Closing the Approximation Gap of Partial AUC Optimization: A Tale of Two Formulations
Yangbangyan Jiang, Qianqian Xu, Huiyang Shao, Zhiyong Yang, Shilong Bao, Xiaochun Cao, Qingming Huang
TL;DR
This work tackles the NP-hard challenge of optimizing Partial AUC by introducing two instance-wise minimax reformulations: an OPAUC path with an asymptotically vanishing bias and an unbiased reformulation for TPAUC. By decoupling pairwise dependencies and replacing top-k selection with differentiable threshold learning, the authors achieve linear per-iteration complexity and a convergence rate of O(ε^{-3}). They provide generalization bounds that illuminate the impact of α/β constraints and demonstrate empirical superiority over strong baselines on long-tailed benchmarks. The proposed ASGDA optimization, along with a rigorous unbiased/path-specific analysis, makes scalable PAUC optimization practical for real-world imbalanced problems.
Abstract
As a variant of the Area Under the ROC Curve (AUC), the partial AUC (PAUC) focuses on a specific range of false positive rate (FPR) and/or true positive rate (TPR) in the ROC curve. It is a pivotal evaluation metric in real-world scenarios with both class imbalance and decision constraints. However, selecting instances within these constrained intervals during its calculation is NP-hard, and thus typically requires approximation techniques for practical resolution. Despite the progress made in PAUC optimization over the last few years, most existing methods still suffer from uncontrollable approximation errors or a limited scalability when optimizing the approximate PAUC objectives. In this paper, we close the approximation gap of PAUC optimization by presenting two simple instance-wise minimax reformulations: one with an asymptotically vanishing gap, the other with the unbiasedness at the cost of more variables. Our key idea is to first establish an equivalent instance-wise problem to lower the time complexity, simplify the complicated sample selection procedure by threshold learning, and then apply different smoothing techniques. Equipped with an efficient solver, the resulting algorithms enjoy a linear per-iteration computational complexity w.r.t. the sample size and a convergence rate of $O(ε^{-1/3})$ for typical one-way and two-way PAUCs. Moreover, we provide a tight generalization bound of our minimax reformulations. The result explicitly demonstrates the impact of the TPR/FPR constraints $α$/$β$ on the generalization and exhibits a sharp order of $\tilde{O}(α^{-1}\n_+^{-1} + β^{-1}\n_-^{-1})$. Finally, extensive experiments on several benchmark datasets validate the strength of our proposed methods.