High dimensional sampling with the Unadjusted Langevin Algorithm

Alain Durmus (LTCI, Telecom ParisTech)
November 23, 2016 — 10:30 — "Salle du conseil du L2S"

Abstract

Recently, the problem of designing MCMC sampler adapted to high- dimensional distributions and with sensible theoretical guarantees has received a lot of interest. The applications are numerous, including large-scale inference in machine learning, Bayesian nonparametrics, Bayesian inverse problem, aggregation of experts among others. When the density is L-smooth (the log-density is continuously differentiable and its derivative is Lipshitz), we will advocate the use of a “rejection- free” algorithm, based on the discretization of the Euler diffusion with either constant or decreasing stepsizes. We will present several new results allowing convergence to stationarity under different conditions for the log-density (from the weakest, bounded oscillations on a compact set and super-exponential in the tails to the log concave). When the density is strongly log-concave, the convergence of an appropriately weighted empirical measure is also investigated and bounds for the mean square error and exponential deviation inequality for Lipschitz functions will be reported. Finally, based on optimzation techniques we will propose new methods to sample from high dimensional distributions. In particular, we will be interested in densities which are not continuously differentiable. Some Monte Carlo experiments will be presented to support our findings.