Theoretical and Computational Guarantees of Mean Field Variational Inference for Community Detection

TL;DR

This paper provides the first theoretical and computational analysis of mean field variational inference for community detection in the stochastic block model, demonstrating linear convergence and optimality within a logarithmic number of iterations.

Contribution

It establishes the convergence rate and optimality of the mean field variational Bayes method for community detection, filling a gap in theoretical understanding.

Findings

Linear convergence rate of the variational algorithm

Convergence to the minimax rate within log n iterations

Optimality results for Gibbs sampling and MLE procedures

Abstract

The mean field variational Bayes method is becoming increasingly popular in statistics and machine learning. Its iterative Coordinate Ascent Variational Inference algorithm has been widely applied to large scale Bayesian inference. See Blei et al. (2017) for a recent comprehensive review. Despite the popularity of the mean field method there exist remarkably little fundamental theoretical justifications. To the best of our knowledge, the iterative algorithm has never been investigated for any high dimensional and complex model. In this paper, we study the mean field method for community detection under the Stochastic Block Model. For an iterative Batch Coordinate Ascent Variational Inference algorithm, we show that it has a linear convergence rate and converges to the minimax rate within $\log n$ iterations. This complements the results of Bickel et al. (2013) which studied the global minimum of the mean field variational Bayes and obtained asymptotic normal estimation of global model parameters. In addition, we obtain similar optimality results for Gibbs sampling and an iterative procedure to calculate maximum likelihood estimation, which can be of independent interest.