High-dimensional normal approximations for sums of Langevin Markov chains

Abstract

Consider the well-known Langevin diffusion on ${\mathbb{R}}^d$ $$\mathrm{d}{X}_t=-\nabla U(X_t)\mathrm{d}t+\sqrt{2}\mathrm{d}{B}_t,$$ and its Euler-Maruyama discretization given by $$X_{k+1}=X_k-\eta \nabla U(X_k)+\sqrt{2\eta }{\xi }_{k+1},$$ where $\eta$ is the step size. Under mild conditions, the Langevin diffusion admits $\pi (\mathrm{d}x)\propto \exp (-U(x))\mathrm{d}x$ as its unique stationary distribution. In this paper, we mainly study the normal approximation of the normalized partial sum $$W_n={\eta }^{1/2}{n}^{-1/2}\left(\sum _{i=0}^{n-1}{X}_i-{\int }_{{\mathbb{R}}^d}x\pi (\mathrm{d}x)\right).$$ To the best of our knowledge, this work provides the first dimension-explicit convergence rates in high-dimensional settings. Our main tool is a novel upper bound for the 1-Wasserstein distance $W_1(W,\gamma )$ via the exchange pair approach, where $W$ is any random vector of interest and $\gamma$ is a $d$-dimensional standard normal random vector.