While federated learning has shown strong results in optimizing a machine learning model without direct access to the original data, its performance may be hindered by intermittent client availability which slows down the convergence and biases the final learned model. There are significant challenges to achieve both stable and bias-free training under arbitrary client availability. To address these challenges, we propose a framework named Federated Graph-based Sampling (FedGS), to stabilize the global model update and mitigate the long-term bias given arbitrary client availability simultaneously. First, we model the data correlations of clients with a Data-Distribution-Dependency Graph (3DG) that helps keep the sampled clients data apart from each other, which is theoretically shown to improve the approximation to the optimal model update. Second, constrained by the far-distance in data distribution of the sampled clients, we further minimize the variance of the numbers of times that the clients are sampled, to mitigate long-term bias. To validate the effectiveness of FedGS, we conduct experiments on three datasets under a comprehensive set of seven client availability modes. Our experimental results confirm FedGS's advantage in both enabling a fair client-sampling scheme and improving the model performance under arbitrary client availability. Our code is available at \url{https://github.com/WwZzz/FedGS}.

Nowadays, the industrial Internet of Things (IIoT) has played an integral role in Industry 4.0 and produced massive amounts of data for industrial intelligence. These data locate on decentralized devices in modern factories. To protect the confidentiality of industrial data, federated learning (FL) was introduced to collaboratively train shared machine learning models. However, the local data collected by different devices skew in class distribution and degrade industrial FL performance. This challenge has been widely studied at the mobile edge, but they ignored the rapidly changing streaming data and clustering nature of factory devices, and more seriously, they may threaten data security. In this paper, we propose FedGS, which is a hierarchical cloud-edge-end FL framework for 5G empowered industries, to improve industrial FL performance on non-i.i.d. data. Taking advantage of naturally clustered factory devices, FedGS uses a gradient-based binary permutation algorithm (GBP-CS) to select a subset of devices within each factory and build homogeneous super nodes participating in FL training. Then, we propose a compound-step synchronization protocol to coordinate the training process within and among these super nodes, which shows great robustness against data heterogeneity. The proposed methods are time-efficient and can adapt to dynamic environments, without exposing confidential industrial data in risky manipulation. We prove that FedGS has better convergence performance than FedAvg and give a relaxed condition under which FedGS is more communication-efficient. Extensive experiments show that FedGS improves accuracy by 3.5% and reduces training rounds by 59% on average, confirming its superior effectiveness and efficiency on non-i.i.d. data.