A Unified View of Decentralized Algorithms for Sparse Linear Regression

We investigate sparse linear regression in high-dimension via the projected LASSO estimator from distributed datasets across a network of agents. This model enables the ambient dimension to scale exponentially with the total sample size. We develop a unified algorithmic framework that encompasses a variety of distributed algorithms, new and old, such as primal, primal-dual, and gradient tracking-based methods, for which we establish both statistical and computational guarantees. Our findings demonstrate that, under standard data generation models, appropriate network connectivity and algorithm tuning, the studied schemes converge to statistically optimal estimates at a linear rate. However, the dependencies of convergence rate on the ambient dimension exhibit a noteworthy difference, ranging from linear to independent scaling. This fact is a new, interesting phenomenon distinctive of the high-dimensional setting and revealed by this study.