This paper considers the problem of fair probabilistic binary classification with binary protected groups. The classifier assigns scores, and a practitioner predicts labels using a certain cut-off threshold based on the desired trade-off between false positives vs. false negatives. It derives these thresholds from the ROC of the classifier. The resultant classifier may be unfair to one of the two protected groups in the dataset. It is desirable that no matter what threshold the practitioner uses, the classifier should be fair to both the protected groups; that is, the $\mathcal{L}_p$ norm between FPRs and TPRs of both the protected groups should be at most $\varepsilon$. We call such fairness on ROCs of both the protected attributes $\varepsilon_p$-Equalized ROC. Given a classifier not satisfying $\varepsilon_1$-Equalized ROC, we aim to design a post-processing method to transform the given (potentially unfair) classifier's output (score) to a suitable randomized yet fair classifier. That is, the resultant classifier must satisfy $\varepsilon_1$-Equalized ROC. First, we introduce a threshold query model on the ROC curves for each protected group. The resulting classifier is bound to face a reduction in AUC. With the proposed query model, we provide a rigorous theoretical analysis of the minimal AUC loss to achieve $\varepsilon_1$-Equalized ROC. To achieve this, we design a linear time algorithm, namely \texttt{FROC}, to transform a given classifier's output to a probabilistic classifier that satisfies $\varepsilon_1$-Equalized ROC. We prove that under certain theoretical conditions, \texttt{FROC}\ achieves the theoretical optimal guarantees. We also study the performance of our \texttt{FROC}\ on multiple real-world datasets with many trained classifiers.

翻译：本文研究具有二元受保护群体的公平概率二分类问题。分类器输出评分，实践者根据假阳性与假阴性之间的权衡需求，采用特定截断阈值预测标签。这些阈值通常从分类器的ROC曲线推导得出。由此得到的分类器可能对数据集中的两个受保护群体之一存在不公平性。理想情况下，无论实践者使用何种阈值，分类器都应对两个受保护群体保持公平；即两个受保护群体的假阳性率（FPR）与真阳性率（TPR）之间的$\mathcal{L}_p$范数距离应不超过$\varepsilon$。我们将这种对两个受保护属性ROC曲线的公平性要求称为$\varepsilon_p$-均衡ROC。针对不满足$\varepsilon_1$-均衡ROC的分类器，我们旨在设计一种后处理方法，将给定（可能存在不公平性）分类器的输出（评分）转换为合适的随机化公平分类器，使最终分类器满足$\varepsilon_1$-均衡ROC。首先，我们针对每个受保护群体的ROC曲线引入阈值查询模型。该模型必然导致所得分类器的AUC指标下降。基于所提出的查询模型，我们严格理论分析了实现$\varepsilon_1$-均衡ROC所需的最小AUC损失。为实现这一目标，我们设计了线性时间算法\texttt{FROC}，将给定分类器的输出转换为满足$\varepsilon_1$-均衡ROC的概率分类器。我们证明在特定理论条件下，\texttt{FROC}能够达到理论最优保证。同时，我们在多个真实数据集上使用多种训练好的分类器验证了\texttt{FROC}的实际性能。