## 2 Background
### Communication
It facilitates rapid and steady knowledge acquisition across distributed devices, particularly in scenarios involving non-identically distributed (non-IID) data or significant volatility. Federated averaging with momentum demonstrates remarkable effectiveness in achieving accurate and efficient model training while safeguarding user privacy and data security. It combines the collaborative capabilities of FL with the stability offered by momentum-based optimization. * The Federated Proximal Algorithm: The Federated Proximal Algorithm represents an advanced iteration in the evolution of FL algorithms, tailored to address the challenges posed by non-IID (independently and identically distributed) data across a network of devices. This algorithm is particularly relevant in scenarios where the data distribution varies significantly among the participating nodes, a common occurrence in real-world applications. The Federated Proximal Algorithm is built upon the foundation of the standard Federated Learning framework but introduces a crucial modification in the optimization process. The key innovation lies in the incorporation of a proximal term to the optimization objective. This term essentially acts as a regularizer that encourages the local models to not deviate significantly from the global model. The mathematical formulation of this algorithm involves adding a proximal term to the local loss function, typically represented as a squared Euclidean distance between the local and global model parameters. * Addressing non-IID data challenges: In standard Federated Learning setups, the assumption is often that the data across devices is identically distributed. However, in many practical situations, this assumption does not hold, leading to significant challenges in model convergence and performance. The Federated Proximal Algorithm mitigates these issues by ensuring that local model updates remain 'proximal' to the global model. This approach effectively handles the statistical heterogeneity of data, ensuring more stable and consistent model training across diverse data distributions. * Optimization process in Federated Proximal Algorithm: During the training process, each participating device computes its local model update by optimizing the modified loss function, which includes the proximal term. Once the local updates are computed, they are sent to a central server where a global aggregation occurs. The server updates the global model by averaging these updates, similar to standard Federated Learning, but with the added nuance provided by the proximal regularization. * Advantages and practical applications: The incorporation of the proximal term offers several advantages. Primarily, it enhances model performance in non-IID data scenarios, which are prevalent in many real-world applications such as healthcare, finance, and mobile services [41]. Additionally, by controlling the extent of deviation of local models from the global model, the Federated Proximal Algorithm promotes more uniform learning across the network, leading to improved overall model accuracy and convergence rates. In summary, the Federated Proximal Algorithm represents a significant advancement in the field of Federated Learning, offering a robust solution to the challenges posed by non-IID data distributions. Its ability to ensure consistent and efficient learning across a decentralized network of devices makes it a valuable tool in the arsenal of Federated Learning algorithms. The optimization algorithms utilized in FLSs undergo continuous development to effectively address the challenges posed by diverse and privacy-sensitive data. The use of these algorithms ensures the efficient generation of precise global models in FL while protecting user privacy. As a result, these algorithms play a critical role in advancing collaborative and privacy-preserving machine learning methodologies. * Personalization algorithms: Personalization algorithms within FLSs play a crucial role in customizing user experiences while preserving data privacy.