V-fold penalization: an alternative to V-fold cross-validation

Sylvain Arlot

Université Paris-Sud


We study the efficiency of $V$-fold cross-validation (VFCV) for model selection from the non-asymptotic viewpoint, and suggest an improvement on it, which we call ``$V$-fold penalization''.

First, considering a particular (though simple) regression problem, we prove that VFCV with a bounded $V$ is suboptimal for model selection. The main reason for this is that VFCV ``overpenalizes'' all the more that $V$ is large. Hence, asymptotic optimality requires $V$ to go to infinity. However, when the signal-to-noise ratio is low, it appears that overpenalizing is necessary, so that the optimal $V$ is not always the larger one, despite of the variability issue. This is confirmed by some simulated data.

In order to improve on the prediction performance of VFCV, we define a new model selection procedure, called ``$V$-fold penalization'' (penVF). It is a $V$-fold subsampling version of Efron's bootstrap penalties, so that it has the same computational cost as VFCV, while being more flexible. In a heteroscedastic regression framework, assuming the models to have a particular structure, we prove that penVF satisfies a non-asymptotic oracle inequality with a leading constant close to 1. In particular, this implies adaptivity to the smoothness of the regression function, even with a highly heteroscedastic noise. Moreover, it is easy to overpenalize with penVF, independently from the $V$ parameter. According to a simulation study, this results in a significant improvement on VFCV in non-asymptotic situations.