Distributionally robust optimization tackles out-of-sample issues like overfitting and distribution shifts by adopting an adversarial approach over a range of possible data distributions, known as the ambiguity set. To balance conservatism and accuracy, these sets must include realistic probability distributions by leveraging information from the nominal distribution. Assuming that nominal distributions arise from a structural causal model with a directed acyclic graph $\mathcal{G}$ and structural equations, previous methods such as adapted and $\mathcal{G}$-causal optimal transport have only utilized causal graph information in designing ambiguity sets. In this work, we propose incorporating structural equations, which include causal graph information, to enhance ambiguity sets, resulting in more realistic distributions. We introduce structural causal optimal transport and its associated ambiguity set, demonstrating their advantages and connections to previous methods. A key benefit of our approach is a relaxed version, where a regularization term replaces the complex causal constraints, enabling an efficient algorithm via difference-of-convex programming to solve structural causal optimal transport. We also show that when structural information is absent and must be estimated, our approach remains effective and provides finite sample guarantees. Lastly, we address the radius of ambiguity sets, illustrating how our method overcomes the curse of dimensionality in optimal transport problems, achieving faster shrinkage with dimension-free order.
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…
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…
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）
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…
Decision Making under the Exponential Family: Distributionally Robust Optimisation with Bayesian Ambiguity Sets
Department of Statistics, University of Warwick, Coventry, UK（威斯敏斯特大学统计学系）；Department of Computer Science, University of Warwick, Coventry, UK（威斯敏斯特大学计算机科学系）；Department of Statistics, University College London, UK（伦敦大学学院统计学系）；The Alan Turing Institute, London, UK（艾伦·图灵研究所）
Carnegie Mellon University（卡内基梅隆大学）；Columbia University（哥伦比亚大学）；University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
One key challenge for multi-task Reinforcement learning (RL) in practice is the absence of task indicators. Robust RL has been applied to deal with task ambiguity, but may result in over-conservative policies. To balance the worst-case (robustness) and average performance, we propose Group Distributionally Robust Markov Decision Process (GDR-MDP), a flexible hierarchical MDP formulation that encodes task groups via a latent mixture model. GDR-MDP identifies the optimal policy that maximizes the expected return under the worst-possible qualified belief over task groups within an ambiguity set. …
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）
We consider machine learning, particularly regression, using locally-differentially private datasets. The Wasserstein distance is used to define an ambiguity set centered at the empirical distribution of the dataset corrupted by local differential privacy noise. The ambiguity set is shown to contain the probability distribution of unperturbed, clean data. The radius of the ambiguity set is a function of the privacy budget, spread of the data, and the size of the problem. Hence, machine learning with locally-differentially private datasets can be rewritten as a distributionally-robust optimizat…
The University of Sydney（悉尼大学）；University of Wisconsin–Madison（威斯康星大学麦迪逊分校）
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, st…
University of California, Santa Barbara（加州大学圣巴巴拉分校）
Building on a recent framework for distributionally robust optimization, we consider estimation of the inverse covariance matrix for multivariate data. We provide a novel notion of a Wasserstein ambiguity set specifically tailored to this estimation problem, leading to a tractable class of regularized estimators. Special cases include penalized likelihood estimators for Gaussian data, specifically the graphical lasso estimator. As a consequence of this formulation, the radius of the Wasserstein ambiguity set is directly related to the regularization parameter in the estimation problem. Using t…
Distributionally Robust Federated Learning with Outlier Resilience
Division of Decision and Control Systems, School of Electrical Engineering and Computer Science, KTH Royal Institute of Technology（瑞典皇家理工学院电气工程与计算机科学系决策与控制系统 division）；Department of Control Science and Engineering, College of Electronics and Information Engineering, Tongji University（同济大学电子信息工程学院控制科学与工程系）；Mechanical Engineering and Material Science, Duke University（达特茅斯大学机械工程与材料科学系）
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…
We propose an end-to-end distributionally robust system for portfolio construction that integrates the asset return prediction model with a distributionally robust portfolio optimization model. We also show how to learn the risk-tolerance parameter and the degree of robustness directly from data. End-to-end systems have an advantage in that information can be communicated between the prediction and decision layers during training, allowing the parameters to be trained for the final task rather than solely for predictive performance. However, existing end-to-end systems are not able to quantify…
