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

Weihang Xu, Maryam Fazel, Simon Shaolei Du

Keywords: over-parameterization, global convergence, non-convex optimization

TL;DR: We give the first global convergence of gradient EM for over-parameterized Gaussian mixture models when the ground truth is single Gaussian.

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.

Primary Area: Learning theory

Submission Number: 15161