## 1 Introduction  
With the development of the internet of things, a huge number of smart devices, such as mobile phones and sensing devices, generate a massive amount of data every day [1; 2; 3]. Relying entirely on a central cloud for global model aggregation will consume considerable bandwidth resources and may result in significant response delays due to the long-haul transmission of massive data [4]. Besides, data privacy is prone to the invasion of unauthorized devices during transmission [5]. Federated edge learning (FEL) is a highly promising paradigm for implementing federated learning (FL)-based solutions in edge computing systems [6]. Compared to FL, FEL enables partial storage and computation resources to be offloaded from the data center (e.g., the cloud platform or one of the edge serves) to the network edge servers (e.g., clients or communication base stations), which reduces the communication frequency and the risk of data leakage [7]. Although FEL has natural advantages in resisting a single point of failure and malicious attacks, the uploaded local models or gradients still pose a risk in terms of privacy leakage. For example, an untrusted cloud platform may infer the sensitive attributes of edge servers from the gradient updates [8], which hinders the willingness of edge servers to participate in FEL. Moreover, the inefficient resource utilization caused by device heterogeneity and the low-quality updates from inactive participants remain open issues in FEL. On the one hand, the performance of FEL systems can be affected by various factors related to both computation and communication [9], such as the data processing and transmission capability of edge servers and the network channel delay of data-collection devices. These factors may prevent each edge server's training progress in global training rounds from being synchronized, which introduces a _synchronization barrier_ in FEL, particularly when requiring uniform update frequencies across all edge servers [4] (_The synchronization barrier_, i.e., the total training delay of an FEL system, is dominated by the slowest edge server, which leads to useless waiting times and the under-utilization of computation resources for efficient edge servers [10]). Some existing efforts have explored strategies, such as dynamic batch-size methods [11] and staleness-bounded updates [12; 13], to alleviate the negative impact of the synchronization barrier. However, a dynamic batch-size method may not adapt well to the heterogeneous computing ability of edge servers, and the staleness-bounded update may have a slower convergence rate due to the potentially stale gradients of stragglers [14]. On the other hand, many existing works that focus on improving system efficiency usually assume unconditional learning resource sharing between the edge servers and the cloud platform [15; 16]. This may not always be possible, as edge servers belong to different entities (e.g., organizations or enterprises) that are unwilling to contribute any computing resource without sufficient financial compensation. It is worth noting that some early and important works incentivize edge servers' effective participation by using auctions [17], contracts [18], and Stackelberg games [19], but they overlook the waiting time of heterogeneous edge servers [18] or focus on motivating low-cost learners [20], which may result in low resource utilization and inefficient model training. In order to address the joint optimization problem of training efficiency and participation incentive in situations where the platform may be untrusted, we need to design a secure and efficient incentive mechanism that considers device heterogeneity and training costs (i.e., computation, communication, and data collection costs), as well as the personal decision-making of strategic edge servers. Nevertheless, edge servers' multi-dimensional private attributes and training strategies make the joint optimization problem nonconvex, and incomplete information further increases the complexity of the analysis.