Statistical guarantees for the EM algorithm: From population to sample-based analysis (1408.2156v1)

Abstract: We develop a general framework for proving rigorous guarantees on the performance of the EM algorithm and a variant known as gradient EM. Our analysis is divided into two parts: a treatment of these algorithms at the population level (in the limit of infinite data), followed by results that apply to updates based on a finite set of samples. First, we characterize the domain of attraction of any global maximizer of the population likelihood. This characterization is based on a novel view of the EM updates as a perturbed form of likelihood ascent, or in parallel, of the gradient EM updates as a perturbed form of standard gradient ascent. Leveraging this characterization, we then provide non-asymptotic guarantees on the EM and gradient EM algorithms when applied to a finite set of samples. We develop consequences of our general theory for three canonical examples of incomplete-data problems: mixture of Gaussians, mixture of regressions, and linear regression with covariates missing completely at random. In each case, our theory guarantees that with a suitable initialization, a relatively small number of EM (or gradient EM) steps will yield (with high probability) an estimate that is within statistical error of the MLE. We provide simulations to confirm this theoretically predicted behavior.

• The paper provides a framework offering non-asymptotic convergence guarantees for the EM algorithm across infinite and finite sample scenarios.
• It establishes conditions under which the algorithm exhibits geometric convergence and contractive behavior toward the global MLE.
• It highlights the practical trade-offs of sample-splitting EM, demonstrating robust performance in complex missing data problems.

Overview of Statistical Guarantees for the EM Algorithm

This paper addresses the performance of the Expectation-Maximization (EM) algorithm and its variant, the gradient EM, by developing a framework that provides theoretical guarantees both at the population level and in finite-sample scenarios. The primary aim is to bridge the gap between the empirical efficacy of the EM algorithm and its theoretical underpinnings, especially when applied to non-convex problems with incomplete data.

The research is divided into two parts: analysis of the performance at the population level, where data is infinite, and analysis based on finite samples. By leveraging the interpretation of EM updates as perturbed likelihood ascent, the authors derive non-asymptotic guarantees for three canonical problems: Gaussian mixture models, mixture of regressions, and linear regression with completely random missing covariates.

Population-Level Analysis

At the population level, the paper provides a domain of attraction for any global maximizer of the likelihood. This is achieved by establishing conditions under which these algorithms are contractive towards the maximum likelihood estimate (MLE). For the Gaussian mixture models and mixture of regressions, the results show that with sufficiently high signal-to-noise ratio, the algorithm exhibits geometric convergence within a prescribed basin of attraction.

For the linear regression with missing covariates, the conditions are more intricate due to the nature of missing data, yet the results affirm that convergence to near-optimal solutions is feasible given bounded signal-to-noise ratios and missingness probabilities.

Sample-Based Analysis

In contrast, at the sample level, the authors extend their analysis to scenarios with finite samples, providing bounds on the estimation error in terms of the sample size. The results show that both EM and gradient EM algorithms require sufficiently large samples to converge within statistical error bounds from the MLE. Remarkably, the sample-splitting version of EM is shown to optimize efficiently by using subsets of the data iteratively, albeit with certain computational trade-offs.

Theoretical Implications

The theoretical findings bridge long-standing gaps in understanding the statistical behavior of EM algorithms, especially in dealing with non-convexity. This is particularly relevant in modern applications involving large datasets with missing or latent information. The developed framework facilitates understanding which initialization strategies may drive EM guarantees, highlighting the importance of proper starting points even in complex landscapes.

Practical Implications and Future Directions

Practically, this work emphasizes the applicability of the EM algorithm in efficiently tackling complex incomplete data problems, reinforcing its versatility and robustness under certain conditions. The simulations provided complement the theoretical results and confirm the expected behavior of the algorithm, validating the role of SNR in determining convergence.

The methodology extends potential insights into other iterative algorithms faced with likelihood-based estimations, prompting further research into areas like robust estimsimulationation methods under model mis-specifications and dependent data scenarios. Future work could focus on exploring generalizations of these results to more diverse models, potentially augmenting the statistical learning toolbox with new strategies for handling latent variable models more effectively.

In conclusion, this paper contributes significantly to the theoretical understanding of the EM algorithm, offering insights that serve both the theoretical explorations in statistical convergence and the practical applications within computational statistics and data science.