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Generalization is a central problem in Machine Learning. Most predictionmethods require careful calibration of hyperparameters carried out on ahold-out \textit{validation} dataset to achieve generalization. The main goalof this paper is to present a novel approach based on a new measure of riskthat allows us to develop novel fully automatic procedures for generalization.We illustrate the pertinence of this new framework in the regression problem.The main advantages of this new approach are: (i) it can simultaneously trainthe model and perform regularization in a single run of a gradient-basedoptimizer on all available data without any previous hyperparameter tuning;(ii) this framework can tackle several additional objectives simultaneously(correlation, sparsity,...) $via$ the introduction of regularizationparameters. Noticeably, our approach transforms hyperparameter tuning as wellas feature selection (a combinatorial discrete optimization problem) into acontinuous optimization problem that is solvable via classical gradient-basedmethods ; (iii) the computational complexity of our methods is $O(npK)$ where$n,p,K$ denote respectively the number of observations, features and iterationsof the gradient descent algorithm. We observe in our experiments asignificantly smaller runtime for our methods as compared to benchmark methodsfor equivalent prediction score. Our procedures are implemented in PyTorch(code is available for replication).

Optimizing generalization on the train set: a novel gradient-based framework to train parameters and hyperparameters simultaneously