The EM Algorithm gives Sample-Optimality for Learning Mixtures of Well-Separated Gaussians

Abstract

We consider the problem of spherical Gaussian Mixture models with $k\ge 3$ components when the components are well separated. A fundamental previous result established that separation of $\Omega (\sqrt{\log k})$ is necessary and sufficient for identifiability of the parameters with polynomial sample complexity (Regev and Vijayaraghavan, 2017). In the same context, we show that $\tilde{O}(kd/{\epsilon }^2)$ samples suffice for any $\epsilon \lesssim 1/k$, closing the gap from polynomial to linear, and thus giving the first optimal sample upper bound for the parameter estimation of well-separated Gaussian mixtures. We accomplish this by proving a new result for the Expectation-Maximization (EM) algorithm: we show that EM converges locally, under separation $\Omega (\sqrt{\log k})$. The previous best-known guarantee required $\Omega (\sqrt{k})$ separation (Yan, et al., 2017). Unlike prior work, our results do not assume or use prior knowledge of the (potentially different) mixing weights or variances of the Gaussian components. Furthermore, our results show that the finite-sample error of EM does not depend on non-universal quantities such as pairwise distances between means of Gaussian components.

arXiv commentAccepted to COLT 2020; Title changed

References in corpus (5)

• Statistical and Computational Guarantees of Lloyd's Algorithm and its Variants
• Global analysis of Expectation Maximization for mixtures of two Gaussians
• Local Maxima in the Likelihood of Gaussian Mixture Models: Structural Results and Algorithmic Consequences
• Near-optimal-sample estimators for spherical Gaussian mixtures
• Statistical Convergence of the EM Algorithm on Gaussian Mixture Models

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• Clustering a Mixture of Gaussians with Unknown Covariance
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• On sampling from data with duplicate records