@inproceedings{
xu2024toward,
title={Toward Global Convergence of Gradient {EM} for Over-Paramterized Gaussian Mixture Models},
author={Weihang Xu and Maryam Fazel and Simon Shaolei Du},
booktitle={The Thirty-eighth Annual Conference on Neural Information Processing Systems},
year={2024},
url={https://openreview.net/forum?id=zv9gYC3xgF}
}
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.
Toward Global Convergence of Gradient EM for Over-Paramterized Gaussian Mixture Models
Weihang Xu Maryam Fazel Simon Shaolei Du