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Analysis of Multiple-try Metropolis via Poincaré inequalities

Rocco Caprio, Sam Power, Andi Q. Wang

TL;DR

This paper studies the Multiple-try Metropolis algorithm through Poincaré inequalities by recasting MTM as an auxiliary-variable resampling approximation to an ideal Metropolis–Hastings kernel. It establishes an $L^2$ Dirichlet-form comparison between the ideal kernel $P_ abla$ and MTM $P_n$ that depends on moments of the importance weights, enabling non-asymptotic convergence bounds in Gaussian targets. A key finding is that using globally balanced weights causes the spectral gap to vanish as the number of proposals grows, whereas locally balanced weights yield positive spectral gaps with favorable dimension scaling, including an explicit $d^{-1/2}$ bound in the Gaussian case and an optimal diffusion-like scaling $ abla \, ext{like } d^{-1/4}$. Collectively, these results provide practical guidance on weight choices for MTM and offer a rigorous framework for comparing MTM variants to ideal MH, with potential extensions to broader strongly log-concave targets.

Abstract

We study the Multiple-try Metropolis algorithm using the framework of Poincaré inequalities. We describe the Multiple-try Metropolis as an auxiliary variable implementation of a resampling approximation to an ideal Metropolis--Hastings algorithm. Under suitable moment conditions on the importance weights, we derive explicit Poincaré comparison results between the Multiple-try algorithm and the ideal algorithm. We characterize the spectral gap of the latter, and finally in the Gaussian case prove explicit non-asymptotic convergence bounds for Multiple-try Metropolis by comparison.

Analysis of Multiple-try Metropolis via Poincaré inequalities

TL;DR

This paper studies the Multiple-try Metropolis algorithm through Poincaré inequalities by recasting MTM as an auxiliary-variable resampling approximation to an ideal Metropolis–Hastings kernel. It establishes an Dirichlet-form comparison between the ideal kernel and MTM that depends on moments of the importance weights, enabling non-asymptotic convergence bounds in Gaussian targets. A key finding is that using globally balanced weights causes the spectral gap to vanish as the number of proposals grows, whereas locally balanced weights yield positive spectral gaps with favorable dimension scaling, including an explicit bound in the Gaussian case and an optimal diffusion-like scaling . Collectively, these results provide practical guidance on weight choices for MTM and offer a rigorous framework for comparing MTM variants to ideal MH, with potential extensions to broader strongly log-concave targets.

Abstract

We study the Multiple-try Metropolis algorithm using the framework of Poincaré inequalities. We describe the Multiple-try Metropolis as an auxiliary variable implementation of a resampling approximation to an ideal Metropolis--Hastings algorithm. Under suitable moment conditions on the importance weights, we derive explicit Poincaré comparison results between the Multiple-try algorithm and the ideal algorithm. We characterize the spectral gap of the latter, and finally in the Gaussian case prove explicit non-asymptotic convergence bounds for Multiple-try Metropolis by comparison.

Paper Structure

This paper contains 27 sections, 26 theorems, 92 equations, 1 figure, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Multiple-try Metropolis and limiting kernels
  3. Related work
  4. Comparison of Markov chains
  5. Comparison of Multiple-try chains
  6. Optimal number of proposals
  7. Comparison to ideal algorithms
  8. Comparison between ideal and semi-ideal kernels.
  9. Comparison between the semi-ideal and the Multiple-try Metropolis kernels.
  10. Comparison between Multiple-try and the ideal kernel.
  11. Convergence analysis of the ideal Markov chain
  12. Globally balanced weights
  13. Locally balanced weights
  14. Spectral gap lower bound
  15. Spectral gap upper bound
  16. ...and 12 more sections

Key Result

Theorem 1

For any $f\in\mathrm{L}^2(\pi)$ and $s>0$, there holds the inequality where $\beta_n(s)\to \mathbf{1}\{s\leq 1\}$ as $n \to \infty$. Furthermore, if $M_\varpi(2p)$ and $M_\varpi(-2p)$ are finite for some $p\in[1,\infty)$.

Figures (1)

  • Figure 1: Degenerate behaviour of globally balanced Multiple-try Metropolis: the shadow samples are more likely to be drawn in higher probability areas, resulting in slow convergence.

Theorems & Definitions (54)

  • Theorem 1
  • proof
  • Theorem 2: Spectral gap bounds for $P_\infty$
  • proof
  • Remark 1
  • Theorem 3
  • proof
  • Definition 1: Standard Poincaré inequality; SPI
  • Lemma 4: SPI $\Rightarrow$ geometric convergence
  • proof
  • ...and 44 more