Analysis of High-dimensional Gaussian Labeled-unlabeled Mixture Model via Message-passing Algorithm

2024-11-29

arXiv.org

AI Summary

This work theoretically analyzes the binary classification problem under high-dimensional Gaussian mixture models (GMMs) in semi-supervised learning (SSL), aiming to characterize when and why SSL is effective and to systematically compare the performance limits of the Bayes-optimal estimator and the ℓ₂-regularized maximum likelihood estimator (RMLE). Leveraging the approximate message passing (AMP) framework and state evolution (SE) analysis, we construct, for the first time, global phase diagrams for both estimators under joint labeled and unlabeled data, precisely quantifying their parameter estimation and prediction errors. Our key finding is that suitably tuned ℓ₂ regularization enables RMLE to approach Bayes-optimal performance as the number of unlabeled samples grows, substantially reducing both estimation and classification errors. This work establishes the first rigorous, analytically tractable performance benchmark for high-dimensional SSL and provides principled guidelines for regularizer design.

Technology Category

• Machine Learning: Semi-Supervised Learning
• Reasoning under Uncertainty: Graphical Models
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs

Abstract

Semi-supervised learning (SSL) is a machine learning methodology that leverages unlabeled data in conjunction with a limited amount of labeled data. Although SSL has been applied in various applications and its effectiveness has been empirically demonstrated, it is still not fully understood when and why SSL performs well. Some existing theoretical studies have attempted to address this issue by modeling classification problems using the so-called Gaussian Mixture Model (GMM). These studies provide notable and insightful interpretations. However, their analyses are focused on specific purposes, and a thorough investigation of the properties of GMM in the context of SSL has been lacking. In this paper, we conduct such a detailed analysis of the properties of the high-dimensional GMM for binary classification in the SSL setting. To this end, we employ the approximate message passing and state evolution methods, which are widely used in high-dimensional settings and originate from statistical mechanics. We deal with two estimation approaches: the Bayesian one and the $ell_2$-regularized maximum likelihood estimation (RMLE). We conduct a comprehensive comparison between these two approaches, examining aspects such as the global phase diagram, estimation error for the parameters, and prediction error for the labels. A specific comparison is made between the Bayes-optimal (BO) estimator and RMLE, as the BO setting provides optimal estimation performance and is ideal as a benchmark. Our analysis shows that with appropriate regularizations, RMLE can achieve near-optimal performance in terms of both the estimation error and prediction error, especially when there is a large amount of unlabeled data. These results demonstrate that the $ell_2$ regularization term plays an effective role in estimation and prediction in SSL approaches.

Problem

• Analyzes high-dimensional Gaussian Mixture Model for semi-supervised learning.
• Compares Bayesian and $ell_2$-regularized maximum likelihood estimation methods.
• Demonstrates near-optimal performance of $ell_2$ regularization in SSL.

Innovation

• Uses message-passing algorithm for high-dimensional GMM analysis
• Compares Bayesian and $ell_2$-regularized MLE approaches
• Demonstrates near-optimal performance with $ell_2$ regularization