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

TL;DR

This paper proves that the classical EM algorithm, when randomly initialized, converges efficiently to near-optimal estimates in symmetric two-component Gaussian mixtures without component separation, given sufficient sample size.

Contribution

It establishes near-optimal convergence rates for EM in Gaussian mixtures with minimal assumptions, improving upon prior results that required strong separation or initialization conditions.

Findings

EM converges in $O(\sqrt{n})$ iterations with high probability.

Estimates are within logarithmic factors of the minimax rate.

Results hold even with zero Fisher information in the worst case.

Abstract

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.