## 1 Introduction  
### Background  
The Space-Air-Ground Integrated Network (SAGIN) [1] is a novel network architecture that combines multiple layers of communication systems, including space (satellites), air (unmanned aerial vehicles and high-altitude platforms), and ground (base stations and user equipment). This architecture enables seamless, continuous, and efficient communication services on a global scale by integrating network nodes at different layers to achieve global coverage capabilities. Due to its multi-layered node structure, the SAGIN can automatically switch to alternative layers when failures occur in one layer, ensuringcommunication continuity and reliability. SAGINs have widespread applications in military operations, disaster emergency response, remote education, and telemedicine, providing critical communication support in areas where traditional networks are unavailable or unstable [2]. SAGINs have seen significant commercial implementations in recent years. Notable examples include Starlink, developed by SpaceX, which provides global broadband internet coverage through thousands of Low Earth Orbit (LEO) satellites integrated with ground stations. Similarly, OneWeb has deployed a satellite constellation aimed at delivering global broadband connectivity, particularly to rural and remote areas. Project Kuiper by Amazon and Telesat's Lightspeed are other emerging SAGIN initiatives addressing the integration of space, air, and ground networks. These commercial implementations highlight the real-world applicability and growing importance of SAGIN technologies in providing global communication services. Despite the SAGIN's enormous potential, its implementation still faces numerous technical challenges: high latency and limited bandwidth, especially in satellite communications where signals must traverse long distances through the atmosphere, resulting in higher communication delays. Geostationary Earth Orbit (GEO) satellites typically exhibit round-trip latencies of 500-700 ms, Medium Earth Orbit (MEO) satellites show 100-250 ms, and Low Earth Orbit (LEO) satellites approximately 20-100 ms. In contrast, ground-based 5G networks achieve latencies as low as 1-10 ms. Simultaneously, the limited bandwidth of satellite channels constrains data transmission speed and quality [3]. Regarding bandwidth, satellite links commonly provide 50-500 Mbps for downlink and 10-50 Mbps for uplink, while aerial networks such as High-Altitude Platform Stations (HAPS) offer 30-100 Mbps. These values are considerably lower than terrestrial fiber networks that deliver 1-100 Gbps. These communication constraints directly impact federated learning performance in SAGINs, as modern deep learning models can range from tens of megabytes for mobile-optimized models (10-50 MB) to several gigabytes for large language models (1-10 GB). Dynamic network topology poses another challenge, as the positions and connection topologies of nodes in space-air-ground networks frequently change, such as satellites moving in orbit and the flight paths of unmanned aerial vehicles. This dynamism challenges network protocol design and the maintenance of model consistency [1]. Heterogeneity and complexity arise because the network comprises various communication technologies and protocols, with significant functional and performance differences among nodes at different layers. Integrating ground, air, and space systems involves different communication protocols and technical standards, requiring effective coordination and management [4]. Security and privacy protection constitute a major concern since the network's extensive coverage and numerous participating devices increase security risks during cross-domain data transmission, making data security and user privacy protection critical issues [5]. Machine learning applications are increasingly deployed across SAGIN environments for various purposes. For instance, in satellite networks, machine learning is used for orbit prediction, constellation management, and anomaly detection in satellite operations.