Geometric ergodicity of SGLD via reflection coupling

TL;DR

This work establishes geometric ergodicity for Stochastic Gradient Langevin Dynamics (SGLD) under local nonconvexity by employing reflection coupling. A Lyapunov-based Kantorovich-Rubinstein distance W_f, defined with a concave function f, captures the contraction despite time discretization and minibatch noise, yielding a contraction rate c = (1/3) e^{-2 c_f R} min( (√{2β^{-1}} c_f R^{-1})/2, κ). Under a constant step size, the analysis yields a unique invariant measure and exponential convergence in W1, with the contraction framework extending to general drift via unbiased minibatch estimates. The methodology suggests broad applicability to other discrete-time, minibatch-based sampling schemes, potentially offering dimension-robust guarantees in practical high-dimensional settings.

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.

Theorems & Definitions (27)

• Remark 2.1
• Lemma 2.1
• Lemma 2.2
• Theorem 2.1
• Corollary 2.1
• proof
• Corollary 2.2
• Lemma 3.1
• proof
• proof : Proof of Theorem \ref{['thm:contraction']}