Regularization for Wasserstein Distributionally Robust Optimization

TL;DR

This paper investigates the regularization of Wasserstein distributionally robust optimization, deriving duality results and approximations, which enhances the understanding and application of optimal transport in machine learning under data uncertainty.

Contribution

It provides a general duality framework for regularized Wasserstein DRO and refines it for entropic regularization, including approximation results as regularization diminishes.

Findings

Established strong duality for regularized Wasserstein DRO

Refined duality results for entropic regularization

Provided approximation bounds as regularization parameters approach zero

Abstract

Optimal transport has recently proved to be a useful tool in various machine learning applications needing comparisons of probability measures. Among these, applications of distributionally robust optimization naturally involve Wasserstein distances in their models of uncertainty, capturing data shifts or worst-case scenarios. Inspired by the success of the regularization of Wasserstein distances in optimal transport, we study in this paper the regularization of Wasserstein distributionally robust optimization. First, we derive a general strong duality result of regularized Wasserstein distributionally robust problems. Second, we refine this duality result in the case of entropic regularization and provide an approximation result when the regularization parameters vanish.