Geometric ergodicity of SGLD via reflection coupling

TL;DR

This paper proves the geometric ergodicity of the SGLD algorithm under certain nonconvex conditions using reflection coupling, addressing discretization and minibatch challenges, and extends results to non-gradient drifts.

Contribution

It establishes Wasserstein contraction and geometric ergodicity of SGLD in nonconvex settings with reflection coupling, including discretization and minibatch considerations.

Findings

SGLD has an invariant distribution under constant step size.

SGLD exhibits geometric ergodicity in Wasserstein-1 distance.

Extension of results to non-gradient drift scenarios.

Abstract

We consider the geometric ergodicity of the Stochastic Gradient Langevin Dynamics (SGLD) algorithm under nonconvexity settings. Via the technique of reflection coupling, we prove the Wasserstein contraction of SGLD when the target distribution is log-concave only outside some compact set. The time discretization and the minibatch in SGLD introduce several difficulties when applying the reflection coupling, which are addressed by a series of careful estimates of conditional expectations. As a direct corollary, the SGLD with constant step size has an invariant distribution and we are able to obtain its geometric ergodicity in terms of $W_1$ distance. The generalization to non-gradient drifts is also included.