## 2 Related Work  
FEL combines the characteristics of privacy protection in FL and the high-energy efficiency of edge intelligence, which can reduce dependence on central aggregation and implement collaborative learning in large network systems [21]. Although promising, the leakage of clients' sensitive attributes due to an untrusted platform, the insufficient resource utilization caused by device heterogeneity, and the economic incentives required by practical deployment are important but still under-explored problems in FEL. Existing solutions have often focused on improving system efficiency by optimizing the allocation scheme of communication and computational resources [15]. For instance, the authors in [22] presented a compromise solution, named stale synchronous parallel (SSP), to create a trade-off between full synchronization and no synchronization, which sets a pre-defined threshold to enable some efficient devices to upload the model updates earlier than the stragglers. The study of [12] suggested that the proportion of time that each device spends on computation and training be increased by introducing the SSP strategy. However, the SSP method can increase the performance gap between efficient devices and straggler devices, leading to the global model being affected by outdated models and causing convergence issues [13]. Some other efforts attempted to leverage the dynamic batch-size method to alleviate the negative impact of a synchronization barrier [11; 23; 24]. As exemplified by [11], the authors proposed an adaptive algorithm to dynamically adjust the batch size based on the predicted loss and training progress during model training. Although these methods can achieve better system efficiency, they cannot adapt well to the heterogeneous computing capabilities of edge servers. By taking device heterogeneity into account, some methods based on device selection were proposed to mitigate the straggler effect [25; 26; 27]. For instance, [PERSON] et al. [25] proposed a device selection scheme to choose more efficient devices without disclosing private device information, thus ensuring a satisfactory convergence rate. Unfortunately, reducing the platform's waiting time for all heterogeneous devices by excluding the engagement of some straggler devices may come at the cost of more epochs for model convergence and greater communication overhead. Moreover, there is still incomplete resource utilization among the selected heterogeneous edge devices. By drawing inspiration from this, our objective is to select edge servers with higher efficiency to improve training efficiency while making the selected edge servers maximize their resource utilization (e.g., training data size or local training rounds) to compensate for the accuracy loss and performance degradation caused by stragglers. Another important practical challenge faced by FEL comes in the form of economic incentives. Many existing works derived their results based on the assumption that edge servers will participate in FL without seeking any rewards, which may not be practical enough since the communication and computation process often causes huge energy con sumption [11; 22; 25]. There are some important and early works that focus on compensating for clients' training costs to incentivize their participation, but they rarely pay extra attention to resource utilization and also have some limitations. For example, the authors in [28] modeled the client's utility based on one-dimensional attributes, i.e., training data size, to incentivize clients' data contribution. [PERSON] et al. [29] assumed a complete information scenario, where the server has full knowledge of clients' private information. The assumption of complete information may pose a risk of privacy leakage when the parameter server is untrusted, while the corresponding solutions are also not easily generalized when clients possess multi-dimensional attributes and keep them private. [PERSON] et al.