Algorithmic Stability and Sanity-Check Bounds for Leave-One-Out Cross-Validation

TLDR

Sanity‑check bounds for leave‑one‑out cross‑validation aim to guarantee its error is not much worse than the training error, a guarantee that has only been established in limited cases and relies on algorithmic stability. The paper seeks to prove such sanity‑check bounds for the leave‑one‑out estimate and to introduce a weaker error‑stability notion that extends these bounds to a broader class of learning algorithms. The authors define a new error‑stability property and use it to derive sanity‑check bounds for leave‑one‑out across training‑error minimization and Bayesian algorithms. They demonstrate that error stability is necessary for these bounds and that, for training‑error minimization algorithms, worst‑case bounds still depend on the VC dimension of the hypothesis class.

Abstract

In this article we prove sanity-check bounds for the error of the leave-oneout 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 fact that although we often expect the leave-one-out estimate to perform considerably better than the training error estimate, we are here only seeking assurance that its performance will not be considerably worse. Perhaps surprisingly, 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 notion of hypothesis stability, whose application was primarily limited to nearest-neighbor and other local algorithms. Here we introduce the new and weaker 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 lower bounds demonstrating the necessity of some form of error stability for proving 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 class.