# Wasserstein Distributionally Robust Regret Optimization

arXiv: https://arxiv.org/abs/2504.10796 · HTML: https://mlsanity.com/paper/2504.10796/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 44 references that are in the map.
- First posted on arXiv 15 Apr 2025.

## Builds on (most cited first, 18 of 18)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Convex Optimization: Algorithms and Complexity](https://mlsanity.com/paper/1405.4980/) (2014)
- [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)
- [A Distributionally Robust Approach to Fair Classification](https://mlsanity.com/paper/2007.09530/) (2020)
- [Minimax Regret Optimization for Robust Machine Learning under Distribution Shift](https://mlsanity.com/paper/2202.05436/) (2022)
- [Optimal Transport Based Distributionally Robust Optimization: Structural Properties and Iterative Schemes](https://mlsanity.com/paper/1810.02403/) (2018)
- [Distributionally Robust Mean-Variance Portfolio Selection with Wasserstein Distances](https://mlsanity.com/paper/1802.04885/) (2018)
- [Distributionally Robust Optimization and Robust Statistics](https://mlsanity.com/paper/2401.14655/) (2024)
- [A Unified Theory of Robust and Distributionally Robust Optimization via the Primal-Worst-Equals-Dual-Best Principle](https://mlsanity.com/paper/2105.00760/) (2021)
- [Sensitivity analysis of Wasserstein distributionally robust optimization problems](https://mlsanity.com/paper/2006.12022/) (2020)
- [On the Impossibility of Statistically Improving Empirical Optimization: A Second-Order Stochastic Dominance Perspective](https://mlsanity.com/paper/2105.13419/) (2021)
- [A Distributionally Robust Approach to Regret Optimal Control using the Wasserstein Distance](https://mlsanity.com/paper/2304.06783/) (2023)
- [Wasserstein Distributionally Robust Regret-Optimal Control under Partial Observability](https://mlsanity.com/paper/2307.04966/) (2023)
- [Wasserstein Distributionally Robust Regret-Optimal Control in the Infinite-Horizon](https://mlsanity.com/paper/2312.17376/) (2023)
- [Continuity of Equilibria for Two-Person Zero-Sum Games with Noncompact Action Sets and Unbounded Payoffs](https://mlsanity.com/paper/1609.03990/) (2016)

## Built on by (most cited first, 2 of 2)

- [Risk-Averse Wasserstein Distributionally Robust Online Learning](https://mlsanity.com/paper/2602.20403/) (2026)
- [Wasserstein Distributionally Robust Regret Optimization for Reinforcement Learning from Human Feedback](https://mlsanity.com/paper/2605.00155/) (2026)

## Abstract (arXiv)

Distributionally robust optimization (DRO) is widely used for decision-making under uncertainty, but its adversarial focus on worst-case loss can lead to overly conservative policies. To mitigate this, we study ex-ante Distributionally Robust Regret Optimization (DRRO) with Wasserstein ambiguity sets, designed to balance robustness with upside potential. We develop a theory of Wasserstein DRRO (WDRRO) paralleling Wasserstein DRO. Under smoothness and regularity, WDRRO selects among ERM optima by a first-order gradient-discrepancy rule. If the ERM optimizer is unique, first-order sensitivity vanishes and a second-order expansion governs deviations. For convex quadratics ERM and DRRO coincide for any radius. We then study regimes where these assumptions fail: nondifferentiable max-affine losses, discrete references, and larger radii, where WDRRO can differ from ERM and WDRO. We show that computing WDRRO regret is NP-hard even without bilinear terms. Nevertheless, we develop exact algorithms, a tractable convex relaxation with guarantees, and experiments showing tightness and loss-dependent behavior.