Sampling from a Log-Concave Distribution with Projected Langevin Monte Carlo

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Abstract

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 \(\widetilde{O}(n^7)\) 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 \(\widetilde{O}(n^4)\) was proved by Lovász and Vempala.

Keywords

Langevin Monte Carlo Sampling and optimization Log-concave measures Rapidly-mixing random walks 

Mathematics Subject Classification

47N10 68W20 68W25 

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Microsoft ResearchRedmondUSA
  2. 2.Faculty of Mathematics and Computer ScienceThe Weizmann Institute of ScienceRehovotIsrael
  3. 3.CEREMADE, Université Paris-DauphineParisFrance

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