## 3 Federated Learning for Space-Air-Ground Integrated Networks
### Adaptive Transfer Learning Strategy
The loss function for domain adaptation \(L_{\text{DA}}\) can be expressed as  
\[L_{\text{DA}}(\theta_{\text{src}},\theta_{\text{target}})=D_{KL}(P_{\text{src }}(X)||P_{\text{target}}(X)) \tag{3}\]  
where \(P_{\text{src}}(X)\) and \(P_{\text{target}}(X)\) represent the data distributions of the source and target tasks, respectively. \(D_{KL}\) denotes the Kullback-Leibler (KL) divergence, which measures the difference between two probability distributions. According to Equation (3), domain adaptation strategies may perform poorly in extreme distribution shift scenarios, such as the following situations:  
(1) Completely disjoint feature spaces: When there is almost no overlap between the feature spaces of the source and target domains, KL divergence measurement becomes less meaningful. (2) Extreme label distribution skew: If the class distribution varies dramatically across different devices (e.g., one device primarily contains samples of class A, while another primarily contains samples of class B), simple KL divergence metrics may fail to adequately capture this imbalance. (3) Multimodal data distributions: KL divergence methods assume relatively simple, continuous distributions. When data present multiple distinct modes or clusters, a single KL divergence metric may oversimplify the actual distribution differences. (4) Insufficient data for distribution estimation: For devices with very limited data volumes, accurately estimating the true data distribution becomes challenging, making KL divergence calculations unreliable. The core of transfer learning lies in accelerating target task training by transferring model knowledge from the source task. Let the model for the source task be \(\theta_{\text{src}}\) and the model for the target task be \(\theta_{\text{target}}\). In space-air-ground integrated networks, due to device and data heterogeneity, the objective of transfer learning is to transfer knowledge from the source task model \(\theta_{\text{src}}\) to the target task, thereby accelerating local training on the device. Transfer Model Update: A key issue in transfer learning is how to update the local model based on device \(i\)'s local data and task, combined with source task knowledge. Let  
function for transfer learning \(L_{\text{TL}}\) combines the target task loss and the knowledge transfer loss from the source task:  
\[L_{\text{TL}}(\theta_{i})=L_{i}(\theta_{i})+\lambda\text{Sim}(\theta_{i},\theta_ {\text{src}})L_{DA} \tag{5}\]  
where \(L_{i}(\theta_{i})\) is the loss function of the local task for device \(i\), \(\lambda\) is a hyperparameter that balances the transfer loss term and the target task loss term, \(\text{Sim}(\theta_{i},\theta_{\text{src}})\) is the similarity between the target task model \(\theta_{i}\) and the source task model \(\theta_{\text{src}}\), measuring the matching degree between source task knowledge and target task knowledge, and \(L_{DA}\) is the domain adaptation loss. For instance, in our experimental setting with 10 terminal nodes, consider a scenario where a ground node with high computational capability (e.g., a base station) shares knowledge with a resource-constrained aerial node (e.g., a UAV). Using Equation (1), if the ground node has 10x the computational capability and 5x the bandwidth of the UAV, its weighting factor would be approximately 50 times larger. Through the transfer learning mechanism described in Equations (4) and (5), the UAV can leverage the ground node's model knowledge while adapting it to its specific data distribution, significantly accelerating convergence compared to training in isolation.