# Nash Equilibria, Regularization and Computation in Optimal Transport-Based Distributionally Robust Optimization

arXiv: https://arxiv.org/abs/2303.03900 · HTML: https://mlsanity.com/paper/2303.03900/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 54 of its 112 references that are in the map.
- First posted on arXiv 7 Mar 2023.

## Builds on (most cited first, 30 of 54)

- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [Intriguing properties of neural networks](https://mlsanity.com/paper/1312.6199/) (2013)
- [Improved Training of Wasserstein GANs](https://mlsanity.com/paper/1704.00028/) (2017)
- [Distillation as a Defense to Adversarial Perturbations against Deep Neural Networks](https://mlsanity.com/paper/1511.04508/) (2015)
- [Generalizing to Unseen Domains via Adversarial Data Augmentation](https://mlsanity.com/paper/1805.12018/) (2018)
- [Transferability in Machine Learning: from Phenomena to Black-Box Attacks using Adversarial Samples](https://mlsanity.com/paper/1605.07277/) (2016)
- [Certifying Some Distributional Robustness with Principled Adversarial Training](https://mlsanity.com/paper/1710.10571/) (2017)
- [Learning Models with Uniform Performance via Distributionally Robust Optimization](https://mlsanity.com/paper/1810.08750/) (2018)
- [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)
- [Formal Guarantees on the Robustness of a Classifier against Adversarial Manipulation](https://mlsanity.com/paper/1705.08475/) (2017)
- [Statistics of Robust Optimization: A Generalized Empirical Likelihood Approach](https://mlsanity.com/paper/1610.03425/) (2016)
- [On the Convergence and Robustness of Adversarial Training](https://mlsanity.com/paper/2112.08304/) (2021)
- [Distributionally Robust Logistic Regression](https://mlsanity.com/paper/1509.09259/) (2015)
- [Stabilizing Training of Generative Adversarial Networks through Regularization](https://mlsanity.com/paper/1705.09367/) (2017)
- [Improving DNN Robustness to Adversarial Attacks using Jacobian Regularization](https://mlsanity.com/paper/1803.08680/) (2018)
- [Regularization via Mass Transportation](https://mlsanity.com/paper/1710.10016/) (2017)
- [Gradient descent GAN optimization is locally stable](https://mlsanity.com/paper/1706.04156/) (2017)
- [Finite-Sample Guarantees for Wasserstein Distributionally Robust Optimization: Breaking the Curse of Dimensionality](https://mlsanity.com/paper/2009.04382/) (2020)
- [Scaleable input gradient regularization for adversarial robustness](https://mlsanity.com/paper/1905.11468/) (2019)
- [A Unified Gradient Regularization Family for Adversarial Examples](https://mlsanity.com/paper/1511.06385/) (2015)
- [From Data to Decisions: Distributionally Robust Optimization is Optimal](https://mlsanity.com/paper/1704.04118/) (2017)
- [Golden Ratio Algorithms for Variational Inequalities](https://mlsanity.com/paper/1803.08832/) (2018)
- [Data-driven Inverse Optimization with Imperfect Information](https://mlsanity.com/paper/1512.05489/) (2015)
- [Recovering Best Statistical Guarantees via the Empirical Divergence-based Distributionally Robust Optimization](https://mlsanity.com/paper/1605.09349/) (2016)
- [A Minimax Approach to Supervised Learning](https://mlsanity.com/paper/1606.02206/) (2016)
- [Theoretical Analysis of Adversarial Learning: A Minimax Approach](https://mlsanity.com/paper/1811.05232/) (2018)
- [Optimal Transport Based Distributionally Robust Optimization: Structural Properties and Iterative Schemes](https://mlsanity.com/paper/1810.02403/) (2018)
- [Wasserstein Distributionally Robust Stochastic Control: A Data-Driven Approach](https://mlsanity.com/paper/1812.09808/) (2018)

## Built on by (most cited first, 8 of 8)

- [Outlier-Robust Wasserstein DRO](https://mlsanity.com/paper/2311.05573/) (2023)
- [Unifying Distributionally Robust Optimization via Optimal Transport Theory](https://mlsanity.com/paper/2308.05414/) (2023)
- [Nonlinear Distributionally Robust Optimization](https://mlsanity.com/paper/2306.03202/) (2023)
- [Provable Robust Overfitting Mitigation in Wasserstein Distributionally Robust Optimization](https://mlsanity.com/paper/2503.04315/) (2025)
- [Risk-Averse Wasserstein Distributionally Robust Online Learning](https://mlsanity.com/paper/2602.20403/) (2026)
- [Generative Neural Networks for Sinkhorn Distributionally Robust Hypothesis Testing](https://mlsanity.com/paper/2608.22746/) (2026)
- [The Value of Human Expertise](https://mlsanity.com/paper/2608.26051/) (2026)
- [Distributionally robust linear regression through the lens of adversarial training](https://mlsanity.com/paper/2609.39449/) (2026)

## Abstract (arXiv)

We study optimal transport-based distributionally robust optimization problems where a fictitious adversary, often envisioned as nature, can choose the distribution of the uncertain problem parameters by reshaping a prescribed reference distribution at a finite transportation cost. In this framework, we show that robustification is intimately related to various forms of variation and Lipschitz regularization even if the transportation cost function fails to be (some power of) a metric. We also derive conditions for the existence and the computability of a Nash equilibrium between the decision-maker and nature, and we demonstrate numerically that nature's Nash strategy can be viewed as a distribution that is supported on remarkably deceptive adversarial samples. Finally, we identify practically relevant classes of optimal transport-based distributionally robust optimization problems that can be addressed with efficient gradient descent algorithms even if the loss function or the transportation cost function are nonconvex (but not both at the same time).