Nonlinear equilibrium transitions in a potential game model for federated learning
| dc.contributor.author | Liu, Kang | |
| dc.contributor.author | Wang, Ziqi | |
| dc.contributor.author | Zuazua, Enrique | |
| dc.date.accessioned | 2026-08-04T06:32:18Z | |
| dc.date.available | 2026-08-04T06:32:18Z | |
| dc.date.issued | 2026-10 | |
| dc.date.updated | 2026-08-04T06:32:18Z | |
| dc.description.abstract | 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. | en |
| dc.description.sponsorship | The work was partially supported by the European Research Council (ERC) under the European Union’s Horizon 2030 research and innovation programme (grant agreement NO: 101096251-CoDeFeL); by the Alexander von Humboldt Professorship program; the European Union’s Horizon Europe MSCA project ModConFlex (HORIZON-MSCA-2021-DN-01 project 101073558); the Transregio 154 Project “Mathematical Modelling, Simulation and Optimization Using the Example of Gas Networks” of the DFG; the AFOSR 24IOE027 project; the SURE-AI Norwegian Centre for Sustainable, Risk-Averse, and Ethical AI grant 357482, Research Council of Norway; by the Grant PID2023-146872OB-I00-DyCMaMod of MICIU (Spain) and by the COST Actions CA24122 - Multiscale Stochastics, Patterns, and Analysis of Combinatorial Environments and CA24136 - Interactions between Control Theory and Machine Learning. | en |
| dc.identifier.citation | Liu, K., Wang, Z., & Zuazua, E. (2026). Nonlinear equilibrium transitions in a potential game model for federated learning. Physica D: Nonlinear Phenomena, 495. https://doi.org/10.1016/J.PHYSD.2026.135288 | |
| dc.identifier.doi | 10.1016/J.PHYSD.2026.135288 | |
| dc.identifier.issn | 0167-2789 | |
| dc.identifier.uri | https://hdl.handle.net/20.500.14454/6454 | |
| dc.language.iso | eng | |
| dc.publisher | Elsevier B.V. | |
| dc.rights | © 2026 The Author(s) | |
| dc.subject.other | Best-response algorithm | |
| dc.subject.other | Federated learning | |
| dc.subject.other | Nash equilibrium | |
| dc.subject.other | Potential game | |
| dc.title | Nonlinear equilibrium transitions in a potential game model for federated learning | en |
| dc.type | journal article | |
| dcterms.accessRights | open access | |
| oaire.citation.title | Physica D: Nonlinear Phenomena | |
| oaire.citation.volume | 495 | |
| oaire.licenseCondition | https://creativecommons.org/licenses/by/4.0/ | |
| oaire.version | VoR |
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