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Expectation maximization (EM) is the default algorithm for fittingprobabilistic models with missing or latent variables, yet we lack a fullunderstanding of its non-asymptotic convergence properties. Previous works showresults along the lines of "EM converges at least as fast as gradient descent"by assuming the conditions for the convergence of gradient descent apply to EM.This approach is not only loose, in that it does not capture that EM can makemore progress than a gradient step, but the assumptions fail to hold fortextbook examples of EM like Gaussian mixtures. In this work we first show thatfor the common setting of exponential family distributions, viewing EM as amirror descent algorithm leads to convergence rates in Kullback-Leibler (KL)divergence. Then, we show how the KL divergence is related to first-orderstationarity via Bregman divergences. In contrast to previous works, theanalysis is invariant to the choice of parametrization and holds with minimalassumptions. We also show applications of these ideas to local linear (andsuperlinear) convergence rates, generalized EM, and non-exponential familydistributions.

Homeomorphic-Invariance of EM: Non-Asymptotic Convergence in KL Divergence for Exponential Families via Mirror Descent