Regularization path

The regularization path is the trajectory that a model's parameters or structure trace as the strength of regularization varies continuously from zero to infinity. In LASSO regression, the path traces which coefficients enter the model, remain active, or shrink to zero as the penalty parameter increases; in gradient boosting, the path describes how the ensemble's complexity evolves as the learning rate, tree depth, and subsampling rates are adjusted. Studying the regularization path reveals the natural hierarchy of feature importance and model complexity that a given dataset demands.

The path is not merely a diagnostic tool — it is a structural map of the model space. Different algorithms produce different path geometries: LASSO paths are piecewise linear in coefficient space, while coordinate descent paths on logistic loss are smooth but nonlinear. The geometry of the regularization path encodes information about feature correlations, loss landscape curvature, and the stability of model selection. A path that changes abruptly at a particular regularization value signals a phase-like transition in the model's representational strategy, analogous to the phase transitions observed in physical systems under changing control parameters.