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We analyze the classical EM algorithm for parameter estimation in thesymmetric two-component Gaussian mixtures in $d$ dimensions. We show that, evenin the absence of any separation between components, provided that the samplesize satisfies $n=\Omega(d \log^3 d)$, the randomly initialized EM algorithmconverges to an estimate in at most $O(\sqrt{n})$ iterations with highprobability, which is at most $O((\frac{d \log^3 n}{n})^{1/4})$ in Euclideandistance from the true parameter and within logarithmic factors of the minimaxrate of $(\frac{d}{n})^{1/4}$. Both the nonparametric statistical rate and thesublinear convergence rate are direct consequences of the zero Fisherinformation in the worst case. Refined pointwise guarantees beyond worst-caseanalysis and convergence to the MLE are also shown under mild conditions.  This improves the previous result of Balakrishnan et al \cite{BWY17} whichrequires strong conditions on both the separation of the components and thequality of the initialization, and that of Daskalakis et al \cite{DTZ17} whichrequires sample splitting and restarting the EM iteration.

Randomly initialized EM algorithm for two-component Gaussian mixture achieves near optimality in $O(\sqrt{n})$ iterations