  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.
