## 2 Related Work  
### Federated Learning  
In traditional centralized cloud services, large amounts of data are pushed from numerous IoT devices to the cloud, a process that can lead to sensitive data leakage and heavy network burden [11]. Federated learning [12], proposed by Google in 2016, effectively solves the conflict between data privacy and data sharing. Federated learning is an emerging machine learning paradigm that enables collaborative training of models by sharing models between multiple data sources without sharing data and completing parameter updates locally based on models published by model owners. In order to achieve the expected model accuracy, a large number of model parameters in federated learning need to carry out multiple rounds of data transmission through a complex network environment. Therefore, mobile devices are faced with the problems of network congestion and communication failure. To address the above problems, [PERSON] et al. [13] proposed a HierFAVG algorithm, which performs two-stage synchronized model aggregation by extending the traditional FedAvg algorithm to a hierarchical setup, proves the convergence of the training process, and confirms that the HierFAVG algorithm is more advantageous compared to existing federated learning schemes. [PERSON] et al. [14] studied the problem of user edge assignment in the presence of non-IID data with the aim of improving performance. These methods improve the convergence speed and reduce the communication cost of end devices in edge computing systems. Although hierarchical federated learning has certain advantages, the above studies did not consider the application of individual rational hierarchical federated learning in edge computing. Existing federated learning approaches mainly utilize a synchronous model aggregation mechanism, in which a central server needs to wait for the slowest device to complete training in each round of communication. Some authors have proposed to improve the efficiency of FL algorithms by changing the FL aggregation method. Several works have proposed asynchronous model aggregation methods in which only one participating device updates the global model at a time [15; 16; 17]. [PERSON] et al. [18] proposed FedAT, a novel federated learning approach that synergistically combines synchronous internal training and asynchronous cross-training to improve convergence speed and reduce communication costs. The authors in [19] proposed a new asynchronous FL mechanism to harmonize the heterogeneity of devices, communication environments, and learning tasks. However, the number of training rounds under the asynchronous approach is higher than the synchronous approach. In addition, gradient aging can be difficult to control due to asynchronism. Therefore, the authors in [20] designed an n-softsync aggregation model that could significantly reduce training time by combining the advantages of synchronous and asynchronous aggregation. In hierarchical federated learning, when mobile devices participate in learning tasks, device resources, including computing and communication resources, are inevitably consumed. Heterogeneous computing and network resources on client devices can lead to stragglers, which will significantly slow down the FL training process [7]. Most federated learning algorithms assume that all workers participate in training unconditionally [21; 22], but this is not consistent with the reality. Selecting the appropriate mobile devices from the candidate clients to participate in distributed learning becomes critical for high-quality federated learning. In ref. [23], [PERSON] et al. suggested excluding clients with irrelevant model updates to reduce communication overhead. [PERSON] et al. [24] proposed a client selection strategy for FL to accelerate the convergence of the training process for non-IID data. [PERSON] et al. [25] considered the client selection problem for FL in mobile edge computing, trying to maximize the number of selected workers under time constraints. [PERSON] et al.