Sampling from a log-concave distribution with Projected Langevin Monte Carlo

Sébastien Bubeck (1) , Ronen Eldan (2) , Joseph Lehec (3)

1 Microsoft Corporation [Redmond]
2 University of Washington [Seattle]
3 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

We extend the Langevin Monte Carlo (LMC) algorithm to compactly supported measures via a projection step, akin to projected Stochastic Gradient Descent (SGD). We show that (projected) LMC allows to sample in polynomial time from a log-concave distribution with smooth potential. This gives a new Markov chain to sample from a log-concave distribution. Our main result shows in particular that when the target distribution is uniform, LMC mixes in O(n 7) steps (where n is the dimension). We also provide preliminary experimental evidence that LMC performs at least as well as hit-and-run, for which a better mixing time of O(n 4) was proved by Lovász and Vempala.

Domaines: Probabilités [math.PR]

Sébastien Bubeck, Ronen Eldan, Joseph Lehec. Sampling from a log-concave distribution with Projected Langevin Monte Carlo. Discrete and Computational Geometry, 2018, 59 (4), pp.757-783. ⟨10.1007/s00454-018-9992-1⟩. ⟨hal-01428950⟩