Oracle inequalities for sparse additive quantile regression in reproducing kernel Hilbert space
针对高维稀疏可加分位数回归，提出一种在再生核希尔伯特空间中的双重Lasso正则化学习方法，建立了无相干条件的非渐近Oracle不等式，并证明了在扩展特征值条件下达到极小化最优速率。
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.