Theory and computation for structured variational inference

2025-11-13

AI Summary

This work addresses fundamental challenges in structured variational inference—namely, the lack of existence, uniqueness, and self-consistency guarantees for approximate posteriors, as well as the absence of quantifiable error bounds—focusing specifically on star-structured models. We propose the first self-consistency theory and error analysis framework for star-structured variational inference, moving beyond the restrictive mean-field assumption. Leveraging optimal transport theory, we design a provably convergent gradient-based algorithm and establish novel stability results for star-decomposable transport maps. Our approach enables the first quantitative error control over posterior approximations for Gaussian measures and hierarchical Bayesian models—including generalized linear models with location priors and debiased spike-and-slab models. Empirical evaluation demonstrates substantial improvements in both approximation accuracy and robustness.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Structured variational inference constitutes a core methodology in modern statistical applications. Unlike mean-field variational inference, the approximate posterior is assumed to have interdependent structure. We consider the natural setting of star-structured variational inference, where a root variable impacts all the other ones. We prove the first results for existence, uniqueness, and self-consistency of the variational approximation. In turn, we derive quantitative approximation error bounds for the variational approximation to the posterior, extending prior work from the mean-field setting to the star-structured setting. We also develop a gradient-based algorithm with provable guarantees for computing the variational approximation using ideas from optimal transport theory. We explore the implications of our results for Gaussian measures and hierarchical Bayesian models, including generalized linear models with location family priors and spike-and-slab priors with one-dimensional debiasing. As a by-product of our analysis, we develop new stability results for star-separable transport maps which might be of independent interest.

Problem

• Extends variational inference theory from mean-field to structured dependencies
• Develops provable algorithms for star-structured variational approximations
• Provides error bounds for posterior approximations in hierarchical models

Innovation

• Star-structured variational inference with interdependent variables
• Gradient-based algorithm with optimal transport guarantees
• Error bounds extended from mean-field to star-structured setting