  In this paper, we show the arc length of the optimal ROC curve is an
$f$-divergence. By leveraging this result, we express the arc length using a
variational objective and estimate it accurately using positive and negative
samples. We show this estimator has a non-parametric convergence rate
$O_p(n^{-\beta/4})$ ($\beta \in (0,1]$ depends on the smoothness). Using the
same technique, we show the surface area between the optimal ROC curve and the
diagonal can be expressed via a similar variational objective. These new
insights lead to a novel classification procedure that maximizes an approximate
lower bound of the maximal AUC. Experiments on CIFAR-10 datasets show the
proposed two-step procedure achieves good AUC performance in imbalanced binary
classification tasks.
