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The expectation-maximization (EM) algorithm has been widely used inminimizing the negative log likelihood (also known as cross entropy) of mixturemodels. However, little is understood about the goodness of the fixed points itconverges to. In this paper, we study the regions where one component ismissing in two-component mixture models, which we call one-cluster regions. Weanalyze the propensity of such regions to trap EM and gradient descent (GD) formixtures of two Gaussians and mixtures of two Bernoullis. In the case ofGaussian mixtures, EM escapes one-cluster regions exponentially fast, while GDescapes them linearly fast. In the case of mixtures of Bernoullis, we find thatthere exist one-cluster regions that are stable for GD and therefore trap GD,but those regions are unstable for EM, allowing EM to escape. Those regions arelocal minima that appear universally in experiments and can be arbitrarily bad.This work implies that EM is less likely than GD to converge to certain badlocal optima in mixture models.

Comparing EM with GD in Mixture Models of Two Components