We consider the parametric learning problem, where the objective of the learner is determined by a parametric loss function. Employing empirical risk minimization with possibly regularization, the inferred parameter vector will be biased toward the training samples. Such bias is measured by the cross validation procedure in practice where the data set is partitioned into a training set used for training and a validation set, which is not used in training and is left to measure the out-of-sample performance. A classical cross validation strategy is the leave-one-out cross validation (LOOCV) where one sample is left out for validation and training is done on the rest of the samples that are presented to the learner, and this process is repeated on all of the samples. LOOCV is rarely used in practice due to the high computational complexity. In this paper, we first develop a computationally efficient approximate LOOCV (ALOOCV) and provide theoretical guarantees for its performance. Then we use ALOOCV to provide an optimization algorithm for finding the regularizer in the empirical risk minimization framework. In our numerical experiments, we illustrate the accuracy and efficiency of ALOOCV as well as our proposed framework for the optimization of the regularizer.

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In contrast, we use 10% of the training set9 for validation, and treat the validation set as apurely held-out test set (this also means that we train on less data).10 Wewillexplainthismoreclearly.30 both spheres are sufficiently tiny (i.e.

Dna demethylation is associated with malignant progression oflower-grade gliomas.Scientific reports,9(1):1-12, 2019.

Previous work showed that despite claims of interpretability, humans are unable to use formal specifications presented in a variety of ways to validate even simple robot behaviors.