## 3 Preliminaries
### Federated Learning  
FL is a distributed ML paradigm that allows multiple clients to collaboratively train a shared model while keeping their data decentralized and private. In FL, clients, such as IoT devices, smartphones, or edge devices, each maintain a local dataset and perform local model training on their respective data. Instead of sending raw data to a central server, clients send only model updates (gradients) to the server, which aggregates these updates to improve the global model. This process preserves data privacy and reduces communication costs associated with centralized data processing [16; 17].  
Mathematically, FL can be formulated as follows:  
\[\min_{\theta}\left(\sum_{k=1}^{K}\frac{n_{k}}{n}F_{k}(\theta)\right) \tag{1}\]  
where  
* \(\theta\) is the global model parameter vector;
* \(F_{k}(\theta)\) is the local objective function (e.g., cross-entropy loss) on client \(k\)'s data;
* \(n_{k}\) is the number of data points on client \(k\);
* \(n=\sum_{k=1}^{K}n_{k}\) is the total number of data points across all \(K\) clients.  
This formulation ensures that each client's contribution is weighted proportionally to its data size. It was introduced in the seminal work by [PERSON] et al. [18] and widely used in the FL literature [16; 17].  
Federated Learning Algorithms  
Several algorithms have been developed to address the core challenges of FL, particularly in handling data distribution, communication efficiency, and system heterogeneity. Two widely used and foundational algorithms are Federated averaging (FedAvg) and FedProx.  
* FedAvg is a basic aggregation method proposed by [PERSON] et al. [18] It works by selecting a subset of clients in each training round. Each selected client performs local model updates on its private data using stochastic gradient descent (SGD) or another optimizer. The clients then send their updated model parameters to a central server, which aggregates them (typically through weighted averaging based on dataset sizes)  
\begin{table}
\begin{tabular}{l l l l l l l} \hline \hline
**Method** & \begin{tabular}{l} **Accuracy-** \\ **Aware** \\ \end{tabular} & \begin{tabular}{l} **Battery-** \\ **Aware** \\ \end{tabular} & \begin{tabular}{l} **Privacy-** \\ **Aware** \\ \end{tabular} & \begin{tabular}{l} **Heterogeneous** \\ **Devices** \\ \end{tabular} & \
\begin{tabular}{l} **Cooperative** \\ **Update** \\ \end{tabular} &
\begin{tabular}{l} **Selective** \\ **Update** \\ \end{tabular} \\ \hline FedCS [6] & ✓ & ✓ & ✗ & ✓ & ✗ & ✗ \\ Oort [7] & ✓ & ✓ & ✗ & ✓ & ✗ & ✗ \\ FRS [8] & ✓ & ✓ & ✗ & ✓ & ✓ & ✗ \\ FedProm [9] & ✓ & ✗ & ✗ & ✗ & ✓ & ✗ \\ FedGCS [10] & ✓ & ✓ & ✗ & ✗ & ✗ & ✗ \\ MDA [11] & ✓ & ✗ & ✗ & ✓ & ✓ & ✗ \\ TIFL [12] & ✓ & ✓ & ✗ & ✓ & ✗ & ✗ \\
**REPS (Ours)** & ✓ & ✓ & ✓ & ✓ & ✓ & ✓ \\ \hline \hline \end{tabular}
\end{table}
Table 1: Comparison of client selection strategies across key criteria. ✓ (green): Feature is addressed by the method; ✗ (red): Feature is not addressed by the method.  
to form the new global model. FedAvg is efficient in terms of communication and computation and serves as the standard baseline in many FL studies.
* FedProx proposed by [PERSON] et al. [19] extends FedAvg by introducing a proximal term to the local objective function. This regularization term penalizes significant divergence of local model parameters from the global model, which helps stabilize training when client data is non-IID (i.e., not independently and identically distributed). FedProx is particularly useful in heterogeneous environments where clients may have diverse data distributions, computation capacities, and participation frequencies.  
These algorithms lay the groundwork for more advanced FL strategies and serve as key benchmarks for evaluating new techniques in federated settings.