Transient regime of piecewise deterministic Monte Carlo algorithms

2025-09-19

AI Summary

This work investigates the transient convergence behavior—i.e., early-stage dynamics from low-probability initial states to the high-probability typical set—of piecewise deterministic Monte Carlo (PDMC) algorithms, such as the Bouncy Particle Sampler and Zig-Zag Sampler, under convex potentials. We introduce a fast-slow variable decomposition framework, revealing a restart mechanism governed solely by slow variables after the event rate vanishes. A fluid limit model is established, and using generator decomposition combined with forward event-chain analysis, we rigorously prove the validity of short microcycle averaging. We further show, for the first time, that Zig-Zag direction selection is equivalent to a box-constrained quadratic programming problem. Theoretically, under Gaussian targets, transient cost matches that of Random Walk Metropolis; for heavy-tailed distributions with subquadratic growth, PDMC achieves superior convergence; and under specific conditions, only a constant number of jumps suffices to reach the typical set.

Technology Category

• Reasoning under Uncertainty: Probabilistic Programming
• Machine Learning: Probabilistic Circuits and Graphical Models
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Security and Privacy: Large-scale security measurements

Abstract

Piecewise Deterministic Markov Processes (PDMPs) such as the Bouncy Particle Sampler and the Zig-Zag Sampler, have gained attention as continuous-time counterparts of classical Markov chain Monte Carlo. We study their transient regime under convex potentials, namely how trajectories that start in low-probability regions move toward higher-probability sets. Using fluid-limit arguments with a decomposition of the generator into fast and slow parts, we obtain deterministic ordinary differential equation descriptions of early-stage behaviour. The fast dynamics alone are non-ergodic because once the event rate reaches zero it does not restart. The slow component reactivates the dynamics, so averaging remains valid when taken over short micro-cycles rather than with respect to an invariant law. Using the expected number of jump events as a cost proxy for gradient evaluations, we find that for Gaussian targets the transient cost of PDMP methods is comparable to that of random-walk Metropolis. For convex heavy-tailed families with subquadratic growth, PDMP methods can be more efficient when event simulation is implemented well. Forward Event-Chain and Coordinate Samplers can, under the same assumptions, reach the typical set with an order-one expected number of jumps. For the Zig-Zag Sampler we show that, under a diagonal-dominance condition, the transient choice of direction coincides with the solution of a box-constrained quadratic program; outside that regime we give a formal derivation and a piecewise-smooth update rule that clarifies the roles of the gradient and the Hessian. These results provide theoretical insight and practical guidance for the use of PDMP samplers in large-scale inference.

Problem

• Studying transient regime of PDMP algorithms under convex potentials
• Analyzing trajectories moving from low to high probability regions
• Comparing computational efficiency with traditional MCMC methods

Innovation

• Uses fluid-limit arguments with generator decomposition
• Employs deterministic ODE descriptions for early-stage behavior
• Implements forward event-chain and coordinate samplers