High-Dimensional Quantile Regression: Convolution Smoothing and Concave Regularization

Abstract ? 1 -penalized quantile regression (QR) is widely used for analysing high-dimensional data with heterogeneity. It now recognized that the ?1-penalty introduces non-negligible estimation bias, while a proper use of concave regularization may lead to estimators refined convergence rates and oracle properties as signal strengthens. Although folded penalized M-estimation strongly convex loss functions have been well studied, extant literature on QR relatively silent. The main difficulty piecewise linear: it non-smooth has curvature concentrated at single point. To overcome lack smoothness strong convexity, we propose study convolution-type smoothed iteratively reweighted ?1-regularization. resulting empirical twice continuously differentiable (provably) locally high probability. We show ?1-penalized estimator, after few iterations, achieves optimal rate convergence, moreover, property under an almost necessary sufficient minimum strength condition. Extensive numerical studies corroborate our theoretical results.