The parallel alternating direction method of multipliers (ADMM) algorithms have become popular in statistics and machine learning due to their ability to efficiently handle large sample data problems. However, the parallel structure of the ADMM algorithms are all based on consensus structure, which can cause too many auxiliary variables for high-dimensional data. In this paper, we focus on nonconvex penalized smooth quantile regression peoblems and develop a parallel linearized ADMM (LADMM) algorithm to solve it. Compared with existing parallel ADMM algorithms, our algorithm does not rely on consensus structure, resulting in a significant reduction in the number of variables that need to be updated at each iteration. It is worth noting that the solution of our algorithm remains unchanged regardless of how the total sample is divided. Furthermore, under some mild assumptions, we prove that the iterative sequence generated by the LADMM converges to a critical point of the nonconvex optimization problem. Numerical experiments on synthetic and real datasets demonstrate the feasibility and validity of the proposed algorithm.

翻译：并行交替方向乘子法（ADMM）算法因其能够高效处理大规模样本数据问题而在统计学和机器学习领域广受欢迎。然而，现有的ADMM算法并行结构均基于一致性架构，这可能导致高维数据中辅助变量过多。本文聚焦于非凸惩罚平滑分位数回归问题，提出了一种并行的线性化ADMM（LADMM）算法进行求解。与现有并行ADMM算法相比，该算法不依赖一致性结构，从而显著减少了每次迭代中需要更新的变量数量。值得注意，无论总体样本如何划分，该算法的求解结果均保持不变。此外，在若干温和假设条件下，我们证明了LADMM生成的迭代序列收敛至非凸优化问题的临界点。基于合成数据集和真实数据集的数值实验表明了所提出算法的可行性与有效性。