Theoretical guarantees for EM under misspecified Gaussian mixture models

Recent years have witnessed substantial progress in understanding the behavior of EM for mixture models that are correctly specified. Given that model misspecification is common in practice, it is important to understand EM in this more general setting. We provide non-asymptotic guarantees for population and sample-based EM for parameter estimation under a few specific univariate settings of misspecified Gaussian mixture models. Due to misspecification, the EM iterates no longer converge to the true model and instead converge to the projection of the true model over the set of models being searched over. We provide two classes of theoretical guarantees: first, we characterize the bias introduced due to the misspecification; and second, we prove that population EM converges at a geometric rate to the model projection under a suitable initialization condition. This geometric convergence rate for population EM imply a statistical complexity of order $1/\sqrt{n}$ when running EM with $n$ samples. We validate our theoretical findings in different cases via several numerical examples.