Complexity Bounds for MCMC via Diffusion Limits

TL;DR

This paper establishes a connection between diffusion limits of MCMC algorithms and their complexity bounds, demonstrating that RWM and MALA algorithms have iteration complexities of O(d) and O(d^{1/3}) respectively in high dimensions.

Contribution

It introduces a novel framework linking diffusion limits to algorithm complexity, providing new bounds for RWM and MALA in high-dimensional settings.

Findings

RWM requires O(d) iterations for convergence

MALA requires O(d^{1/3}) iterations for convergence

Diffusion limit results imply complexity bounds for MCMC algorithms

Abstract

We connect known results about diffusion limits of Markov chain Monte Carlo (MCMC) algorithms to the Computer Science notion of algorithm complexity. Our main result states that any diffusion limit of a Markov process implies a corresponding complexity bound (in an appropriate metric). We then combine this result with previously-known MCMC diffusion limit results to prove that under appropriate assumptions, the Random-Walk Metropolis (RWM) algorithm in $d$ dimensions takes $O(d)$ iterations to converge to stationarity, while the Metropolis-Adjusted Langevin Algorithm (MALA) takes $O(d^{1/3})$ iterations to converge to stationarity.