Reducing rejection exponentially improves Markov chain Monte Carlo sampling

TL;DR

This paper introduces a rejection control transition kernel for Markov chain Monte Carlo that significantly reduces autocorrelation times by exponentially decreasing rejection rates, enhancing sampling efficiency.

Contribution

The authors propose a novel one-parameter rejection control kernel that optimizes MCMC sampling efficiency across various models by controlling rejection probabilities.

Findings

Reducing rejection rate exponentially decreases autocorrelation time.

The new kernel outperforms conventional algorithms in efficiency.

Autocorrelation times align on a single curve as a function of rejection rate.

Abstract

The choice of transition kernel critically influences the performance of the Markov chain Monte Carlo method. Despite the importance of kernel choice, guiding principles for optimal kernels have not been established. Here, we propose a one-parameter rejection control transition kernel that can be applied to various Monte Carlo samplings and demonstrate that the rejection process plays a major role in determining the sampling efficiency. Varying the rejection probability, we examine the autocorrelation time of the order parameter in the two- and three-dimensional ferromagnetic Potts models. Our results reveal that reducing the rejection rate leads to an exponential decrease in autocorrelation time in sequential spin updates and an algebraic reduction in random spin updates. The autocorrelation times of conventional algorithms almost fall on a single curve as a function of the rejection rate. The present transition kernel with an optimal parameter provides one of the most efficient samplers for general cases of discrete variables.