Large-scale Optimization of Partial AUC in a Range of False Positive Rates

TL;DR

This paper introduces a scalable optimization algorithm for partial AUC in specific FPR ranges, suitable for deep learning and large datasets, addressing limitations of previous methods.

Contribution

It formulates partial AUC optimization as a non-smooth DC program and develops an efficient stochastic gradient method with theoretical complexity guarantees.

Findings

Effective partial AUC maximization demonstrated on large datasets.

Algorithm outperforms existing methods in scalability and efficiency.

Applicable to deep neural networks and ranked range loss minimization.

Abstract

The area under the ROC curve (AUC) is one of the most widely used performance measures for classification models in machine learning. However, it summarizes the true positive rates (TPRs) over all false positive rates (FPRs) in the ROC space, which may include the FPRs with no practical relevance in some applications. The partial AUC, as a generalization of the AUC, summarizes only the TPRs over a specific range of the FPRs and is thus a more suitable performance measure in many real-world situations. Although partial AUC optimization in a range of FPRs had been studied, existing algorithms are not scalable to big data and not applicable to deep learning. To address this challenge, we cast the problem into a non-smooth difference-of-convex (DC) program for any smooth predictive functions (e.g., deep neural networks), which allowed us to develop an efficient approximated gradient descent method based on the Moreau envelope smoothing technique, inspired by recent advances in non-smooth DC optimization. To increase the efficiency of large data processing, we used an efficient stochastic block coordinate update in our algorithm. Our proposed algorithm can also be used to minimize the sum of ranked range loss, which also lacks efficient solvers. We established a complexity of $\tilde O(1/\epsilon^6)$ for finding a nearly $\epsilon$-critical solution. Finally, we numerically demonstrated the effectiveness of our proposed algorithms for both partial AUC maximization and sum of ranked range loss minimization.