Adaptive schemes for piecewise deterministic Monte Carlo algorithms

TL;DR

This paper introduces adaptive schemes for Bouncy Particle and Zig-Zag samplers that learn target distribution covariance and tune refreshment rates, improving convergence speed and robustness.

Contribution

It proposes novel adaptive algorithms that automatically adjust covariance and refreshment rates, with proven ergodicity and demonstrated numerical benefits.

Findings

Adaptive schemes improve convergence speed.

Algorithms are ergodic with proven theoretical guarantees.

Numerical simulations show enhanced performance.

Abstract

The Bouncy Particle sampler (BPS) and the Zig-Zag sampler (ZZS) are continuous time, non-reversible Monte Carlo methods based on piecewise deterministic Markov processes. Experiments show that the speed of convergence of these samplers can be affected by the shape of the target distribution, as for instance in the case of anisotropic targets. We propose an adaptive scheme that iteratively learns all or part of the covariance matrix of the target and takes advantage of the obtained information to modify the underlying process with the aim of increasing the speed of convergence. Moreover, we define an adaptive scheme that automatically tunes the refreshment rate of the BPS or ZZS. We prove ergodicity and a law of large numbers for all the proposed adaptive algorithms. Finally, we show the benefits of the adaptive samplers with several numerical simulations.