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

TL;DR

This paper provides a theoretical analysis of the EM algorithm's convergence behavior and statistical accuracy in overspecified mixed linear regression, highlighting differences based on initial weight balance and extending to low SNR scenarios.

Contribution

It characterizes the convergence rates and statistical accuracy of EM in overspecified 2MLR, revealing how initial weight balance affects efficiency and accuracy, and extends analysis to low SNR regimes.

Findings

Linear convergence with unbalanced weights at population level

Sublinear convergence with balanced weights at population level

Statistical accuracy depends on weight balance and sample size

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.