To address the challenges posed by the heterogeneity inherent in federated learning (FL) and to attract high-quality clients, various incentive mechanisms have been employed. However, existing incentive mechanisms are typically utilized in conventional synchronous aggregation, resulting in significant straggler issues. In this study, we propose a novel asynchronous FL framework that integrates an incentive mechanism based on contract theory. Within the incentive mechanism, we strive to maximize the utility of the task publisher by adaptively adjusting clients' local model training epochs, taking into account factors such as time delay and test accuracy. In the asynchronous scheme, considering client quality, we devise aggregation weights and an access control algorithm to facilitate asynchronous aggregation. Through experiments conducted on the MNIST dataset, the simulation results demonstrate that the test accuracy achieved by our framework is 3.12% and 5.84% higher than that achieved by FedAvg and FedProx without any attacks, respectively. The framework exhibits a 1.35% accuracy improvement over the ideal Local SGD under attacks. Furthermore, aiming for the same target accuracy, our framework demands notably less computation time than both FedAvg and FedProx.

Federated learning makes it possible for all parties with data isolation to train the model collaboratively and efficiently while satisfying privacy protection. To obtain a high-quality model, an incentive mechanism is necessary to motivate more high-quality workers with data and computing power. The existing incentive mechanisms are applied in offline scenarios, where the task publisher collects all bids and selects workers before the task. However, it is practical that different workers arrive online in different orders before or during the task. Therefore, we propose a reverse auction-based online incentive mechanism for horizontal federated learning with budget constraint. Workers submit bids when they arrive online. The task publisher with a limited budget leverages the information of the arrived workers to decide on whether to select the new worker. Theoretical analysis proves that our mechanism satisfies budget feasibility, computational efficiency, individual rationality, consumer sovereignty, time truthfulness, and cost truthfulness with a sufficient budget. The experimental results show that our online mechanism is efficient and can obtain high-quality models.