On Stein's Identity and Near-Optimal Estimation in High-dimensional Index Models

We consider estimating the parametric components of semi-parametric multiple index models in a high-dimensional non-Gaussian setting. Our estimators leverage the score function based second-order Stein's lemma and do not require Gaussian or elliptical symmetry assumptions made in the literature. Moreover, to handle score functions and response variables that are heavy-tailed, our estimators are constructed via carefully thresholding their empirical counterparts. We show that our estimator achieves near- optimal statistical rate of convergence in several settings. We supplement our theoretical results via simulation experiments that confirm the theory.