Exponential convergence of Langevin distributions and their discrete approximations

Abstract

In this paper we consider a continuous-time method of approximating a given distribution $\pi$ using the Langevin diffusion $d{L}_t=d{W}_t+\frac{1}{2}\nabla log\pi \left(L_t\right)dt$. We find conditions under which this diffusion converges exponentially quickly to $\pi$ or does not: in one dimension, these are essentially that for distributions with exponential tails of the form $\pi \left(x\right)\propto exp(-\gamma |x{|}^{\beta })$, $0<\beta <\mathrm{\infty }$, exponential convergence occurs if and only if $\beta \ge 1$. We then consider conditions under which the discrete approximations to the diffusion converge. We first show that even when the diffusion itself converges, naive discretizations need not do so. We then consider a 'Metropolis-adjusted' version of the algorithm, and find conditions under which this also converges at an exponential rate: perhaps surprisingly, even the Metropolized version need not converge exponentially fast even if the diffusion does. We briefly discuss a truncated form of the algorithm which, in practice, should avoid the difficulties of the other forms.