Geometric ergodicity of the bouncy particle sampler

TL;DR

This paper proves the geometric ergodicity of the Bouncy Particle Sampler under weaker conditions, providing new insights into its convergence properties and applications to simulated annealing.

Contribution

It establishes geometric ergodicity of BPS with weaker assumptions and introduces a new coupling method for analyzing convergence.

Findings

Proved geometric ergodicity under weaker conditions.

Developed a new coupling for quantitative convergence analysis.

Analyzed convergence dependency on dimension in a toy model.

Abstract

The Bouncy Particle Sampler (BPS) is a Monte Carlo Markov Chain algorithm to sample from a target density known up to a multiplicative constant. This method is based on a kinetic piecewise deterministic Markov process for which the target measure is invariant. This paper deals with theoretical properties of BPS. First, we establish geometric ergodicity of the associated semi-group under weaker conditions than in [10] both on the target distribution and the velocity probability distribution. This result is based on a new coupling of the process which gives a quantitative minorization condition and yields more insights on the convergence. In addition, we study on a toy model the dependency of the convergence rates on the dimension of the state space. Finally, we apply our results to the analysis of simulated annealing algorithms based on BPS.