A U-statistic estimator for the variance of resampling-based error estimators
This work provides a method for more reliable error estimation in machine learning, which is incremental but improves statistical rigor in algorithm comparison.
The paper tackles the problem of estimating the variance of resampling-based error estimators in binary classification by proposing a U-statistic estimator, which is unbiased and asymptotically normally distributed, enabling an asymptotically exact hypothesis test for comparing two learning algorithms.
We revisit resampling procedures for error estimation in binary classification in terms of U-statistics. In particular, we exploit the fact that the error rate estimator involving all learning-testing splits is a U-statistic. Thus, it has minimal variance among all unbiased estimators and is asymptotically normally distributed. Moreover, there is an unbiased estimator for this minimal variance if the total sample size is at least the double learning set size plus two. In this case, we exhibit such an estimator which is another U-statistic. It enjoys, again, various optimality properties and yields an asymptotically exact hypothesis test of the equality of error rates when two learning algorithms are compared. Our statements apply to any deterministic learning algorithms under weak non-degeneracy assumptions.