Nonparametric regression with adaptive truncation via a convex hierarchical penalty

Asad Haris, Ali Shojaie and Noah Simon

Biometrika, 2019, vol. 106, issue 1, 87-107

Abstract: SUMMARY We consider the problem of nonparametric regression with a potentially large number of covariates. We propose a convex, penalized estimation framework that is particularly well suited to high-dimensional sparse additive models and combines the appealing features of finite basis representation and smoothing penalties. In the case of additive models, a finite basis representation provides a parsimonious representation for fitted functions but is not adaptive when component functions possess different levels of complexity. In contrast, a smoothing spline-type penalty on the component functions is adaptive but does not provide a parsimonious representation. Our proposal simultaneously achieves parsimony and adaptivity in a computationally efficient way. We demonstrate these properties through empirical studies and show that our estimator converges at the minimax rate for functions within a hierarchical class. We further establish minimax rates for a large class of sparse additive models. We also develop an efficient algorithm that scales similarly to the lasso with the number of covariates and sample size.

Keywords: Additive model; High-dimensional data; Minimax estimation; Nonparametric regression; Sparsity (search for similar items in EconPapers)
Date: 2019
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