  The composite quantile regression (CQR) was introduced by Zou and Yuan [Ann.
Statist. 36 (2008) 1108--1126] as a robust regression method for linear models
with heavy-tailed errors while achieving high efficiency. Its penalized
counterpart for high-dimensional sparse models was recently studied in Gu and
Zou [IEEE Trans. Inf. Theory 66 (2020) 7132--7154], along with a specialized
optimization algorithm based on the alternating direct method of multipliers
(ADMM). Compared to the various first-order algorithms for penalized least
squares, ADMM-based algorithms are not well-adapted to large-scale problems. To
overcome this computational hardness, in this paper we employ a
convolution-smoothed technique to CQR, complemented with iteratively reweighted
$\ell_1$-regularization. The smoothed composite loss function is convex, twice
continuously differentiable, and locally strong convex with high probability.
We propose a gradient-based algorithm for penalized smoothed CQR via a variant
of the majorize-minimization principal, which gains substantial computational
efficiency over ADMM. Theoretically, we show that the iteratively reweighted
$\ell_1$-penalized smoothed CQR estimator achieves near-minimax optimal
convergence rate under heavy-tailed errors without any moment constraint, and
further achieves near-oracle convergence rate under a weaker minimum signal
strength condition than needed in Gu and Zou (2020). Numerical studies
demonstrate that the proposed method exhibits significant computational
advantages without compromising statistical performance compared to two
state-of-the-art methods that achieve robustness and high efficiency
simultaneously.
