The Annals of Statistics
- Ann. Statist.
- Volume 46, Number 2 (2018), 781-813.
Oracle inequalities for sparse additive quantile regression in reproducing kernel Hilbert space
This paper considers the estimation of the sparse additive quantile regression (SAQR) in high-dimensional settings. Given the nonsmooth nature of the quantile loss function and the nonparametric complexities of the component function estimation, it is challenging to analyze the theoretical properties of ultrahigh-dimensional SAQR. We propose a regularized learning approach with a two-fold Lasso-type regularization in a reproducing kernel Hilbert space (RKHS) for SAQR. We establish nonasymptotic oracle inequalities for the excess risk of the proposed estimator without any coherent conditions. If additional assumptions including an extension of the restricted eigenvalue condition are satisfied, the proposed method enjoys sharp oracle rates without the light tail requirement. In particular, the proposed estimator achieves the minimax lower bounds established for sparse additive mean regression. As a by-product, we also establish the concentration inequality for estimating the population mean when the general Lipschitz loss is involved. The practical effectiveness of the new method is demonstrated by competitive numerical results.
Ann. Statist., Volume 46, Number 2 (2018), 781-813.
Received: February 2016
Revised: January 2017
First available in Project Euclid: 3 April 2018
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Lv, Shaogao; Lin, Huazhen; Lian, Heng; Huang, Jian. Oracle inequalities for sparse additive quantile regression in reproducing kernel Hilbert space. Ann. Statist. 46 (2018), no. 2, 781--813. doi:10.1214/17-AOS1567. https://projecteuclid.org/euclid.aos/1522742436
- Supplement to “Oracle inequalities for sparse additive quantile regression in reproducing kernel Hilbert space”. To highlight the nature and usefulness of Assumptions 3–4, we state some simple sufficient conditions to verify them respectively in the Supplementary Material. Besides, due to space limitation, we also give the proofs of Theorem 1 and Lemma 2 in the Supplementary Material.