## 2 Related Work  
### Federated Learning  
Federated learning is an emerging distributed machine learning paradigm aimed at enabling collaborative training of a global model across multiple participants while preserving data privacy [25]. The fundamental principle involves performing model training locally at each participant's site and sharing only model parameters or gradients instead of raw data, thereby mitigating the risk of data breaches [26; 27]. The core mechanism of federated learning can be summarized in three stages: local model training, global model aggregation, and parameter updates. In federated learning, suppose there are \(K\) participants, each with a local dataset \(\mathcal{D}_{k}\). The objective is to collaboratively train a global model \(w\) that minimizes the loss function over all datasets. The global optimization objective is formulated as  
\[\min_{w}F(w)=\sum_{k=1}^{K}\frac{n_{k}}{n}F_{k}(w),\]  
current global model parameters \(w_{t}\). For each participant, the local optimization problem is expressed as  
\[w_{k}^{(t+1)}=\arg\min_{w}F_{k}(w). \tag{2}\]  
Typically, \(F_{k}(w)\) is optimized using gradient descent, with the update rule given by  
\[w_{k}^{(t+1)}=w_{k}^{(t)}-\eta\
abla F_{k}(w_{k}^{(t)}), \tag{3}\]  
where \(\eta\) is the learning rate, and \(\
abla F_{k}(w_{k}^{(t)})\) represents the gradient of the local loss function with respect to \(w_{k}\). After completing local training, federated learning integrates information across participants through global model aggregation. A commonly used method is weighted averaging, where the global model parameters \(w\) are updated as  
\[w^{(t+1)}=\sum_{k=1}^{K}\frac{n_{k}}{n}w_{k}^{(t+1)}. \tag{4}\]  
This formula indicates that the global model parameters are a weighted average of the local model parameters, with weights determined by the sample size of each participant's dataset. This mechanism ensures model sharing without transmitting raw data, effectively preserving data privacy. The success of federated learning lies in balancing data privacy and model performance. However, certain challenges persist in practical applications. One major issue is communication overhead. Frequent transmission of model parameters between participants and the central server, especially in large-scale distributed environments, can become a bottleneck for system performance. To address this, model compression techniques such as quantization and pruning have been proposed to reduce the volume of transmitted parameters. Another critical challenge is the non-IID (non-independent and identically distributed) nature of data among participants, which can adversely affect the performance of the global model. When data distributions vary significantly, the aggregated global model may fail to generalize effectively in specific scenarios. Solutions such as personalized model training, adaptive aggregation strategies, and adversarial training have been developed to mitigate these issues. In financial forecasting, federated learning provides a novel solution for collaborative modeling across institutions. For instance, multiple banks or financial organizations can collaboratively train market prediction models without exposing customer data, thereby enhancing both prediction accuracy and generalization [28]. Similarly, in medical imaging analysis, federated learning facilitates cross-institutional collaboration, enabling multiple hospitals to jointly train tumor detection models without sharing sensitive patient data. By combining local training and global aggregation, federated learning not only improves model performance but also meets stringent privacy protection requirements. In scenarios involving time-series data (e.g., stock prices) and spatial data (e.g., medical images or autonomous driving image data), federated learning demonstrates significant advantages. Traditional centralized data processing methods often expose privacy risks when handling large-scale heterogeneous data, while federated learning mitigates these issues through distributed training.