New perspectives in cross-validation

Appealing due to its universality, cross-validation is an ubiquitous tool for model tuning and selection. At its core, cross-validation proposes to split the data (potentially several times), and alternatively use some of the data for fitting a model and the rest for testing the model. This produces a reliable estimate of the risk, although many questions remain concerning how best to compare such estimates across different models. Despite its widespread use, many theoretical problems remain unanswered for cross-validation, particularly in high-dimensional regimes where bias issues are non-negligible. We first provide an asymptotic analysis of the cross-validated risk in relation to the train-test split risk for a large class of estimators under stability conditions. This asymptotic analysis is expressed in the form of a central limit theorem, and allows us to characterize the speed-up of the cross-validation procedure for general parametric M-estimators. In particular, we show that when the loss used for fitting differs from that used for evaluation, k-fold cross-validation may offer a reduction in variance less (or greater) than k. We then turn our attention to the high-dimensional regime (where the number of parameters is comparable to the number of observations). In such a regime, k-fold cross-validation presents asymptotic bias, and hence increasing the number of folds is of interest. We study the extreme case of leave-one-out cross-validation, and show that, for generalized linear models under smoothness conditions, it is a consistent estimate of the risk at the optimal rate. Given the large computational requirements of leave-one-out cross-validation, we finally consider the problem of obtaining a fast approximate version of the leave-one-out cross-validation (ALO) estimator. We propose a general strategy for deriving formulas for such ALO estimators for penalized generalized linear models, and apply it to many common estimators such as the LASSO, SVM, nuclear norm minimization. The performance of such approximations are evaluated on simulated and real datasets

High-dimensional variable selection

This paper explores the following question: what kind of statistical guarantees can be given when doing variable selection in high-dimensional models? In particular, we look at the error rates and power of some multi-stage regression methods. In the first stage we fit a set of candidate models. In the second stage we select one model by cross-validation. In the third stage we use hypothesis testing to eliminate some variables. We refer to the first two stages as "screening" and the last stage as "cleaning." We consider three screening methods: the lasso, marginal regression, and forward stepwise regression. Our method gives consistent variable selection under certain conditions.Comment: Published in at http://dx.doi.org/10.1214/08-AOS646 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Concentration inequalities of the cross-validation estimate for stable predictors

In this article, we derive concentration inequalities for the cross-validation estimate of the generalization error for stable predictors in the context of risk assessment. The notion of stability has been first introduced by \cite{DEWA79} and extended by \cite{KEA95}, \cite{BE01} and \cite{KUNIY02} to characterize class of predictors with infinite VC dimension. In particular, this covers $k$-nearest neighbors rules, bayesian algorithm (\cite{KEA95}), boosting,... General loss functions and class of predictors are considered. We use the formalism introduced by \cite{DUD03} to cover a large variety of cross-validation procedures including leave-one-out cross-validation, $k$-fold cross-validation, hold-out cross-validation (or split sample), and the leave-$\upsilon$-out cross-validation. In particular, we give a simple rule on how to choose the cross-validation, depending on the stability of the class of predictors. In the special case of uniform stability, an interesting consequence is that the number of elements in the test set is not required to grow to infinity for the consistency of the cross-validation procedure. In this special case, the particular interest of leave-one-out cross-validation is emphasized