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arXiv · 1507.02564

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

Sébastien Bubeck·Ronen Eldan·Joseph Lehec
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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 $\tilde{O}(n^7)$ 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 $\tilde{O}(n^4)$ was proved by Lov{\'a}sz and Vempala.

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Sébastien Bubeck, Ronen Eldan, Joseph Lehec. 2015-07-09. Sampling from a log-concave distribution with Projected Langevin Monte Carlo. https://arxiv.org/abs/1507.02564

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