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arXiv · 1611.09972

Nonparametric Regression with Adaptive Truncation via a Convex Hierarchical Penalty

Asad Haris·Ali Shojaie·Noah Simon
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Abstract

We consider the problem of non-parametric regression with a potentially large number of covariates. We propose a convex, penalized estimation framework that is particularly well-suited for high-dimensional sparse additive models. The proposed approach combines appealing features of finite basis representation and smoothing penalties for non-parametric estimation. In particular, in the case of additive models, a finite basis representation provides a parsimonious representation for fitted functions but is not adaptive when component functions posses different levels of complexity. On the other hand, a smoothing spline type penalty on the component functions is adaptive but does not offer a parsimonious representation of the estimated function. The proposed approach simultaneously achieves parsimony and adaptivity in a computationally efficient framework. We demonstrate these properties through empirical studies on both real and simulated datasets. We 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. The proposed method is implemented using an efficient algorithm that scales similarly to the Lasso with the number of covariates and samples size.

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Asad Haris, Ali Shojaie, Noah Simon. 2019-06-18. Nonparametric Regression with Adaptive Truncation via a Convex Hierarchical Penalty. https://doi.org/10.1093/biomet%2Fasy056

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