Analysis of Langevin Monte Carlo via convex optimization

02/26/2018

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by   Alain Durmus, et al.

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ens-cachan.fr

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University of Warsaw

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Zimbra, Inc.

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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 R^d. Our proofs are then easily extended to the Stochastic Gradient Langevin Dynamics, which is a popular extension of the Unadjusted Langevin Algorithm. Finally, this interpretation leads to a new methodology to sample from a non-smooth target distribution, for which a similar study is done.