## 2 Related Works  
In the context of three-tier FL architectures, researchers have primarily explored three categories of methods to address the inefficiency of training caused by resource heterogeneity across clients [5]. The first category involves introducing asynchronous or semi-asynchronous update mechanisms, whereby the cloud or edges are not required to wait for all slow clients to complete their local training before performing aggregation. This approach reduces latency associated with synchronization delays. For example, [PERSON] et al. [6] proposed an asynchronous FL algorithm that allows clients to upload updates without global synchronization, significantly alleviating the bottleneck caused by straggling devices. However, asynchronous updates may introduce the problem of stale gradients, where model updates from slow clients are based on outdated global models, thus degrading convergence accuracy. To address this, some studies propose weighted asynchronous aggregation, dynamically adjusting each client's contribution to the global model based on its computing capability, thereby mitigating the negative impact of delayed updates from low-resource clients [7, 8]. The second category of methods aims to accommodate client heterogeneity by reducing the computational burden on clients, either by shrinking the model size or by updating only a portion of the model parameters. For instance, model dropout mechanisms have been employed to reduce the workload for weak clients. [PERSON] et al. [9] proposed an asynchronous distributed dropout approach, in which only a subset of neurons participates in local updates at each client during training rounds. Specifically, the model is partitioned by layers, and different subsets of neurons in each layer are assigned to different clients. High-capability clients update the less frequently used (and more computationally intensive) parameters, while low-capability clients are responsible for updating the frequently used and critical parts, thereby accelerating convergence. Similarly, other studies allow clients to train sub-models of varying complexity (e.g., shallow or compact models), which are then fused using zero-padding or knowledge distillation during aggregation, balancing participation despite capability differences [3]. However, under the three-tier architecture, the number and computational distribution of clients served by different edge nodes may vary significantly, resulting in imbalanced update frequencies and quality at the edge. The third category addresses the challenge from a system scheduling perspective by incorporating resource-aware coordination in the cloud-edge-client collaborative aggregation process. For example, some studies propose selecting clients for each training round based on indicators such as computing power, communication bandwidth, and data volume [10]. The earlier FedCS algorithm [11] follows this idea by filtering out clients that do not meet resource requirements, thereby excluding slow devices from aggregation to reduce per-round latency. However, naively eliminating weak clients may compromise model generalization. To overcome this, more refined scheduling approaches have emerged in recent years. [PERSON] et al. [12] proposed a joint optimization of client selection and wireless bandwidth allocation by formulating a Markov decision process (MDP) model, which derives the optimal allocation strategy while ensuring data diversity across devices, thereby simultaneously reducing training latency and preserving model accuracy. Additionally, [PERSON] et al. [13] investigated resource-aware strategies for optimizing both communication and computation in wireless IoT networks, dynamically adjusting the global aggregation frequency and local resource allocation based on network conditions to improve FL efficiency under heterogeneous settings. While these strategies alleviate inefficiencies caused by client heterogeneity at the system level, they often reduce or suppress the participation of low-performance clients [14].