Regularized EM Algorithms: A Unified Framework and Statistical Guarantees
This work addresses a computational bottleneck in latent variable models for machine learning practitioners, offering a novel method with theoretical backing, though it is incremental in building on prior EM convergence results.
The authors tackled the challenge of applying the EM algorithm in high-dimensional settings where the M-step is ill-defined, by introducing a regularized EM framework that balances optimization and statistical errors. They demonstrated the approach on sparse Gaussian mixture models, high-dimensional mixed regression, and regression with missing variables, providing statistical guarantees for these cases.
Latent variable models are a fundamental modeling tool in machine learning applications, but they present significant computational and analytical challenges. The popular EM algorithm and its variants, is a much used algorithmic tool; yet our rigorous understanding of its performance is highly incomplete. Recently, work in Balakrishnan et al. (2014) has demonstrated that for an important class of problems, EM exhibits linear local convergence. In the high-dimensional setting, however, the M-step may not be well defined. We address precisely this setting through a unified treatment using regularization. While regularization for high-dimensional problems is by now well understood, the iterative EM algorithm requires a careful balancing of making progress towards the solution while identifying the right structure (e.g., sparsity or low-rank). In particular, regularizing the M-step using the state-of-the-art high-dimensional prescriptions (e.g., Wainwright (2014)) is not guaranteed to provide this balance. Our algorithm and analysis are linked in a way that reveals the balance between optimization and statistical errors. We specialize our general framework to sparse gaussian mixture models, high-dimensional mixed regression, and regression with missing variables, obtaining statistical guarantees for each of these examples.