mlsanityData to 7 Oct 2026

On Linear Optimization over Wasserstein Balls

arXiv 2004.07162 ↗First posted 15 Apr 2020math.OC cs.LGcited by 23

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Abstract (arXiv)

Wasserstein balls, which contain all probability measures within a pre-specified Wasserstein distance to a reference measure, have recently enjoyed wide popularity in the distributionally robust optimization and machine learning communities to formulate and solve data-driven optimization problems with rigorous statistical guarantees. In this technical note we prove that the Wasserstein ball is weakly compact under mild conditions, and we offer necessary and sufficient conditions for the existence of optimal solutions. We also characterize the sparsity of solutions if the Wasserstein ball is centred at a discrete reference measure. In comparison with the existing literature, which has proved similar results under different conditions, our proofs are self-contained and shorter, yet mathematically rigorous, and our necessary and sufficient conditions for the existence of optimal solutions are easily verifiable in practice.

Key facts. Cited by 23 papers in the map (data through 7 Oct 2026). First posted on arXiv 15 Apr 2020.

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Citing papers per year: 2021: 2; 2022: 4; 2023: 5; 2024: 7; 2025: 4; 2026: 1.

23 papers.

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  1. 2022The Many Faces of Adversarial Riskcited by 28
  2. 2021Statistical Analysis of Wasserstein Distributionally Robust Estimatorscited by 27
  3. 2021Mixed Nash Equilibria in the Adversarial Examples Gamecited by 22
  4. 2023Algorithms for mean-field variational inference via polyhedral optimization in the Wasserstein spacecited by 20
  5. 2023Unifying Distributionally Robust Optimization via Optimal Transport Theorycited by 12
  6. 2023Distributionally Robust Statistical Verification with Imprecise Neural Networkscited by 12
  7. 2022Distributionally Robust Fair Principal Components via Geodesic Descentscited by 9
  8. 2023Nash Equilibria, Regularization and Computation in Optimal Transport-Based Distributionally Robust Optimizationcited by 8
  9. 2024Stochastic Inverse Problem: stability, regularization and Wasserstein gradient flowcited by 8
  10. 2024Distributionally Robust Optimization via Iterative Algorithms in Continuous Probability Spacescited by 8
  11. 2024Sampling from the Mean-Field Stationary Distributioncited by 7
  12. 2025Worst-case generation via minimax optimization in Wasserstein spacecited by 7
11 more
  1. 2023Learning Against Distributional Uncertainty: On the Trade-off Between Robustness and Specificitycited by 4
  2. 2022Markov Decision Processes under Model Uncertaintycited by 2
  3. 2022Distributional Robustness Bounds Generalization Errorscited by 2
  4. 2024Wasserstein Distributionally Robust Shallow Convex Neural Networkscited by 2
  5. 2024Automatic Outlier Rectification via Optimal Transportcited by 1
  6. 2024Global Optimisation of Black-Box Functions with Generative Models in the Wasserstein Spacecited by 1
  7. 2025A stochastic smoothing framework for nonconvex-nonconcave minEmax problems with applications to Wasserstein distributionally robust optimizationcited by 1
  8. 2025Distributionally Robust Deep Q-Learningcited by 1
  9. 2024Distributionally and Adversarially Robust Logistic Regression via Intersecting Wasserstein Ballscited by 0
  10. 2025Provable Robust Overfitting Mitigation in Wasserstein Distributionally Robust Optimizationcited by 0
  11. 2026Data-driven robust Markov decision processes on Borel spaces: performance guarantees via an axiomatic approachcited by 0