The EM Algorithm is Adaptively-Optimal for Unbalanced Symmetric Gaussian Mixtures

Abstract

This paper studies the problem of estimating the means $\pm\theta_{*}\in\mathbb{R}^{d}$ of a symmetric two-component Gaussian mixture $\delta_{*}\cdot N(\theta_{*},I)+(1-\delta_{*})\cdot N(-\theta_{*},I)$ where the weights $\delta_{*}$ and $1-\delta_{*}$ are unequal. Assuming that $\delta_{*}$ is known, we show that the population version of the EM algorithm globally converges if the initial estimate has non-negative inner product with the mean of the larger weight component. This can be achieved by the trivial initialization $\theta_{0}=0$. For the empirical iteration based on $n$ samples, we show that when initialized at $\theta_{0}=0$, the EM algorithm adaptively achieves the minimax error rate $\tilde{O}\Big(\min\Big\{\frac{1}{(1-2\delta_{*})}\sqrt{\frac{d}{n}},\frac{1}{\|\theta_{*}\|}\sqrt{\frac{d}{n}},\left(\frac{d}{n}\right)^{1/4}\Big\}\Big)$ in no more than $O\Big(\frac{1}{\|\theta_{*}\|(1-2\delta_{*})}\Big)$ iterations (with high probability). We also consider the EM iteration for estimating the weight $\delta_{*}$, assuming a fixed mean $\theta$ (which is possibly mismatched to $\theta_{*}$). For the empirical iteration of $n$ samples, we show that the minimax error rate $\tilde{O}\Big(\frac{1}{\|\theta_{*}\|}\sqrt{\frac{d}{n}}\Big)$ is achieved in no more than $O\Big(\frac{1}{\|\theta_{*}\|^{2}}\Big)$ iterations. These results robustify and complement recent results of Wu and Zhou obtained for the equal weights case $\delta_{*}=1/2$.