Regularized EM Algorithms: A Unified Framework and Statistical Guarantees

TL;DR

This paper introduces a unified regularized EM framework for high-dimensional latent variable models, balancing optimization and statistical errors, with guarantees demonstrated on various models like sparse Gaussian mixtures and high-dimensional regression.

Contribution

It develops a general regularized EM algorithm framework that addresses high-dimensional challenges and provides statistical guarantees for multiple models.

Findings

The framework achieves linear local convergence in high-dimensional settings.

Regularization balances optimization progress and structure identification.

Statistical guarantees are established for Gaussian mixtures, mixed regression, and missing data models.

Abstract

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.