Universality of the Langevin diffusion as scaling limit of a family of Metropolis-Hastings processes I: fixed dimension

TL;DR

This paper demonstrates that different Metropolis-Hastings processes, when scaled appropriately in fixed dimensions, universally converge to Langevin diffusions, revealing fundamental properties of these stochastic algorithms.

Contribution

It introduces new universal properties of two types of Metropolis-Hastings processes and proves their convergence to Langevin diffusions in fixed dimensions, extending prior results.

Findings

$M_2$ processes share universal properties with $M_1$ processes.

All considered processes converge to Langevin diffusion under scaling.

New results for $M_2$ and convex combinations are established.

Abstract

Given a target distribution $\mu$ on a general state space $\mathcal{X}$ and a proposal Markov jump process with generator $Q$, the purpose of this paper is to investigate two universal properties enjoyed by two types of Metropolis-Hastings (MH) processes with generators $M_1(Q,\mu)$ and $M_2(Q,\mu)$ respectively. First, we motivate our study of $M_2$ by offering a geometric interpretation of $M_1$, $M_2$ and their convex combinations as $L^1$ minimizers between $Q$ and the set of $\mu$-reversible generators of Markov jump processes. Second, specializing into the case of $\mathcal{X} = \mathbb{R}^d$ along with a Gaussian proposal with vanishing variance and Gibbs target distribution, we prove that, upon appropriate scaling in time, the family of Markov jump processes corresponding to $M_1$, $M_2$ or their convex combinations all converge weakly to an universal Langevin diffusion. While $M_1$ and $M_2$ are seemingly different stochastic dynamics, it is perhaps surprising that they share these two universal properties. These two results are known for $M_1$ in Billera and Diaconis (2001) and Gelfand and Mitter (1991), and the counterpart results for $M_2$ and their convex combinations are new.