Information-Theoretic Generalization Bounds for SGLD via Data-Dependent Estimates

TL;DR

This paper introduces improved data-dependent mutual information bounds for Stochastic Gradient Langevin Dynamics, linking them to flatness of the empirical risk surface and demonstrating significantly tighter bounds than previous methods.

Contribution

It develops novel data-dependent estimates for mutual information bounds in SGLD, enhancing the theoretical understanding of its generalization performance.

Findings

Bounds are orders of magnitude smaller than previous gradient-norm-based bounds

The approach applies broadly within existing information-theoretic frameworks

The bounds relate to the flatness of the empirical risk surface

Abstract

In this work, we improve upon the stepwise analysis of noisy iterative learning algorithms initiated by Pensia, Jog, and Loh (2018) and recently extended by Bu, Zou, and Veeravalli (2019). Our main contributions are significantly improved mutual information bounds for Stochastic Gradient Langevin Dynamics via data-dependent estimates. Our approach is based on the variational characterization of mutual information and the use of data-dependent priors that forecast the mini-batch gradient based on a subset of the training samples. Our approach is broadly applicable within the information-theoretic framework of Russo and Zou (2015) and Xu and Raginsky (2017). Our bound can be tied to a measure of flatness of the empirical risk surface. As compared with other bounds that depend on the squared norms of gradients, empirical investigations show that the terms in our bounds are orders of magnitude smaller.