Multivariate approximations in Wasserstein distance by Stein's method and Bismut's formula

TL;DR

This paper develops a new method combining Stein's approach and Bismut's formula to improve multivariate probability approximations in Wasserstein distance, with applications to Langevin algorithms.

Contribution

It introduces a general theorem for multivariate approximations using Malliavin calculus and Stein's exchangeable pair method, providing near optimal error bounds.

Findings

Established a general theorem for multivariate Wasserstein approximations

Applied the theorem to analyze the unadjusted Langevin algorithm

Achieved near optimal error bounds in Wasserstein distance

Abstract

Stein's method has been widely used for probability approximations. However, in the multi-dimensional setting, most of the results are for multivariate normal approximation or for test functions with bounded second- or higher-order derivatives. For a class of multivariate limiting distributions, we use Bismut's formula in Malliavin calculus to control the derivatives of the Stein equation solutions by the first derivative of the test function. Combined with Stein's exchangeable pair approach, we obtain a general theorem for multivariate approximations with near optimal error bounds on the Wasserstein distance.We apply the theorem to the unadjusted Langevin algorithm.