# Distributionally Robust Stochastic Optimization with Wasserstein Distance

arXiv: https://arxiv.org/abs/1604.02199 · HTML: https://mlsanity.com/paper/1604.02199/

## Key facts

- Cited by 185 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Apr 2016.

## Built on by (most cited first, 30 of 185)

- [Examining and Combating Spurious Features under Distribution Shift](https://mlsanity.com/paper/2106.07171/) (2021)
- [Improved Sample Complexity Bounds for Distributionally Robust Reinforcement Learning](https://mlsanity.com/paper/2303.02783/) (2023)
- [Sinkhorn Distributionally Robust Optimization](https://mlsanity.com/paper/2109.11926/) (2021)
- [A Unified Wasserstein Distributional Robustness Framework for Adversarial Training](https://mlsanity.com/paper/2202.13437/) (2022)
- [A Finite Sample Complexity Bound for Distributionally Robust Q-learning](https://mlsanity.com/paper/2302.13203/) (2023)
- [Towards Theoretical Understandings of Robust Markov Decision Processes: Sample Complexity and Asymptotics](https://mlsanity.com/paper/2105.03863/) (2021)
- [Statistical Analysis of Wasserstein Distributionally Robust Estimators](https://mlsanity.com/paper/2108.02120/) (2021)
- [Outlier-Robust Optimal Transport: Duality, Structure, and Statistical Analysis](https://mlsanity.com/paper/2111.01361/) (2021)
- [Integrated Conditional Estimation-Optimization](https://mlsanity.com/paper/2110.12351/) (2021)
- [Estimate-Then-Optimize versus Integrated-Estimation-Optimization versus Sample Average Approximation: A Stochastic Dominance Perspective](https://mlsanity.com/paper/2304.06833/) (2023)
- [Model-Free Robust Average-Reward Reinforcement Learning](https://mlsanity.com/paper/2305.10504/) (2023)
- [An improved central limit theorem and fast convergence rates for entropic transportation costs](https://mlsanity.com/paper/2204.09105/) (2022)
- [Holistic Robust Data-Driven Decisions](https://mlsanity.com/paper/2207.09560/) (2022)
- [Robust LLM Alignment via Distributionally Robust Direct Preference Optimization](https://mlsanity.com/paper/2502.01930/) (2025)
- [Distributionally Robust Policy Learning via Adversarial Environment Generation](https://mlsanity.com/paper/2107.06353/) (2021)
- [Wasserstein distributional robustness of neural networks](https://mlsanity.com/paper/2306.09844/) (2023)
- [Outlier-Robust Wasserstein DRO](https://mlsanity.com/paper/2311.05573/) (2023)
- [AdvST: Revisiting Data Augmentations for Single Domain Generalization](https://mlsanity.com/paper/2312.12720/) (2023)
- [Smooth p-Wasserstein Distance: Structure, Empirical Approximation, and Statistical Applications](https://mlsanity.com/paper/2101.04039/) (2021)
- [On the Importance of Feature Separability in Predicting Out-Of-Distribution Error](https://mlsanity.com/paper/2303.15488/) (2023)
- [Poisoning Attack against Estimating from Pairwise Comparisons](https://mlsanity.com/paper/2107.01854/) (2021)
- [Distributionally Robust Optimization via Ball Oracle Acceleration](https://mlsanity.com/paper/2203.13225/) (2022)
- [Distributed Distributionally Robust Optimization with Non-Convex Objectives](https://mlsanity.com/paper/2210.07588/) (2022)
- [Instance-Dependent Generalization Bounds via Optimal Transport](https://mlsanity.com/paper/2211.01258/) (2022)
- [Unifying Distributionally Robust Optimization via Optimal Transport Theory](https://mlsanity.com/paper/2308.05414/) (2023)
- [Flow-based Distributionally Robust Optimization](https://mlsanity.com/paper/2310.19253/) (2023)
- [Fundamental Tradeoffs in Distributionally Adversarial Training](https://mlsanity.com/paper/2101.06309/) (2021)
- [Robustifying Conditional Portfolio Decisions via Optimal Transport](https://mlsanity.com/paper/2103.16451/) (2021)
- [A Short and General Duality Proof for Wasserstein Distributionally Robust Optimization](https://mlsanity.com/paper/2205.00362/) (2022)
- [Global-Local Regularization Via Distributional Robustness](https://mlsanity.com/paper/2203.00553/) (2022)

## Abstract (arXiv)

Distributionally robust stochastic optimization (DRSO) is an approach to optimization under uncertainty in which, instead of assuming that there is a known true underlying probability distribution, one hedges against a chosen set of distributions. In this paper we first point out that the set of distributions should be chosen to be appropriate for the application at hand, and that some of the choices that have been popular until recently are, for many applications, not good choices. We next consider sets of distributions that are within a chosen Wasserstein distance from a nominal distribution. Such a choice of sets has two advantages: (1) The resulting distributions hedged against are more reasonable than those resulting from other popular choices of sets. (2) The problem of determining the worst-case expectation over the resulting set of distributions has desirable tractability properties. We derive a strong duality reformulation of the corresponding DRSO problem and construct approximate worst-case distributions explicitly via the first-order optimality conditions of the dual problem. Our contributions are four-fold. (i) We identify necessary and sufficient conditions for the existence of a worst-case distribution, which are naturally related to the growth rate of the objective function. (ii) We show that the worst-case distributions resulting from an appropriate Wasserstein distance have a concise structure and a clear interpretation. (iii) Using this structure, we show that data-driven DRSO problems can be approximated to any accuracy by robust optimization problems, and thereby many DRSO problems become tractable by using tools from robust optimization. (iv) Our strong duality result holds in a very general setting. As examples, we show that it can be applied to infinite-dimensional process control and intensity estimation for point processes.