Stability of Mean-Field Variational Inference

• URL: http://arxiv.org/abs/2506.07856v1
• Date: Mon, 09 Jun 2025 15:21:37 GMT
• Title: Stability of Mean-Field Variational Inference
• Authors: Shunan Sheng, Bohan Wu, Alberto González-Sanz, Marcel Nutz,
• Abstract summary: Mean-field inference (MFVI) is a widely used method for approxing high-dimensional probability distributions by product measures.<n>We show that the MFVI depends differentiably on the target potential and characterize the derivative by a partial differential equation.
• Score: 3.5729687931166136
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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