  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.
