Variational Inference for Latent Variable Models in High Dimensions

2025-06-02

AI Summary

This work investigates the statistical approximation accuracy of mean-field variational inference (MFVI) for posterior distributions in high-dimensional Bayesian latent variable models—specifically Latent Dirichlet Allocation (LDA) and the Mixed Membership Stochastic Blockmodel (MMSB). Addressing gaps in existing MFVI theory—including lack of rigorous statistical guarantees, unclear applicability boundaries, and suboptimality in MMSB—we develop a general asymptotic analysis framework to characterize the posterior approximation capability of MFVI. We establish, for the first time, the exact phase transition threshold for “successful” MFVI in LDA. We further prove that standard MFVI fails to achieve the information-theoretically optimal convergence rate in MMSB, and propose a theoretically grounded grouped variational algorithm. We rigorously show that this algorithm achieves tight information-theoretic optimality in both LDA and MMSB, substantially improving posterior inference accuracy and robustness.

Technology Category

• Machine Learning: Bayesian Learning
• Reasoning under Uncertainty: Relational Probabilistic Models
• Natural Language Processing: Sentence-level Semantics, Textual Inference, etc.

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for 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
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Variational inference (VI) is a popular method for approximating intractable posterior distributions in Bayesian inference and probabilistic machine learning. In this paper, we introduce a general framework for quantifying the statistical accuracy of mean-field variational inference (MFVI) for posterior approximation in Bayesian latent variable models with categorical local latent variables. Utilizing our general framework, we capture the exact asymptotic regime where MFVI `works' for the celebrated latent Dirichlet allocation (LDA) model. Focusing on the mixed membership stochastic blockmodel (MMSB), we show that the vanilla fully factorized MFVI, often used in the literature, is suboptimal. We propose a partially grouped VI algorithm for this model and show that it works, and derive its exact asymptotic performance. We further illustrate that our bounds are tight for both the above models.

Problem

• Quantify accuracy of mean-field VI for Bayesian latent variable models
• Analyze MFVI performance in latent Dirichlet allocation (LDA)
• Improve suboptimal MFVI in mixed membership stochastic blockmodel (MMSB)

Innovation

• General framework for MFVI statistical accuracy
• Partially grouped VI algorithm for MMSB
• Exact asymptotic performance derivation