## 3 Federated Learning for Space-Air-Ground Integrated Networks
### Privacy-Preserving Federated Learning
#### 3.5.1 Local Training at Terminal Nodes
Differential privacy ensures that individual training samples do not significantly influence the final model results by adding noise, thereby ensuring the privacy of individual data is not leaked. In privacy-preserving federated learning for space-air-ground integrated networks, each device adds differential privacy noise when uploading model parameters to prevent leakage of sensitive local data. When device \(i\) uploads its local model parameters \(\theta_{i}\), differential privacy noise is added to the model updates. The differential privacy noise \(\mathcal{N}_{i}\) comes from the Laplace mechanism or Gaussian mechanism, ensuring that the final uploaded model parameters do not contain information that can be directly associated with individual training data. Assuming that device \(i\) has updated model parameters \(\theta_{i}\) in the local training process, to ensure differential privacy, the device adds noise \(\mathcal{N}_{i}\) before uploading the local model parameters to the edge server as follows:  
\[\hat{\theta}_{i}=\theta_{i}+\mathcal{N}_{i} \tag{32}\]  
where \(\hat{\theta}_{i}\) is the model update with differential privacy noise, and \(\mathcal{N}_{i}\) is the noise from the Laplace distribution or Gaussian distribution, with intensity controlled by the privacy budget \(\epsilon_{i}\). The \(\mathcal{N}_{i}\) based on the Laplace distribution is represented as follows:  
\[\mathcal{N}_{i}\sim\mathcal{L}(0,\frac{\Delta f_{i}}{\epsilon_{i}}) \tag{33}\]  
where \(\Delta f_{i}\) is the sensitivity of the model update in device \(i\)'s local training process, representing the minimum response to changes in local data points. \(\epsilon_{i}\) is the privacy budget of device \(i\), controlling the strength of privacy protection. One advantage of Laplace noise is that it provides strict \(\epsilon\)-differential privacy, suitable for scenarios with extremely high privacy protection requirements. Compared to Gaussian noise, Laplace noise has higher computational efficiency, making it suitable for terminal devices with limited computational capabilities. One disadvantage of Laplace noise is its relatively large amplitude, which may significantly impact model performance, especially in high-dimensional data. In highly dynamic networks, larger noise may lead to slower model convergence. Unlike Laplace noise, Gaussian noise has a smaller amplitude, less impact on model performance, and is suitable for high-dimensional data and complex models. In highly dynamic networks, smaller noise helps the model converge quickly. Its weakness is that it provides weaker privacy guarantees and has a relatively higher computational complexity, which may not be friendly to terminal devices with limited computational capabilities. In this paper, when adding differential privacy noise to terminal device model updates, either Laplace noise or Gaussian noise is chosen depending on the specific application scenario to better highlight the advantages of both noise addition methods. In space-air-ground integrated networks, device \(i\) may have different computational capabilities and bandwidth limitations, so the privacy budget \(\epsilon_{i}\) needs to be dynamically allocated according to the characteristics of the device. For example, devices with stronger computational capabilities can be allocated a larger privacy budget, thereby reducing the impact of noise on model accuracy. The privacy budget \(\epsilon_{i}\) controls the intensity of differential privacy noise, directly affecting the privacy protection effect of the model parameters uploaded by the device and the accuracy of model training. A larger privacy budget \(\epsilon_{i}\) will reduce noise, thereby improving model accuracy, but may lead to a higher risk of privacy leakage; a smaller privacy budget enhances privacy protection but may affect model accuracy. Therefore, how to dynamically allocate privacy budgets based on the device's computational resources (such as CPU/GPU performance, memory, etc.