Algorithmic stability and sanity-check bounds for leave-one-out cross-validation

In this article we prove sanity-check bounds for the error of the leave-one -out cross-validation estimate of the generalization error: that is, bounds showing that the worst-case error of this estimate is not much worse than that of the training error estimate. The name sanity check refers to the fa ct that although we often expect the leave-one-out estimate to perform cons iderably better than the training error estimate, we are here only seeking assurance that its performance will not be considerably worse. Perhaps surp risingly, such assurance has been given only for limited cases in the prior literature on cross-validation. Any nontrivial bound on the error of leave-one-out must rely on some notion of algorithmic stability. Previous bounds relied on the rather strong noti on of hypothesis stability, whose application was primarily limited to near est-neighbor and other local algorithms. Here we introduce the new and weak er notion of error stability and apply it to obtain sanity-check bounds for leave-one-out for other classes of learning algorithms, including training error minimization procedures and Bayesian algorithms. We also provide low er bounds demonstrating the necessity of some form of error stability for p roving bounds on the error of the leave-one-out estimate, and the fact that for training error minimization algorithms, in the worst case such bounds must still depend on the Vapnik-Chervonenkis dimension of the hypothesis cl ass.