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.

中文翻译：

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用于社区检测的平均场变分推理的理论和计算保证

平均场变分贝叶斯方法在统计学和机器学习中越来越流行。其迭代坐标上升变分推理算法已广泛应用于大规模贝叶斯推理。见布莱等人。(2017) 进行了最近的全面审查。尽管平均场方法很流行，但几乎没有基本的理论依据。据我们所知，迭代算法从未被研究过用于任何高维和复杂模型。在本文中，我们研究了随机块模型下社区检测的平均场方法。对于迭代批坐标上升变分推理算法，我们表明它具有线性收敛率，并在 $\log n$ 次迭代内收敛到极大极小值。这补充了 Bickel 等人的结果。(2013) 研究了平均场变分贝叶斯的全局最小值，并获得了全局模型参数的渐近正态估计。此外，我们获得了类似的 Gibbs 采样的最优性结果和一个迭代过程来计算最大似然估计，这可能是独立的兴趣。