NONASYMPTOTIC CONVERGENCE ANALYSIS FOR THE UNADJUSTED LANGEVIN ALGORITHM

• Published in: The Annals of applied probability Vol. 27; no. 3; pp. 1551 - 1587
• Main Authors: Durmus, Alain, Moulines, Éric
• Format: Journal Article
• Language: English
• Published: Hayward Institute of Mathematical Statistics 01-06-2017; Institute of Mathematical Statistics (IMS)
• Subjects: Algorithms; Convergence; Differential equations; Discretization; Mathematics; Probability; Statistics; Stochastic models
• ISSN: 1050-5164; 2168-8737
• DOI: 10.1214/16-aap1238

In this paper, we study a method to sample from a target distribution π over ℝd 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 π. For both constant and decreasing step sizes in the Euler discretization, we obtain nonasymptotic bounds for the convergence to the target distribution π 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.