Sampling from a log-concave distribution with compact support with proximal Langevin Monte Carlo

Nicolas Brosse (1, 2) , Alain Durmus (3) , Éric Moulines (1, 2) , Marcelo Pereyra (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 Department of Actuarial Mathematics and Statistics [Edinburgh]

This paper presents a detailed theoretical analysis of the Langevin Monte Carlo sampling algorithm recently introduced in [DMP16] when applied to log-concave probability distributions that are restricted to a convex body K. This method relies on a regularisation procedure involving the Moreau-Yosida envelope of the indicator function associated with K. Explicit convergence bounds in total variation norm and in Wasserstein distance of order 1 are established. In particular, we show that the complexity of this algorithm given a first order oracle is polynomial in the dimension of the state space. Finally, some numerical experiments are presented to compare our method with competing MCMC approaches from the literature.

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

Nicolas Brosse, Alain Durmus, Éric Moulines, Marcelo Pereyra. Sampling from a log-concave distribution with compact support with proximal Langevin Monte Carlo. Proceedings of Machine Learning Research, 2017, 65, pp.319-342. ⟨hal-01648665⟩