The tamed unadjusted Langevin algorithm

• Published in: Stochastic processes and their applications Vol. 129; no. 10; pp. 3638 - 3663
• Main Authors: Brosse, Nicolas, Durmus, Alain, Moulines, Éric, Sabanis, Sotirios
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.10.2019; Elsevier
• Subjects: Applications; Machine Learning; Markov chain Monte Carlo; Methodology; Statistics; Statistics Theory; Tamed unadjusted Langevin algorithm; Total variation distance; Wasserstein distance; Tamed unadjusted Langevin algorithm; Markov chain Monte Carlo; Wasserstein distance; 65C05; 60F05; Total variation distance; 62L10
• ISSN: 0304-4149; 1879-209X
• DOI: 10.1016/j.spa.2018.10.002

In this article, we consider the problem of sampling from a probability measure π having a density on Rd proportional to x↦e−U(x). The Euler discretization of the Langevin stochastic differential equation (SDE) is known to be unstable, when the potential U is superlinear. Based on previous works on the taming of superlinear drift coefficients for SDEs, we introduce the Tamed Unadjusted Langevin Algorithm (TULA) and obtain non-asymptotic bounds in V-total variation norm and Wasserstein distance of order 2 between the iterates of TULA and π, as well as weak error bounds. Numerical experiments are presented which support our findings.