We consider the vulnerability of fairness-constrained learning to small amounts of malicious noise in the training data. Konstantinov and Lampert (2021) initiated the study of this question and presented negative results showing there exist data distributions where for several fairness constraints, any proper learner will exhibit high vulnerability when group sizes are imbalanced. Here, we present a more optimistic view, showing that if we allow randomized classifiers, then the landscape is much more nuanced. For example, for Demographic Parity we show we can incur only a $\Theta(\alpha)$ loss in accuracy, where $\alpha$ is the malicious noise rate, matching the best possible even without fairness constraints. For Equal Opportunity, we show we can incur an $O(\sqrt{\alpha})$ loss, and give a matching $\Omega(\sqrt{\alpha})$lower bound. In contrast, Konstantinov and Lampert (2021) showed for proper learners the loss in accuracy for both notions is $\Omega(1)$. The key technical novelty of our work is how randomization can bypass simple "tricks" an adversary can use to amplify his power. We also consider additional fairness notions including Equalized Odds and Calibration. For these fairness notions, the excess accuracy clusters into three natural regimes $O(\alpha)$,$O(\sqrt{\alpha})$ and $O(1)$. These results provide a more fine-grained view of the sensitivity of fairness-constrained learning to adversarial noise in training data.

翻译：本文研究了公平约束学习对训练数据中小量恶意噪声的脆弱性。Konstantinov与Lampert（2021）首次探讨了该问题，并给出消极结论：在特定数据分布下，当群体规模失衡时，任何恰当学习器（proper learner）对于若干公平约束均表现出高度脆弱性。本文提出更乐观的观点，表明若允许随机分类器存在，则情形更为复杂。例如，对于人口统计均等（Demographic Parity），我们仅需承受$\Theta(\alpha)$的准确率损失（其中$\alpha$为恶意噪声率），这与无公平约束时的最优结果一致。对于机会均等（Equal Opportunity），我们可承受$O(\sqrt{\alpha})$的损失，并给出匹配的下界$\Omega(\sqrt{\alpha})$。与之对比，Konstantinov与Lampert（2021）证明恰当学习器在这两种约束下的准确率损失均为$\Omega(1)$。本文的关键技术贡献在于揭示随机化如何规避对手放大能力的简单"诡计"。我们还考虑了其他公平性概念，包括几率均等（Equalized Odds）与校准（Calibration）。对于这些公平性概念，超额准确率损失自然归入三个区间：$O(\alpha)$、$O(\sqrt{\alpha})$和$O(1)$。这些结果为公平约束学习对训练数据中对抗噪声的敏感性提供了更细致的分析视角。