Cross-validation: what does it estimate and how well does it do it?

TL;DR

This paper investigates the true nature of cross-validation error estimates, revealing they measure the error of models trained on different data sets, not the same data, and proposes a nested scheme for better confidence intervals.

Contribution

It demonstrates that cross-validation estimates the prediction error of models trained on other datasets, not the same, and introduces a nested cross-validation method for more accurate confidence intervals.

Findings

Cross-validation estimates the error of models trained on different datasets.

Standard confidence intervals often have lower than desired coverage.

Nested cross-validation improves the accuracy of variance estimation.

Abstract

Cross-validation is a widely-used technique to estimate prediction error, but its behavior is complex and not fully understood. Ideally, one would like to think that cross-validation estimates the prediction error for the model at hand, fit to the training data. We prove that this is not the case for the linear model fit by ordinary least squares; rather it estimates the average prediction error of models fit on other unseen training sets drawn from the same population. We further show that this phenomenon occurs for most popular estimates of prediction error, including data splitting, bootstrapping, and Mallow's Cp. Next, the standard confidence intervals for prediction error derived from cross-validation may have coverage far below the desired level. Because each data point is used for both training and testing, there are correlations among the measured accuracies for each fold, and so the usual estimate of variance is too small. We introduce a nested cross-validation scheme to estimate this variance more accurately, and we show empirically that this modification leads to intervals with approximately correct coverage in many examples where traditional cross-validation intervals fail.