On explicit $L^2$-convergence rate estimate for piecewise deterministic Markov processes in MCMC algorithms

TL;DR

This paper derives explicit $L^2$-convergence rates for three key piecewise deterministic Markov processes used in MCMC sampling, enhancing the quantitative understanding of their efficiency.

Contribution

It introduces a variational hypocoercivity framework to obtain explicit $L^2$ convergence rates for these processes, improving upon previous less quantitative results.

Findings

Established $L^2$-exponential convergence rates for three PDMP-based MCMC methods.

Provided more explicit and quantitative convergence estimates than prior work.

Applied a novel variational hypocoercivity approach combining Poincaré inequalities and energy estimates.

Abstract

We establish $L^2$-exponential convergence rate for three popular piecewise deterministic Markov processes for sampling: the randomized Hamiltonian Monte Carlo method, the zigzag process, and the bouncy particle sampler. Our analysis is based on a variational framework for hypocoercivity, which combines a Poincar\'{e}-type inequality in time-augmented state space and a standard $L^2$ energy estimate. Our analysis provides explicit convergence rate estimates, which are more quantitative than existing results.