Minimax Statistical Learning with Wasserstein Distances

TL;DR

This paper introduces a minimax framework for statistical learning using Wasserstein distance-based ambiguity sets, providing generalization bounds and applications to domain adaptation with unlabeled data.

Contribution

It develops a novel minimax approach with Wasserstein ambiguity sets and derives generalization bounds linked to ERM complexity measures.

Findings

Provides generalization guarantees for Wasserstein-based domain adaptation.

Establishes bounds involving covering numbers of ERM problems.

Demonstrates reliable estimation of Wasserstein distances from unlabeled samples.

Abstract

As opposed to standard empirical risk minimization (ERM), distributionally robust optimization aims to minimize the worst-case risk over a larger ambiguity set containing the original empirical distribution of the training data. In this work, we describe a minimax framework for statistical learning with ambiguity sets given by balls in Wasserstein space. In particular, we prove generalization bounds that involve the covering number properties of the original ERM problem. As an illustrative example, we provide generalization guarantees for transport-based domain adaptation problems where the Wasserstein distance between the source and target domain distributions can be reliably estimated from unlabeled samples.