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 $h\downarrow0$, 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.

💡 Research Summary

The paper investigates the convergence behavior of Metropolis‑adjusted Langevin algorithm (MALA) and a related Metropolis‑Hastings (MH) scheme that uses Ornstein‑Uhlenbeck (OU) proposals when the target distribution is a log‑concave perturbation of a Gaussian reference measure in high dimensions. The authors consider a target density of the form
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