Following Curved Regularized Optimization Solution Paths

Regularization plays a central role in the analysis of modern data, where non-regularized fitting is likely to lead to over-fitted models, useless for both prediction and interpretation. We consider the design of incremental algorithms which follow paths of regularized solutions, as the regularization varies. These approaches often result in methods which are both efficient and highly flexible. We suggest a general path-following algorithm based on second-order approximations, prove that under mild conditions it remains “very close” to the path of optimal solutions and illustrate it with examples.