On the entropic convergence for piecewise deterministic samplers: speedup and obstruction

2026-06-24

AI Summary

This study investigates whether piecewise-deterministic Markov processes—such as Randomized Hamiltonian Monte Carlo (RHMC), the Bouncy Particle Sampler (BPS), and the Zig-Zag Process (ZZP)—exhibit quadratic acceleration in convergence relative to overdamped Langevin diffusion when measured in terms of relative entropy. By combining relative entropy analysis, L² hypocoercivity theory, and ergodicity arguments, the work establishes for the first time that RHMC indeed achieves quadratic acceleration in entropy. In contrast, both BPS and ZZP fail to attain exponential entropy decay rates, even for a standard Gaussian target distribution. These findings reveal fundamental differences in the information-geometric convergence behavior among non-diffusive samplers, highlighting that not all piecewise-deterministic dynamics benefit from accelerated entropy contraction.

Abstract

For piecewise deterministic samplers such as Randomized Hamiltonian Monte Carlo (RHMC), Bouncy Particle Sampler (BPS) or Zig-Zag Process (ZZP), long-time exponential convergence rates have been established in previous works using Harris or $L^2$ hypocoercivity approaches. In particular, in the $L^2$ framework, a so-called \emph{diffusive-to-ballistic} speedup was known for log-concave targets, according to which the convergence rates of these samplers, with suitable parameters, are quadratically improved with respect to the standard overdamped Langevin diffusion process. A recent work by Jianfeng Lu showed that this speedup also holds for the kinetic Langevin diffusion process when the convergence is stated in terms of relative entropy, raising the question whether this also holds for piecewise deterministic samplers. The present work provides a positive and a negative answer to this: first, we show that the speedup holds in entropy for RHMC; second, we show that for BPS or ZZS, even for a standard Gaussian target, a similar result cannot hold, and even that exponential convergence (at any rate) in entropy fails.

Problem

• piecewise deterministic samplers
• entropic convergence
• speedup
• relative entropy
• log-concave targets

Innovation

• piecewise deterministic samplers
• entropic convergence
• diffusive-to-ballistic speedup
• hypocoercivity
• relative entropy