Sparse Composite Quantile Regression in Ultrahigh Dimensions With Tuning Parameter Calibration
Sparse Composite Quantile Regression in Ultrahigh Dimensions With Tuning Parameter Calibration
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DOI：
10.1109/tit.2020.3001090
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发表时间：
2020-11-01
影响因子：
2.5
通讯作者：
Zou, Hui
中科院分区：
文献类型：
--
作者：
Gu, Yuwen;Zou, Hui
当估计线性模型中的系数时，（稀疏）复合分位数回归首先在Zou和Yuan（2008）中提出，作为（稀疏）最小二乘的有效替代方案来处理任意误差分布。稀疏复合分位数回归中复合损失的高度非光滑性质使得其理论分析和数值计算比最小二乘法更具挑战性。Zou和Yuan（2008）中的理论在固定维渐近下得到了证明，并且通过线性规划计算估计量，该线性规划在高维情况下不能很好地扩展。本文主要研究了高维下的稀疏复合分位数回归问题，并做了三个方面的工作。首先，我们提供了一个非渐近分析的套索和折叠凹惩罚复合分位数回归，这揭示了一个实用的方法来实现预言估计。其次，我们构造了折叠凹惩罚复合分位数回归中正则化参数选择的新信息准则，并证明了其选择的一致性。第三，我们利用复合损失的结构，并设计了一个专门的优化算法，通过交替方向法的乘数计算惩罚复合分位数回归。我们进行了大量的模拟来说明理论结果。我们的分析提供了一个统一的处理涉及的综合损失的浓度不等式。这些不平等可能是独立的利益。
When estimating coefficients in a linear model, the (sparse) composite quantile regression was first proposed in Zou and Yuan (2008) as an efficient alternative to the (sparse) least squares to handle arbitrary error distribution. The highly nonsmooth nature of the composite loss in the sparse composite quantile regression makes its theoretical analysis as well as numerical computation much more challenging than the least squares method. The theory in Zou and Yuan (2008) was proven under fixed-dimension asymptotics and the estimator was computed via linear programming that does not scale well with high dimensions. In this paper, we study the sparse composite quantile regression under ultrahigh dimensionality and make three contributions. First, we provide a non-asymptotic analysis of both the lasso and the folded concave penalized composite quantile regression, which reveals a practical way of achieving the oracle estimator. Second, we construct a novel information criterion for selecting the regularization parameter in the folded concave penalized composite quantile regression and prove its selection consistency. Third, we exploit the structure of the composite loss and design a specialized optimization algorithm for computing the penalized composite quantile regression via the alternating direction method of multipliers. We conduct extensive simulations to illustrate the theoretical results. Our analysis provides a unified treatment of the concentration inequalities involving the composite loss. Those inequalities could be of independent interest.