Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithm

Alain Durmus (1) , Éric Moulines (2, 3)

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

In this paper, we study a method to sample from a target distribution $\pi$ over $\mathbb{R}^d$ 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 non-asymptotic 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 2014).

Mots clés: Markov Chain Monte Carlo; Metropolis Adjusted Langevin Algorithm; total variation distance; Langevin diffusion; Rate of convergence

Domaines: Théorie [stat.TH]; Calcul [stat.CO]; Méthodologie [stat.ME]

Alain Durmus, Éric Moulines. Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithm. 2016. ⟨hal-01176132v3⟩