Local Maxima in the Likelihood of Gaussian Mixture Models: Structural Results and Algorithmic Consequences (1609.00978v1)

Abstract: We provide two fundamental results on the population (infinite-sample) likelihood function of Gaussian mixture models with $M \geq 3$ components. Our first main result shows that the population likelihood function has bad local maxima even in the special case of equally-weighted mixtures of well-separated and spherical Gaussians. We prove that the log-likelihood value of these bad local maxima can be arbitrarily worse than that of any global optimum, thereby resolving an open question of Srebro (2007). Our second main result shows that the EM algorithm (or a first-order variant of it) with random initialization will converge to bad critical points with probability at least $1-e^{-\Omega(M)}$. We further establish that a first-order variant of EM will not converge to strict saddle points almost surely, indicating that the poor performance of the first-order method can be attributed to the existence of bad local maxima rather than bad saddle points. Overall, our results highlight the necessity of careful initialization when using the EM algorithm in practice, even when applied in highly favorable settings.

• The paper demonstrates that GMMs with three or more components can have severe local maxima where the likelihood is arbitrarily lower than the global optimum.
• It shows that the EM algorithm, even with random initialization, frequently converges to these poor local optima, challenging common convergence assumptions.
• The study confirms that first-order EM avoids strict saddle points almost surely, emphasizing the importance of improved initialization strategies for robust parameter estimation.

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

This paper presents critical insights into the behavior of the population likelihood function of Gaussian Mixture Models (GMMs) with three or more components. The authors address the existence of problematic local maxima and the implications for algorithms like the Expectation-Maximization (EM) algorithm. The research systematically refutes the conjecture by Srebro (2007), which posited that with a sufficient sample size, local maxima would align with global ones in uniformly weighted, well-separated GMMs.

Key Findings

1. Existence of Bad Local Maxima: The researchers construct examples of GMMs with $M \geq 3$ components where the population likelihood function exhibits local maxima with significantly lower values than the global maxima. In particular, they show the likelihood at these local maxima can be arbitrarily worse than at global optima. This finding highlights the necessity for caution when estimating GMM parameters, as popular belief about the absence of bad local optima in such models does not hold.
2. Algorithmic Consequences for EM and Its Variants: It's shown that the EM algorithm, even under favorable parameter settings and random initialization, converges to suboptimal critical points with high probability. Specifically, it converges to bad local maxima with probability at least $1 - e^{-\Omega(M)}$. This probability behavior underscores the inadequacy of traditional random initialization schemes for ensuring EM's global convergence.
3. Robustness Against Saddle Points in First-order EM: For the first-order variant of EM, it is demonstrated that the algorithm does not converge to strict saddle points almost surely. By leveraging properties studied in dynamical systems theory, the paper confirms that first-order EM converges to critical points, which are almost surely not strict saddles, reinforcing that the primary hindrance stems from bad local maxima rather than saddles.

Implications

This research implies that careful initialization strategies are vital for practical effectiveness when using the EM algorithm to estimate parameters in GMMs. The findings accentuate that robust estimation methods, possibly integrating initial estimates from alternative strategies such as those based on the method of moments, are essential to circumvent convergence to suboptimal points. Such approaches could involve combining EM with algorithmic schemes that offer better coverage of the parameter space or using modifications that enhance convergence to more optimal solutions.

The results presented also have theoretical implications. They suggest that, even in well-separated GMM configurations, there is a complex interplay of likelihood surfaces that challenges traditional algorithmic assumptions. Consequently, this calls for a deeper exploration of non-convex optimization landscapes, especially in high-dimensional settings, to develop a more comprehensive understanding of algorithmic behaviors and possible improvements.

Speculation on Future Developments

The insights from this paper are likely to stimulate further research into improving convergence guarantees of not only EM but other heuristic algorithms employed in probabilistic modeling and clustering tasks. Future work may explore adaptive initialization strategies statistically tailored to different applications or hybrid methods that combine the strengths of multiple estimation approaches. Furthermore, there is potential for advances in understanding the geometrical and topological properties of likelihood surfaces, which could lead to more robust algorithmic solutions in artificial intelligence and machine learning contexts.