# A Discrete Bouncy Particle Sampler

arXiv: https://arxiv.org/abs/1707.05200 · HTML: https://mlsanity.com/paper/1707.05200/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Jul 2017.

## Built on by (most cited first, 4 of 4)

- [Delayed rejection Hamiltonian Monte Carlo for sampling multiscale distributions](https://mlsanity.com/paper/2110.00610/) (2021)
- [The Application of Zig-Zag Sampler in Sequential Markov Chain Monte Carlo](https://mlsanity.com/paper/2111.10210/) (2021)
- [Effect of Random Learning Rate: Theoretical Analysis of SGD Dynamics in Non-Convex Optimization via Stationary Distribution](https://mlsanity.com/paper/2406.16032/) (2024)
- [Bouncy particle sampler with infinite exchanging parallel tempering](https://mlsanity.com/paper/2509.02003/) (2025)

## Abstract (arXiv)

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.