Federated learning (FL) is capable of performing large distributed machine learning tasks across multiple edge users by periodically aggregating trained local parameters. To address key challenges of enabling FL over a wireless fog-cloud system (e.g., non-i.i.d. data, users’ heterogeneity), we first propose an efficient FL algorithm based on Federated Averaging (called

$\mathsf {FedFog}$

) to perform the local aggregation of gradient parameters at fog servers and global training update at the cloud. Next, we employ

$\mathsf {FedFog}$

in wireless fog-cloud systems by investigating a novel network-aware FL optimization problem that strikes the balance between the global loss and completion time. An iterative algorithm is then developed to obtain a precise measurement of the system performance, which helps design an efficient stopping criteria to output an appropriate number of global rounds. To mitigate the straggler effect, we propose a flexible user aggregation strategy that trains fast users first to obtain a certain level of accuracy before allowing slow users to join the global training updates. Extensive numerical results using several real-world FL tasks are provided to verify the theoretical convergence of

$\mathsf {FedFog}$

. We also show that the proposed co-design of FL and communication is essential to substantially improve resource utilization while achieving comparable accuracy of the learning model.