High-Dimensional Quantile Regression: Convolution Smoothing and Concave Regularization
针对高维数据中分位数回归因损失函数非光滑导致的估计偏差问题，提出卷积平滑与迭代加权L1正则化方法，实现最优收敛速度和强Oracle性质。
Abstract ℓ 1 -penalized quantile regression (QR) is widely used for analysing high-dimensional data with heterogeneity. It is now recognized that the ℓ1-penalty introduces non-negligible estimation bias, while a proper use of concave regularization may lead to estimators with refined convergence rates and oracle properties as the signal strengthens. Although folded concave penalized M-estimation with strongly convex loss functions have been well studied, the extant literature on QR is relatively silent. The main difficulty is that the quantile loss is piecewise linear: it is non-smooth and has curvature concentrated at a single point. To overcome the lack of smoothness and strong convexity, we propose and study a convolution-type smoothed QR with iteratively reweighted ℓ1-regularization. The resulting smoothed empirical loss is twice continuously differentiable and (provably) locally strongly convex with high probability. We show that the iteratively reweighted ℓ1-penalized smoothed QR estimator, after a few iterations, achieves the optimal rate of convergence, and moreover, the oracle rate and the strong oracle property under an almost necessary and sufficient minimum signal strength condition. Extensive numerical studies corroborate our theoretical results.