On geometric convergence for the Metropolis-adjusted Langevin algorithm under simple conditions

• 成果类型: Article
• 署名作者: Oliviero-Durmus, Alain; Moulines, Eric
• 署名单位: Institut Polytechnique de Paris; ENSTA Paris; Ecole Polytechnique
• 刊物名称: BIOMETRIKA
• ISSN/ISSBN: 0006-3444
• DOI: 10.1093/biomet/asad060
• 发表日期: 2024
• 关键词: approximations; ergodicity; hastings

摘要

While the Metropolis-adjusted Langevin algorithm is a popular and widely used Markov chain Monte Carlo method, very few papers derive conditions that ensure its convergence. In particular, to the authors' knowledge, assumptions that are both easy to verify and guarantee geometric convergence, are still missing. In this work, we establish V-uniformly geometric convergence for the Metropolis-adjusted Langevin algorithm under mild assumptions about the target distribution. Unlike previous work, we only consider tail and smoothness conditions for the potential associated with the target distribution. These conditions are quite common in the Markov chain Monte Carlo literature. Finally, we pay special attention to the dependence of the bounds we derive on the step size of the Euler-Maruyama discretization, which corresponds to the proposed Markov kernel of the Metropolis-adjusted Langevin algorithm.