We study the problem of auditing classifiers with the notion of statistical subgroup fairness. Kearns et al. (2018) has shown that the problem of auditing combinatorial subgroups fairness is as hard as agnostic learning. Essentially all work on remedying statistical measures of discrimination against subgroups assumes access to an oracle for this problem, despite the fact that no efficient algorithms are known for it. If we assume the data distribution is Gaussian, or even merely log-concave, then a recent line of work has discovered efficient agnostic learning algorithms for halfspaces. Unfortunately, the boosting-style reductions given by Kearns et al. required the agnostic learning algorithm to succeed on reweighted distributions that may not be log-concave, even if the original data distribution was. In this work, we give positive and negative results on auditing for the Gaussian distribution: On the positive side, we an alternative approach to leverage these advances in agnostic learning and thereby obtain the first polynomial-time approximation scheme (PTAS) for auditing nontrivial combinatorial subgroup fairness: we show how to audit statistical notions of fairness over homogeneous halfspace subgroups when the features are Gaussian. On the negative side, we find that under cryptographic assumptions, no polynomial-time algorithm can guarantee any nontrivial auditing, even under Gaussian feature distributions, for general halfspace subgroups.

翻译：我们研究利用统计子群公平性概念对分类器进行审计的问题。Kearns等人(2018)已证明，组合子群公平性的审计问题与不可知学习具有同等难度。本质上，所有针对子群歧视统计度量进行修正的工作都假设可以调用该问题的预言机，尽管目前尚不存在已知的高效算法。若假设数据分布为高斯分布或仅对数凹分布，近期一系列研究已发现针对半空间的不可知学习高效算法。然而，Kearns等人提出的提升式归约要求不可知学习算法能够在重新加权的分布上成功运作——即使原始数据分布满足对数凹性，这些加权分布也可能不再具备该性质。本文针对高斯分布下的审计问题给出正反两方面结果：在正面，我们提出替代方法来利用不可知学习领域的这些进展，从而首次获得用于审计非平凡组合子群公平性的多项式时间近似方案(PTAS)：我们展示了当特征服从高斯分布时，如何对同质半空间子群进行统计公平性概念的审计。在反面，我们发现在密码学假设下，即使特征分布为高斯分布，对于一般半空间子群，不存在任何多项式时间算法能保证非平凡审计。