Toward Global Convergence of Gradient EM for Over-Parameterized Gaussian Mixture Models

2024-06-29

Neural Information Processing Systems

AI Summary

This work addresses the global convergence of gradient EM for overparameterized Gaussian mixture models (GMMs) with $n > 2$ components fitted to data generated from a single Gaussian—where existing analyses suffer from sublinear convergence rates, non-monotonic iterates, and lack of global guarantees. Method: We develop the first nonconvex convergence analysis framework grounded in the likelihood function, explicitly accounting for the geometry of the overparameterized regime. Contribution/Results: We establish the first rigorous global convergence guarantee for gradient EM in *any* $n > 2$ overparameterized GMM setting, with a convergence rate of $O(1/sqrt{t})$. Furthermore, we characterize a class of “bad local regions” that induce exponential slowdown, revealing their geometric origin in parameter space. This work fills a long-standing gap by providing the first global convergence theory—including explicit rate and identification of fundamental optimization barriers—for overparameterized GMMs beyond the $n=2$ case.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Learning with Manifolds
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: Large Language Models (LLM) for user modeling and recommendation
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We study the gradient Expectation-Maximization (EM) algorithm for Gaussian Mixture Models (GMM) in the over-parameterized setting, where a general GMM with $n>1$ components learns from data that are generated by a single ground truth Gaussian distribution. While results for the special case of 2-Gaussian mixtures are well-known, a general global convergence analysis for arbitrary $n$ remains unresolved and faces several new technical barriers since the convergence becomes sub-linear and non-monotonic. To address these challenges, we construct a novel likelihood-based convergence analysis framework and rigorously prove that gradient EM converges globally with a sublinear rate $O(1/sqrt{t})$. This is the first global convergence result for Gaussian mixtures with more than $2$ components. The sublinear convergence rate is due to the algorithmic nature of learning over-parameterized GMM with gradient EM. We also identify a new emerging technical challenge for learning general over-parameterized GMM: the existence of bad local regions that can trap gradient EM for an exponential number of steps.

Problem

• Global convergence analysis for over-parameterized GMM with gradient EM
• Sublinear and non-monotonic convergence challenges for arbitrary n components
• Identifying bad local regions trapping gradient EM in over-parameterized GMM

Innovation

• Gradient EM for over-parameterized GMM
• Novel likelihood-based convergence framework
• Sublinear global convergence rate O(1/sqrt(t))