Error bounds for Metropolis-Hastings algorithms applied to perturbations of Gaussian measures in high dimensions

TL;DR

This paper establishes dimension-independent error bounds for the convergence rates of the Metropolis-Hastings algorithms, specifically MALA and Ornstein-Uhlenbeck proposals, applied to high-dimensional Gaussian-related measures.

Contribution

It provides new dimension-independent bounds for MALA's contraction rates in high dimensions, approaching optimal rates as step size diminishes.

Findings

Bounds are dimension-independent for regular densities.

Bounds approach optimal rates as step size decreases.

MALA outperforms Ornstein-Uhlenbeck proposals in convergence efficiency.

Abstract

The Metropolis-adjusted Langevin algorithm (MALA) is a Metropolis-Hastings method for approximate sampling from continuous distributions. We derive upper bounds for the contraction rate in Kantorovich-Rubinstein-Wasserstein distance of the MALA chain with semi-implicit Euler proposals applied to log-concave probability measures that have a density w.r.t. a Gaussian reference measure. For sufficiently "regular" densities, the estimates are dimension-independent, and they hold for sufficiently small step sizes $h$ that do not depend on the dimension either. In the limit $h\downarrow0$, the bounds approach the known optimal contraction rates for overdamped Langevin diffusions in a convex potential. A similar approach also applies to Metropolis-Hastings chains with Ornstein-Uhlenbeck proposals. In this case, the resulting estimates are still independent of the dimension but less optimal, reflecting the fact that MALA is a higher order approximation of the diffusion limit than Metropolis-Hastings with Ornstein-Uhlenbeck proposals.