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

arXiv: https://arxiv.org/abs/2311.12319 · HTML: https://mlsanity.com/paper/2311.12319/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 90 references that are in the map.
- First posted on arXiv 21 Nov 2023.

## Builds on (most cited first, 7 of 7)

- [Semismooth Newton Coordinate Descent Algorithm for Elastic-Net Penalized Huber Loss Regression and Quantile Regression](https://mlsanity.com/paper/1509.02957/) (2015)
- [An efficient Hessian based algorithm for solving large-scale sparse group Lasso problems](https://mlsanity.com/paper/1712.05910/) (2017)
- [Adaptive sparse group LASSO in quantile regression](https://mlsanity.com/paper/1911.01081/) (2019)
- [The flare Package for High Dimensional Linear Regression and Precision Matrix Estimation in R](https://mlsanity.com/paper/2006.15419/) (2020)
- [Background Subtraction via Generalized Fused Lasso Foreground Modeling](https://mlsanity.com/paper/1504.03707/) (2015)
- [High-dimensional Fused Lasso Regression using Majorization-Minimization and Parallel Processing](https://mlsanity.com/paper/1306.1970/) (2013)
- [Convex Fused Lasso Denoising with Non-Convex Regularization and its use for Pulse Detection](https://mlsanity.com/paper/1509.02811/) (2015)

## Built on by (most cited first, 2 of 2)

- [Leveraging Noisy Manual Labels as Useful Information: An Information Fusion Approach for Enhanced Variable Selection in Penalized Logistic Regression](https://mlsanity.com/paper/2504.16585/) (2025)
- [Parallel Algorithms for Structured Sparse Support Vector Machines: Application in Music Genre Classification](https://mlsanity.com/paper/2512.07463/) (2025)

## Abstract (arXiv)

The parallel alternating direction method of multipliers (ADMM) algorithm is widely recognized for its effectiveness in handling large-scale datasets stored in a distributed manner, making it a popular choice for solving statistical learning models. However, there is currently limited research on parallel algorithms specifically designed for high-dimensional regression with combined (composite) regularization terms. These terms, such as elastic-net, sparse group lasso, sparse fused lasso, and their nonconvex variants, have gained significant attention in various fields due to their ability to incorporate prior information and promote sparsity within specific groups or fused variables. The scarcity of parallel algorithms for combined regularizations can be attributed to the inherent nonsmoothness and complexity of these terms, as well as the absence of closed-form solutions for certain proximal operators associated with them. In this paper, we propose a unified constrained optimization formulation based on the consensus problem for these types of convex and nonconvex regression problems and derive the corresponding parallel ADMM algorithms. Furthermore, we prove that the proposed algorithm not only has global convergence but also exhibits linear convergence rate. Extensive simulation experiments, along with a financial example, serve to demonstrate the reliability, stability, and scalability of our algorithm. The R package for implementing the proposed algorithms can be obtained at https://github.com/xfwu1016/CPADMM.