# Wasserstein Distributionally Robust Optimization: Theory and Applications in Machine Learning

arXiv: https://arxiv.org/abs/1908.08729 · HTML: https://mlsanity.com/paper/1908.08729/

## Key facts

- Cited by 170 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Aug 2019.

## Built on by (most cited first, 30 of 170)

- [Unbalanced minibatch Optimal Transport; applications to Domain Adaptation](https://mlsanity.com/paper/2103.03606/) (2021)
- [Recent Advances in Optimal Transport for Machine Learning](https://mlsanity.com/paper/2306.16156/) (2023)
- [Twice regularized MDPs and the equivalence between robustness and regularization](https://mlsanity.com/paper/2110.06267/) (2021)
- [Distributionally Robust Learning](https://mlsanity.com/paper/2108.08993/) (2021)
- [Diagnosing Model Performance Under Distribution Shift](https://mlsanity.com/paper/2303.02011/) (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)
- [Minimax Regret Optimization for Robust Machine Learning under Distribution Shift](https://mlsanity.com/paper/2202.05436/) (2022)
- [Learning to Solve Routing Problems via Distributionally Robust Optimization](https://mlsanity.com/paper/2202.07241/) (2022)
- [Statistical Analysis of Wasserstein Distributionally Robust Estimators](https://mlsanity.com/paper/2108.02120/) (2021)
- [Noisy Feature Mixup](https://mlsanity.com/paper/2110.02180/) (2021)
- [Algorithms for mean-field variational inference via polyhedral optimization in the Wasserstein space](https://mlsanity.com/paper/2312.02849/) (2023)
- [Adaptive Preference Scaling for Reinforcement Learning with Human Feedback](https://mlsanity.com/paper/2406.02764/) (2024)
- [Counterfactual Plans under Distributional Ambiguity](https://mlsanity.com/paper/2201.12487/) (2022)
- [Sobolev Transport: A Scalable Metric for Probability Measures with Graph Metrics](https://mlsanity.com/paper/2202.10723/) (2022)
- [An improved central limit theorem and fast convergence rates for entropic transportation costs](https://mlsanity.com/paper/2204.09105/) (2022)
- [Robust LLM Alignment via Distributionally Robust Direct Preference Optimization](https://mlsanity.com/paper/2502.01930/) (2025)
- [AdvST: Revisiting Data Augmentations for Single Domain Generalization](https://mlsanity.com/paper/2312.12720/) (2023)
- [Sequential Domain Adaptation by Synthesizing Distributionally Robust Experts](https://mlsanity.com/paper/2106.00322/) (2021)
- [Data-Driven Sample Average Approximation with Covariate Information](https://mlsanity.com/paper/2207.13554/) (2022)
- [Distributed Distributionally Robust Optimization with Non-Convex Objectives](https://mlsanity.com/paper/2210.07588/) (2022)
- [Algorithmic Bias and Data Bias: Understanding the Relation between Distributionally Robust Optimization and Data Curation](https://mlsanity.com/paper/2106.09467/) (2021)
- [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)
- [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)
- [An Optimal Transport Approach to Personalized Federated Learning](https://mlsanity.com/paper/2206.02468/) (2022)
- [Distributionally Robust Recourse Action](https://mlsanity.com/paper/2302.11211/) (2023)
- [Bridging Distributionally Robust Learning and Offline RL: An Approach to Mitigate Distribution Shift and Partial Data Coverage](https://mlsanity.com/paper/2310.18434/) (2023)

## Abstract (arXiv)

Many decision problems in science, engineering and economics are affected by uncertain parameters whose distribution is only indirectly observable through samples. The goal of data-driven decision-making is to learn a decision from finitely many training samples that will perform well on unseen test samples. This learning task is difficult even if all training and test samples are drawn from the same distribution -- especially if the dimension of the uncertainty is large relative to the training sample size. Wasserstein distributionally robust optimization seeks data-driven decisions that perform well under the most adverse distribution within a certain Wasserstein distance from a nominal distribution constructed from the training samples. In this tutorial we will argue that this approach has many conceptual and computational benefits. Most prominently, the optimal decisions can often be computed by solving tractable convex optimization problems, and they enjoy rigorous out-of-sample and asymptotic consistency guarantees. We will also show that Wasserstein distributionally robust optimization has interesting ramifications for statistical learning and motivates new approaches for fundamental learning tasks such as classification, regression, maximum likelihood estimation or minimum mean square error estimation, among others.