Randomly initialized EM algorithm for two-component Gaussian mixture achieves near optimality in O(√(n)) iterations
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We analyze the classical EM algorithm for parameter estimation in the symmetric two-component Gaussian mixtures in d dimensions. We show that, even in the absence of any separation between components, provided that the sample size satisfies n=Ω(d log^3 d), the randomly initialized EM algorithm converges to an estimate in at most O(√(n)) iterations with high probability, which is at most O((d log^3 n/n)^1/4) in Euclidean distance from the true parameter and within logarithmic factors of the minimax rate of (d/n)^1/4. Both the nonparametric statistical rate and the sublinear convergence rate are direct consequences of the zero Fisher information in the worst case. Refined pointwise guarantees beyond worst-case analysis and convergence to the MLE are also shown under mild conditions. This improves the previous result of Balakrishnan et al <cit.> which requires strong conditions on both the separation of the components and the quality of the initialization, and that of Daskalakis et al <cit.> which requires sample splitting and restarting the EM iteration.
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