Stability and Instability of Mean Field Variational Inference
Mean-field variational inference (MFVI) is a workhorse of modern Bayesian computation, yet its fundamental properties remain poorly understood. This talk presents a unified geometric analysis addressing three questions: When is MFVI stable? Why does it collapse modes? And how can we fix it? First, we establish that for strongly log-concave targets, the MFVI optimizer depends Lipschitz continuously on the target distribution in the 2-Wasserstein distance - with a dimension-free constant - and characterize its derivative via a PDE. These results yield new guarantees for robust Bayes and empirical Bayes. Second, we provide the first theoretical explanation of mode collapse: when targeting mixtures, MFVI concentrates mass on a single component. We quantify this phenomenon via a notion of δ-separateness and show that collapse depends critically on component geometry. Finally, we propose Rotational Variational Inference (RoVI) , which augments MFVI with a learnable rotation to capture correlation while retaining tractability. Numerical examples illustrate the theory and the benefits of RoVI.
Palabras clave: Mean Field Variational Inference Stability Wasserstein Geometry