A Discrete Bouncy Particle Sampler

TL;DR

This paper introduces the Discrete Bouncy Particle Sampler, a new non-reversible MCMC algorithm that simplifies implementation by only requiring point-wise evaluations, and provides theoretical analysis and empirical results demonstrating its efficiency.

Contribution

It presents a novel discrete-time algorithm that generalizes the Bouncy Particle Sampler, with theoretical efficiency analysis and practical extensions for cases with unavailable exact gradients.

Findings

The Discrete Bouncy Particle Sampler is easier to implement than the original Bouncy Particle Sampler.

Theoretical efficiency of the algorithm is characterized as dimension increases.

Empirical results show competitive performance across various target distributions.

Abstract

Most Markov chain Monte Carlo methods operate in discrete time and are reversible with respect to the target probability. Nevertheless, it is now understood that the use of non-reversible Markov chains can be beneficial in many contexts. In particular, the recently-proposed Bouncy Particle Sampler leverages a continuous-time and non-reversible Markov process and empirically shows state-of-the-art performances when used to explore certain probability densities; however, its implementation typically requires the computation of local upper bounds on the gradient of the log target density. We present the Discrete Bouncy Particle Sampler, a general algorithm based upon a guided random walk, a partial refreshment of direction, and a delayed-rejection step. We show that the Bouncy Particle Sampler can be understood as a scaling limit of a special case of our algorithm. In contrast to the Bouncy Particle Sampler, implementing the Discrete Bouncy Particle Sampler only requires point-wise evaluation of the target density and its gradient. We propose extensions of the basic algorithm for situations when the exact gradient of the target density is not available. In a Gaussian setting, we establish a scaling limit for the radial process as dimension increases to infinity. We leverage this result to obtain the theoretical efficiency of the Discrete Bouncy Particle Sampler as a function of the partial-refreshment parameter, which leads to a simple and robust tuning criterion. A further analysis in a more general setting suggests that this tuning criterion applies more generally. Theoretical and empirical efficiency curves are then compared for different targets and algorithm variations.