Diffusive Scaling Limits of Forward Event-Chain Monte Carlo: Provably Efficient Exploration with Partial Refreshment

TL;DR

This paper provides a theoretical analysis of Forward Event-Chain Monte Carlo (FECMC), demonstrating its superior efficiency over Bouncy Particle Sampler (BPS) through high-dimensional scaling limits and diffusion coefficient comparisons.

Contribution

It develops a high-dimensional scaling analysis for FECMC, deriving explicit diffusion coefficients and establishing its asymptotic efficiency advantages over BPS.

Findings

FECMC's diffusion coefficient is larger than BPS's, indicating higher efficiency.

Optimal FECMC occurs at zero global refreshment, maximizing efficiency.

Numerical experiments confirm theoretical efficiency gains across various targets.

Abstract

Piecewise deterministic Markov process samplers are attractive alternatives to Metropolis--Hastings algorithms. A central design question is how to incorporate partial velocity refreshment to ensure ergodicity without injecting excessive noise. Forward Event-Chain Monte Carlo (FECMC) is a generalization of the Bouncy Particle Sampler (BPS) that addresses this issue through a stochastic reflection mechanism, thereby reducing reliance on global refreshment moves. Despite promising empirical performance, its theoretical efficiency remains largely unexplored. We develop a high-dimensional scaling analysis for standard Gaussian targets and prove that the negative log-density (or potential) process of FECMC converges to an Ornstein--Uhlenbeck diffusion, under the same scaling as BPS. We derive closed-form expressions for the limiting diffusion coefficients of both methods by analyzing their associated radial momentum processes and solving the corresponding Poisson equations. These expressions yield a sharp efficiency comparison: the diffusion coefficient of FECMC is strictly larger than that of optimally tuned BPS, and the optimum for FECMC is attained at zero global refreshment. Specifically, they imply an approximately eightfold increase in effective sample size per event over optimal BPS. Numerical experiments confirm the predicted diffusion coefficients and show that the resulting efficiency gains remain substantial for a range of non-Gaussian targets. Finally, as an application of these results, we propose an asymptotic variance estimator for Piecewise deterministic Markov processes that becomes increasingly efficient in high dimensions by extracting information from the velocity variable.