  $\ell_1$-penalized quantile regression is widely used for analyzing
high-dimensional data with heterogeneity. It is now recognized that the
$\ell_1$-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 quantile regression 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 quantile regression with iteratively reweighted
$\ell_1$-regularization. The resulting smoothed empirical loss is twice
continuously differentiable and (provably) locally strongly convex with high
probability. We show that the iteratively reweighted $\ell_1$-penalized
smoothed quantile regression 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.
