GeoEvo: Identity-Aware Potential Game with Geometric Evolution for Personalized Multimodal Federated Learning

We reconceptualize Personalized Multimodal Federated Learning (PMFL) by treating missing modalities as intrinsic structural identities that constrain each client to a distinct Riemannian submanifold, rather than as deficiencies to be compensated. To reconcile the tension between identity preservation and cross-client collaboration, we cast PMFL as an identity-aware potential game and seek a geometry-consistent equilibrium instead of a monolithic full-modality optimum. We propose GeoEvo, a federated approximate solver that combines curvature-adaptive Fisher descent with manifold-lifted evolutionary search: Natural Evolution Strategies for basin escape and Particle Swarm updates anchored to a server-broadcast Fréchet prototype for cross-client transfer. A monotone selection rule guarantees surrogate descent and an O(1/ √ T ) stationarity rate; controlled lift residuals further imply expected potential dissipation and, when vanishing, convergence toward first-order Nash equilibria. Empirically, GeoEvo improves personalization and robustness across diverse missing-modality patterns.