Optimal Ricci curvature Markov chain Monte Carlo methods on finite states

TL;DR

This paper introduces an optimal MCMC method on finite states with proven convergence rates, independent of the target distribution, and demonstrates its effectiveness through numerical examples.

Contribution

It develops a new MCMC algorithm with optimal acceptance-rejection functions and establishes its convergence properties, generalizing the Metropolis-Hastings method.

Findings

Convergence rate is at least one-half in $L^1$ distance.

Method recovers Metropolis-Hastings on two-point states.

Numerical examples show the algorithm's effectiveness.

Abstract

We construct a new Markov chain Monte Carlo method on finite states with optimal choices of acceptance-rejection ratio functions. We prove that the constructed continuous time Markov jumping process has a global in-time convergence rate in $L^1$ distance. The convergence rate is no less than one-half and is independent of the target distribution. For example, our method recovers the Metropolis-Hastings algorithm on a two-point state. And it forms a new algorithm for sampling general target distributions. Numerical examples are presented to demonstrate the effectiveness of the proposed algorithm.