Non-stationary phase of the MALA algorithm

TL;DR

This paper analyzes the non-stationary phase of the MALA algorithm in high dimensions, revealing that its computational cost scales as N^(1/2), which is higher than the N^(1/3) cost in stationary regimes.

Contribution

It extends previous diffusion limit analyses of MALA to non-stationary, non-product target measures, providing new insights into its computational complexity in high-dimensional settings.

Findings

Cost scales as N^(1/2) in non-stationary regime

Cost scales as N^(1/3) in stationary regime

Analyzes non-product target measures in high dimensions

Abstract

The Metropolis-Adjusted Langevin Algorithm (MALA) is a Markov Chain Monte Carlo method which creates a Markov chain reversible with respect to a given target distribution, pi^N, with Lebesgue density on R^N; it can hence be used to approximately sample the target distribution. When the dimension N is large a key question is to determine the computational cost of the algorithm as a function of N. One approach to this question, which we adopt here, is to derive diffusion limits for the algorithm. The family of target measures that we consider in this paper are, in general, in non-product form and are of interest in applied problems as they arise in Bayesian nonparametric statistics and in the study of conditioned diffusions. Furthermore, we study the situation, which arises in practice, where the algorithm is started out of stationarity. We thereby significantly extend previous works which consider either only measures of product form, when the Markov chain is started out of stationarity, or measures defined via a density with respect to a Gaussian, when the Markov chain is started in stationarity. We prove that, in the non-stationary regime, the computational cost of the algorithm is of the order N^(1/2) with dimension, as opposed to what is known to happen in the stationary regime, where the cost is of the order N^(1/3).