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Data-driven Distributionally Robust Optimization Using the Wasserstein Metric: Performance Guarantees and Tractable Reformulations

arXiv 1505.05116 ↗First posted 19 May 2015math.OC stat.COcited by 291

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Abstract (arXiv)

We consider stochastic programs where the distribution of the uncertain parameters is only observable through a finite training dataset. Using the Wasserstein metric, we construct a ball in the space of (multivariate and non-discrete) probability distributions centered at the uniform distribution on the training samples, and we seek decisions that perform best in view of the worst-case distribution within this Wasserstein ball. The state-of-the-art methods for solving the resulting distributionally robust optimization problems rely on global optimization techniques, which quickly become computationally excruciating. In this paper we demonstrate that, under mild assumptions, the distributionally robust optimization problems over Wasserstein balls can in fact be reformulated as finite convex programs---in many interesting cases even as tractable linear programs. Leveraging recent measure concentration results, we also show that their solutions enjoy powerful finite-sample performance guarantees. Our theoretical results are exemplified in mean-risk portfolio optimization as well as uncertainty quantification.

Key facts. Cited by 291 papers in the map (data through 14 Oct 2026). First posted on arXiv 19 May 2015.

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Citing papers per year: 2021: 52; 2022: 42; 2023: 43; 2024: 48; 2025: 53; 2026: 53.

291 papers.

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  9. 2023A Finite Sample Complexity Bound for Distributionally Robust Q-learningcited by 36
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  11. 2021Statistical Analysis of Wasserstein Distributionally Robust Estimatorscited by 27
  12. 2021Outlier-Robust Optimal Transport: Duality, Structure, and Statistical Analysiscited by 27
38 more
  1. 2021Disentangled Recurrent Wasserstein Autoencodercited by 22
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  4. 2023Estimate-Then-Optimize versus Integrated-Estimation-Optimization versus Sample Average Approximation: A Stochastic Dominance Perspectivecited by 20
  5. 2021Fast Distributionally Robust Learning with Variance Reduced Min-Max Optimizationcited by 19
  6. 2021Robust Risk-Aware Reinforcement Learningcited by 18
  7. 2022Holistic Robust Data-Driven Decisionscited by 18
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  10. 2021Learning Representations that Support Robust Transfer of Predictorscited by 17
  11. 2022Partial Identification of Treatment Effects with Implicit Generative Modelscited by 17
  12. 2022Near-Optimal Algorithms for Group Distributionally Robust Optimization and Beyondcited by 17
  13. 2023Robust Control for Dynamical Systems With Non-Gaussian Noise via Formal Abstractionscited by 16
  14. 2023Wasserstein distributional robustness of neural networkscited by 16
  15. 2023Outlier-Robust Wasserstein DROcited by 16
  16. 2021Smooth p-Wasserstein Distance: Structure, Empirical Approximation, and Statistical Applicationscited by 15
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  18. 2024A Short Survey on Importance Weighting for Machine Learningcited by 15
  19. 2024Score-based generative models are provably robust: an uncertainty quantification perspectivecited by 15
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  27. 2023Distributionally Robust Optimization with Probabilistic Groupcited by 14
  28. 2021Semi-Discrete Optimal Transport: Hardness, Regularization and Numerical Solutioncited by 13
  29. 2021Sampling-Based Robust Control of Autonomous Systems with Non-Gaussian Noisecited by 13
  30. 2022Instance-Dependent Generalization Bounds via Optimal Transportcited by 13
  31. 2023Unifying Distributionally Robust Optimization via Optimal Transport Theorycited by 13
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  33. 2021Robustifying Conditional Portfolio Decisions via Optimal Transportcited by 12
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  37. 2022A Short and General Duality Proof for Wasserstein Distributionally Robust Optimizationcited by 12
  38. 2023Bridging Distributionally Robust Learning and Offline RL: An Approach to Mitigate Distribution Shift and Partial Data Coveragecited by 12

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