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

arXiv: https://arxiv.org/abs/2110.09651 · HTML: https://mlsanity.com/paper/2110.09651/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 47 references that are in the map.
- First posted on arXiv 18 Oct 2021.

## Builds on (most cited first, 7 of 7)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [f-GAN: Training Generative Neural Samplers using Variational Divergence Minimization](https://mlsanity.com/paper/1606.00709/) (2016)
- [PacGAN: The power of two samples in generative adversarial networks](https://mlsanity.com/paper/1712.04086/) (2017)
- [One-Pass AUC Optimization](https://mlsanity.com/paper/1305.1363/) (2013)
- [Density Estimation in Infinite Dimensional Exponential Families](https://mlsanity.com/paper/1312.3516/) (2013)
- [Stochastic Proximal AUC Maximization](https://mlsanity.com/paper/1906.06053/) (2019)

## Abstract (arXiv)

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 Op(n-β/4) (β ∈ (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.