# Unregularized limit of stochastic gradient method for Wasserstein distributionally robust optimization

arXiv: https://arxiv.org/abs/2506.04948 · HTML: https://mlsanity.com/paper/2506.04948/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 44 references that are in the map.
- First posted on arXiv 5 Jun 2025.

## Builds on (most cited first, 14 of 14)

- [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)
- [Stochastic model-based minimization of weakly convex functions](https://mlsanity.com/paper/1803.06523/) (2018)
- [Distributionally Robust Logistic Regression](https://mlsanity.com/paper/1509.09259/) (2015)
- [Finite-Sample Guarantees for Wasserstein Distributionally Robust Optimization: Breaking the Curse of Dimensionality](https://mlsanity.com/paper/2009.04382/) (2020)
- [Distributionally Robust Optimization and Robust Statistics](https://mlsanity.com/paper/2401.14655/) (2024)
- [Graphical Convergence of Subgradients in Nonconvex Optimization and Learning](https://mlsanity.com/paper/1810.07590/) (2018)
- [A Short and General Duality Proof for Wasserstein Distributionally Robust Optimization](https://mlsanity.com/paper/2205.00362/) (2022)
- [Exact Generalization Guarantees for (Regularized) Wasserstein Distributionally Robust Models](https://mlsanity.com/paper/2305.17076/) (2023)
- [Annealed Sinkhorn for Optimal Transport: convergence, regularization path and debiasing](https://mlsanity.com/paper/2408.11620/) (2024)
- [Universal generalization guarantees for Wasserstein distributionally robust models](https://mlsanity.com/paper/2402.11981/) (2024)
- [DRO: A Python Library for Distributionally Robust Optimization in Machine Learning](https://mlsanity.com/paper/2505.23565/) (2025)
- [textttskwdro: a library for Wasserstein distributionally robust machine learning](https://mlsanity.com/paper/2410.21231/) (2024)
- [A stochastic smoothing framework for nonconvex-nonconcave minEmax problems with applications to Wasserstein distributionally robust optimization](https://mlsanity.com/paper/2502.17602/) (2025)

## Abstract (arXiv)

Wasserstein distributionally robust optimization offers a framework for model fitting in machine learning under potential shifts in the data distribution. We study a regularized variant of this problem in which entropic smoothing produces a sampled approximation of the original objective. We establish convergence of the approximate gradients to subgradients of the unregularized objective as the regularization parameter vanishes, enabling convergence guarantees for stochastic gradient methods. We obtain qualitative convergence results under general assumptions, then we provide convergence rates under additional regularity. In particular, we prove rates for the convergence of the unregularized objective values, up to sampling errors, when the regularization level is decreased across iterations. Our analysis yields byproducts of independent interest, including approximation results for smoothing of maximum functions subdifferentials and empirical lower bounds for dual solutions of Wasserstein distributionally robust optimization.