The parallel alternating direction method of multipliers (ADMM) algorithms have gained popularity in statistics and machine learning for their efficient handling of large sample data problems. However, the parallel structure of these algorithms is based on the consensus problem, which can lead to an excessive number of auxiliary variables for high-dimensional data. In this paper, we propose a partition-insensitive parallel framework based on the linearized ADMM (LADMM) algorithm and apply it to solve nonconvex penalized smooth quantile regression problems. Compared to existing parallel ADMM algorithms, our algorithm does not rely on the consensus problem, 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, which is also known as partition-insensitivity. Furthermore, under some mild assumptions, we prove that the iterative sequence generated by the parallel LADMM algorithm 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）算法因其高效处理大规模样本数据问题的能力，在统计学和机器学习领域广受欢迎。然而，此类算法的并行结构基于共识问题，这可能导致高维数据中出现过多的辅助变量。本文提出一种基于线性化ADMM（LADMM）算法的分割不敏感并行框架，并将其应用于求解非凸惩罚光滑分位数回归问题。与现有并行ADMM算法相比，所提算法不依赖共识问题，从而显著减少了每次迭代需要更新的变量数量。值得注意的是，无论总样本如何分割，该算法的解均保持不变——这一特性被称为"分割不敏感性"。此外，在温和假设条件下，我们证明了并行LADMM算法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验证了所提算法的可行性与有效性。