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arXiv · 1802.09188

Analysis of Langevin Monte Carlo via convex optimization

Alain Durmus·Szymon Majewski·Błażej Miasojedow
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Abstract

In this paper, we provide new insights on the Unadjusted Langevin Algorithm. We show that this method can be formulated as a first order optimization algorithm of an objective functional defined on the Wasserstein space of order $2$. Using this interpretation and techniques borrowed from convex optimization, we give a non-asymptotic analysis of this method to sample from logconcave smooth target distribution on $\mathbb{R}^d$. Based on this interpretation, we propose two new methods for sampling from a non-smooth target distribution, which we analyze as well. Besides, these new algorithms are natural extensions of the Stochastic Gradient Langevin Dynamics (SGLD) algorithm, which is a popular extension of the Unadjusted Langevin Algorithm. Similar to SGLD, they only rely on approximations of the gradient of the target log density and can be used for large-scale Bayesian inference.

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Alain Durmus, Szymon Majewski, Błażej Miasojedow. 2018-03-28. Analysis of Langevin Monte Carlo via convex optimization. https://arxiv.org/abs/1802.09188

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