Providing various machine learning (ML) applications in the real world, concerns about discrimination hidden in ML models are growing, particularly in high-stakes domains. Existing techniques for assessing the discrimination level of ML models include commonly used group and individual fairness measures. However, these two types of fairness measures are usually hard to be compatible with each other, and even two different group fairness measures might be incompatible as well. To address this issue, we investigate to evaluate the discrimination level of classifiers from a manifold perspective and propose a "harmonic fairness measure via manifolds (HFM)" based on distances between sets. Yet the direct calculation of distances might be too expensive to afford, reducing its practical applicability. Therefore, we devise an approximation algorithm named "Approximation of distance between sets (ApproxDist)" to facilitate accurate estimation of distances, and we further demonstrate its algorithmic effectiveness under certain reasonable assumptions. Empirical results indicate that the proposed fairness measure HFM is valid and that the proposed ApproxDist is effective and efficient.

翻译：随着机器学习（ML）在现实世界中的广泛应用，人们对ML模型中隐藏的歧视问题日益关注，尤其是在高风险领域。现有评估ML模型歧视程度的技术包括常用的群体公平性与个体公平性度量。然而，这两类公平性度量通常难以相互兼容，甚至两种不同的群体公平性度量也可能存在冲突。为解决这一问题，我们从流形视角研究分类器歧视程度的评估方法，并提出一种基于集合间距离的"流形调和公平性度量（HFM）"。但距离的直接计算可能成本过高，降低了其实用性。因此，我们设计了一种名为"集合间距离近似算法（ApproxDist）"的近似方法，以促进距离的精确估计，并在合理假设下进一步证明了其算法有效性。实验结果表明，所提出的公平性度量HFM是有效的，而ApproxDist算法兼具高效性与有效性。