Semi-supervised linear regression: enhancing efficiency and robustness in high dimensions

In semi-supervised learning, the prevailing understanding suggests that observing additional unlabeled samples improves estimation accuracy for linear parameters only in the case of model misspecification. In this work, we challenge such a claim and show that additional unlabeled samples are beneficial in high-dimensional settings. Initially focusing on a dense scenario, we introduce robust semi-supervised estimators for the regression coefficient without relying on sparse structures in the population slope. Even when the true underlying model is linear, we show that leveraging information from large-scale unlabeled data helps reduce estimation bias, thereby improving both estimation accuracy and inference robustness. Moreover, we propose semi-supervised methods with further enhanced efficiency in scenarios with a sparse linear slope. The performance of the proposed methods is demonstrated through extensive numerical studies.