## 4 Federated Lemus Optimization Algorithm (FedLO)  
Federated LO (FedLO) combines the principles of the LO with the FL paradigm to address global optimization problems in a distributed manner. Inspired by the agile and coordinated behaviors of lemurs, LO is adapted to operate in an FL setting where data are distributed among multiple clients. Each client, analogous to a lemur in the natural environment, contributes to the global optimization process. The initialization of FedLO involves randomly placing a population of lemurs (representing potential solutions) within the variable bounds of the search space. In this context, the evaluation process involves discerning the global best (_gbest_) and the best among nearest neighbors (_nbest_). To facilitate this, lemurs engage in a strategic exploration phase, employing inspired long jumps reminiscent of their natural leaping behavior. Mathematically, the objective function is expressed as follows:  
\[\max_{\theta}\sum_{i=1}^{N}\frac{1}{|\mathcal{D}_{i}|}\sum_{(x,y)\in\mathcal{D }_{i}}\mathbb{I}(f_{\theta}(x_{i})=y_{i}) \tag{5}\]  
where \(\theta\) represents the parameters of the global model to be optimized, and \(N\) is the total number of participating clients. \(\mathcal{D}_{i}\) denotes the local dataset of client \(i\). \((x,y)\) represents a data point with input \(x\) and true label \(y\). \(f_{\theta}(x_{i})\) is the prediction made by the global model with parameters \(\theta\) for input \(x_{i}\). \(\mathbb{I}(\cdot)\) is the indicator function, equal to 1 if the condition inside is true, and 0 otherwise. In essence, the objective function aims to maximize the average accuracy across all clients, considering the predictions made by the global model on their respective local datasets. Optimization involves identifying the parameters \(\theta\) that maximize this accuracy measure. In doing so, we enhance the overall performance of the federated model, ensuring a collective improvement in the accuracy and efficacy of the global model as it collaboratively learns from diverse local datasets. This comprehensive approach reflects the intricacies of the fitness evaluation system, contributing to the robustness and adaptability of the federated model in a distributed learning environment. In the exploitation phase, resembling dance-hopping, which involves collaboration between lemurs. In FedLO, this corresponds to clients sharing their knowledge with the central server, which updates the global best position for the next iteration. Additionally, FedLO incorporates client collaboration and information exchange mechanisms during the optimization process. After each exploration and exploitation phase, clients with improved lemurs, reflecting better potential solutions, share their findings with the central server. This collaboration enables the server to aggregate and update the global best position, promoting knowledge transfer across the federated network. The collaboration step enhances FedLO's ability to adapt to diverse local landscapes encountered by individual clients. Lemurs exploring different regions of the search space contribute their unique insights, creating a collective intelligence that guides the optimization process. This decentralized collaboration aligns with the FL philosophy of leveraging local knowledge for global model improvement. Furthermore, FedLO introduces adaptive mechanisms for adjusting exploration and exploitation parameters to accommodate the dynamic nature of federated environments. The risk parameter FRR, responsible for controlling the balance between exploration and exploitation, is continuously adapted based on the current iteration and the specifiedmaximum iteration count. This adaptability ensures that FedLO can respond effectively to changes in the optimization landscape throughout FL. Inspired by leaping behavior, the exploration phase allows lemurs to take agile, long jumps towards potentially promising regions. This translates to clients exploring diverse areas of their local search spaces in a federated context.