User-friendly guarantees for the Langevin Monte Carlo

Abstract: In this talk, I will revisit the recently established theoretical guarantees for the convergence of the Langevin Monte Carlo algorithm of sampling from a smooth and (strongly) log-concave density. I will discuss the existing results when the accuracy of sampling is measured in the Wasserstein distance and provide further insights on relations between, on the one hand, the Langevin Monte Carlo for sampling and, on the other hand, the gradient descent for optimization. I will also present non-asymptotic guarantees for the accuracy of a version of the Langevin Monte Carlo algorithm that is based on inaccurate evaluations of the gradient. Finally, I will propose a variable-step version of the Langevin Monte Carlo algorithm that has two advantages. First, its step-sizes are independent of the target accuracy and, second, its rate provides a logarithmic improvement over the constant-step Langevin Monte Carlo algorithm.
This is a joint work with A. Karagulyan

Biography: 

Personal Website: http://www.arnak-dalalyan.fr/

Arnak Dalalyan is a full professor of Statistics at ENSAE-CREST. He obtained his PhD (2001) from Le Mans University on Statistics for Random Processes. He was a postdoctoral fellow (2002–03) at the Humboldt University of Berlin, an assistant professor (2003–08) at Paris 6 University and a research professor at ENPC (2008–2011). Arnak’s research focuses on high dimensional statistics, statistics of diffusion processes and statistical learning theory. Presently, he is an associate editor of Electronic Journal of Statistics, Statistical Inference for Stochastic Processes and Journal of the Japan Statistical Society. Arnak is also regularly serving in the programme committees of machine learning conferences COLT and NIPS. He is a member of the Bernoulli Society council.