Wasserstein distance estimates for the distributions of numerical approximations to ergodic stochastic differential equations

TL;DR

This paper develops a non-asymptotic framework to estimate the Wasserstein distance between the invariant distribution of ergodic SDEs and their numerical approximations, covering various integrators and introducing a new splitting method.

Contribution

It introduces a unified non-asymptotic analysis framework for Wasserstein distances in ergodic SDEs and proposes a novel efficient splitting method for underdamped Langevin dynamics.

Findings

The framework applies to multiple existing integrators.

The new splitting method requires only one gradient evaluation per step.

The method achieves near-optimal sample complexity under smoothness assumptions.

Abstract

We present a framework that allows for the non-asymptotic study of the $2$-Wasserstein distance between the invariant distribution of an ergodic stochastic differential equation and the distribution of its numerical approximation in the strongly log-concave case. This allows us to study in a unified way a number of different integrators proposed in the literature for the overdamped and underdamped Langevin dynamics. In addition, we analyse a novel splitting method for the underdamped Langevin dynamics which only requires one gradient evaluation per time step. Under an additional smoothness assumption on a $d$--dimensional strongly log-concave distribution with condition number $\kappa$, the algorithm is shown to produce with an $\mathcal{O}\big(\kappa^{5/4} d^{1/4}\epsilon^{-1/2} \big)$ complexity samples from a distribution that, in Wasserstein distance, is at most $\epsilon>0$ away from the target distribution.