On Generalization and Regularization via Wasserstein Distributionally Robust Optimization

TL;DR

This paper broadens the theoretical understanding of Wasserstein distributionally robust optimization (DRO), showing its generalization bounds and regularization equivalence hold in more general settings, including arbitrary decision criteria and risk measures.

Contribution

It extends Wasserstein DRO's theoretical guarantees to broader settings with general decision criteria and risk measures, and establishes its equivalence to regularization as exact and dimension-independent.

Findings

Generalization bounds are dimension-independent.

Wasserstein DRO is exactly equivalent to regularization.

Coincides with max-sliced Wasserstein DRO under affine rules.

Abstract

Wasserstein distributionally robust optimization (DRO) has gained prominence in operations research and machine learning as a powerful method for achieving solutions with favorable out-of-sample performance. Two compelling explanations for its success are the generalization bounds derived from Wasserstein DRO and its equivalence to regularization schemes commonly used in machine learning. However, existing results on generalization bounds and regularization equivalence are largely limited to settings where the Wasserstein ball is of a specific type, and the decision criterion takes certain forms of expected functions. In this paper, we show that generalization bounds and regularization equivalence can be obtained in a significantly broader setting, where the Wasserstein ball is of a general type and the decision criterion accommodates any form, including general risk measures. This not only addresses important machine learning and operations management applications but also expands to general decision-theoretical frameworks previously unaddressed by Wasserstein DRO. Our results are strong in that the generalization bounds do not suffer from the curse of dimensionality and the equivalency to regularization is exact. As a by-product, we show that Wasserstein DRO coincides with the recent max-sliced Wasserstein DRO for {\it any} decision criterion under affine decision rules -- resulting in both being efficiently solvable as convex programs via our general regularization results. These general assurances provide a strong foundation for expanding the application of Wasserstein DRO across diverse domains of data-driven decision problems.