  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.
