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

TL;DR

This paper proves the first global convergence guarantee for gradient EM algorithms applied to over-parameterized Gaussian Mixture Models with more than two components, revealing sublinear convergence behavior.

Contribution

It introduces a novel likelihood-based analysis framework and establishes the first global convergence result for arbitrary GMMs with over-parameterization.

Findings

Gradient EM converges globally with a sublinear rate of O(1/√t)

First proof of global convergence for GMMs with more than 2 components

Identifies exponential trapping regions in over-parameterized GMMs

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.