# A Short and General Duality Proof for Wasserstein Distributionally Robust Optimization

arXiv: https://arxiv.org/abs/2205.00362 · HTML: https://mlsanity.com/paper/2205.00362/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 30 references that are in the map.
- First posted on arXiv 30 Apr 2022.

## Builds on (most cited first, 7 of 7)

- [Certifying Some Distributional Robustness with Principled Adversarial Training](https://mlsanity.com/paper/1710.10571/) (2017)
- [Data-driven Distributionally Robust Optimization Using the Wasserstein Metric: Performance Guarantees and Tractable Reformulations](https://mlsanity.com/paper/1505.05116/) (2015)
- [Distributionally Robust Stochastic Optimization with Wasserstein Distance](https://mlsanity.com/paper/1604.02199/) (2016)
- [Wasserstein Distributionally Robust Optimization: Theory and Applications in Machine Learning](https://mlsanity.com/paper/1908.08729/) (2019)
- [Statistical Analysis of Wasserstein Distributionally Robust Estimators](https://mlsanity.com/paper/2108.02120/) (2021)
- [On Distributionally Robust Chance Constrained Programs with Wasserstein Distance](https://mlsanity.com/paper/1806.07418/) (2018)
- [A Unified Theory of Robust and Distributionally Robust Optimization via the Primal-Worst-Equals-Dual-Best Principle](https://mlsanity.com/paper/2105.00760/) (2021)

## Built on by (most cited first, 12 of 12)

- [Unifying Distributionally Robust Optimization via Optimal Transport Theory](https://mlsanity.com/paper/2308.05414/) (2023)
- [Nash Equilibria, Regularization and Computation in Optimal Transport-Based Distributionally Robust Optimization](https://mlsanity.com/paper/2303.03900/) (2023)
- [Wasserstein Distributionally Robust Nonparametric Regression](https://mlsanity.com/paper/2505.07967/) (2025)
- [Optimal Transport Regularized Divergences: Application to Adversarial Robustness](https://mlsanity.com/paper/2309.03791/) (2023)
- [Certifiably Robust Model Evaluation in Federated Learning under Meta-Distributional Shifts](https://mlsanity.com/paper/2410.20250/) (2024)
- [Enhancing Distributional Robustness in Principal Component Analysis by Wasserstein Distances](https://mlsanity.com/paper/2503.02494/) (2025)
- [Unregularized limit of stochastic gradient method for Wasserstein distributionally robust optimization](https://mlsanity.com/paper/2506.04948/) (2025)
- [Concave Certificates: Geometric Framework for Distributionally Robust Risk and Complexity Analysis](https://mlsanity.com/paper/2601.01311/) (2026)
- [Statistical Guarantees for Distributionally Robust Optimization with Optimal Transport and OT-Regularized Divergences](https://mlsanity.com/paper/2603.27871/) (2026)
- [Brenier Meets Adversarial Training: Optimal Transport Geometry for Robust Learning](https://mlsanity.com/paper/2609.31363/) (2026)
- [Distributionally robust linear regression through the lens of adversarial training](https://mlsanity.com/paper/2609.39449/) (2026)
- [Distributionally Robust Schr"odinger Bridge](https://mlsanity.com/paper/2610.02043/) (2026)

## Abstract (arXiv)

We present a general duality result for Wasserstein distributionally robust optimization that holds for any Kantorovich transport cost, measurable loss function, and nominal probability distribution. Assuming an interchangeability principle inherent in existing duality results, our proof only uses one-dimensional convex analysis. Furthermore, we demonstrate that the interchangeability principle holds if and only if certain measurable projection and weak measurable selection conditions are satisfied. To illustrate the broader applicability of our approach, we provide a rigorous treatment of duality results in distributionally robust Markov decision processes and distributionally robust multistage stochastic programming. Additionally, we extend our analysis to other problems such as infinity-Wasserstein distributionally robust optimization, risk-averse optimization, and globalized distributionally robust counterpart.