Optimal and Safe Estimation for High-Dimensional Semi-Supervised Learning.

Area of Science:

• Statistics
• Machine Learning

Background:

• High-dimensional data analysis presents challenges for accurate parameter estimation.
• Linear models are frequently used but may be misspecified in real-world applications.
• Semi-supervised learning offers a framework to leverage both labeled and unlabeled data.

Purpose of the Study:

• To investigate the conditions and methods for effectively utilizing unlabeled data in high-dimensional semi-supervised learning.
• To develop estimators that improve upon supervised methods when linear models are potentially misspecified.
• To establish theoretical performance bounds for parameter estimation in this setting.

Main Methods:

• Establishing the minimax lower bound for parameter estimation in the semi-supervised context.
• Proposing an optimal semi-supervised estimator designed to achieve this lower bound.
• Developing a 'safe' semi-supervised estimator that guarantees performance at least as good as supervised estimators.
• Extending the methodology to aggregate multiple estimators for robustness against different model misspecifications.

Main Results:

• Demonstrating that supervised estimators alone cannot achieve the established minimax lower bound.
• Showing that the proposed optimal semi-supervised estimator can attain the lower bound under specific conditions.
• Confirming that the safe semi-supervised estimator provides a performance guarantee.
• Illustrating the effectiveness of the proposed methods through extensive simulations and real data analysis.

Conclusions:

• Unlabeled data can significantly improve parameter estimation in high-dimensional semi-supervised learning, especially when models are misspecified.
• The proposed optimal and safe semi-supervised estimators offer theoretical and practical advantages over purely supervised approaches.
• The developed aggregation strategy enhances robustness in the presence of model uncertainty.