mlsanityData to 14 Oct 2026

Stability of Mean-Field Variational Inference

arXiv 2506.07856 ↗First posted 9 Jun 2025math.PR math.FA math.STcited by 3

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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.

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

13 of 83 references.

References · 13

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  5. 2019Model-Free Mean-Field Reinforcement Learning: Mean-Field MDP and Mean-Field Q-Learningcited by 59
  6. 2017Covariances, Robustness, and Variational Bayescited by 40
  7. 2017Using stacking to average Bayesian predictive distributionscited by 39
  8. 2019Multivariate Rank-based Distribution-free Nonparametric Testing using Measure Transportationcited by 24
  9. 2023Algorithms for mean-field variational inference via polyhedral optimization in the Wasserstein spacecited by 20
  10. 2024Convergence of coordinate ascent variational inference for log-concave measures via optimal transportcited by 14
  11. 2021Variational Inference in high-dimensional linear regressioncited by 10
  12. 2023Mean-field variational inference with the TAP free energy: Geometric and statistical properties in linear modelscited by 5
1 more
  1. 2023A Mean Field Approach to Empirical Bayes Estimation in High-dimensional Linear Regressioncited by 4

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2025: 2; 2026: 1.

3 papers.

  1. 2025Multi-modal Bayesian Neural Network Surrogates with Conjugate Last-Layer Estimationcited by 0
  2. 2025Mode Collapse of Mean-Field Variational Inferencecited by 0
  3. 2026Stability of Finite-Batch Particle Mean-Field Variational Inference Beyond Strong Convexitycited by 0