Appropriate State-Dependent Friction Coefficient Accelerates Kinetic Langevin Dynamics

Abstract

We consider the convergence of kinetic Langevin dynamics to its ergodic invariant measure, which is Gibbs distribution. Instead of the standard setup where the friction coefficient is a constant scalar, we investigate position-dependent friction coefficient and the possible accelerated convergence it enables. We show that by choosing this coefficient matrix to be $2\sqrt{\text{Hess}V}$, convergence is accelerated in the sense that no constant scalar friction coefficient can lead to faster convergence for a large subset of (nonlinear) strongly-convex potential $V$'s. The speed of convergence is quantified in terms of chi-square divergence from the target distribution, and proved using a Lyapunov approach, based on viewing sampling as optimization in the infinite dimensional space of probability distributions.