User-friendly guarantees for the Langevin Monte Carlo with inaccurate gradient

摘要

In this paper, we revisit the recently established theoretical guarantees forthe convergence of the Langevin Monte Carlo algorithm of sampling from a smoothand (strongly) log-concave density. We improve, in terms of constants, theexisting results when the accuracy of sampling is measured in the Wassersteindistance and provide further insights on relations between, on the one hand,the Langevin Monte Carlo for sampling and, on the other hand, the gradientdescent for optimization. More importantly, we establish non-asymptoticguarantees for the accuracy of a version of the Langevin Monte Carlo algorithmthat is based on inaccurate evaluations of the gradient. Finally, we propose avariable-step version of the Langevin Monte Carlo algorithm that has twoadvantages. First, its step-sizes are independent of the target accuracy and,second, its rate provides a logarithmic improvement over the constant-stepLangevin Monte Carlo algorithm.