## 3 Federated Learning for Space-Air-Ground Integrated Networks
### Adaptive Transfer Learning Strategy  
Devices in space-air-ground integrated networks (such as terrestrial devices, unmanned aerial vehicles, satellites, etc.) exhibit significant heterogeneity in terms of computing capabilities, storage capacities, communication bandwidth, and other aspects. The local data on each device may also have different distributions (i.e., data heterogeneity), which affects the convergence speed and accuracy of the global model in federated learning. To  
Figure 2: Federated learning framework of a space–air–ground integrated network. overcome these challenges, transfer learning, as an effective knowledge transfer method, can enhance model training performance by sharing knowledge between different devices. Transfer learning can leverage knowledge from source tasks (e.g., models or tasks already existing on other devices) to accelerate learning for target tasks, especially when devices have limited computational capabilities or incomplete data. Suppose there are \(N\) devices in the space-air-ground integrated network, and each device \(i\) contributes differently to model updates based on its computational capability and bandwidth. To comprehensively consider these factors, a weighting factor \(w_{i}\) is introduced for each device, representing its contribution to the global model training. This factor can be dynamically adjusted according to the device's computational capability \(C_{i}\) and bandwidth \(B_{i}\):  
\[w_{i}=\frac{C_{i}\cdot B_{i}}{\sum\limits_{j=1}^{N}C_{j}\cdot B_{j}} \tag{1}\]  
where \(C_{i}\) and \(B_{i}\) represent the computational capability (FLOPS-Floating Point Operations Per Second) and bandwidth of device \(i\), respectively, and \(N\) is the total number of devices. A larger weighting factor \(w_{i}\) indicates a greater contribution of device \(i\) to the global model. Figure 3 illustrates knowledge transfer between heterogeneous devices in a SAGIN. The diagram employs two distinct arrow types to represent knowledge flow patterns within the network. Intra-layer knowledge transfer is depicted by solid horizontal arrows, showing knowledge sharing between devices within the same layer. Cross-layer knowledge transfer is represented by dashed vertical arrows, indicating knowledge transmission between different layers. Examples include transfers from LEO satellites in the space layer to UAVs in the aerial layer, or from HAPS in the aerial layer to IoT devices in the ground layer. Devices in space-air-ground integrated networks (such as ground base stations, satellites, UAVs, etc.) typically face task heterogeneity during execution. Task matching and domain adaptation are key issues in transfer learning, especially in environments with heterogeneous devices and inconsistent data distributions. The effectiveness of knowledge transfer can be measured by calculating the similarity between source and target tasks. Assuming the feature spaces of the source and target tasks are \(X_{\text{src}}\) and \(X_{\text{target}}\), respec  
Figure 3: Schematic diagram of knowledge transfer between heterogeneous devices in a SAGIN. tively, and the data distributions of the source and target tasks are \(P_{\text{src}}(X)\) and \(P_{\text{target}}(X)\), respectively, the task matching degree \(\text{Sim}(\cdot)\) can be calculated using the following formula  
\[\text{Sim}(X_{\text{src}},X_{\text{target}})=\int_{X}P_{\text{src}}(X)\cdot P _{\text{target}}(X)dX \tag{2}\]  
where \(X\) represents the input space of the tasks, and \(\text{Sim}(\cdot)\) is the matching degree between the source and target tasks. The objective of domain adaptation is to bridge the data distribution gap between source and target tasks by transferring model knowledge from the source task to the target task. Assuming the knowledge of the source task is \(\theta_{\text{src}}\) and the knowledge of the target task is \(\theta_{\text{target}}\), the goal of domain adaptation is to minimize the distribution difference between the source and target tasks.