Abstract: Federated Learning (FL) is an emerging collaborative machine learning framework where multiple clients train the global model without sharing their own datasets. In FL, the model inconsistency caused by the local data heterogeneity across clients results in the near-orthogonality of client updates, which leads to the global update norm reduction and slows down the convergence. Most previous works focus on eliminating the difference of parameters (or gradients) between the local and global models, which may fail to reflect the model inconsistency due to the complex structure of the machine learning model and the Euclidean space's limitation in meaningful geometric representations.In this paper, we propose FedMRUR by adopting the manifold model fusion scheme and a new global optimizer to alleviate the negative impacts.Concretely, FedMRUR adopts a hyperbolic graph manifold regularizer enforcing the representations of the data in the local and global models are close to each other in a low-dimensional subspace. Because the machine learning model has the graph structure, the distance in hyperbolic space can reflect the model bias better than the Euclidean distance.In this way, FedMRUR exploits the manifold structures of the representations to significantly reduce the model inconsistency.FedMRUR also aggregates the client updates norms as the global update norm, which can appropriately enlarge each client's contribution to the global update, thereby mitigating the norm reduction introduced by the near-orthogonality of client updates.Furthermore, we theoretically prove that our algorithm can achieve a linear speedup property $\mathcal{O}(\frac{1}{\sqrt{SKT}})$ for non-convex setting under partial client participation, where $S$ is the participated clients number, $K$ is the local interval and $T$ is the total number of communication rounds.Experiments demonstrate that FedMRUR can achieve a new state-of-the-art (SOTA) accuracy with less communication.