Tighter expected generalization error bounds via Wasserstein distance

TL;DR

This paper develops new expected generalization error bounds using Wasserstein distance, bridging geometric and entropy-based approaches, and introduces bounds based on various information measures.

Contribution

It introduces novel generalization bounds based on Wasserstein distance that unify geometric and entropy-based methods, including bounds with different information measures.

Findings

Bounds are tighter than existing entropy-based bounds when loss is bounded.

New bounds based on Wasserstein distance can be derived for various information measures.

The bounds recover from below, providing non-vacuous estimates in certain settings.

Abstract

This work presents several expected generalization error bounds based on the Wasserstein distance. More specifically, it introduces full-dataset, single-letter, and random-subset bounds, and their analogues in the randomized subsample setting from Steinke and Zakynthinou [1]. Moreover, when the loss function is bounded and the geometry of the space is ignored by the choice of the metric in the Wasserstein distance, these bounds recover from below (and thus, are tighter than) current bounds based on the relative entropy. In particular, they generate new, non-vacuous bounds based on the relative entropy. Therefore, these results can be seen as a bridge between works that account for the geometry of the hypothesis space and those based on the relative entropy, which is agnostic to such geometry. Furthermore, it is shown how to produce various new bounds based on different information measures (e.g., the lautum information or several $f$-divergences) based on these bounds and how to derive similar bounds with respect to the backward channel using the presented proof techniques.