NONASYMPTOTIC CONVERGENCE ANALYSIS FOR THE UNADJUSTED LANGEVIN ALGORITHM

• 成果类型: Article
• 署名作者: Durmus, Alain; Moulines, Eric
• 署名单位: IMT - Institut Mines-Telecom; Institut Polytechnique de Paris; Telecom Paris; Centre National de la Recherche Scientifique (CNRS); Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Institut Polytechnique de Paris; ENSTA Paris; Ecole Polytechnique
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/16-AAP1238
• 发表日期: 2017
• 页码: 1551-1587
• 关键词: INEQUALITIES; equilibrium; lyapunov; mcmc

摘要

In this paper, we study a method to sample from a target distribution pi over R-d having a positive density with respect to the Lebesgue measure, known up to a normalisation factor. This method is based on the Euler discretization of the overdamped Langevin stochastic differential equation associated with pi. For both constant and decreasing step sizes in the Euler discretization, we obtain nonasymptotic bounds for the convergence to the target distribution pi in total variation distance. A particular attention is paid to the dependency on the dimension d, to demonstrate the applicability of this method in the high-dimensional setting. These bounds improve and extend the results of Dalalyan