Consistency of Local and Global Flatness for Federated Learning

In federated learning (FL), multi-step local updates and data heterogeneity usually lead to sharper global minima, which degrades the performance of the global model. Popular FL algorithms integrate sharpness-aware minimization (SAM) into local training to address this issue. However, in the high data heterogeneity setting, the flatness in local training does not imply the flatness of the global model. Therefore, minimizing the sharpness of the local loss surfaces on the client data does not enable the effectiveness of SAM in FL to improve the generalization ability of the global model. We define the \textbf{flatness distance} to explain this phenomenon. By rethinking the SAM in FL and theoretically analyzing the \textbf{flatness distance}, we propose a novel \textbf{FedNSAM} algorithm that accelerates the SAM algorithm by introducing global Nesterov momentum into the local update to harmonize the consistency of global and local flatness. \textbf{FedNSAM} uses the global Nesterov momentum as the direction of local estimation of client global perturbations and extrapolation. Theoretically, we prove a tighter convergence bound than FedSAM by Nesterov extrapolation. Empirically, we conduct comprehensive experiments on CNN and Transformer models to verify the superior performance and efficiency of \textbf{FedNSAM}. The code is available at https://github.com/junkangLiu0/FedNSAM.