Characterizing Evolution in Expectation-Maximization Estimates for Overspecified Mixed Linear Regression

Abstract

Mixture models have attracted significant attention due to practical effectiveness and comprehensive theoretical foundations. A persisting challenge is model misspecification, which occurs when the model to be fitted has more mixture components than those in the data distribution. In this paper, we develop a theoretical understanding of the Expectation-Maximization (EM) algorithm's behavior in the context of targeted model misspecification for overspecified two-component Mixed Linear Regression (2MLR) with unknown $d$-dimensional regression parameters and mixing weights. In Theorem 5.1 at the population level, with an unbalanced initial guess for mixing weights, we establish linear convergence of regression parameters in $O(\log(1/\epsilon))$ steps. Conversely, with a balanced initial guess for mixing weights, we observe sublinear convergence in $O(\epsilon^{-2})$ steps to achieve the $\epsilon$-accuracy at Euclidean distance. In Theorem 6.1 at the finite-sample level, for mixtures with sufficiently unbalanced fixed mixing weights, we demonstrate a statistical accuracy of $O((d/n)^{1/2})$, whereas for those with sufficiently balanced fixed mixing weights, the accuracy is $O((d/n)^{1/4})$ given $n$ data samples. Furthermore, we underscore the connection between our population level and finite-sample level results: by setting the desired final accuracy $\epsilon$ in Theorem 5.1 to match that in Theorem 6.1 at the finite-sample level, namely letting $\epsilon = O((d/n)^{1/2})$ for sufficiently unbalanced fixed mixing weights and $\epsilon = O((d/n)^{1/4})$ for sufficiently balanced fixed mixing weights, we intuitively derive iteration complexity bounds $O(\log (1/\epsilon))=O(\log (n/d))$ and $O(\epsilon^{-2})=O((n/d)^{1/2})$ at the finite-sample level for sufficiently unbalanced and balanced initial mixing weights. We further extend our analysis in overspecified setting to low SNR regime.