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We propose SDORE, a Semi-supervised Deep Sobolev Regressor, for thenonparametric estimation of the underlying regression function and itsgradient. SDORE employs deep ReQU neural networks to minimize the empiricalrisk with gradient norm regularization, allowing the approximation of theregularization term by unlabeled data. Our study includes a thorough analysisof the convergence rates of SDORE in $L^{2}$-norm, achieving the minimaxoptimality. Further, we establish a convergence rate for the associated plug-ingradient estimator, even in the presence of significant domain shift. Thesetheoretical findings offer valuable insights for selecting regularizationparameters and determining the size of the neural network, while showcasing theprovable advantage of leveraging unlabeled data in semi-supervised learning. Tothe best of our knowledge, SDORE is the first provable neural network-basedapproach that simultaneously estimates the regression function and itsgradient, with diverse applications such as nonparametric variable selection.The effectiveness of SDORE is validated through an extensive range of numericalsimulations.

Semi-Supervised Deep Sobolev Regression: Estimation and Variable Selection by ReQU Neural Network