Two Facets of SDE Under an Information-Theoretic Lens: Generalization of SGD via Training Trajectories and via Terminal States

TL;DR

This paper explores how stochastic differential equations (SDEs) can be used to understand the generalization behavior of stochastic gradient descent (SGD) by analyzing training trajectories and terminal states through an information-theoretic lens.

Contribution

It introduces novel generalization bounds for SGD by approximating its dynamics with SDEs, using information-theoretic methods for both trajectory and terminal state analyses.

Findings

Trajectory-based bounds outperform previous results.

Terminal-state bounds decay rapidly, similar to stability bounds.

SDE approximation effectively captures SGD's generalization behavior.

Abstract

Stochastic differential equations (SDEs) have been shown recently to characterize well the dynamics of training machine learning models with SGD. When the generalization error of the SDE approximation closely aligns with that of SGD in expectation, it provides two opportunities for understanding better the generalization behaviour of SGD through its SDE approximation. Firstly, viewing SGD as full-batch gradient descent with Gaussian gradient noise allows us to obtain trajectory-based generalization bound using the information-theoretic bound from Xu and Raginsky [2017]. Secondly, assuming mild conditions, we estimate the steady-state weight distribution of SDE and use information-theoretic bounds from Xu and Raginsky [2017] and Negrea et al. [2019] to establish terminal-state-based generalization bounds. Our proposed bounds have some advantages, notably the trajectory-based bound outperforms results in Wang and Mao [2022], and the terminal-state-based bound exhibits a fast decay rate comparable to stability-based bounds.