While it has long been empirically observed that adversarial robustness may be at odds with standard accuracy and may have further disparate impacts on different classes, it remains an open question to what extent such observations hold and how the class imbalance plays a role within. In this paper, we attempt to understand this question of accuracy disparity by taking a closer look at linear classifiers under a Gaussian mixture model. We decompose the impact of adversarial robustness into two parts: an inherent effect that will degrade the standard accuracy on all classes due to the robustness constraint, and the other caused by the class imbalance ratio, which will increase the accuracy disparity compared to standard training. Furthermore, we also show that such effects extend beyond the Gaussian mixture model, by generalizing our data model to the general family of stable distributions. More specifically, we demonstrate that while the constraint of adversarial robustness consistently degrades the standard accuracy in the balanced class setting, the class imbalance ratio plays a fundamentally different role in accuracy disparity compared to the Gaussian case, due to the heavy tail of the stable distribution. We additionally perform experiments on both synthetic and real-world datasets to corroborate our theoretical findings. Our empirical results also suggest that the implications may extend to nonlinear models over real-world datasets. Our code is publicly available on GitHub at https://github.com/Accuracy-Disparity/AT-on-AD.

翻译：尽管长期以来经验观察表明，对抗鲁棒性可能与标准准确性存在矛盾，并可能对不同类别产生进一步的差异化影响，但此类观察结论在多大程度上成立、以及类别不平衡在其中扮演何种角色，仍是悬而未决的问题。本文通过深入分析高斯混合模型下的线性分类器，试图理解这一准确性差异问题。我们将对抗鲁棒性的影响分解为两部分：一是固有效应，即由于鲁棒性约束导致所有类别的标准准确性下降；二是由类别不平衡比例引起的效应，相较于标准训练，该效应将加剧准确性差异。此外，通过将数据模型推广至稳定分布族，我们进一步证明此类影响并不仅限于高斯混合模型。具体而言，我们证明：虽然平衡类别设置下对抗鲁棒性的约束会持续降低标准准确性，但由于稳定分布的重尾特性，类别不平衡比例在准确性差异中扮演的角色与高斯情形存在根本性不同。我们还在合成数据集和真实数据集上进行了实验，以验证理论发现。实证结果同时表明，这些结论可能推广至真实数据集上的非线性模型。我们的代码已在GitHub上公开，地址为：https://github.com/Accuracy-Disparity/AT-on-AD。