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The parallel alternating direction method of multipliers (ADMM) algorithm iswidely recognized for its effectiveness in handling large-scale datasets storedin a distributed manner, making it a popular choice for solving statisticallearning models. However, there is currently limited research on parallelalgorithms specifically designed for high-dimensional regression with combined(composite) regularization terms. These terms, such as elastic-net, sparsegroup lasso, sparse fused lasso, and their nonconvex variants, have gainedsignificant attention in various fields due to their ability to incorporateprior information and promote sparsity within specific groups or fusedvariables. The scarcity of parallel algorithms for combined regularizations canbe attributed to the inherent nonsmoothness and complexity of these terms, aswell as the absence of closed-form solutions for certain proximal operatorsassociated with them. In this paper, we propose a unified constrainedoptimization formulation based on the consensus problem for these types ofconvex and nonconvex regression problems and derive the corresponding parallelADMM algorithms. Furthermore, we prove that the proposed algorithm not only hasglobal convergence but also exhibits linear convergence rate. Extensivesimulation experiments, along with a financial example, serve to demonstratethe reliability, stability, and scalability of our algorithm. The R package forimplementing the proposed algorithms can be obtained athttps://github.com/xfwu1016/CPADMM.

A unified consensus-based parallel ADMM algorithm for high-dimensional regression with combined regularizations