Blum & Stangl (2019) propose a data bias model to simulate
under-representation and label bias in underprivileged population. For a
stylized data distribution with i.i.d. label noise, under certain simple
conditions on the bias parameters, they show that fair classification with
equal opportunity constraints even on extremely biased distribution can recover
an optimally accurate and fair classifier on the original distribution.
Although their distribution is stylized, their result is interesting because it
demonstrates that fairness constraints can implicitly rectify data bias and
simultaneously overcome a perceived fairness-accuracy trade-off. In this paper,
we give an alternate proof of their result using threshold-based
characterization of optimal fair classifiers. Moreover, we show that their
conditions on the bias parameters are both necessary and sufficient for their
recovery result. Our technique is arguably more flexible, as it readily extends
to more general distributions, e.g., when the labels in the original
distribution have Massart noise instead of i.i.d. noise. Finally, we prove that
for any data distribution, if the optimally accurate classifier in a hypothesis
class is fair and robust, then it can be recovered through fair classification
on the biased distribution, whenever the bias parameters satisfy certain simple
conditions.

通过阈值为基础的最优公平分类器的特征，本文给出了 Blum & Stangl (2019) 结果的另一证明，并证明了其对于偏倚参数的条件既是必要的又是充分的。此外，我们还证明了对于任意数据分布，只要优化准确分类器在假设类别中是公平和稳健的，那么在偏倚分布上进行公平分类就可以恢复该分类器，前提是偏倚参数满足一定的简单条件。