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We provide two fundamental results on the population (infinite-sample)likelihood function of Gaussian mixture models with $M \geq 3$ components. Ourfirst main result shows that the population likelihood function has bad localmaxima even in the special case of equally-weighted mixtures of well-separatedand spherical Gaussians. We prove that the log-likelihood value of these badlocal maxima can be arbitrarily worse than that of any global optimum, therebyresolving an open question of Srebro (2007). Our second main result shows thatthe EM algorithm (or a first-order variant of it) with random initializationwill converge to bad critical points with probability at least$1-e^{-\Omega(M)}$. We further establish that a first-order variant of EM willnot converge to strict saddle points almost surely, indicating that the poorperformance of the first-order method can be attributed to the existence of badlocal maxima rather than bad saddle points. Overall, our results highlight thenecessity of careful initialization when using the EM algorithm in practice,even when applied in highly favorable settings.

Local Maxima in the Likelihood of Gaussian Mixture Models: Structural Results and Algorithmic Consequences