Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Towards practical PDMP sampling: Metropolis adjustments, locally adaptive step-sizes, and NUTS-based time lengths

Augustin Chevallier, Sam Power, Matthew Sutton

TL;DR

The paper advances practical PDMP-based MCMC by introducing a Metropolis-adjusted PDMP framework that removes the need for analytical bounds, together with local adaptivity for both step size and trajectory length via a NUTS-inspired stopping mechanism. By leveraging time-reversal and skew-reversibility, it constructs exact or bias-free Metropolis corrections for PDMP approximations and extends the approach to BPS and ZigZag processes. The proposed doubly-adaptive PDMP samplers demonstrate robust performance and favorable dimension-scaling, particularly on challenging distributions such as Neal’s funnel, and they often outperform HMC in these regimes. While offering practical benefits, the work notes computational trade-offs (notably the $O(n^2)$ cost of the stopping criterion) and outlines future work on faster stopping rules and mass-matrix adaptation to further enhance scalability and efficiency.

Abstract

Piecewise-Deterministic Markov Processes (PDMPs) hold significant promise for sampling from complex probability distributions. However, their practical implementation is hindered by the need to compute model-specific bounds. Conversely, while Hamiltonian Monte Carlo (HMC) offers a generally efficient approach to sampling, its inability to adaptively tune step sizes impedes its performance when sampling complex distributions like funnels. To address these limitations, we introduce three innovative concepts: (a) a Metropolis-adjusted approximation for PDMP simulation that eliminates the need for explicit bounds without compromising the invariant measure, (b) an adaptive step size mechanism compatible with the Metropolis correction, and (c) a No U-Turn Sampler (NUTS)-inspired scheme for dynamically selecting path lengths in PDMPs. These three ideas can be seamlessly integrated into a single, `doubly-adaptive' PDMP sampler with favourable robustness and efficiency properties.

Towards practical PDMP sampling: Metropolis adjustments, locally adaptive step-sizes, and NUTS-based time lengths

TL;DR

The paper advances practical PDMP-based MCMC by introducing a Metropolis-adjusted PDMP framework that removes the need for analytical bounds, together with local adaptivity for both step size and trajectory length via a NUTS-inspired stopping mechanism. By leveraging time-reversal and skew-reversibility, it constructs exact or bias-free Metropolis corrections for PDMP approximations and extends the approach to BPS and ZigZag processes. The proposed doubly-adaptive PDMP samplers demonstrate robust performance and favorable dimension-scaling, particularly on challenging distributions such as Neal’s funnel, and they often outperform HMC in these regimes. While offering practical benefits, the work notes computational trade-offs (notably the cost of the stopping criterion) and outlines future work on faster stopping rules and mass-matrix adaptation to further enhance scalability and efficiency.

Abstract

Piecewise-Deterministic Markov Processes (PDMPs) hold significant promise for sampling from complex probability distributions. However, their practical implementation is hindered by the need to compute model-specific bounds. Conversely, while Hamiltonian Monte Carlo (HMC) offers a generally efficient approach to sampling, its inability to adaptively tune step sizes impedes its performance when sampling complex distributions like funnels. To address these limitations, we introduce three innovative concepts: (a) a Metropolis-adjusted approximation for PDMP simulation that eliminates the need for explicit bounds without compromising the invariant measure, (b) an adaptive step size mechanism compatible with the Metropolis correction, and (c) a No U-Turn Sampler (NUTS)-inspired scheme for dynamically selecting path lengths in PDMPs. These three ideas can be seamlessly integrated into a single, `doubly-adaptive' PDMP sampler with favourable robustness and efficiency properties.

Paper Structure

This paper contains 28 sections, 16 theorems, 69 equations, 4 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Concepts
  3. Piecewise-Deterministic Markov Processes
  4. Kinetic Methods for MCMC Sampling
  5. Hamiltonian Monte Carlo, and the No U-Turn Criterion
  6. Practical Simulation of PDMPs through Metropolis Adjustment
  7. Time-reversal and skew-reversibility
  8. Path space
  9. Approximation of a PDMP
  10. Piecewise constant approximation
  11. Piecewise linear approximation
  12. Local adaptation of the step size
  13. Metropolis-Adjustment of PDMP Discretisations
  14. BPS and ZigZag
  15. BPS
  16. ...and 13 more sections

Key Result

Proposition 1

Assuming that $Z_t$ is a PDMP with deterministic flow $\phi_t$, jump rate $\lambda$ and jump kernel $Q(z,\cdot) = \delta_{F(z)}$ where $F$ is an injection leaving the Lebesgue measure invariant, and that $\mu$ is an invariant distribution with a density, the time-reversal is the following PDMP:

Figures (4)

  • Figure 1: Gaussian targets, order 1 approximation, 1000 NUTS-BPS steps, for various dimensions.
  • Figure 2: (a) Optimal scaling for the stepsize $h$. (b) Acceptance rate for the optimal stepsize $h$. (c) Scaling of the ESS/complexity for the optimal value of $h$.
  • Figure 3: An example trajectory in a funnel for MHBPS-NUTS, with the location of gradient evaluations displayed in blue and red. The effect of the adaptivity on the stepsize can clearly be seen.
  • Figure 4: Comparison between HMC with different step sizes, and MHBPS-NUTS for the funnel.

Theorems & Definitions (36)

  • Proposition 1: Practical time-reversal
  • Proposition 2: skew-reversibility
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Remark 1
  • Proposition 5: scale invariance
  • proof
  • ...and 26 more