Exponential Convergence of Langevin Diffusions and Their Discrete
Approximations

G. O. Roberts and  R. L. Tweedie

University of Cambridge and Colorado State University 


ABSTRACT

In this paper we consider a continous time method of approximating a
given distribution $\pi$ using the Langevin diffusion  $
d{\bf L}_t=d\W_t+ \half{\nabla \log \pi ({\bf L}_t) }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   $\pi(x)
\propto \exp({-\gamma |x|^{\beta}})$, $0 < \beta < \infty$,
exponential convergence occurs if and only if $\beta \geq 1$.

We then consider conditions under which the discrete approximations to
the diffusion converge. We first show that even when the diffusion
itself converges, naive discretisations 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 Metropolised 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.