High-dimensional penalty selection via minimum description length principle

We tackle the problem of penalty selection for regularization on the basis of the minimum description length (MDL) principle. In particular, we consider that the design space of the penalty function is high-dimensional. In this situation, the luckiness-normalized-maximum-likelihood (LNML)-minimization approach is favorable, because LNML quantifies the goodness of regularized models with any forms of penalty functions in view of the MDL principle, and guides us to a good penalty function through the high-dimensional space. However, the minimization of LNML entails two major challenges: (1) the computation of the normalizing factor of LNML and (2) its minimization in high-dimensional spaces. In this paper, we present a novel regularization selection method (MDL-RS), in which a tight upper bound of LNML (uLNML) is minimized with local convergence guarantee. Our main contribution is the derivation of uLNML, which is a uniform-gap upper bound of LNML in an analytic expression. This solves the above challenges in an approximate manner because it allows us to accurately approximate LNML and then efficiently minimize it. The experimental results show that MDL-RS improves the generalization performance of regularized estimates specifically when the model has redundant parameters.