  In the regression setting, given a set of hyper-parameters, a
model-estimation procedure constructs a model from training data. The optimal
hyper-parameters that minimize generalization error of the model are usually
unknown. In practice they are often estimated using split-sample validation. Up
to now, there is an open question regarding how the generalization error of the
selected model grows with the number of hyper-parameters to be estimated. To
answer this question, we establish finite-sample oracle inequalities for
selection based on a single training/test split and based on cross-validation.
We show that if the model-estimation procedures are smoothly parameterized by
the hyper-parameters, the error incurred from tuning hyper-parameters shrinks
at nearly a parametric rate. Hence for semi- and non-parametric
model-estimation procedures with a fixed number of hyper-parameters, this
additional error is negligible. For parametric model-estimation procedures,
adding a hyper-parameter is roughly equivalent to adding a parameter to the
model itself. In addition, we specialize these ideas for penalized regression
problems with multiple penalty parameters. We establish that the fitted models
are Lipschitz in the penalty parameters and thus our oracle inequalities apply.
This result encourages development of regularization methods with many penalty
parameters.
