Piecewise deterministic simulated annealing

TL;DR

This paper introduces a piecewise deterministic Markov process for sampling Gibbs measures, providing theoretical results on exit times and convergence conditions for simulated annealing in various dimensions.

Contribution

It develops a novel piecewise deterministic approach for Gibbs sampling and derives conditions for convergence and exit times, extending classical diffusion results.

Findings

Eyring-Kramers formula for exit times in 1D

Necessary and sufficient cooling schedule condition in 1D

Sufficient condition for convergence in higher dimensions

Abstract

Given an energy potential on the Euclidian space, a piecewise deterministic Markov process is designed to sample the corresponding Gibbs measure. In dimension one an Eyring-Kramers formula is obtained for the exit time of the domain of a local minimum at low temperature, and a necessary and sufficient condition is given on the cooling schedule in a simulated annealing algorithm to ensure the process converges to the set of global minima. This condition is similar to the classical one for diffusions and involves the critical depth of the potential. In higher dimension a non optimal sufficient condition is obtained.