## 2 Materials and Methods  
### Exploring Issues of Client Selection Methods  
In the context of Federated Learning (FL), the selection of diverse clients, combined with limited data samples and suboptimal data quality, can impede convergence rates and contribute to the development of an inaccurate global model. Numerous studies have proposed client selection methodologies designed to address the statistical discrepancies encountered during the global model aggregation process. The challenge of statistical heterogeneity originates from multiple sources, including variations in data distributions, inconsistent sampling practices, and Non-IID data or imbalanced datasets [13]. To manage this heterogeneity, it is crucial to implement client selection strategies during the training process that ensure convergence across the system, considering the diversity of statistical heterogeneity [14]. Furthermore, additional complexities arise from variations in model architectures, network conditions, and hardware capabilities among participating clients. These factors collectively exacerbate the existing challenges, diminishing the server's ability to perform effective model aggregation and resulting in a global model with compromised precision [15]. A comprehensive review and analysis of the challenges and limitations associated with these algorithms can be conducted as follows:  
FedAvg represents a straightforward approach that involves the random selection of clients for local model updates, specifically employing Convolutional Neural Networks (CNNs) to distribute computational tasks among clients and address challenges associated with centralized machine learning. This method applies gradient descent to perform local updates on the models of a randomly selected subset of clients, which are subsequently aggregated to create a global model. Although FedAvg averages the updates from individual client models, it may encounter significant divergence from the optimal model, particularly when the training data distribution of a single client markedly differs from the heterogeneous and non-IID data distribution [16; 17]. The ideal model equation for FedAvg can be decomposed into two primary components: (1) Local Updates: Each client k refines its model \(w_{t+1}^{k}\) through gradient descent, employing a step size determined by \(\eta\), (the learning rate). This process is influenced by the client's local dataset and the gradient \(g_{k}\) of the loss function. (2) Global aggregation: The locally updated models from the clients are amalgamated to form the global model \(w_{t+1}\), with the aggregation weighted proportionally to the size of each client's dataset \(n_{k}\) relative to the total dataset size \(n\), as shown in Equation (1). \[ \begin{split}\forall k,\ w_{t+1}^{k}&\leftarrow\ w_{t}-ng_{k}\\ w_{t+1}&\leftarrow\ \sum_{k=1}^{K}\frac{n_{k}}{n}w_{t+1} ^{k}\end{split}  \tag{1}\]  
The problems with FedAvg can be demonstrated through the following example, as shown in Figure 1.  
In Figure 1, it is assumed that three clients (K = 3), represented as k\({}_{1}\), k\({}_{2}\), and k\({}_{3}\), are each responsible for updating their local models (\(w_{k1}\), \(w_{k2}\), and \(w_{k3}\)) with a local dataset (d\({}_{k}\)) and then contribute to aggregating a global model (\(W_{t}\)) in the communication round (t). In the random client selection process, the probability of selecting a client (p\({}_{k}\)) for local updates is  
Figure 1: Divergence problem of random client selection in FedAvg. equally distributed, with a probability value of \(\rm p_{k}=1/K\) (0.33). We assume that \(\rm k_{1}\) has 100 samples with 3 labels (\(\rm n_{k}=100\), \(\rm L_{k}=3\)), \(\rm k_{2}\) has 50 samples with 1 label (\(\rm n_{k}=50\), \(\rm L_{k}=2\)), and \(\rm k_{3}\) has 20 samples with 1 label (\(\rm n_{k}=20\), \(\rm L_{k}=1\)). If clients with low data quantity and few labels (\(\rm k_{2}\) and \(\rm k_{3}\)) are frequently selected for updates, this can cause significant divergence.