  Generalization is a central problem in Machine Learning. Indeed most
prediction methods require careful calibration of hyperparameters usually
carried out on a hold-out \textit{validation} dataset to achieve
generalization. The main goal of this paper is to introduce a novel approach to
achieve generalization without any data splitting, which is based on a new risk
measure which directly quantifies a model's tendency to overfit. To fully
understand the intuition and advantages of this new approach, we illustrate it
in the simple linear regression model ($Y=X\beta+\xi$) where we develop a new
criterion. We highlight how this criterion is a good proxy for the true
generalization risk. Next, we derive different procedures which tackle several
structures simultaneously (correlation, sparsity,...). Noticeably, these
procedures \textbf{concomitantly} train the model and calibrate the
hyperparameters. In addition, these procedures can be implemented via classical
gradient descent methods when the criterion is differentiable w.r.t. the
hyperparameters. Our numerical experiments reveal that our procedures are
computationally feasible and compare favorably to the popular approach (Ridge,
LASSO and Elastic-Net combined with grid-search cross-validation) in term of
generalization. They also outperform the baseline on two additional tasks:
estimation and support recovery of $\beta$. Moreover, our procedures do not
require any expertise for the calibration of the initial parameters which
remain the same for all the datasets we experimented on.
