Fast convergence of the Expectation Maximization algorithm under a logarithmic Sobolev inequality (2407.17949v1)

Abstract: By utilizing recently developed tools for constructing gradient flows on Wasserstein spaces, we extend an analysis technique commonly employed to understand alternating minimization algorithms on Euclidean space to the Expectation Maximization (EM) algorithm via its representation as coordinate-wise minimization on the product of a Euclidean space and a space of probability distributions due to Neal and Hinton (1998). In so doing we obtain finite sample error bounds and exponential convergence of the EM algorithm under a natural generalisation of a log-Sobolev inequality. We further demonstrate that the analysis technique is sufficiently flexible to allow also the analysis of several variants of the EM algorithm.

• The paper establishes finite-sample error bounds and exponential convergence for the EM algorithm using a generalized log-Sobolev inequality.
• The study applies Wasserstein gradient flows to connect EM iterations with optimal transport geometries for robust analysis.
• The work extends its findings to EM variants like First-order and Langevin EM, offering both theoretical and practical insights.

Fast Convergence of the Expectation Maximization Algorithm under a Logarithmic Sobolev Inequality

The research paper presents a novel analysis of the Expectation Maximization (EM) algorithm, demonstrating finite sample error bounds and exponential convergence under a generalization of the log-Sobolev inequality. By leveraging recent advancements in gradient flow methods on Wasserstein spaces, the authors extend techniques traditionally used for alternating minimization algorithms in Euclidean spaces to the EM algorithm formulated as coordinate-wise minimization in a product space of Euclidean and probability distribution spaces.

Overview

The EM algorithm, widely utilized for maximizing the marginal likelihood in models with latent variables, is formulated as an alternating procedure under a free energy functional. This paper formalizes and harnesses the relationship between the EM algorithm and optimal transport geometries, specifically through Wasserstein gradient flows. The primary contributions include:

1. Finite Sample Error Bounds and Exponential Convergence:
   • Utilization of Wasserstein gradient flows to derive non-asymptotic error bounds.
   • Demonstration of exponential convergence of the EM algorithm when the underlying model satisfies a log-Sobolev inequality (LSI).
2. Analysis of EM Variants:
   • Extension of the method to several EM variants, including First-order EM and Langevin EM algorithms.
   • Detailed comparison of convergence properties and numerical implications.

Technical Details and Results

The EM algorithm iterates between maximizing the expected log-likelihood over latent variables (M-step) and computing the posterior distribution (E-step). By framing the EM algorithm in a product space of Euclidean and probability distributions, and using the Wasserstein geometry, the authors derive:

• Free Energy Functional: The free energy functional $F(\theta, q)$ combines the model parameters $\theta$ with the distribution $q$ over latent variables.
• Gradient Expression: Derivation of the gradient $\textup{grad}_{\mathcal{M}_2}F$ in the Wasserstein space, allowing an analysis of free energy dissipation across EM iterations.
• Log-Sobolev Inequality (xLSI): A generalized LSI is proposed, which facilitates the characterization of exponential convergence by relating the free energy to its gradient in the product space.
• Bounded Convergence Rates: Utilizing the xLSI, non-asymptotic bounds are established for the convergence rate, explicitly showing how the free energy and iterates converge to the optimal set $\mathcal{M}_\star$.

Practical and Theoretical Implications

Practical Implications

• Robust Error Bounds: The provided non-asymptotic error bounds are particularly relevant for practitioners who require guarantees on the convergence rate of the EM algorithm in finite samples.
• Algorithm Alternatives: By analyzing First-order and Langevin EM, the research offers comprehensive insights into substitutable techniques when either the M-step or E-step or both are computationally prohibitive.

Theoretical Implications

• Functional Inequalities and Optimal Transport: The integration of advanced tools from these domains provides a unified framework for analyzing EM, suggesting potential extensions and more generalized results for multi-dimensional latent variable models.
• Further Extensions: Given the relevance of LSI in other computational methods, future research could explore its implications for broader classes of iterative algorithms in statistical learning and beyond.

Speculation on Future AI Developments

Considering the advances in understanding EM through Wasserstein and log-Sobolev lenses, future developments could include:

• Adaptive EM Algorithms: Tailoring EM steps dynamically based on gradient information in the Wasserstein space might lead to more efficient, adaptive algorithms.
• Connections to Langevin Monte Carlo: Exploring further interplay between EM and Langevin Monte Carlo methods could yield hybrid algorithms leveraging strengths from both areas, especially in high-dimensional settings.

Conclusion

This paper significantly contributes to the theoretical foundation of the EM algorithm by embedding it in the rich framework of Wasserstein spaces and functional inequalities. The detailed examination of convergence properties under a generalized log-Sobolev inequality offers not only robust theoretical underpinnings but also practical algorithms amenable to modern computational challenges. The proposed methods and results promise to inform future advancements in algorithmic statistics and machine learning, paving the way for more refined and adaptive inferential techniques in complex, high-dimensional latent variable models.