  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.
