A piecewise deterministic scaling limit of lifted Metropolis–Hastings in the Curie–Weiss model

Abstract

In Turitsyn, Chertkov and 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 and Vucelja (2011). The scaling limit of LMH turns out to be a non-reversible piecewise deterministic exponentially ergodic 'zig-zag' Markov process. Primary 60F05; secondary 65C05. Keywords and phrases: weak convergence, Markov chain Monte Carlo, piecewise deterministic Markov process, phase transition, exponential ergodicity. * Both authors would like to thank the EPSRC for support under grants EP/D002060/1 (CRiSM) and EP/K014463/1 (iLike) 1 imsart-generic ver. 2011/11/15 file: spin.tex date: June 3, 2016 J. Bierkens and G. Roberts/Piecewise deterministic scaling limit of Lifted Metropolis-Hastings 2 'lifting approach' is not the only way of obtaining non-reversible Markov processes. For interesting approaches to constructing and analyzing the benefits of non-reversible Markov processes, see e.g. Hwang, Duncan, Lelièvre and Pavliotis (2015).It is the goal of this paper to shed light on the general theory of lifted non-reversible Markov chains, and in particular on the recent experimental result of Turitsyn, Chertkov and Vucelja (2011) mentioned above. This is achieved by obtaining a scaling limit of Lifted Metropolis-Hastings, in its application to the Curie-Weiss model. This scaling limit may be compared to a similar scaling limit for (classical) Metropolis-Hastings.Initiated by Roberts, Gelman and Gilks (1997), a large amount of understanding of particular Markov Chain Monte Carlo (MCMC) algorithms has been obtained by identifying a suitable diffusion limit: Given a sequence of Markov chains of increasing size or dimensionality n, a suitable scaling of the state space and of the amount of steps per unit time interval is determined. As n tends towards infinity, the scaled Markov process converges (in the sense of weak convergence on Skorohod path space) to a diffusion process, which is often of an elementary nature. In particular the required number of Markov chain transitions per unit time interval as a function of n provides a fundamental measure of the speed of the Markov chain.The Curie-Weiss model is an exchangeable probability distribution on {−1, 1} n which depends on two parameters, the 'external field' h, and the 'inverse temperature' β (which describes interactions between components). At inverse temperature β = 1 the model undergoes a phase transition. This results in differences in behaviour for β < 1, β = 1 and β > 1, and we shall analyse the behaviour of both standard Metropolis-Hastings and Lifted Me...

Ergodicity of the zigzag process

“…However, this class of functions does not contain the full space L 2 (π). Indeed, by the proof of [6,Theorem 5], the function V is growing at a certain exponential rate as |x| → ∞, putting a restriction on the growth of the functions f (x, θ).…”

“…A particular instance of such a process is the zigzag process, described already in e.g. [13,21] and given its name in [6]. The zigzag process is a variant of, and extends the telegraph process [17] and is intimately related to the Bouncy Particle Sampler [22,7].…”

“…[6, Theorem 5]). Suppose U is continuously differentiable, λ refr is continuous, and for λ ± (x) = (±U (x) ∨ 0) + λ refr (x), we have thatlim inf x→∞ λ + (x) > lim sup x→∞ λ − (x) and lim inf x→−∞ λ − (x) > lim sup x→∞ λ + (x).Then there are constants c > 0, α > 0 and a continuous norm-like functionV ∈ L 2 (µ), V > 0 on E, such that for all (x, θ) ∈ E, |P (t)f (x, θ) − π(f )| ≤ c(1 + V (x, θ))e −αt , t ≥ 0,for all measurable f : E → R satisfying |f (x, θ)| ≤ 1 + V (x, θ) on E.…”

On the Convergence Time of Some Non-Reversible Markov Chain Monte Carlo Methods

“…Initialize in X 0 ∼ µ 0 and let X t = x Propose Y ∼ Q(x, · ) y Set X t+1 = y with probability A(x, y) = 1 ∧ R(x, y) where R(x, y) := π(y)Q(y, x) π(x)Q(x, y) if π(x)Q(x, y) = 0 1 otherwise (4) If the proposal is rejected, set X t+1 = x Nevertheless, as the DBC (2) imposes that the joint probabilities Pr(X t ∈ A, X t+1 ∈ B) and Pr(X t ∈ B, X t+1 ∈ A) are equal, reversibility may prevent the Markov chain from roaming efficiently through the state space, especially when π's topology is irregular. This is illustrated by the following example.…”

“…The main reason for their popularity is perhaps the property that a π-reversible Markov chain is necessarily π-invariant. Hence, constructing a Markov chain satisfying (2) avoids further questions regarding the existence of a stationary distribution.…”