Is $\sqrt{d}$ Separation Necessary for Gradient EM to Learn Gaussian Mixtures in High Dimensions?
Year: 2026
Type: preprint
Abstract: Learning Gaussian mixture models (GMMs) using the Expectation-Maximization (EM) algorithm and its gradient-based variants is a fundamental problem in machine learning. It is known that randomly initialized (gradient) EM fails to learn multi-component GMMs in the exact-parameterized setting, where the number of components matches that of the ground-truth GMM. Recently, global convergence of gradient EM has been established in the over-parameterized setting, where more components are used, provided that the ground-truth components are well separated. In particular, the minimum separation between ground-truth components is required to scale as $Ω(\sqrt{d})$, where $d$ is the dimension. In this paper, we show that this dimensional dependence is unavoidable in high-dimensional settings. Specifically, we consider a hybrid EM algorithm that uses standard EM updates for the mixing weights and gradient EM updates for the component means. For any $ε> 0$, we prove that when the dimension is sufficiently large, in the worst case a separation of order $Ω(d^{0.5-ε})$ is insufficient to guarantee global convergence of population gradient EM in sub-exponential time under random initialization, even in the over-parameterized regime. Our result establishes an almost optimal worst-case lower bound on the ground-truth separation required for learning Gaussian mixtures via gradient EM in high dimensions.
Source: arXiv (Cornell University)
Institutions: University of California, Berkeley, University of Washington
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Funder: National Science Foundation
Awards: Collaborative Research: AF: Medium: Machine Learning Markets: Dynamics, Competition, and Interventions, CAREER: Toward a Foundation of Over-Parameterization, TRIPODS: Institute for Foundations of Data Science, Collaborative Research: SCALE MoDL: Adaptivity of Deep Neural Networks, Collaborative Research: CIF: Medium: MoDL:Toward a Mathematical Foundation of Deep Reinforcement Learning, AI Institute for Societal Decision Making (AI-SDM)
OpenAlex ID: W7220909375