Sharpened Generalization Bounds based on Conditional Mutual Information and an Application to Noisy, Iterative Algorithms

TL;DR

This paper develops sharper generalization bounds using conditional mutual information, improving upon previous bounds, and applies these to analyze noisy, iterative algorithms like Langevin dynamics.

Contribution

It introduces tighter generalization bounds based on conditional mutual information and applies them to analyze Langevin dynamics algorithms.

Findings

Conditional mutual information bounds are tighter than unconditional ones.

Bounds leverage information from optimization trajectories.

Application to Langevin dynamics shows improved generalization analysis.

Abstract

The information-theoretic framework of Russo and J. Zou (2016) and Xu and Raginsky (2017) provides bounds on the generalization error of a learning algorithm in terms of the mutual information between the algorithm's output and the training sample. In this work, we study the proposal, by Steinke and Zakynthinou (2020), to reason about the generalization error of a learning algorithm by introducing a super sample that contains the training sample as a random subset and computing mutual information conditional on the super sample. We first show that these new bounds based on the conditional mutual information are tighter than those based on the unconditional mutual information. We then introduce yet tighter bounds, building on the "individual sample" idea of Bu, S. Zou, and Veeravalli (2019) and the "data dependent" ideas of Negrea et al. (2019), using disintegrated mutual information. Finally, we apply these bounds to the study of Langevin dynamics algorithm, showing that conditioning on the super sample allows us to exploit information in the optimization trajectory to obtain tighter bounds based on hypothesis tests.