A sharp uniform-in-time error estimate for Stochastic Gradient Langevin Dynamics

Abstract

We establish a sharp uniform-in-time error estimate for the Stochastic Gradient Langevin Dynamics (SGLD), which is a popular sampling algorithm. Under mild assumptions, we obtain a uniform-in-time O(η 2 ) bound for the KL-divergence between the SGLD iteration and the Langevin diffusion, where η is the step size (or learning rate). Our analysis is also valid for varying step sizes. Based on this, we are able to obtain an O(η) bound for the distance between the SGLD iteration and the invariant distribution of the Langevin diffusion, in terms of Wasserstein or total variation distances.

Geometric ergodicity of SGLD via reflection coupling

“…Under Assumption 2.1, π satisfies the log-Sobolev inequality, and one can get a uniform in time error estimate using KL divergence in [16]. We are then able to estimate the W 1 distance between the target distribution π and the invariant measure π of the SGLD algorithm.…”

“…We are then able to estimate the W 1 distance between the target distribution π and the invariant measure π of the SGLD algorithm. In fact, for constant step size η, by [16,Theorem 3.2], the discretization error in terms of KL-divergence is given by…”

“…Consider the SGLD with constant step size η. Under Assumption 2.1, for the step size η small enough (with the restrictions in Theorem 2.1 and in Theorem 3.2 of [16]), the SGLD iteration has a unique invariant measure π satisfying for some…”

“…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.…”

Central Limit Theorem for Temporal Average of Backward Euler-Maruyama Method

“…When the inertia of the particle is negligible compared with the damping force due to the friction, the trajectory of the Langevin equation is approximately described by (1.1) (see e.g. [17,28,29]). It is known that (1.1) admits a unique invariant measure (thus is ergodic) π(dq) = Z −1 e −βV (q) dq with Z = R e −βV (q) dq.…”