Geometric ergodicity of SGLD via reflection coupling

The Stochastic Gradient Langevin Dynamics (SGLD), first introduced by Welling and Teh [25], has attracted a lot of attention in various areas [18, 26, 4]. The SGLD algorithm and its variants have fantastic performance when dealing with many practical sampling or optimization tasks. Recent decades have witnessed great development of theoretical research for SGLD, where most researchers focus on its discretization error, namely, the "distance" between the SGLD algorithm and the corresponding Langevin diffusion in terms of the time step (or learning rate) η [12, 18, 26, 16]. The SGLD itself can be regarded as a stochastic process and the ergodicity is also of great importance. So far, the justification of the geometric ergodicity of SGLD mostly relies on the strong convexity conditions, namely, the strong log-concaveness of the target distribution. In [4], under strong convexity settings, the authors considered the Synchronous coupling and established the geometric ergodicity of SGLD and some other numerical schemes in terms of Wasserstein-2 distance. However, the strong convexity assumption seems to limit the applicability of the result, and the ergodicity of the SGLD algorithm in a general setting and the existence of an invariant measure are still unclear. In our work, we aim to study the geometric ergodicity under locally nonconvex setting in this paper. The main technique we apply is reflection coupling [8], which was originally designed earlier to study the contraction property of many continuous SDEs.