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Does Egalitarian Fairness Lead to Instability? The Fairness Bounds in Stable Federated Learning Under Altruistic Behaviors
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Theoretically characterizes egalitarian fairness vs. core stability in federated learning under altruistic client behaviors, modeled as altruism coalition formation games, and derives fairness bounds as functions of dataset-size disparities, validated on mean-estimation and linear-regression simulations.
Problem
Whether pursuing egalitarian fairness (uniform performance across clients) in federated learning destabilizes coalitions, since data-rich clients may leave; the achievable fairness level under core stability is unknown, especially when clients are altruistic toward friends.
Approach
Models FL with altruistic friend networks as altruism coalition formation games (ACFGs) where utilities depend on friends across coalitions, and derives tight egalitarian fairness bounds under selfish, welfare-altruistic, equal-altruistic and friendly-altruistic behaviors, showing fairness does not inevitably cause instability.
Key results
- Theoretical egalitarian fairness bounds align with empirical fairness achieved in the core-stable grand coalition under various client behaviors and friend networks (Figures 2-3).
- Under purely equal altruistic or friendly altruistic behavior with a fully connected network, the grand coalition is core-stable with fairness bound 1.306/0.734 ≈ 1.78, whereas under purely selfish behavior the grand coalition is not core-stable.
- Corollary 2 and 3 show strict egalitarian fairness (λ=1) is asymptotically achievable under equal dataset sizes (selfish) or when all clients are friends with the smallest-data client and Ng→∞ (welfare altruistic).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Replace purely selfish client behavior with altruistic behavior in a fully connected friend network | Federated learning coalition · Mean estimation, 4 clients, fully connected network (Relation I) | egalitarian fairness bound (λ): 2.06 -> 1.78 | Helped |
| Change the friend network from fully connected (Relation I) to partially connected (Relation II) | Altruistic federated learning coalition · Mean estimation, 4 clients | egalitarian fairness bound (λ): 1.78 -> 2.06 | Hurt |
| Use equal local dataset sizes across clients | Purely selfish federated learning coalition · Core-stable grand coalition | egalitarian fairness bound (λ): λ = 1 | Helped |
| Make every client friends with the client holding the smallest dataset and let total dataset size grow | Purely welfare-altruistic federated learning coalition · Core-stable grand coalition | egalitarian fairness bound (λ): null | Helped |
| Account for client altruism and friend-network topology when setting fairness | Federated learning coalition · Fair federated learning | core stability: null | Mixed |
Surprising findings
- Egalitarian fairness does not necessarily cause instability; the achievable fairness depends on altruism type and friend-network topology.
- The AHG formulation can yield core-stable structures that are not Pareto-optimal in local errors, whereas the ACFG grand coalition is Pareto-optimal.
Limitations
- Analysis relies on the stylized exact-error mean estimation model of Donahue et al.; does not cover overfitting or realistic training.
- Only egalitarian fairness considered; proportional fairness left for future work.
- Experiments are small synthetic settings with 3-4 clients.
Experimental setup
Datasets: synthetic mean estimation data synthetic linear regression data
Benchmarks: synthetic mean estimation task synthetic linear regression task
Metrics: egalitarian fairness bound (error ratio) empirically achieved egalitarian fairness
Baselines: altruistic hedonic games (AHG) purely selfish clients
Builds on: FedAvg altruistic hedonic games hedonic game formulation of FL (Donahue et al.)
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