Is Cross-validation the Gold Standard to Estimate Out-of-sample Model Performance?

Cross-Validation (CV) is the default choice for estimate the out-of-sample performance of machine learning models. Despite its wide usage, their statistical benefits have remained half-understood, especially in challenging nonparametric regimes. In this paper we fill in this gap and show that, in terms of estimating the out-of-sample performances, for a wide spectrum of models, CV does not statistically outperform the simple ``plug-in'' approach where one reuses training data for testing evaluation. Specifically, in terms of both the asymptotic bias and coverage accuracy of the associated interval for out-of-sample evaluation, $K$-fold CV provably cannot outperform plug-in regardless of the rate at which the parametric or nonparametric models converge. Leave-one-out CV can have a smaller bias as compared to plug-in; however, this bias improvement is negligible compared to the variability of the evaluation, and in some important cases leave-one-out again does not outperform plug-in once this variability is taken into account. We obtain our theoretical comparisons via a novel higher-order Taylor analysis that dissects the limit theorems of testing evaluations, which applies to model classes that are not amenable to previously known sufficient conditions. Our numerical results demonstrate that plug-in performs indeed no worse than CV in estimating model performance across a wide range of examples.