Abstract

Recent years have witnessed substantial progress in understanding the behavior of EM for mixture models that are correctly specified. Given that model misspecificationiscommoninpractice,itisimportanttounderstandEMinthismore general setting. We provide non-asymptotic guarantees for the population and sample-based EM algorithms when used to estimate parameters of certain misspecified Gaussian mixture models. Due to mis-specification, the EM iterates no longer converge to the true model and instead converge to the projection of the truemodelontothefittedmodelclass. Weprovidetwoclassesoftheoreticalguarantees: (a) a characterization of the bias introduced due to the mis-specification; and (b) guarantees of geometric convergence of the population EM to the model projectiongivenasuitableinitialization. ThisgeometricconvergencerateforpopulationEMimpliesthattheEMalgorithmbasedon n samplesconvergestoanestimatewith accuracy. Wevalidateourtheoreticalfindingsindifferentcases via several numerical examples.