Speeding up Langevin Dynamics by Mixing

Abstract

We study an overdamped Langevin equation on the $d$-dimensional torus with stationary distribution proportional to~$p = e^{-U / \kappa}$. When~$U$ has multiple wells the mixing time of the associated process is exponentially large (of size~$e^{O(1/\kappa)}$). We add a drift to the Langevin dynamics (without changing the stationary distribution) and obtain quantitative estimates on the mixing time. We show that an exponentially mixing drift can be rescaled to make the mixing time of the Langevin system arbitrarily small. For numerical purposes, it is useful to keep the size of the imposed drift small, and we show that the smallest allowable rescaling ensures that the mixing time is $O( d/\kappa^2)$, which is an order of magnitude smaller than~$e^{O(1/\kappa)}$. We provide one construction of an exponentially mixing drift, although with rate constants whose~$\kappa$-dependence is unknown.Heuristics (from discrete time) suggest that $\kappa$-dependence of the mixing rate is such that the imposed drift is of size~$O(d / \kappa^3)$. The large amplitude of the imposed drift increases the numerical complexity, and thus we expect this method will be most useful in the initial phase of Monte Carlo methods to rapidly explore the state space.