Efficient Group Lasso Regularized Rank Regression with Data-Driven Parameter Determination
This addresses robust regression for statisticians and data scientists dealing with noisy, high-dimensional data, representing an incremental improvement combining existing techniques.
The paper tackles robust high-dimensional regression with heavy-tailed noise and outliers by developing a group lasso regularized rank regression method with a data-driven tuning rule, achieving improved robustness and scalability in numerical experiments.
High-dimensional regression often suffers from heavy-tailed noise and outliers, which can severely undermine the reliability of least-squares based methods. To improve robustness, we adopt a non-smooth Wilcoxon score based rank objective and incorporate structured group sparsity regularization, a natural generalization of the lasso, yielding a group lasso regularized rank regression method. By extending the tuning-free parameter selection scheme originally developed for the lasso, we introduce a data-driven, simulation-based tuning rule and further establish a finite-sample error bound for the resulting estimator. On the computational side, we develop a proximal augmented Lagrangian method for solving the associated optimization problem, which eliminates the singularity issues encountered in existing methods, thereby enabling efficient semismooth Newton updates for the subproblems. Extensive numerical experiments demonstrate the robustness and effectiveness of our proposed estimator against alternatives, and showcase the scalability of the algorithm across both simulated and real-data settings.