Optimal scaling and diffusion limits for the Langevin algorithm in high dimensions

TL;DR

This paper analyzes the efficiency of the Metropolis-adjusted Langevin algorithm (MALA) in high-dimensional settings, showing it converges faster than random walk Metropolis and providing diffusion limit results for infinite-dimensional target measures.

Contribution

It establishes the diffusion limit of MALA in infinite dimensions and quantifies its optimal scaling, demonstrating improved efficiency over random walk Metropolis in high dimensions.

Findings

MALA converges in (N^{1/3}) steps for high-dimensional problems.

Compared to random walk Metropolis, MALA requires fewer steps ((N)) in high dimensions.

Theoretical diffusion limits are derived for MALA on infinite-dimensional target measures.

Abstract

The Metropolis-adjusted Langevin (MALA) algorithm is a sampling algorithm which makes local moves by incorporating information about the gradient of the logarithm of the target density. In this paper we study the efficiency of MALA on a natural class of target measures supported on an infinite dimensional Hilbert space. These natural measures have density with respect to a Gaussian random field measure and arise in many applications such as Bayesian nonparametric statistics and the theory of conditioned diffusions. We prove that, started in stationarity, a suitably interpolated and scaled version of the Markov chain corresponding to MALA converges to an infinite dimensional diffusion process. Our results imply that, in stationarity, the MALA algorithm applied to an N-dimensional approximation of the target will take $\mathcal{O}(N^{1/3})$ steps to explore the invariant measure, comparing favorably with the Random Walk Metropolis which was recently shown to require $\mathcal{O}(N)$ steps when applied to the same class of problems.