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

TL;DR

This paper establishes non-asymptotic convergence rates for EM algorithms in exponential families by framing EM as a mirror descent method, providing invariant analysis in KL divergence with minimal assumptions.

Contribution

It introduces a novel mirror descent perspective for EM, yielding invariant convergence results in KL divergence applicable to exponential families and beyond.

Findings

EM converges at quantifiable rates in KL divergence

Analysis is invariant to parameterization

Applications include local convergence and generalized EM

Abstract

Expectation maximization (EM) is the default algorithm for fitting probabilistic models with missing or latent variables, yet we lack a full understanding of its non-asymptotic convergence properties. Previous works show results 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 make more progress than a gradient step, but the assumptions fail to hold for textbook examples of EM like Gaussian mixtures. In this work we first show that for the common setting of exponential family distributions, viewing EM as a mirror descent algorithm leads to convergence rates in Kullback-Leibler (KL) divergence. Then, we show how the KL divergence is related to first-order stationarity via Bregman divergences. In contrast to previous works, the analysis is invariant to the choice of parametrization and holds with minimal assumptions. We also show applications of these ideas to local linear (and superlinear) convergence rates, generalized EM, and non-exponential family distributions.