There is an increasing interest in a fast-growing machine learning technique called Federated Learning, in which the model training is distributed over mobile user equipments (UEs), exploiting UEs local computation and training data. Despite its advantages in data privacy-preserving, Federated Learning (FL) still has challenges in heterogeneity across UEs data and physical resources. We first propose a FL algorithm which can handle the heterogeneous UEs data challenge without further assumptions except strongly convex and smooth loss functions. We provide the convergence rate characterizing the trade-off between local computation rounds of UE to update its local model and global communication rounds to update the FL global model. We then employ the proposed FL algorithm in wireless networks as a resource allocation optimization problem that captures the trade-off between the FL convergence wall clock time and energy consumption of UEs with heterogeneous computing and power resources. Even though the wireless resource allocation problem of FL is non-convex, we exploit this problems structure to decompose it into three sub-problems and analyze their closed-form solutions as well as insights to problem design. Finally, we illustrate the theoretical analysis for the new algorithm with Tensorflow experiments and extensive numerical results for the wireless resource allocation sub-problems. The experiment results not only verify the theoretical convergence but also show that our proposed algorithm outperforms the vanilla FedAvg algorithm in terms of convergence rate and testing accuracy.
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