  We study the convergence rates of the EM algorithm for learning two-component
mixed linear regression under all regimes of signal-to-noise ratio (SNR). We
resolve a long-standing question that many recent results have attempted to
tackle: we completely characterize the convergence behavior of EM, and show
that the EM algorithm achieves minimax optimal sample complexity under all SNR
regimes. In particular, when the SNR is sufficiently large, the EM updates
converge to the true parameter $\theta^{*}$ at the standard parametric
convergence rate $\mathcal{O}((d/n)^{1/2})$ after $\mathcal{O}(\log(n/d))$
iterations. In the regime where the SNR is above $\mathcal{O}((d/n)^{1/4})$ and
below some constant, the EM iterates converge to a $\mathcal{O}({\rm SNR}^{-1}
(d/n)^{1/2})$ neighborhood of the true parameter, when the number of iterations
is of the order $\mathcal{O}({\rm SNR}^{-2} \log(n/d))$. In the low SNR regime
where the SNR is below $\mathcal{O}((d/n)^{1/4})$, we show that EM converges to
a $\mathcal{O}((d/n)^{1/4})$ neighborhood of the true parameters, after
$\mathcal{O}((n/d)^{1/2})$ iterations. Notably, these results are achieved
under mild conditions of either random initialization or an efficiently
computable local initialization. By providing tight convergence guarantees of
the EM algorithm in middle-to-low SNR regimes, we fill the remaining gap in the
literature, and significantly, reveal that in low SNR, EM changes rate,
matching the $n^{-1/4}$ rate of the MLE, a behavior that previous work had been
unable to show.
