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

arXiv: https://arxiv.org/abs/2011.14185 · HTML: https://mlsanity.com/paper/2011.14185/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Nov 2020.

## Built on by (most cited first, 10 of 10)

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- [SADA: Safe and Adaptive Aggregation of Multiple Black-Box Predictions in Semi-Supervised Learning](https://mlsanity.com/paper/2509.21707/) (2025)
- [High-dimensional Semi-supervised Classification via the Fermat Distance](https://mlsanity.com/paper/2604.23573/) (2026)
- [Inference with non-differentiable surrogate loss in a general high-dimensional classification framework](https://mlsanity.com/paper/2405.11723/) (2024)
- [Optimal Sampling for Generalized Linear Model under Measurement Constraint with Surrogate Variables](https://mlsanity.com/paper/2501.00972/) (2025)
- [Rescuing double robustness: safe estimation under complete misspecification](https://mlsanity.com/paper/2509.22446/) (2025)
- [Statistical Learning for Latent Embedding Alignment with Application to Brain Encoding and Decoding](https://mlsanity.com/paper/2603.21042/) (2026)
- [Semi-Supervised Conditional Diffusion via Label Augmentation](https://mlsanity.com/paper/2607.16685/) (2026)

## Abstract (arXiv)

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.