Sparse estimation and inference for prediction-powered semi-supervised linear regression

Semi-supervised learning has become increasingly prevalent in recent years. In the semi-supervised setting, most of the data are unlabeled since the acquisition of high-quality labels requires expensive scientific measurements and/or laborious human labeling. Hence, it is common to employ some black-box machine learning methods as predictive models to generate outcomes on unlabeled data for subsequent statistical inference. In this paper, we consider sparse regression in semi-supervised setting with prediction assistance. Although the predictions may be imperfect and/or noisy, the empirical risk based on a rectified loss function, is unbiased with respect to the population risk, thereby yielding a relatively safe imputation strategy. Under some regularity conditions, the near-optimal statistical rate is established. It is interesting to derive the prediction error bound on the unlabeled data, rather than the labeled data. More importantly, the asymptotic normality and confidence intervals are also studied via debiasing strategy. With mild conditions on predictive models, it is shown that our proposed method, integrating unlabeled data for combined analysis, is asymptotically more efficient than supervised sparse regression. Both numerical evidence and real-world data analysis demonstrate the effectiveness of our method.

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