Sharp Analysis of Expectation-Maximization for Weakly Identifiable Models
This work addresses a theoretical bottleneck in statistical learning for researchers dealing with non-standard mixture models, showing that EM can be inefficient in such settings, which is incremental but provides rigorous analysis.
The paper tackled the problem of slow convergence in Expectation-Maximization (EM) algorithms for weakly identifiable mixture models, proving that EM converges in order n^(3/4) steps and yields parameter estimates with errors of order n^(-1/8) for location and n^(-1/4) for scale in univariate Gaussian mixtures, with similar slow rates demonstrated in multivariate cases.
We study a class of weakly identifiable location-scale 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 Expectation-Maximization (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 epoch-based 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 the identity.