Sampling from a log-concave distribution with Projected Langevin Monte Carlo
Bubeck, Sébastien; Eldan, Ronen; Lehec, Joseph (2018), Sampling from a log-concave distribution with Projected Langevin Monte Carlo, Discrete and Computational Geometry, 59, 4, p. 757–783. 10.1007/s00454-018-9992-1
TypeArticle accepté pour publication ou publié
Journal nameDiscrete and Computational Geometry
MetadataShow full item record
Microsoft Corporation [Redmond]
The Weizmann Institute of Science
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Abstract (EN)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.
Subjects / KeywordsLangevin Monte Carlo algorithm; Stochastic Gradient Descent; Sampling and optimization; Log-concave measures; Rapidly-mixing random walks
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