Distributionally Robust Optimization and Generalization in Kernel Methods

Abstract

Distributionally robust optimization (DRO) has attracted attention in machine learning due to its connections to regularization, generalization, and robustness. Existing work has considered uncertainty sets based on phi-divergences and Wasserstein distances, each of which have drawbacks. In this paper, we study DRO with uncertainty sets measured via maximum mean discrepancy (MMD). We show that MMD DRO is roughly equivalent to regularization by the Hilbert norm and, as a byproduct, reveal deep connections to classic results in statistical learning. In particular, we obtain an alternative proof of a generalization bound for Gaussian kernel ridge regression via a DRO lense. The proof also suggests a new regularizer. Our results apply beyond kernel methods: we derive a generically applicable approximation of MMD DRO, and show that it generalizes recent work on variance-based regularization.