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

08/13/2018
by   George Deligiannidis, et al.
0

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.

READ FULL TEXT
POST COMMENT

Comments

There are no comments yet.

Authors

page 1

page 2

page 3

page 4

08/26/2018

Hypercoercivity of Piecewise Deterministic Markov Process-Monte Carlo

In this paper we derive spectral gap estimates for several Piecewise Det...
09/29/2020

Couplings for Andersen Dynamics

Andersen dynamics is a standard method for molecular simulations, and a ...
09/03/2016

Stochastic Bouncy Particle Sampler

We introduce a novel stochastic version of the non-reversible, rejection...
11/02/2017

Binary Bouncy Particle Sampler

The Bouncy Particle Sampler is a novel rejection-free non-reversible sam...
01/29/2016

On the Geometric Ergodicity of Hamiltonian Monte Carlo

We establish general conditions under which Markov chains produced by th...
04/06/2022

Strongly convergent homogeneous approximations to inhomogeneous Markov jump processes and applications

The study of inhomogeneous Markov jump processes is a traditional topic ...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.