Discrimination in machine learning often arises along multiple dimensions (a.k.a. protected attributes); it is then desirable to ensure \emph{intersectional fairness} -- i.e., that no subgroup is discriminated against. It is known that ensuring \emph{marginal fairness} for every dimension independently is not sufficient in general. Due to the exponential number of subgroups, however, directly measuring intersectional fairness from data is impossible. In this paper, our primary goal is to understand in detail the relationship between marginal and intersectional fairness through statistical analysis. We first identify a set of sufficient conditions under which an exact relationship can be obtained. Then, we prove bounds (easily computable through marginal fairness and other meaningful statistical quantities) in high-probability on intersectional fairness in the general case. Beyond their descriptive value, we show that these theoretical bounds can be leveraged to derive a heuristic improving the approximation and bounds of intersectional fairness by choosing, in a relevant manner, protected attributes for which we describe intersectional subgroups. Finally, we test the performance of our approximations and bounds on real and synthetic data-sets.

翻译：机器学习中的歧视往往沿着多个维度（即受保护属性）产生；此时，确保*交叉公平性*——即没有任何子群体受到歧视——是可取的。已知仅独立确保每个维度的*边际公平性*通常是不够的。然而，由于子群体数量呈指数级增长，直接从数据中衡量交叉公平性是不可能的。本文的主要目标是通过统计分析详细理解边际公平性与交叉公平性之间的关系。我们首先识别出一组充分条件，在此条件下可以获得精确的关系。然后，我们证明了在一般情况下，通过边际公平性和其他有意义的统计量可以轻松计算的交叉公平性高概率边界。除了其描述性价值外，我们还展示了这些理论边界可用于推导启发式方法，通过以相关方式选择描述交叉子群体的受保护属性，从而改进交叉公平性的近似和边界。最后，我们在真实和合成数据集上测试了我们的近似和边界性能。