GeoFed: Geometry-Aware Byzantine Robust Federated Learning on SPD Manifolds in Heterogeneous Environments

Federated learning (FL) has been increasingly applied in the Internet of Things (IoT), leveraging its decentralized nature to facilitate collaboration among clients and enable resource-constrained clients to jointly train a globally optimal model based on consensus. However, it is difficult to confirm data authenticity and participant integrity due to the unobservability of local training procedures and the inaccessibility of local training data. As a result, FL is highly susceptible to Byzantine attacks, including data poisoning and model poisoning, which can manipulate the training process and degrade model performance. Moreover, IoT data is often highly heterogeneous and high-dimensional, rendering most existing Byzantine-robust FL approaches ineffective in practical scenarios. To address this challenge, we propose GeoFed, which iteratively filters out malicious clients based on the geodesic distance between clients. This geodesic distance is measured on the Riemannian manifold spanned by the covariance of local gradient update. To further mitigate the impact of data heterogeneity, GeoFed assigns a weight factor to each client after removing Byzantine attackers, optimizing the accuracy and flexibility of global model aggregation according to the quality of client data. We conduct extensive experimental evaluations of GeoFed under various Byzantine attack scenarios and highly heterogeneous data environments. To validate the efficacy of GeoFed, we provide a theoretical analysis of its convergence properties. The results demonstrate that GeoFed outperforms state-of-the-art Byzantine-robust FL approaches in heterogeneous IoT settings. Especially, under different Byzantine attacks, the accuracy of detecting malicious clients on the heterogeneous MNIST dataset approaches 100%.