METASTABLE MIXING OF MARKOV CHAINS: EFFICIENTLY SAMPLING LOW TEMPERATURE EXPONENTIAL RANDOM GRAPHS

• 成果类型: Article
• 署名作者: Bresler, Guy; Nagaraj, Dheeraj; Nichani, Eshaan
• 署名单位: Massachusetts Institute of Technology (MIT); Alphabet Inc.; Google Incorporated; Princeton University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/23-AAP1971
• 发表日期: 2024
• 页码: 517-554
• 关键词: monte-carlo; glauber dynamics; distributions; time

摘要

In this paper, we consider the problem of sampling from the low temperature exponential random graph model (ERGM). The usual approach is via Markov chain Monte Carlo, but Bhamidi et al. showed that any local Markov chain suffers from an exponentially large mixing time due to metastable states. We instead consider metastable mixing, a notion of approximate mixing relative to the stationary distribution, for which it turns out to suffice to mix only within a collection of metastable states. We show that the Glauber dynamics for the ERGM at any temperature except at a lower-dimensional critical set of parameters when initialized at G(n, p) for the right choice of p has a metastable mixing time of O(n(2) log n) to within total variation distance exp(-Omega(n)).