Estimating the Arc Length of the Optimal ROC Curve and Lower Bounding the Maximal AUC

TL;DR

This paper introduces a method to estimate the arc length of the optimal ROC curve as an $f$-divergence, providing a way to lower bound the maximal AUC and improve classification performance.

Contribution

The paper establishes the arc length of the optimal ROC curve as an $f$-divergence and develops a non-parametric estimator with a convergence rate, leading to a new AUC maximization procedure.

Findings

Estimator achieves a convergence rate of $O_p(n^{-eta/4})$

Proposed method effectively bounds the maximal AUC from below

Experiments demonstrate improved AUC on CIFAR-10 in imbalanced settings

Abstract

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.