[Paper Review] Randomized Hamiltonian Monte Carlo as Scaling Limit of the Bouncy Particle Sampler and Dimension-Free Convergence Rates

George Deligiannidis, Daniel Paulin+2 more|arXiv (Cornell University)|Aug 13, 2018

Markov Chains and Monte Carlo Methods4 citations

TL;DR

This paper establishes that the Bouncy Particle Sampler (BPS) converges weakly to Randomized Hamiltonian Monte Carlo (RHMC) in the high-dimensional limit, demonstrating dimension-free convergence rates for RHMC under strong log-concavity with bounded Hessians using coupling and hypocoercivity techniques.

ABSTRACT

The Bouncy Particle Sampler is a Markov chain Monte Carlo method based on a nonreversible piecewise deterministic Markov process. In this scheme, a particle explores the state space of interest by evolving according to a linear dynamics which is altered by bouncing on the hyperplane tangent to the gradient of the negative log-target density at the arrival times of an inhomogeneous Poisson Process (PP) and by randomly perturbing its velocity at the arrival times of an homogeneous PP. Under regularity conditions, we show here that the process corresponding to the first component of the particle and its corresponding velocity converges weakly towards a Randomized Hamiltonian Monte Carlo (RHMC) process as the dimension of the ambient space goes to infinity. RHMC is another piecewise deterministic non-reversible Markov process where a Hamiltonian dynamics is altered at the arrival times of a homogeneous PP by randomly perturbing the momentum component. We then establish dimension-free convergence rates for RHMC for strongly log-concave targets with bounded Hessians using coupling ideas and hypocoercivity techniques.

Motivation & Objective

• To understand the asymptotic behavior of the Bouncy Particle Sampler (BPS) as the dimension of the state space increases.
• To establish a rigorous connection between BPS and Randomized Hamiltonian Monte Carlo (RHMC) as a scaling limit.
• To derive dimension-free convergence rates for RHMC under strong log-concave targets with bounded Hessians.
• To leverage coupling and hypocoercivity techniques to analyze convergence in high-dimensional settings.
• To provide theoretical justification for the efficiency of nonreversible MCMC methods in high dimensions.

Proposed method

• Uses a piecewise deterministic Markov process (PDMP) framework for both BPS and RHMC, where the particle evolves via linear dynamics and is perturbed at random times.
• Models BPS as a process with velocity jumps at inhomogeneous Poisson process arrivals based on the gradient of the log-target, and momentum resets at homogeneous Poisson process arrivals.
• Applies weak convergence theory to show that the first component of the BPS process converges to the corresponding component of RHMC as dimension tends to infinity.
• Employs coupling techniques to compare paths of the RHMC process and bound the convergence rate to the target distribution.
• Utilizes hypocoercivity methods to establish convergence rates independent of dimension under strong log-concavity and bounded Hessian conditions.
• Analyzes the generator and invariant measure of the PDMPs to ensure ergodicity and convergence to the target distribution.

Experimental results

Research questions

• RQ1 Does the Bouncy Particle Sampler converge to a known MCMC method in the high-dimensional limit?
• RQ2 What is the relationship between the Bouncy Particle Sampler and Randomized Hamiltonian Monte Carlo in the scaling limit?
• RQ3 Can dimension-free convergence rates be established for RHMC under strong log-concavity with bounded Hessians?
• RQ4 How do coupling and hypocoercivity techniques contribute to proving convergence in high-dimensional settings?
• RQ5 What theoretical guarantees can be given for nonreversible PDMP-based samplers in high dimensions?

Key findings

• The Bouncy Particle Sampler converges weakly to Randomized Hamiltonian Monte Carlo as the dimension of the state space tends to infinity.
• The limiting process, RHMC, is a nonreversible PDMP where momentum is randomly perturbed at homogeneous Poisson process arrival times.
• Dimension-free convergence rates are established for RHMC under strong log-concavity and bounded Hessian conditions.
• The convergence rate is bounded independently of the dimension, indicating scalability in high-dimensional target distributions.
• Coupling techniques combined with hypocoercivity provide a rigorous framework for analyzing convergence in high-dimensional nonreversible MCMC.
• The theoretical results justify the empirical efficiency of nonreversible samplers like BPS and RHMC in high-dimensional Bayesian inference.