A Unified Semismooth Newton Approach to Multitask and Multivariate Square-Root Lasso Problems

We study scalable second-order algorithms for high-dimensional square-root Lasso problems with multiple responses. The resulting estimators preserve the scale-free tuning property of their single-response counterpart, but accurate large-scale computation is challenging because both the loss and penalty terms are nonsmooth. In this paper, we propose a computational framework that universally applies to multitask and multivariate square-root Lasso models (which includes the single-response version) based on the Douglas-Rachford splitting. We solve the nonsmooth equation induced by the splitting by a regularized semismooth Newton method with projection and fixed-point safeguards. Despite the nonsmoothness of the problem, the structure of the proximal maps and generalized Jacobian lead to a reduced Newton system in a much smaller dimension than that in the original space, promoting scalability. Our method enjoys global and locally quadratic convergence under mild conditions. Experiments on synthetic regression problems and large-scale multi-omic data show that the proposed method reaches the same fitted models and predictive performance as latest solvers tailored to individual models, while requiring fewer iterations and less wall-clock time in large-scale, high-dimensional settings.