Hierarchical Federated Learning (HFL) is a distributed machine learning paradigm tailored for multi-tiered computation architectures, which supports massive access of devices' models simultaneously. To enable efficient HFL, it is crucial to design suitable incentive mechanisms to ensure that devices actively participate in local training. However, there are few studies on incentive mechanism design for HFL. In this paper, we design two-level incentive mechanisms for the HFL with a two-tiered computing structure to encourage the participation of entities in each tier in the HFL training. In the lower-level game, we propose a coalition formation game to joint optimize the edge association and bandwidth allocation problem, and obtain efficient coalition partitions by the proposed preference rule, which can be proven to be stable by exact potential game. In the upper-level game, we design the Stackelberg game algorithm, which not only determines the optimal number of edge aggregations for edge servers to maximize their utility, but also optimize the unit reward provided for the edge aggregation performance to ensure the interests of cloud servers. Furthermore, numerical results indicate that the proposed algorithms can achieve better performance than the benchmark schemes.

翻译：分层联邦学习（HFL）是一种面向多层计算架构的分布式机器学习范式，可同时支持海量设备模型的接入。为实现高效的HFL，设计合适的激励机制以确保设备积极参与本地训练至关重要。然而，目前关于HFL激励机制设计的研究较少。本文针对具有两层计算结构的HFL，设计了两级激励机制以鼓励各层实体参与HFL训练。在底层博弈中，我们提出一种联盟形成博弈来联合优化边缘关联与带宽分配问题，并通过所提出的偏好规则获得有效的联盟划分，该划分可通过精确势博弈证明其稳定性。在顶层博弈中，我们设计了Stackelberg博弈算法，该算法不仅能为边缘服务器确定最优边缘聚合次数以最大化其效用，还能优化边缘聚合性能的单元奖励，从而保障云服务器的利益。此外，数值结果表明，所提算法相较于基准方案能够实现更优的性能。