The Discrete Bouncy Particle Sampler

The Bouncy Particle Sampler (BPS) is a continuous-time, non-reversible MCMC algorithm that shows great promise in efficiently sampling from certain high-dimensional distributions; a particle moves with a fixed velocity except that occasionally it "bounces" off the hyperplane perpendicular to the gradient of the target density. One practical difficulty is that for each specific target distribution, a locally-valid upper bound on the component of the gradient in the direction of movement must be found so as to allow for simulation of the bounce times via Poisson thinning; for efficient implementation this bound should also be tight. In dimension $d=1$, the discrete-time version of the Bouncy Particle Sampler (and, equivalently, of the Zig-Zag sampler, another continuous-time, non-reversible algorithm) is known to consist of fixing a time step, $\Delta t$, and proposing a shift of $v \Delta t$ which is accepted with a probability dependent on the ratio of target evaluated at the proposed and current positions; on rejection the velocity is reversed. We present a discrete-time version of the BPS that is valid in any dimension $d\ge 1$ and the limit of which (as $\Delta t\downarrow 0$) is the BPS, which is rejection free. The Discrete BPS has the advantages of non-reversible algorithms in terms of mixing, but does not require an upper bound on a Poisson intensity and so is straightforward to apply to complex targets, such as those which can be evaluated pointwise but for which general properties, such as local or global Lipshitz bounds on derivatives, cannot be obtained. [Joint work with Dr. Alex Thiery].