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

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 Mark...

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Bibliographic Details
Published inDiscrete & computational geometry Vol. 59; no. 4; pp. 757 - 783
Main Authors Bubeck, Sébastien, Eldan, Ronen, Lehec, Joseph
Format Journal Article
LanguageEnglish
Published New York Springer US 01.06.2018
Springer Nature B.V
Springer Verlag
Subjects
Combinatorics
Computational Mathematics and Numerical Analysis
Markov chains
Mathematics
Mathematics and Statistics
Monte Carlo simulation
Probability
68W20
Langevin Monte Carlo
Rapidly-mixing random walks
Log-concave measures
Sampling and optimization
68W25
47N10
Online AccessGet full text
ISSN0179-5376
1432-0444
DOI10.1007/s00454-018-9992-1

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Summary: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.
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ISSN:0179-5376
1432-0444
DOI:10.1007/s00454-018-9992-1