Pillai, Natesh S., Stuart, A. M. and Thiéry, Alexandre H. (2012) Optimal scaling and diffusion limits for the Langevin algorithm in high dimensions. The Annals of Applied Probability, Vol.22 (No.6). pp. 2320-2356. doi:10.1214/11-AAP828 ISSN 1050-5164.

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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 O(N1/3) steps to explore the invariant measure, comparing favorably with the Random Walk Metropolis which was recently shown to require O(N) steps when applied to the same class of problems. As a by-product of the diffusion limit, it also follows that the MALA algorithm is optimized at an average acceptance probability of 0.574. Previous results were proved only for targets which are products of one-dimensional distributions, or for variants of this situation, limiting their applicability. The correlation in our target means that the rescaled MALA algorithm converges weakly to an infinite dimensional Hilbert space valued diffusion, and the limit cannot be described through analysis of scalar diffusions. The limit theorem is proved by showing that a drift-martingale decomposition of the Markov chain, suitably scaled, closely resembles a weak Euler–Maruyama discretization of the putative limit. An invariance principle is proved for the martingale, and a continuous mapping argument is used to complete the proof.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science, Engineering and Medicine > Science > Mathematics Faculty of Science, Engineering and Medicine > Science > Statistics
Library of Congress Subject Headings (LCSH):	Langevin equations, Distribution (Probability theory), Sampling (Statistics)
Journal or Publication Title:	The Annals of Applied Probability
Publisher:	Institute of Mathematical Statistics
ISSN:	1050-5164
Official Date:	2012
Dates:	Date Event 2012 Published
Volume:	Vol.22
Number:	No.6
Page Range:	pp. 2320-2356
DOI:	10.1214/11-AAP828
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access
Date of first compliant deposit:	23 December 2015
Date of first compliant Open Access:	23 December 2015
Funder:	National Science Foundation (U.S.) (NSF), Engineering and Physical Sciences Research Council (EPSRC), European Research Council (ERC)
Grant number:	DMS-11-07070 (NSF)

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