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

Jesus Maria Sanz-Serna, Konstantinos C. Zygalakis

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
κ, the algorithm is shown to produce with an O􀀀κ5/4d1/4ǫ−1/2
complexity samples from a distribution that, in
Wasserstein distance, is at most ǫ > 0 away from the target distribution.