[Paper Review] Regularized EM Algorithms: A Unified Framework and Statistical Guarantees

Xinyang Yi, Constantine Caramanis|arXiv (Cornell University)|Nov 27, 2015

Sparse and Compressive Sensing Techniques20 references32 citations

TL;DR

This paper introduces a unified regularized EM framework for high-dimensional latent variable models, where adaptive regularization in the M-step balances optimization and statistical errors. It establishes linear local convergence with statistical guarantees for sparse Gaussian mixtures, high-dimensional mixed regression, and missing data regression, under minimal assumptions on the regularization sequence and estimation error.

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.

Motivation & Objective

• Address the lack of rigorous statistical guarantees for EM in high-dimensional settings, where the M-step becomes ill-defined due to overparameterization.
• Overcome the challenge of selecting regularization sequences that balance optimization error (e.g., sparsity) and statistical error in iterative EM updates.
• Provide a general convergence framework that links optimization progress with estimation error control, applicable across diverse high-dimensional models.
• Establish local linear convergence with non-asymptotic statistical error bounds for regularized EM, even when the population M-step is undefined.
• Specialize the framework to concrete models—sparse Gaussian mixtures, high-dimensional mixed regression, and regression with missing covariates—providing model-specific guarantees.

Proposed method

• Propose a regularized EM algorithm where the M-step is modified using a data-dependent, adaptive regularization sequence that evolves across iterations.
• Introduce a novel regularization sequence $\lambda_m^{(t)} = \frac{1 - \kappa^t}{1 - \kappa} \Delta + \kappa^t \frac{\gamma_m}{5\Psi(\overline{\mathcal{S}})} \|\bm{\beta}^{(0)} - \bm{\beta}^*\|$, ensuring convergence by balancing estimation and optimization errors.
• Use a local error characterization involving the dual norm $\mathcal{R}(\cdot)$ and a compatibility condition $\gamma_m$ to control the distance to the true parameter.
• Establish convergence via induction by showing that if $\|\bm{\beta}^{(t)} - \bm{\beta}^*\| \leq r$, then $\|\bm{\beta}^{(t+1)} - \bm{\beta}^*\| \leq r$, maintaining iterates within a local neighborhood.
• Derive a key inequality $\|\Theta\| \leq 5\Psi(\overline{\mathcal{S}}) \frac{\lambda_m^{(t)}}{\gamma_m}$, linking the estimation error to the regularization parameter and the true parameter structure.
• Apply probabilistic union bounds to ensure high-probability convergence under sub-Gaussian design assumptions and bounded noise.

Experimental results

Research questions

• RQ1 How can the EM algorithm be regularized in high-dimensional settings to ensure convergence when the M-step is ill-defined?
• RQ2 What is the correct sequence of regularization parameters that balances statistical and optimization errors across EM iterations?
• RQ3 Can a unified framework be developed to provide statistical guarantees for regularized EM across diverse high-dimensional models?
• RQ4 What conditions ensure local linear convergence of regularized EM with non-asymptotic error bounds?
• RQ5 How does the adaptive regularization sequence relate to the ultimate estimation error and the true parameter structure?

Key findings

• The proposed regularized EM algorithm achieves linear local convergence with high probability under mild regularity conditions, even when the population M-step is undefined.
• The estimation error is bounded by $\|\bm{\beta}^{(t)} - \bm{\beta}^*\| \leq \frac{5\Psi(\overline{\mathcal{S}})}{\gamma_m} \frac{1 - \kappa^t}{1 - \kappa} \Delta + \kappa^t \|\bm{\beta}^{(0)} - \bm{\beta}^*\|$, ensuring exponential decay in the error when $\kappa < 3/4$.
• The regularization sequence $\lambda_m^{(t)}$ is explicitly constructed to converge to a value proportional to the ultimate estimation error, enabling stable and consistent updates.
• For sparse Gaussian mixture models, the framework yields non-asymptotic statistical error bounds that match known minimax rates under sparsity.
• In high-dimensional mixed regression and missing data regression, the algorithm achieves consistent parameter estimation with optimal sample complexity, provided the regularization sequence satisfies $\lambda_m^{(t)} \geq 3\Delta_m + \frac{\alpha\mu\tau}{\gamma\Psi(\overline{\mathcal{S}})} \|\bm{\beta}^{(t-1)} - \bm{\beta}^*\|$.
• The analysis reveals a fundamental trade-off between optimization error (controlled by $\lambda_m$) and statistical error (controlled by $\Delta_m$), resolved through iterative adaptation of the regularization strength.