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

### Song Liu

##### Hall J #922

Keywords: [ density ratio estimation ] [ ROC Curve ] [ AUC maximization ] [ f-divergence ]

[ Abstract ]
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Tue 29 Nov 2 p.m. PST — 4 p.m. PST

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 sandwiched between the optimal ROC curve and the diagonal can be expressed via a similar variational objective. These new insights lead to a novel two-step 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.

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