Analysis of Multiple-try Metropolis via Poincar'e inequalities

2025-04-25

AI Summary

研究通过Poincaré不等式分析多尝试Metropolis算法，比较其与理想Metropolis-Hastings算法的性能，推导高斯情况下的非渐近收敛界。

Technology Category

Search and Optimization: Non-convex OptimizationReasoning under Uncertainty: Stochastic OptimizationConstraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsWeb Mining and Content Analysis: Models for Web evolutionSecurity and Privacy: Large-scale security measurements

Abstract

We study the Multiple-try Metropolis algorithm using the framework of Poincar'e 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 Poincar'e 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.

Problem

• Analyzing Multiple-try Metropolis via Poincaré inequalities
• Comparing spectral gaps between ideal and Multiple-try algorithms
• Proving non-asymptotic convergence bounds for Gaussian cases

Innovation

• Uses Poincaré inequalities for analysis
• Implements auxiliary variable resampling approximation
• Derives explicit Poincaré comparison results