ERROR BOUNDS FOR METROPOLIS-HASTINGS ALGORITHMS APPLIED TO PERTURBATIONS OF GAUSSIAN MEASURES IN HIGH DIMENSIONS

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 down arrow 0, 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.