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 deficiencies to be compensated. To resolve 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 single full-modality global optimum. We propose GeoEvo, which realizes this equilibrium via Fisher--Riemannian evolutionary dynamics: Natural Evolution Strategies for curvature-adaptive local exploration and subspace-constrained particle swarm updates for symbiotic knowledge transfer. GeoEvo admits a Lyapunov potential and, with a monotone acceptance rule, guarantees potential dissipation; in non-convex regimes it achieves an $O(1/\sqrt{T})$ stationarity rate, implying convergence to first-order Nash equilibria, and empirically improves personalization and robustness across diverse modality-missing identities.