Hard-thresholding regularization method for high-dimensional heterogeneous models

Heterogeneity is often natural in many contemporary applications involving massive data, as exemplified by domains such as personalized education and tailored pricing strategy. In this article, we propose a new methodology for subgroup analysis via a unified penalization approach, which combines the concave pairwise fusion penalty and an additional hard-thresholding penalty to concurrently estimate the heterogeneous effects and coefficients for high-dimensional covariates. Furthermore, we show that the oracle least squares estimator based on a prior knowledge of the true grouping structure and the index set of nonzero true coefficients enjoys appealing statistical properties including oracle inequalities under various prediction and estimation losses. Moreover, we provide theoretical guarantees for the proposed approach by establishing the joint estimation error bound for both heterogeneous effects and coefficients of high-dimensional covariates. The practical utility of our proposed method is illustrated through simulation and real data examples.