Correction to "Wasserstein distance estimates for the distributions of numerical approximations to ergodic stochastic differential equations"
Daniel Paulin, Peter A. Whalley
TL;DR
This note identifies and fixes a dimension-dependence issue in the non-asymptotic Wasserstein-2 analysis of the UBU integrator for ergodic SDEs. By adopting a stronger, strongly Hessian Lipschitz assumption, it reconciles theory with observed high-dimensional behavior, correcting a local-error bound and providing an explicit revised version of Theorem 25 with new constants that ensure controlled dimension dependence ($O(d^{1/4})$ scaling) in many practical models. The corrections establish that the UBU method retains strong order and dimension-aware guarantees under the strengthened smoothness condition, with concrete constants $C_0$, $C_1$, and $C_2$ that depend on problem parameters and dimension as detailed. The results bolster reliability of non-asymptotic convergence guarantees for high-dimensional Bayesian inference and related problems using ergodic SDE discretizations.
Abstract
A method for analyzing non-asymptotic guarantees of numerical discretizations of ergodic SDEs in Wasserstein-2 distance is presented by Sanz-Serna and Zygalakis in ``Wasserstein distance estimates for the distributions of numerical approximations to ergodic stochastic differential equations". They analyze the UBU integrator which is strong order two and only requires one gradient evaluation per step, resulting in desirable non-asymptotic guarantees, in particular $\mathcal{O}(d^{1/4}ε^{-1/2})$ steps to reach a distance of $ε> 0$ in Wasserstein-2 distance away from the target distribution. However, there is a mistake in the local error estimates in Sanz-Serna and Zygalakis (2021), in particular, a stronger assumption is needed to achieve these complexity estimates. This note reconciles the theory with the dimension dependence observed in practice in many applications of interest.