Quotient EM under Misspecification:Tight Local Rates and Finite-Sample Bounds in General Integral Probability Metrics

TL;DR

This paper analyzes the EM algorithm's convergence and finite-sample performance under model misspecification and nonidentifiability, providing sharp local rates and bounds in the context of integral probability metrics.

Contribution

It introduces a quotient space formulation of EM and derives tight local convergence rates and finite-sample bounds under general misspecification and IPM measures.

Findings

Sharp local linear convergence rate governed by spectral radius

Finite-sample bounds via perturbed contraction inequalities

Applicability to general integral probability metrics

Abstract

We study the expectation-maximization (EM) algorithm for general latent-variable models under (i) distributional misspecification and (ii) nonidentifiability induced by a group action. We formulate EM on the quotient parameter space and measure error using an arbitrary integral probability metric (IPM). Our main results give (a) a sharp local linear convergence rate for population EM governed by the spectral radius of the linearization on a local slice, and (b) tight finite-sample bounds for sample EM obtained via perturbed contraction inequalities and generic chaining/entropy control of EM-induced empirical processes.