Nonlinear Equilibrium Transitions in a Potential Game Model for Federated Learning
Authors: Kang Liu, Ziqi Wang, Enrique Zuazua
Organizations: Institut de Mathématiques de Bourgogne, Université Bourgogne Europe, CNRS, Dijon, 21000, France · Chair for Dynamics, Control, Machine Learning and Numerics – Alexander von Humboldt Professorship, Department of Mathematics, Friedrich-Alexander-Universität Erlangen-Nürnberg, Cauerstrasse 11, Erlangen, 91058, Germany · Chair of Computational Mathematics, Fundación Deusto, Avenida de las Universidades, 24, Bilbao, 48007, Basque Country, Spain · Departamento de Matemáticas, Universidad Autónoma de Madrid, Ciudad Universitaria de Cantoblanco, Madrid, 28049, Spain
In federated learning (FL), a central server typically allocates training efforts to clients. However, from a market-oriented perspective, clients may independently choose their training efforts based on rational self-interest. To study this setting, we propose a potential game framework in which each client's payoff is determined by its individual effort and the rewards provided by the server. The rewards are influenced by the collective efforts of all clients and can be modulated by a reward factor. We first establish the existence of Nash equilibria (NEs) and then investigate their uniqueness in a stationary setting. We show that the NEs depend nonlinearly on the reward factor and exhibit a nonsmooth transition at a critical value, where the stationary potential loses strict curvature, leading to nonunique NEs and a jump between low-effort and high-effort branches. Furthermore, we prove the convergence of the best-response algorithm for computing NEs in our FL game. Finally, we apply the clients' rational efforts derived from the NEs to FL training with various datasets and models, thereby validating the effectiveness of the identified critical reward factor. The source code is available at https://github.com/DCN-FAU-AvH/FL-Potential-Game
Figures & tables
Figure 1 : In the FL game (left), each client i maximizes the payoff Pi , and the NE s∗ satisfies Pi(si∗,s−i∗)≥Pi(si,s−i∗) for all si∈Si . The equilibrium efforts s∗ ( e.g. , local epochs) then drive the FL training stage (right), where the model parameters are updated according to θit+1=Updatei(θt,si∗) .
Figure 2 : Illustration of the nonlinear threshold structure and jump phenomenon.
Symbol
Name
Eq.
Definition and role
λ∗
Jump point
( 3.3 )
The critical reward factor where non-uniqueness occurs.
λ1
Activation point
( 3.8 )
If λ∈(0,λ1) , the unique NE is at minimum efforts.
λ2
Saturation point
( 3.8 )
If λ∈(λ2,+∞) , the unique NE is at maximum efforts.
c1
Lower jump bound
( 3.4 )
If λ∈(λ1,λ∗) , the unique NE satisfies sˉ∗<c1 .
c2
Upper jump bound
( 3.4 )
If λ∈(λ∗,λ2) , the unique NE satisfies sˉ∗>c2 .
Table 1 : Summary of the critical thresholds, see Theorem 3.2 and Corollary 3.7 for more details.
Figure 3 : Two-player stationary potential landscape of Example 3.6 . At λ=λ∗ , a flat ridge appears along s1=s2 , yielding infinitely many NEs on the feasible diagonal segment. In contrast, for λ<λ∗ and λ>λ∗ , the potential has a unique maximizer, corresponding to a unique NE.
Figure 4 : Average training effort sˉ∗ as a function of the server’s reward factor λ . The activation point λ1 , jump point λ∗ , and saturation point λ2 are indicated in the legends.
Balanced, non-IID, m=1,000
Imbalanced, non-IID, m=3,000
λ1=1.998,λ∗=2.904,λ2=4.685
λ1=2.003,λ∗=2.886,λ2=4.790
NE
Test accuracy
NE
Test accuracy
Case
Regime
λ
sˉ∗
CIFAR-10
CIFAR-100
IMDB
λ
sˉ∗
FEMNIST
Case 1
pre-activation
1.99
1.00
0.434
0.120
0.535
2.00
1.00
0.124
Case 2
pre-jump
2.90
1.32
0.477
0.142
0.523
2.88
1.32
0.232
Case 3
post-jump
2.91
16.41
0.558
0.208
0.796
2.89
15.94
0.665
Table 2 : Summary of experimental cases. Cases 1–4 correspond, respectively, to the pre-activation, pre-jump, post-jump, and post-saturation regimes, that is, before λ1 , immediately before λ∗ , immediately after λ∗ , and after λ2 . Test accuracies are averaged across different FL algorithms.
Figure 5 : Results of FL training under four NE cases across different datasets and algorithms. The largest gain is observed when moving from Case 2 to Case 3 in all experiments.
Clustered federated learning benefits from organizing heterogeneous participants into coalitions that train coalition-specific models, but such clustering is sustainable only if participants prefer their assigned coalition and the required transfers are affordable. We develop a transferable-surplus model separating learning benefit, system cost, participant cost, and monetary transfers; an allocation rule converts coalition surplus into hedonic preferences, and weak budget feasibility guarantees nonnegative retained coordinator surplus. For symmetric pairwise allocations the induced game is an exact potential game: a Nash-stable partition exists, every strict better-response process converges, and with destination consent accepted better responses reach an individually stable partition. We characterize feasibility of bounded pair incentives and verify the exponentially many budget constraints in polynomial oracle time when retained slack is submodular. Decomposing welfare into participant potential and retained slack yields additive and multiplicative price-of-stability guarantees, the latter asymptotically tight; exact balance gives welfare-optimal stability only on the pairwise-representable class, and budget feasibility alone permits unbounded welfare loss. Global potential maximization equals weighted maximum-agreement correlation clustering, and approximation followed by stabilization satisfies an end-to-end welfare bound governed by retained slack and negative-edge mass, attained by an explicit construction. In a preregistered five-seed CIFAR-10 study the mechanism reaches the certified estimated-table welfare optimum on every primary instance, equal-surplus sharing has no Nash-stable outcome on three, and pairwise validation gain gives far more reliable pair signs than gradient alignment.
Federated learning enables collaborative model training across distributed clients without centralising their data, yet privacy remains a persistent concern because the shared model updates can leak information about local datasets. Existing privacy-preserving methods either inject calibrated noise into client updates, limiting their composition guarantees, or formulate client privacy choices as a multi-agent game whose Nash equilibrium becomes intractable as the number of clients grows. We bridge these two lines of work by formulating privacy-preserving federated learning as a mean-field privacy game: each client strategically chooses its own privacy budget while interacting with the population only through a single mean-field statistic. The mean-field limit yields a tractable equilibrium for arbitrarily many clients, accommodates heterogeneous client preferences, and inherits an exponentially decaying privacy guarantee through a log-Sobolev contraction. The framework recovers the entropic privacy baseline as the homogeneous special case and the multi-agent privacy game as the finite-population case. Experiments on quadratic regression, logistic regression, and MNIST demonstrate that the proposed framework attains the privacy-utility trade-off of the entropic baseline while delivering a personalized privacy guarantee that the homogeneous baseline cannot express.
Federated Learning (FL) algorithms implicitly assume that clients passively comply with server-side orchestration by sharing local model updates upon server request. However, this overlooks an important aspect in real-world cross-silo environments: clients are often rational agents who may prioritize their utilities such as local model performance over that of the global model. In settings with significant statistical heterogeneity, rational clients may opt out of the federation if the perceived benefits of collaboration fail to meet their local utility thresholds. Such attrition degrades the global model performance and can lead to the collapse of the federated training process. In this work, we introduce FedUCA, (Federated Learning by Utility-Constrained Stochastic Aggregation for Improving Rational Participation), a framework that formalizes the server's role as an optimizer seeking to maximize global model performance by sustaining client participation. We substantiate our framework through extensive experiments on standard datasets demonstrating that by prioritizing participation feasibility, FedUCA achieves significantly higher client retention and, consequently, a superior global model performance.
M Yashwanth, Arunabh Singh, Ashok Nayak +2
Indian Institute of Science, Bangalore · Indian Institute of Technology Bombay · IIIT Hyderabad