Online Hyperparameter Search Interleaved with Proximal Parameter Updates

There is a clear need for efficient hyperparameter optimization (HO) algorithms for statistical learning, since commonly applied search methods (such as grid search with N-fold cross-validation) are inefficient and/or approximate. Previously existing gradient-based HO algorithms that rely on the smoothness of the cost function cannot be applied in problems such as Lasso regression. In this contribution, we develop a HO method that relies on the structure of proximal gradient methods and does not require a smooth cost function. Such a method is applied to Leave-one-out (LOO)-validated Lasso and Group Lasso, and an online variant is proposed. Numerical experiments corroborate the convergence of the proposed methods to stationary points of the LOO validation error curve, and the improved efficiency and stability of the online algorithm.