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

arXiv: https://arxiv.org/abs/1710.11268 · HTML: https://mlsanity.com/paper/1710.11268/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Oct 2017.

## Built on by (most cited first, 19 of 19)

- [Algorithms for mean-field variational inference via polyhedral optimization in the Wasserstein space](https://mlsanity.com/paper/2312.02849/) (2023)
- [Mean-field Variational Inference via Wasserstein Gradient Flow](https://mlsanity.com/paper/2207.08074/) (2022)
- [On the Convergence of Coordinate Ascent Variational Inference](https://mlsanity.com/paper/2306.01122/) (2023)
- [Local convexity of the TAP free energy and AMP convergence for Z2-synchronization](https://mlsanity.com/paper/2106.11428/) (2021)
- [Convergence of coordinate ascent variational inference for log-concave measures via optimal transport](https://mlsanity.com/paper/2404.08792/) (2024)
- [Variational Inference in high-dimensional linear regression](https://mlsanity.com/paper/2104.12232/) (2021)
- [Statistical and Computational Trade-offs in Variational Inference: A Case Study in Inferential Model Selection](https://mlsanity.com/paper/2207.11208/) (2022)
- [Particle Mean Field Variational Bayes](https://mlsanity.com/paper/2303.13930/) (2023)
- [A Mean Field Approach to Empirical Bayes Estimation in High-dimensional Linear Regression](https://mlsanity.com/paper/2309.16843/) (2023)
- [Bayesian Model Selection via Mean-Field Variational Approximation](https://mlsanity.com/paper/2312.10607/) (2023)
- [Exponential Convergence of CAVI for Bayesian PCA](https://mlsanity.com/paper/2505.16145/) (2025)
- [Nearly Dimension-Independent Convergence of Mean-Field Black-Box Variational Inference](https://mlsanity.com/paper/2505.21721/) (2025)
- [Stability of Sequential and Parallel Coordinate Ascent Variational Inference](https://mlsanity.com/paper/2603.20929/) (2026)
- [Variational Inference: Posterior Threshold Improves Network Clustering Accuracy in Sparse Regimes](https://mlsanity.com/paper/2301.04771/) (2023)
- [Revisiting Instance-Optimal Cluster Recovery in the Labeled Stochastic Block Model](https://mlsanity.com/paper/2306.12968/) (2023)
- [The surrogate Gibbs-posterior of a corrected stochastic MALA: Towards uncertainty quantification for neural networks](https://mlsanity.com/paper/2310.09335/) (2023)
- [Selecting the Number of Communities for Weighted Degree-Corrected Stochastic Block Models](https://mlsanity.com/paper/2406.05340/) (2024)
- [Model-free algorithms for fast node clustering in SBM type graphs and application to social role inference in animals](https://mlsanity.com/paper/2509.15989/) (2025)
- [Multi-modal Bayesian Neural Network Surrogates with Conjugate Last-Layer Estimation](https://mlsanity.com/paper/2509.21711/) (2025)

## Abstract (arXiv)

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.