A piecewise deterministic scaling limit of Lifted Metropolis-Hastings in the Curie-Weiss model

TL;DR

This paper derives a scaling limit for the Lifted Metropolis-Hastings algorithm applied to the Curie-Weiss model, revealing a non-reversible 'zig-zag' process that outperforms traditional methods in high-temperature regimes.

Contribution

It introduces a piecewise deterministic scaling limit for LMH in the Curie-Weiss model, highlighting its efficiency and non-reversible dynamics compared to standard MH.

Findings

LMH requires a jump rate of n^{1/2} at high temperature, versus n for MH.

At critical temperature, LMH's jump rate is n^{3/4}, lower than n^{3/2} for MH.

The scaling limit is a non-reversible 'zig-zag' Markov process.

Abstract

In Turitsyn, Chertkov, Vucelja (2011) a non-reversible Markov Chain Monte Carlo (MCMC) method on an augmented state space was introduced, here referred to as Lifted Metropolis-Hastings (LMH). A scaling limit of the magnetization process in the Curie-Weiss model is derived for LMH, as well as for Metropolis-Hastings (MH). The required jump rate in the high (supercritical) temperature regime equals $n^{1/2}$ for LMH, which should be compared to $n$ for MH. At the critical temperature the required jump rate equals $n^{3/4}$ for LMH and $n^{3/2}$ for MH, in agreement with experimental results of Turitsyn, Chertkov, Vucelja (2011). The scaling limit of LMH turns out to be a non-reversible piecewise deterministic exponentially ergodic `zig-zag' Markov process.