Generalized adaptive bridge regression: a unified framework for high-dimensional architecture discovery
In high-dimensional regression, the choice of regularization penalty typically forces a rigid assumption upon the underlying signal structure, dichotomizing data into strictly sparse (Lasso, q=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q=1$$\end{document}) or entirely dense (Ridge, q=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q=2$$\end{document}) regimes. However, real-world data generating mechanisms frequently exist on a continuum between these extremes, requiring flexible geometries to handle varying degrees of sparsity and considerable multicollinearity. In this work, we propose a data-driven framework to learn the optimal regularization norm by elevating the Lq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_q$$\end{document} exponent (q∈(0,2]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q \in (0, 2]$$\end{document}) from a discrete choice to a strictly continuous, learnable hyper-parameter. To overcome the computational bottleneck of evaluating non-convex and non-smooth penalty landscapes, we develop a universal proximal coordinate descent solver that utilizes a safeguarded jumping threshold operator and a novel empirical Karush-Kuhn-Tucker (KKT) verification strategy. This solver is coupled with a stochastic Tree-structured Parzen Estimator (TPE) utilizing randomized internal validation splits, enabling the rapid discovery of optimal penalty geometries without over-fitting. We evaluate the framework on simulated architectures, demonstrating its dynamic adaptivity to structural sparsity, collinearity, and varying signal-to-noise ratios. Applied to four high-dimensional genomic datasets (scaling up to P≈50,000\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P \approx 50,000$$\end{document} features), our generalized adaptive bridge regression (GABR) framework successfully identifies optimal, off-grid grouping architectures (q≈1.63\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q \approx 1.63$$\end{document} to 1.80), outperforming purely sparse and purely dense alternatives. These results demonstrate that the exact regression geometry can be efficiently learned from the data, enabling a unified approach to high-dimensional inference without the computational restrictions of exhaustive discrete grid searches.