Asymptotic Analysis of the Random-Walk Metropolis Algorithm on Ridged Densities

TL;DR

This paper analyzes the asymptotic behavior of the Random-Walk Metropolis algorithm on high-dimensional, ridged probability densities, revealing that traditional step-size adaptation may be sub-optimal in certain concentrated scenarios.

Contribution

It provides the first analytical diffusion limit for the algorithm on ridged densities with multiple scales, showing non-constant diffusion coefficients and implications for step-size tuning.

Findings

Diffusion limit derived for target measures on sub-manifolds.

The limiting SDE often has a non-constant diffusion coefficient.

Optimal acceptance probability approaches zero in some cases.

Abstract

In this paper we study the asymptotic behavior of the Random-Walk Metropolis algorithm on probability densities with two different `scales', where most of the probability mass is distributed along certain key directions with the `orthogonal' directions containing relatively less mass. Such class of probability measures arise in various applied contexts including Bayesian inverse problems where the posterior measure concentrates on a sub-manifold when the noise variance goes to zero. When the target measure concentrates on a linear sub-manifold, we derive analytically a diffusion limit for the Random-Walk Metropolis Markov chain as the scale parameter goes to zero. In contrast to the existing works on scaling limits, our limiting Stochastic Differential Equation does not in general have a constant diffusion coefficient. Our results show that in some cases, the usual practice of adapting the step-size to control the acceptance probability might be sub-optimal as the optimal acceptance probability is zero (in the limit).