## 2 Materials and Methods
### Exploring Issues of Client Selection Methods
Upon examining client selection methodologies, such as the Power-of-Choice (PoC) algorithm and the FedChoice algorithm, it becomes evident that bias values are allocated to clients based on the volume of training samples and elevated local loss. A significant concern emerges when a client with a substantial number of samples is selected, particularly if these samples are predominantly from a single label. This scenario detrimentally affects the diversity of datasets utilized in model updates and global aggregation [18]. The issue of label scarcity further compounds this challenge, as certain clients may lack adequate labeled data due to resource constraints or insufficient motivation for labeling tasks. Such complications contribute to suboptimal model performance and impede the development of a well-generalized model across varied datasets. Ideally, client selection should encompass a range of labels alongside a considerable number of samples [19; 20]. This approach ensures that local model training effectively captures diverse patterns, thereby enhancing the global model's generalization [21]. When addressing the bias gap arising from an exclusive reliance on large sample sizes for client selection in local loss estimation and model updates within the PoC and FedChoice frameworks, as well as the conventional practice of utilizing all clients for local model performance assessment in Fed-RHLP, it becomes apparent that these approaches may result in a deficiency of local dataset diversity and significant execution time delays. This issue is evident, as shown in Figure 2.  
In Figure 2, a comparison of client selection methods for local model updates in the PoC, FedChoice, and traditional Fed-RHLP algorithms is shown. In the example, there are three clients (\(\rm k_{1}\), \(\rm k_{2}\), and \(\rm k_{3}\)). Figure 2: Concepts of client selection methods in PoC, FedChoice, and traditional Fed-RHLP. The PoC and FedChoice algorithms assume that the server selects only two clients as candidates for local loss estimation in each communication round. These algorithms prioritize clients with larger local datasets for participation. The probability of selecting each client (k) based on its local dataset size is calculated as \(\mathrm{P_{k}}=\frac{\mathrm{P_{k}}}{\sum_{k=1}^{\mathrm{L}}\mathrm{n_{k}}}\). For k\({}_{1}\), with a local dataset size of \(\mathrm{n_{k1}}\) = 300, the probability is calculated as \(\mathrm{P_{k1}}=300/700=0.43\). For k\({}_{2}\), with \(\mathrm{n_{k2}}\) = 250, \(\mathrm{P_{k2}}\) = 250/700 = 0.36. Finally, for k\({}_{3}\), with \(\mathrm{n_{k3}}\) = 150, \(\mathrm{P_{k3}}\) = 150/700 = 0.21. As a result, k\({}_{1}\) has the highest probability of being selected, followed by k\({}_{2}\), which has a higher probability than k\({}_{3}\). This selection process often leads to clients with a single label, such as k\({}_{1}\) and k\({}_{2}\), being selected repeatedly for local model training. Consequently, this reduces the diversity of the aggregated global model. The traditional Fed-RHLP algorithm, proposed in [12], uses all clients (k\({}_{1}\), k\({}_{2}\), and k\({}_{3}\)) to estimate the clients' local model performance (\(\mathrm{LP_{k}}\)) by sending the global model (W) to each client for estimating with the local dataset (d\({}_{\mathrm{k}}\)). Subsequently, the \(\mathrm{LP_{k}}\) of each client is converted into a weight for local model performance (\(\mathrm{WLP_{k}}\)), which is added to a biased roulette wheel for randomly selecting model updates and aggregation based on clients' local model performance, as shown in Equations (2)-(4). \[\mathrm{LP_{k}}=\mathrm{f(W;\ d_{k})} \tag{2}\]  
\[\mathrm{TotalLP}=\sum_{k=1}^{\mathrm{N}}\mathrm{LP_{k}} \tag{3}\]  
\[\mathrm{WLP_{k}}=\frac{\mathrm{LP_{k}}}{\mathrm{TotalLP}} \tag{4}\]  
The conventional Fed-RHLP algorithm is marked by significant computational demands, particularly in situations where certain clients are unavailable for processing at specific times.