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”. Considering a particular (though simple) regression problem, we prove that VFCV with a bounded V is suboptimal for model selection, because it “overpenalizes” all the more that V is large. Hence, asymptotic opti...