The Tamed Unadjusted Langevin Algorithm

Nicolas Brosse (1, 2) , Alain Durmus (3) , Éric Moulines (1, 2) , Sotirios Sabanis (4)

1 XPOP - Modélisation en pharmacologie de population
2 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
3 LTCI - Laboratoire Traitement et Communication de l'Information
4 Edin. - University of Edinburgh

In this article, we consider the problem of sampling from a probability measure π having a density on R d known up to a normalizing constant, $x → e −U (x) / R d e −U (y) dy$. The Euler discretization of the Langevin stochastic differential equation (SDE) is known to be unstable in a precise sense, when the potential U is superlinear, i.e. lim inf $x→+∞ U (x) / x = +∞$. 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.

Domaines: Méthodologie [stat.ME]; Applications [stat.AP]; Théorie [stat.TH]; Machine Learning [stat.ML]

Nicolas Brosse, Alain Durmus, Éric Moulines, Sotirios Sabanis. The Tamed Unadjusted Langevin Algorithm. Stochastic Processes and their Applications, 2018, ⟨10.1016/j.spa.2018.10.002⟩. ⟨hal-01648667⟩