Comparing EM with GD in Mixture Models of Two Components
This work addresses the problem of algorithm selection for practitioners in machine learning by revealing EM's superior robustness in avoiding poor solutions in mixture models, though it is incremental as it builds on existing theoretical analyses.
The paper analyzes the convergence behavior of the Expectation-Maximization (EM) algorithm and gradient descent (GD) in two-component mixture models, finding that EM escapes problematic one-cluster regions exponentially fast for Gaussian mixtures and avoids stable traps for Bernoulli mixtures, while GD escapes linearly or gets trapped, implying EM is less prone to bad local optima.
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.