiguity set centered at the empirical distribution to model distributional uncertainty and derive a conservative reformulation for the worst-case equal opportunity unfairness measure. We establish that the model is equivalent to a mixed binary optimization problem, which can be solved by standard off-the-shelf solvers. To improve scalability, we further propose a convex, hinge-loss-b…
Leibniz Institute of Agricultural Engineering and Bioeconomy e.V.（莱布尼茨农业工程与生物经济研究所）；University of Potsdam（波茨坦大学）
Deep metric learning employs deep neural networks to embed instances into a metric space such that distances between instances of the same class are small and distances between instances from different classes are large. In most existing deep metric learning techniques, the embedding of an instance is given by a feature vector produced by a deep neural network and Euclidean distance or cosine similarity defines distances between these vectors. In this paper, we study deep distributional embeddings of sequences, where the embedding of a sequence is given by the distribution of learned deep feat…
Stanford University（斯坦福大学）；Seoul National University（首尔国立大学）
Wasserstein distributionally robust optimization (WDRO) attempts to learn a model that minimizes the local worst-case risk in the vicinity of the empirical data distribution defined by Wasserstein ball. While WDRO has received attention as a promising tool for inference since its introduction, its theoretical understanding has not been fully matured. Gao et al. (2017) proposed a minimizer based on a tractable approximation of the local worst-case risk, but without showing risk consistency. In this paper, we propose a minimizer based on a novel approximation theorem and provide the correspondin…
Wasserstein distributionally robust optimization (DRO) aims to find robust and generalizable solutions by hedging against data perturbations in Wasserstein distance. Despite its recent empirical success in operations research and machine learning, existing performance guarantees for generic loss functions are either overly conservative due to the curse of dimensionality, or plausible only in large sample asymptotics. In this paper, we develop a non-asymptotic framework for analyzing the out-of-sample performance for Wasserstein robust learning and the generalization bound for its related Lipsc…
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）
Stanford University（斯坦福大学）；EPFL（洛桑联邦理工学院）；Delft University of Technology（代尔夫特理工大学）
We introduce a distributionally robust minimium mean square error estimation model with a Wasserstein ambiguity set to recover an unknown signal from a noisy observation. The proposed model can be viewed as a zero-sum game between a statistician choosing an estimator -- that is, a measurable function of the observation -- and a fictitious adversary choosing a prior -- that is, a pair of signal and noise distributions ranging over independent Wasserstein balls -- with the goal to minimize and maximize the expected squared estimation error, respectively. We show that if the Wasserstein balls are…
VinAI Research（VinAI研究院）；University of Texas, Austin（得克萨斯大学奥斯汀分校）
Sliced-Wasserstein distance (SW) and its variant, Max Sliced-Wasserstein distance (Max-SW), have been used widely in the recent years due to their fast computation and scalability even when the probability measures lie in a very high dimensional space. However, SW requires many unnecessary projection samples to approximate its value while Max-SW only uses the most important projection, which ignores the information of other useful directions. In order to account for these weaknesses, we propose a novel distance, named Distributional Sliced-Wasserstein distance (DSW), that finds an optimal dist…
CSIRO’s Data61（澳大利亚联邦科学与工业研究组织数据61中心）；University of Melbourne（墨尔本大学）
We use distributionally-robust optimization for machine learning to mitigate the effect of data poisoning attacks. We provide performance guarantees for the trained model on the original data (not including the poison records) by training the model for the worst-case distribution on a neighbourhood around the empirical distribution (extracted from the training dataset corrupted by a poisoning attack) defined using the Wasserstein distance. We relax the distributionally-robust machine learning problem by finding an upper bound for the worst-case fitness based on the empirical sampled-averaged f…
We propose a distributionally robust return-risk model for Markov decision processes (MDPs) under risk and reward ambiguity. The proposed model optimizes the weighted average of mean and percentile performances, and it covers the distributionally robust MDPs and the distributionally robust chance-constrained MDPs (both under reward ambiguity) as special cases. By considering that the unknown reward distribution lies in a Wasserstein ambiguity set, we derive the tractable reformulation for our model. In particular, we show that that the return-risk model can also account for risk from uncertain…
LIPN-CNRS（法国国家科学研究中心计算机科学实验室）；Université Sorbonne Paris Nord（索邦巴黎北大学）
This paper deals with a clustering algorithm for histogram data based on a Self-Organizing Map (SOM) learning. It combines a dimension reduction by SOM and the clustering of the data in a reduced space. Related to the kind of data, a suitable dissimilarity measure between distributions is introduced: the $L_2$ Wasserstein distance. Moreover, the number of clusters is not fixed in advance but it is automatically found according to a local data density estimation in the original space. Applications on synthetic and real data sets corroborate the proposed strategy.
Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）；University of Freiburg（弗赖堡大学）
In order to anticipate rare and impactful events, we propose to quantify the worst-case risk under distributional ambiguity using a recent development in kernel methods -- the kernel mean embedding. Specifically, we formulate the generalized moment problem whose ambiguity set (i.e., the moment constraint) is described by constraints in the associated reproducing kernel Hilbert space in a nonparametric manner. We then present the tractable approximation and its theoretical justification. As a concrete application, we numerically test the proposed method in characterizing the worst-case constrai…
Stanford University（斯坦福大学）；Singapore University of Technology and Design（新加坡科技设计大学）；VinAI Research（VinAI研究院）
We consider statistical methods which invoke a min-max distributionally robust formulation to extract good out-of-sample performance in data-driven optimization and learning problems. Acknowledging the distributional uncertainty in learning from limited samples, the min-max formulations introduce an adversarial inner player to explore unseen covariate data. The resulting Distributionally Robust Optimization (DRO) formulations, which include Wasserstein DRO formulations (our main focus), are specified using optimal transportation phenomena. Upon describing how these infinite-dimensional min-max…
Distributional Reduction: Unifying Dimensionality Reduction and Clustering with Gromov-Wasserstein
ENS de Lyon, CNRS, UMPA UMR 5669（里昂大学、国家科学研究中心、UMPA UMR 5669）；EPFL, LTS4（苏黎世联邦理工学院、LTS4）；Université Bretagne Sud, IRISA UMR 6074（布列塔尼南部大学、IRISA UMR 6074）；École polytechnique, Institut Polytechnique de Paris, CMAP UMR 7641（巴黎高等学院、巴黎理工学院、CMAP UMR 7641）；Inria, ENS de Lyon, CNRS, Université Claude Bernard Lyon 1, LIP UMR 5668（法国国家科学研究中心、里昂大学、 Claude Bernard Lyon 1 大学、LIP UMR 5668）
Wasserstein Distributionally Robust Nash Equilibrium Seeking with Heterogeneous Data: A Lagrangian Approach
Division of Decision and Control Systems, School of Electrical Engineering and Computer Science, KTH Royal Institute of Technology（决策与控制系统系，电气工程与计算机科学学院，皇家理工学院）；Delft Center for Systems and Control, Delft University of Technology（系统与控制中心，代尔夫特理工大学）；Department of Mechanical Engineering and Materials Science, Duke University（机械工程与材料科学系，杜克大学）
As opposed to standard empirical risk minimization (ERM), distributionally robust optimization aims to minimize the worst-case risk over a larger ambiguity set containing the original empirical distribution of the training data. In this work, we describe a minimax framework for statistical learning with ambiguity sets given by balls in Wasserstein space. In particular, we prove generalization bounds that involve the covering number properties of the original ERM problem. As an illustrative example, we provide generalization guarantees for transport-based domain adaptation problems where the Wa…
A stochastic smoothing framework for nonconvex-nonconcave minEmax problems with applications to Wasserstein distributionally robust optimization
Department of Applied Mathematics, Hong Kong Polytechnic University, Hong Kong, China（应用数学系，香港理工大学）；Institute for Math & AI, Wuhan (IMAI), Wuhan university, China（数学与人工智能研究院，武汉大学）；Department of Mathematical Sciences Rensselaer Polytechnic Institute, Troy, NY, USA（数学科学系，伦斯勒理工学院）
Wasserstein multivariate auto-regressive models for modeling distributional time series
Université Grenoble Alpes, CNRS, Inria, Grenoble INP, LJK（格勒诺布尔阿尔卑斯大学、国家科学研究中心、法国国家信息与自动化研究所、格勒诺布尔INP、LJK研究中心）；Institut de Mathématiques de Bordeaux, Université de Bordeaux（波尔多数学研究所、波尔多大学）
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…
FDR-SVM: A Federated Distributionally Robust Support Vector Machine via a Mixture of Wasserstein Balls Ambiguity Set
H. Milton Stewart School of Industrial and Systems Engineering（H. Milton Stewart工业与系统工程学院）；Georgia Institute of Technology（佐治亚理工学院）；Department of Electrical Engineering（电气工程系）；University of Chile（智利大学）
Columbia University（哥伦比亚大学）；Northwestern University（西北大学）；Columbia Mailman School of Public Health（哥伦比亚大学梅尔曼公共卫生学院）；Hong Kong University of Science and Technology（香港科技大学）
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…
