Scaling Results for Piecewise Deterministic Monte Carlo : A Survey

2026-07-24

AI Summary

This study investigates the asymptotic behavior and computational efficiency of Piecewise Deterministic Monte Carlo (PDMC) algorithms—specifically the Bouncy Particle Sampler and the Zig-Zag Process—in complex settings characterized by high dimensionality, anisotropy, and large data volumes. By modeling these algorithms as continuous-time Markov processes and leveraging scaling limit analysis together with functional central limit theorems, the work systematically examines their theoretical properties. The analysis clarifies convergence characteristics and dimension dependence across a range of challenging scenarios, elucidating how anisotropy and dataset size influence algorithmic efficiency. These findings provide a rigorous theoretical foundation to guide the selection and refinement of PDMC methods in practical applications.

Abstract

Piecewise Deterministic Monte Carlo (PDMC) algorithms utilize continuous time Markov processes to generate samples from continuous distributions, and provide a modern alternative to discrete time Markov chain Monte Carlo algorithms. In this work we provide a survey of recent results on scaling limit arguments to understand the behaviour and efficiency of two often-used Piecewise Deterministic Monte Carlo algorithms: the Bouncy Particle Sampler and the Zig-Zag Process. In particular we discuss a Functional Central Limit Theorem, scaling in the high-dimensional regime, scaling under anisotropy, and scaling in the big data regime. This work is intended as part of the proceedings of the 2024 Isaac Newton Institute Programme "Stochastic systems for anomalous diffusion".

Problem

• Piecewise Deterministic Monte Carlo
• scaling limits
• high-dimensional
• anisotropy
• big data

Innovation

• Piecewise Deterministic Monte Carlo
• scaling limits
• Functional Central Limit Theorem
• high-dimensional sampling
• big data regime