Convergence of Stochastic Gradient Langevin Dynamics in the Lazy Training Regime

TL;DR

This paper provides a non-asymptotic convergence analysis of stochastic gradient Langevin dynamics (SGLD) in the lazy training regime, revealing conditions for exponential convergence and finite-time bounds, supported by numerical experiments.

Contribution

It establishes the first finite-time convergence guarantees for SGLD in the lazy training regime under regularity conditions.

Findings

SGLD achieves exponential convergence to the empirical risk minimizer.

SGLD maintains a non-degenerate kernel throughout training.

Finite-time and finite-width bounds on the optimality gap are derived.

Abstract

Continuous-time models provide important insights into the training dynamics of optimization algorithms in deep learning. In this work, we establish a non-asymptotic convergence analysis of stochastic gradient Langevin dynamics (SGLD), which is an It\^o stochastic differential equation (SDE) approximation of stochastic gradient descent in continuous time, in the lazy training regime. We show that, under regularity conditions on the Hessian of the loss function, SGLD with multiplicative and state-dependent noise (i) yields a non-degenerate kernel throughout the training process with high probability, and (ii) achieves exponential convergence to the empirical risk minimizer in expectation, and we establish finite-time and finite-width bounds on the optimality gap. We corroborate our theoretical findings with numerical examples in the regression setting.