This work presents an information-theoretic perspective to group fairness trade-offs in federated learning (FL) with respect to sensitive attributes, such as gender, race, etc. Existing works often focus on either $\textit{global fairness}$ (overall disparity of the model across all clients) or $\textit{local fairness}$ (disparity of the model at each client), without always considering their trade-offs. There is a lack of understanding regarding the interplay between global and local fairness in FL, particularly under data heterogeneity, and if and when one implies the other. To address this gap, we leverage a body of work in information theory called partial information decomposition (PID), which first identifies three sources of unfairness in FL, namely, $\textit{Unique Disparity}$, $\textit{Redundant Disparity}$, and $\textit{Masked Disparity}$. We demonstrate how these three disparities contribute to global and local fairness using canonical examples. This decomposition helps us derive fundamental limits on the trade-off between global and local fairness, highlighting where they agree or disagree. We introduce the $\textit{Accuracy and Global-Local Fairness Optimality Problem (AGLFOP)}$, a convex optimization that defines the theoretical limits of accuracy and fairness trade-offs, identifying the best possible performance any FL strategy can attain given a dataset and client distribution. We also present experimental results on synthetic datasets and the ADULT dataset to support our theoretical findings.

翻译：本文提出了一种信息论视角，用于分析联邦学习（FL）中与性别、种族等敏感属性相关的群体公平性权衡问题。现有研究通常聚焦于全局公平性（模型在所有客户端上的整体差异）或局部公平性（模型在每个客户端上的差异），却往往忽视两者间的权衡关系。尤其在数据异构场景下，全局与局部公平性之间的相互作用机制尚不明确，且缺乏对两者相互蕴含关系的研究。为填补这一空白，我们借鉴信息论中的部分信息分解（PID）方法，首先识别出联邦学习中公平性问题的三种根源：唯一差异、冗余差异与掩蔽差异。通过典型示例论证了这三种差异如何共同影响全局与局部公平性。该分解方法帮助我们推导出全局与局部公平性权衡的理论极限，阐明两者在何种情形下一致或矛盾。我们提出精度与全局-局部公平性最优问题（AGLFOP），这是一个凸优化问题，定义了精度与公平性权衡的理论边界，能够确定给定数据集与客户端分布下任何联邦学习策略所能达到的最佳性能。最后通过合成数据集与ADULT数据集的实验结果验证理论发现。