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

• 成果类型: Article
• 署名作者: Eberle, Andreas
• 署名单位: University of Bonn
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/13-AAP926
• 发表日期: 2014
• 页码: 337-377
• 关键词: diffusion limits; volume; CONVERGENCE

摘要

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.