# Comparing EM with GD in Mixture Models of Two Components

## TL;DR

This paper compares the behavior of EM and gradient descent in two-component mixture models, showing EM generally escapes problematic regions faster and is less prone to trapping in bad local minima than GD.

## Contribution

It provides a detailed analysis of how EM and GD differ in escaping one-cluster regions in Gaussian and Bernoulli mixture models, highlighting EM's advantages.

## Key findings

- EM escapes one-cluster regions exponentially fast in Gaussian mixtures
- GD escapes linearly fast and can get trapped in stable regions
- EM can escape certain bad local minima that trap GD

## Abstract

The expectation-maximization (EM) algorithm has been widely used in minimizing the negative log likelihood (also known as cross entropy) of mixture models. However, little is understood about the goodness of the fixed points it converges to. In this paper, we study the regions where one component is missing in two-component mixture models, which we call one-cluster regions. We analyze the propensity of such regions to trap EM and gradient descent (GD) for mixtures of two Gaussians and mixtures of two Bernoullis. In the case of Gaussian mixtures, EM escapes one-cluster regions exponentially fast, while GD escapes them linearly fast. In the case of mixtures of Bernoullis, we find that there 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 are local 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 bad local optima in mixture models.