Theoretical guarantees for lifted samplers

TL;DR

This work establishes a general, Tierney-based framework unifying generalized Metropolis–Hastings and lifted samplers across arbitrary state spaces. It derives a sharp Peskun-type bound showing the asymptotic variance of lifted estimators cannot exceed twice that of the generalized MH estimators, with an intermediate reversible counterpart lying between; the bound is demonstrated to be essentially optimal. The authors provide concrete illustrations on both partially ordered finite spaces and totally ordered continuous spaces, highlighting when lifting yields improvements and how the choice of directional splitting impacts performance. The results offer a principled guide for deploying lifting in challenging Bayesian settings, indicating that, at worst, lifting is not harmful and often yields substantial gains without extra computational cost.

Abstract

Lifted samplers form a class of Markov chain Monte Carlo methods which has drawn a lot attention in recent years due to superior performance in challenging Bayesian applications. A canonical example of such sampler is the one that is derived from a random walk Metropolis algorithm for a totally-ordered state space such as the integers or the real numbers. The lifted sampler is derived by splitting into two the proposal distribution: one part in the increasing direction, and the other part in the decreasing direction. It keeps following a direction, until a rejection occurs, upon which it flips the direction. In terms of asymptotic variances, it outperforms the random walk Metropolis algorithm, regardless of the target distribution, at no additional computational cost. Other studies show, however, that beyond this simple case, lifted samplers do not always outperform their Metropolis counterparts. In this paper, we leverage the celebrated work of Tierney (1998) to provide an analysis in a general framework encompassing a broad class of lifted samplers. Our finding is that, essentially, the asymptotic variances cannot increase by a factor of more than 2, regardless of the target distribution, the way the directions are induced, and the type of algorithm from which the lifted sampler is derived (be it a Metropolis--Hastings algorithm, a reversible jump algorithm, etc.). This result indicates that, while there is potentially a lot to gain from lifting a sampler, there is not much to lose.

Figures (3)

• Figure 1: Asymptotic variance of the MCMC estimator of the standardized version of the mapping $x \mapsto \sum_{i=1}^n x_i$ for the MH algorithm, the lifted sampler and the reversible counterpart of the latter, all using the Barker proposal distribution when: (a) $\eta$ increases from 10 to 50 and the other parameters are kept fixed ($\mu = 1$ and $\lambda = 0.5$); (b) $\mu$ increases from 1 to 3.5 and the other parameters are kept fixed ($\eta = 10$ and $\lambda = 0.5$); in (b), the upper bound provided in \ref{['thm2']} is also presented.
• Figure 2: Trace plots for the MH algorithm on the left panel and the lifted sampler on the right panel, both with the Barker proposal distribution.
• Figure 3: Illustration of the MH chain (left) and its lifted counterpart (right); the colour of the arrows indicates the level of the transition probabilities (darker is higher).

Theorems & Definitions (18)

• Proposition 1: tierney1998note
• Theorem 1: tierney1998note
• Lemma 1
• Lemma 2
• proof
• Proposition 2
• Proposition 3
• Theorem 2
• proof
• proof : Proof of \ref{['lemma:varphi']}