  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 $n=\Omega(d \log^3 d)$, the randomly initialized EM algorithm
converges to an estimate in at most $O(\sqrt{n})$ iterations with high
probability, which is at most $O((\frac{d \log^3 n}{n})^{1/4})$ in Euclidean
distance from the true parameter and within logarithmic factors of the minimax
rate of $(\frac{d}{n})^{1/4}$. 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 \cite{BWY17} which
requires strong conditions on both the separation of the components and the
quality of the initialization, and that of Daskalakis et al \cite{DTZ17} which
requires sample splitting and restarting the EM iteration.
