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The Metropolis-adjusted Langevin algorithm (MALA) is a Metropolis-Hastingsmethod for approximate sampling from continuous distributions. We derive upperbounds for the contraction rate in Kantorovich-Rubinstein-Wasserstein distanceof the MALA chain with semi-implicit Euler proposals applied to log-concaveprobability 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 thedimension either. In the limit $h\downarrow0$, the bounds approach the knownoptimal contraction rates for overdamped Langevin diffusions in a convexpotential. A similar approach also applies to Metropolis-Hastings chains withOrnstein-Uhlenbeck proposals. In this case, the resulting estimates are stillindependent of the dimension but less optimal, reflecting the fact that MALA isa higher order approximation of the diffusion limit than Metropolis-Hastingswith Ornstein-Uhlenbeck proposals.

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