SOLIDARITY OF GIBBS SAMPLERS: THE SPECTRAL GAP

• 成果类型: Article
• 署名作者: Chlebicka, Iwona; Latuszynski, Krzysztof; Miasojedow, Blaej
• 署名单位: University of Warsaw; University of Warwick
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/24-AAP2111
• 发表日期: 2025
• 页码: 142-157
• 关键词: geometric ergodicity; markov-chains; stochastic relaxation; glauber dynamics; inequalities; CONVERGENCE; parallel; bounds

摘要

Gibbs samplers are preeminent Markov chain Monte Carlo algorithms used in computational physics and statistical computing. Yet, their most fun- damental properties, such as relations between convergence characteristics of their various versions, are not well understood. In this paper we prove the solidarity principle of the spectral gap for the Gibbs sampler: if any of the random scan or d! deterministic scans has a spec- tral gap then all of them have. Our methods rely on geometric interpretation of the Gibbs samplers as alternating projection algorithms and analysis of the rate of convergence in the von Neumann-Halperin method of cyclic al- ternating projections. As a byproduct of our analysis, we also establish that deterministic scan Gibbs, despite being nonreversible, share many robustness properties with reversible chains, including exponential inequalities and cen- tral limit theorems under the same conditions. In addition, we provide a quantitative result: if the spectral gap of the random scan Gibbs sampler decays with dimension at a polynomial rate beta , then the rate is no worse than 2 beta + 2 for any deterministic scan.