Exponential Convergence of Langevin Distributions and Their Discrete Approximations

Abstract

In this paper we consider a continuous-time method of approximating a given distribution using the Langevin diusion dL t dW t 1 2 r log (L t )dt.We ®nd conditions under this diusion converges exponentially quickly to or does not: in one dimension, these are essentially that for distributions with exponential tails of the form (x) / exp (ÿ|x| , 0<<1, exponential convergence occurs if and only if 1.We then consider conditions under which the discrete approximations to the diusion converge.We ®rst show that even when the diusion itself converges, naive discretizations need not do so.We then consider a `Metropolis-adjusted' version of the algorithm, and ®nd conditions under which this also converges at an exponential rate: perhaps surprisingly, even the Metropolized version need not converge exponentially fast even if the diusion does.We brie¯y discuss a truncated form of the algorithm which, in practice, should avoid the diculties of the other forms.