Adaptive schemes for piecewise deterministic Monte Carlo algorithms

Abstract

We introduce a novel class of generative models based on piecewise deterministic Markov processes (PDMPs), a family of non-diffusive stochastic processes consisting of deterministic motion and random jumps at random times. Similarly to diffusions, such Markov processes admit time reversals that turn out to be PDMPs as well. We apply this observation to three PDMPs considered in the literature: the Zig-Zag process, Bouncy Particle Sampler, and Randomised Hamiltonian Monte Carlo. For these three particular instances, we show that the jump rates and kernels of the corresponding time reversals admit explicit expressions depending on some conditional densities of the PDMP under consideration before and after a jump. Based on these results, we propose efficient training procedures to learn these characteristics and consider methods to approximately simulate the reverse process. Finally, we provide bounds in the total variation distance between the data distribution and the resulting distribution of our model in the case where the base distribution is the standard d-dimensional Gaussian distribution. Promising numerical simulations support further investigations into this class of models.

Splitting schemes for second order approximations of piecewise-deterministic Markov processes

“…What we observe is that the adjusted ZZS defined in Algorithm 3 is exact for targets with diagonal covariance as expected, but the number of rejections increases with the correlation between components of the target. It is well known that the continuous time ZZS has lower efficiency for correlated targets (see [2]), and in the case of Algorithm 3 this is seen as a large number of reflections. On the other hand, the adjusted BPS given by Algorithm 4 appears to suffer when σ 2 is small, while the number of rejections remains controlled for large correlation ρ.…”

“…As expected, ZZS is sensitive to high correlation between components and its error increases with ρ. It is possible to improve in these cases by applying the adaptive schemes proposed in [2], which learn the covariance structure of the target and use this information to tune the set of velocities of the ZZS suitably. It seems also clear that the schemes based on ZZS are more robust when the target is very narrow in some components.…”

“…In particular, for g = ϕ, using the rotation-invariance of the uniform measure on the sphere and the fact that ϕ is increasing and with values in [1,2], we bound…”

“…Two other works ( [38] and [14]) focus on approximate simulation of the Zig-Zag sampler, which is one of our two main examples. In [38] the authors suggest to approximate event times by using numerical approximations of the integral of the rates along the dynamics (2), as well as a root finding algorithm. In [14], the authors suggest Algorithm 1: Splitting scheme DBD for ZZS Input : Number of iterations N , initial condition (x, v), step size δ.…”

Sticky PDMP samplers for sparse and local inference problems

“…If the upper bound is not tight to the Poisson rate λ, the procedure induce extra computational costs that can deteriorate the performance of the sampler. Several numerical schemes have been recently proposed trying to address this issue, see for example Pagani et al, 2020, Corbella, Spencer, and Roberts (2022),Bertazzi and Bierkens (2022) andSutton and Fearnhead (2021).…”