# Stability of Mean-Field Variational Inference

arXiv: https://arxiv.org/abs/2506.07856 · HTML: https://mlsanity.com/paper/2506.07856/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 83 references that are in the map.
- First posted on arXiv 9 Jun 2025.

## Builds on (most cited first, 13 of 13)

- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [An Introduction to Variational Autoencoders](https://mlsanity.com/paper/1906.02691/) (2019)
- [High-dimensional Bayesian inference via the Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1605.01559/) (2016)
- [Frequentist Consistency of Variational Bayes](https://mlsanity.com/paper/1705.03439/) (2017)
- [Model-Free Mean-Field Reinforcement Learning: Mean-Field MDP and Mean-Field Q-Learning](https://mlsanity.com/paper/1910.12802/) (2019)
- [Covariances, Robustness, and Variational Bayes](https://mlsanity.com/paper/1709.02536/) (2017)
- [Using stacking to average Bayesian predictive distributions](https://mlsanity.com/paper/1704.02030/) (2017)
- [Multivariate Rank-based Distribution-free Nonparametric Testing using Measure Transportation](https://mlsanity.com/paper/1909.08733/) (2019)
- [Algorithms for mean-field variational inference via polyhedral optimization in the Wasserstein space](https://mlsanity.com/paper/2312.02849/) (2023)
- [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)
- [Mean-field variational inference with the TAP free energy: Geometric and statistical properties in linear models](https://mlsanity.com/paper/2311.08442/) (2023)
- [A Mean Field Approach to Empirical Bayes Estimation in High-dimensional Linear Regression](https://mlsanity.com/paper/2309.16843/) (2023)

## Built on by (most cited first, 3 of 3)

- [Multi-modal Bayesian Neural Network Surrogates with Conjugate Last-Layer Estimation](https://mlsanity.com/paper/2509.21711/) (2025)
- [Mode Collapse of Mean-Field Variational Inference](https://mlsanity.com/paper/2510.17063/) (2025)
- [Stability of Finite-Batch Particle Mean-Field Variational Inference Beyond Strong Convexity](https://mlsanity.com/paper/2608.11486/) (2026)

## Abstract (arXiv)

Mean-field variational inference (MFVI) is a widely used method for approximating high-dimensional probability distributions by product measures. This paper studies the stability properties of the mean-field approximation when the target distribution varies within the class of strongly log-concave measures. We establish dimension-free Lipschitz continuity of the MFVI optimizer with respect to the target distribution, measured in the 2-Wasserstein distance, with Lipschitz constant inversely proportional to the log-concavity parameter. Under additional regularity conditions, we further show that the MFVI optimizer depends differentiably on the target potential and characterize the derivative by a partial differential equation. Methodologically, we follow a novel approach to MFVI via linearized optimal transport: the non-convex MFVI problem is lifted to a convex optimization over transport maps with a fixed base measure, enabling the use of calculus of variations and functional analysis. We discuss several applications of our results to robust Bayesian inference and empirical Bayes, including a quantitative Bernstein--von Mises theorem for MFVI, as well as to distributed stochastic control.