Hypocoercivity of Piecewise Deterministic Markov Process-Monte Carlo

TL;DR

This paper proves exponential convergence in L^2 for a broad class of Piecewise Deterministic Markov Processes used in Monte Carlo methods, providing theoretical insights into their efficiency and scalability.

Contribution

It develops a unifying hypocoercivity framework for these processes, offering explicit convergence estimates and scaling analysis with problem dimension.

Findings

Established L^2-exponential convergence for the processes

Derived explicit constants for convergence rates

Analyzed the algorithms' scaling with problem dimension

Abstract

In this work, we establish $\mathrm{L}^2$-exponential convergence for a broad class of Piecewise Deterministic Markov Processes recently proposed in the context of Markov Process Monte Carlo methods and covering in particular the Randomized Hamiltonian Monte Carlo, the Zig-Zag process and the Bouncy Particle Sampler. The kernel of the symmetric part of the generator of such processes is non-trivial, and we follow the ideas recently introduced by (Dolbeault et al., 2009, 2015) to develop a rigorous framework for hypocoercivity in a fairly general and unifying set-up, while deriving tractable estimates of the constants involved in terms of the parameters of the dynamics. As a by-product we characterize the scaling properties of these algorithms with respect to the dimension of classes of problems, therefore providing some theoretical evidence to support their practical relevance.