Large deviations for the empirical measure of the zig-zag process

Abstract

The zig-zag process is a piecewise deterministic Markov process in position and velocity space. The process can be designed to have an arbitrary Gibbs type marginal probability density for its position coordinate, which makes it suitable for Monte Carlo simulation of continuous probability distributions. An important question in assessing the efficiency of this method is how fast the empirical measure converges to the stationary distribution of the process. In this paper we provide a partial answer to this question by characterizing the large deviations of the empirical measure from the stationary distribution. Based on the Feng-Kurtz approach, we develop an abstract framework aimed at encompassing piecewise deterministic Markov processes in position-velocity space. We derive explicit conditions for the zig-zag process to allow the Donsker-Varadhan variational formulation of the rate function, both for a compact setting (the torus) and one-dimensional Euclidean space. Finally we derive an explicit expression for the Donsker-Varadhan functional for the case of a compact state space and use this form of the rate function to address a key question concerning the optimal choice of the switching rate of the zig-zag process.

Complexity of zigzag sampling algorithm for strongly log-concave distributions

“…which violates (6). On the other hand, for the same choice of µ 0 , as long as U satisfies Assumption 1, one can estimate…”

“…Exponential convergence of the zigzag process is established in [7,31] using a Lyapunov function argument. A central limit theorem of the zigzag process is established in [2], and a large deviation principle is established for the empirical measure in [6]. The spectrum of the zigzag process has been studied in [5,32].…”

“…Recall the assumption ε ě expp´d 4Kκ log d q as well as (6), which implies that ? LT ď d 2 log d for our choice of T as in (9).…”

Large Deviations of Irreversible Processes

“…Chapter 4: Large Deviations of Empirical Measures. This chapter is based on a joint work with Joris Bierkens and Pierre Nyquist [BNS19]. Joris Bierkens and Gareth Roberts discovered the zig-zag process as a scaling limit of the Lifted Metropolis-Hastings [BR17].…”