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

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.