Sharp Analysis of ExpectationMaximization for Weakly Identifiable Models
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Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics, PMLR 108:18661876, 2020.
Abstract
We study a class of weakly identifiable locationscale mixture models for which the maximum likelihood estimates based on $n$ i.i.d. samples are known to have lower accuracy than the classical $n^{ \frac{1}{2}}$ error. We investigate whether the ExpectationMaximization (EM) algorithm also converges slowly for these models. We provide a rigorous characterization of EM for fitting a weakly identifiable Gaussian mixture in a univariate setting where we prove that the EM algorithm converges in order $n^{\frac{3}{4}}$ steps and returns estimates that are at a Euclidean distance of order ${ n^{ \frac{1}{8}}}$ and ${ n^{\frac{1} {4}}}$ from the true location and scale parameter respectively. Establishing the slow rates in the univariate setting requires a novel localization argument with two stages, with each stage involving an epochbased argument applied to a different surrogate EM operator at the population level. We demonstrate several multivariate ($d \geq 2$) examples that exhibit the same slow rates as the univariate case. We also prove slow statistical rates in higher dimensions in a special case, when the fitted covariance is constrained to be a multiple of identity.
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