# Convergence Rates of Variational Posterior Distributions

arXiv: https://arxiv.org/abs/1712.02519 · HTML: https://mlsanity.com/paper/1712.02519/

## Key facts

- Cited by 45 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Dec 2017.

## Built on by (most cited first, 30 of 45)

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- [Structured Optimal Variational Inference for Dynamic Latent Space Models](https://mlsanity.com/paper/2209.15117/) (2022)
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- [Pointwise uncertainty quantification for sparse variational Gaussian process regression with a Brownian motion prior](https://mlsanity.com/paper/2310.00097/) (2023)
- [Convergence of Statistical Estimators via Mutual Information Bounds](https://mlsanity.com/paper/2412.18539/) (2024)
- [Globally Convergent Variational Inference](https://mlsanity.com/paper/2501.08201/) (2025)
- [Robust and Scalable Variational Bayes](https://mlsanity.com/paper/2504.12528/) (2025)
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## Abstract (arXiv)

We study convergence rates of variational posterior distributions for nonparametric and high-dimensional inference. We formulate general conditions on prior, likelihood, and variational class that characterize the convergence rates. Under similar "prior mass and testing" conditions considered in the literature, the rate is found to be the sum of two terms. The first term stands for the convergence rate of the true posterior distribution, and the second term is contributed by the variational approximation error. For a class of priors that admit the structure of a mixture of product measures, we propose a novel prior mass condition, under which the variational approximation error of the mean-field class is dominated by convergence rate of the true posterior. We demonstrate the applicability of our general results for various models, prior distributions and variational classes by deriving convergence rates of the corresponding variational posteriors.