We consider the estimation problem in high-dimensional semi-supervised learning. Our goal is to investigate when and how the unlabeled data can be exploited to improve the estimation of the regression parameters of linear model in light of the fact that such linear models may be misspecified in data analysis. We first establish the minimax lower bound for parameter estimation in the semi-supervised setting, and show that this lower bound cannot be achieved by supervised estimators using the labeled data only. We propose an optimal semi-supervised estimator that can attain this lower bound and therefore improves the supervised estimators, provided that the conditional mean function can be consistently estimated with a proper rate. We further propose a safe semi-supervised estimator. We view it safe, because this estimator is always at least as good as the supervised estimators. We also extend our idea to the aggregation of multiple semi-supervised estimators caused by different misspecifications of the conditional mean function. Extensive numerical simulations and a real data analysis are conducted to illustrate our theoretical results.

翻译：我们考虑高维半监督学习中的估计问题。鉴于线性模型在数据分析中可能存在设定偏误，本研究旨在探讨何时以及如何利用无标签数据来改善线性模型回归参数的估计。首先，我们建立了半监督设定下参数估计的极小极大下界，并证明仅使用标签数据的监督估计量无法达到该下界。我们提出了一种最优半监督估计量，在条件均值函数能够以适当速率进行一致估计的前提下，该估计量可达到该下界，从而优于监督估计量。进一步地，我们提出了一种安全半监督估计量，之所以称其为"安全"，是因为该估计量始终不劣于监督估计量。此外，我们将思想拓展至因条件均值函数存在不同设定偏误而产生的多个半监督估计量的聚合问题。通过大量数值模拟与一项实际数据分析，我们验证了理论结果的正确性。