This paper investigates quantile regression in the presence of non-convex and non-smooth sparse penalties, such as the minimax concave penalty (MCP) and smoothly clipped absolute deviation (SCAD). The non-smooth and non-convex nature of these problems often leads to convergence difficulties for many algorithms. While iterative techniques like coordinate descent and local linear approximation can facilitate convergence, the process is often slow. This sluggish pace is primarily due to the need to run these approximation techniques until full convergence at each step, a requirement we term as a \emph{secondary convergence iteration}. To accelerate the convergence speed, we employ the alternating direction method of multipliers (ADMM) and introduce a novel single-loop smoothing ADMM algorithm with an increasing penalty parameter, named SIAD, specifically tailored for sparse-penalized quantile regression. We first delve into the convergence properties of the proposed SIAD algorithm and establish the necessary conditions for convergence. Theoretically, we confirm a convergence rate of $o\big({k^{-\frac{1}{4}}}\big)$ for the sub-gradient bound of augmented Lagrangian. Subsequently, we provide numerical results to showcase the effectiveness of the SIAD algorithm. Our findings highlight that the SIAD method outperforms existing approaches, providing a faster and more stable solution for sparse-penalized quantile regression.

翻译：本文研究了在非凸且非光滑稀疏惩罚（如极小极大凹惩罚（MCP）和光滑削边绝对偏差（SCAD））下的分位数回归问题。这些问题的非光滑和非凸特性常导致许多算法收敛困难。尽管坐标下降和局部线性逼近等迭代技术有助于促进收敛，但过程通常较慢。这种缓慢的主要原因是每次迭代需要运行这些逼近技术直至完全收敛，我们将此要求称为*二次收敛迭代*。为加速收敛速度，我们采用交替方向乘子法（ADMM），并提出一种新颖的单循环平滑ADMM算法（含递增惩罚参数），命名为SIAD，专门针对稀疏惩罚分位数回归。我们首先深入探讨了所提SIAD算法的收敛性质，并建立了收敛的必要条件。理论上，我们确认了增广拉格朗日子梯度界以$o\big({k^{-\frac{1}{4}}}\big)$的速率收敛。随后，我们通过数值结果展示了SIAD算法的有效性。研究结果表明，SIAD方法优于现有方法，为稀疏惩罚分位数回归提供了更快且更稳定的解。