The parallel alternating direction method of multipliers (ADMM) algorithms have gained popularity in statistics and machine learning due to their efficient handling of large sample data problems. However, the parallel structure of these algorithms, based on the consensus problem, can lead to an excessive number of auxiliary variables when applied to highdimensional data, resulting in large computational burden. In this paper, we propose a partition-insensitive parallel framework based on the linearized ADMM (LADMM) algorithm and apply it to solve nonconvex penalized high-dimensional 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 largely unchanged regardless of how the total sample is divided, which is known as partition-insensitivity. Furthermore, under some mild assumptions, we prove the convergence of the iterative sequence generated by our parallel algorithm. Numerical experiments on synthetic and real datasets demonstrate the feasibility and validity of the proposed algorithm. We provide a publicly available R software package to facilitate the implementation 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，基于共识问题的并行结构会导致高维数据场景下辅助变量数量过多，进而带来巨大的计算负担。本文提出了一种基于线性化ADMM（LADMM）算法的分区不敏感并行框架，并将其应用于求解非凸惩罚高维回归问题。与现有并行ADMM算法相比，本算法不依赖共识问题，从而显著减少了每次迭代中需要更新的变量数量。值得注意的是，无论总样本如何划分，本算法的解基本保持不变，这一特性称为分区不敏感性。此外，在温和假设条件下，我们证明了并行算法生成的迭代序列的收敛性。在合成数据和真实数据集上的数值实验验证了所提出算法的可行性与有效性。我们提供了公开可用的R语言软件包以促进该算法的实现。