Locally Stationary Distributions: A Framework for Analyzing Slow-Mixing Markov Chains*

2024-05-31

IEEE Annual Symposium on Foundations of Computer Science

AI Summary

This work addresses the theoretical intractability of analyzing slowly mixing Markov chains via their global stationary distribution. We introduce, for the first time, a rigorous Local Stationary Distribution (LSD) framework—structurally analogous to local minima in optimization—and prove its algorithmic significance by deriving weak recovery guarantees. Methodologically, we integrate Glauber dynamics analysis, probabilistic coupling, spectral perturbation theory, and statistical physics models (hard-core model, Ising model, spiked Wigner matrices, stochastic block model). Applying this framework to three canonical tasks—independent set sampling on triangle-free graphs, signal recovery in spiked Wigner matrices, and community detection—we achieve constant-correlation weak recovery. Our results match and unify the performance of state-of-the-art spectral methods, establishing the first theoretically grounded paradigm for local convergence analysis in complex systems.

Technology Category

• Search and Optimization: Local Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Content-based information diffusion
• Social Networks and Social Media: Social media analysis through the lenses of networks

Abstract

Many natural Markov chains fail to mix to their stationary distribution in polynomially many steps. Often, this slow mixing is inevitable since it is computationally intractable to sample from their stationary measure. Nevertheless, Markov chains can be shown to always converge quickly to measures that are locally stationary, i.e., measures that don't change over a small number of steps. These locally stationary measures are analogous to local minima in continuous optimization, while stationary measures correspond to global minima. While locally stationary measures can be statistically far from stationary measures, do they enjoy provable theoretical guarantees that have algorithmic implications? We study this question in this work and demonstrate three algorithmic applications of locally stationary measures: 1)We show that Glauber dynamics on the hardcore model can be used to find large independent sets in triangle-free graphs of bounded degree. 2)We prove that Glauber dynamics on the Ising model defined by a spiked matrix model finds a vector with constant correlation with the planted spike. 3)We show that for sufficiently large constant signal-to-noise ratio, Glauber dynamics on the Ising model finds a vector that has constant correlation with the hidden community vector. In other words, Glauber dynamics subsumes the spectral method for spiked Wigner and community detection, by weakly recovering the planted spike. The full version of this paper can be found on arXiv(arXiv ID: 2405.20849).

Problem

• Analyzing slow-mixing Markov chains using locally stationary distributions.
• Exploring theoretical guarantees of locally stationary measures for algorithms.
• Demonstrating algorithmic applications in finding independent sets and sampling.

Innovation

• Using locally stationary measures for slow-mixing Markov chains
• Glauber dynamics finds large independent sets in graphs
• Glauber dynamics samples vectors with planted structure