Optimal design of the Barker proposal and other locally-balanced Metropolis-Hastings algorithms

TL;DR

This paper analyzes a class of locally-balanced Metropolis-Hastings algorithms, establishing optimal acceptance rates, scaling, and efficiency, and proposes improvements like a bi-modal noise distribution for enhanced practical performance.

Contribution

It provides a theoretical framework for optimal parameter choices within the class, including noise distribution and balancing functions, and introduces a more efficient Barker proposal variant.

Findings

Universal 57% acceptance rate at optimal scaling

Explicit asymptotic efficiency formulas derived

Bi-modal noise distribution improves Barker proposal performance

Abstract

We study the class of first-order locally-balanced Metropolis--Hastings algorithms introduced in Livingstone & Zanella (2021). To choose a specific algorithm within the class the user must select a balancing function $g:\mathbb{R} \to \mathbb{R}$ satisfying $g(t) = tg(1/t)$, and a noise distribution for the proposal increment. Popular choices within the class are the Metropolis-adjusted Langevin algorithm and the recently introduced Barker proposal. We first establish a universal limiting optimal acceptance rate of 57% and scaling of $n^{-1/3}$ as the dimension $n$ tends to infinity among all members of the class under mild smoothness assumptions on $g$ and when the target distribution for the algorithm is of the product form. In particular we obtain an explicit expression for the asymptotic efficiency of an arbitrary algorithm in the class, as measured by expected squared jumping distance. We then consider how to optimise this expression under various constraints. We derive an optimal choice of noise distribution for the Barker proposal, optimal choice of balancing function under a Gaussian noise distribution, and optimal choice of first-order locally-balanced algorithm among the entire class, which turns out to depend on the specific target distribution. Numerical simulations confirm our theoretical findings and in particular show that a bi-modal choice of noise distribution in the Barker proposal gives rise to a practical algorithm that is consistently more efficient than the original Gaussian version.