This work addresses the key challenges of applying federated learning to large-scale deep neural networks, particularly the issue of client drift due to data heterogeneity across clients and the high costs of communication, computation, and memory. We propose FedSub, an efficient subspace algorithm for federated learning on heterogeneous data. Specifically, FedSub utilizes subspace projection to guarantee local updates of each client within low-dimensional subspaces, thereby reducing communication, computation, and memory costs. Additionally, it incorporates low-dimensional dual variables to mitigate client drift. We provide convergence analysis that reveals the impact of key factors such as step size and subspace projection matrices on convergence. Experimental results demonstrate its efficiency.

Personalized Federated Learning is essential in AI-driven ubiquitous systems, supporting the distributed development of models able to adapt to diverse and evolving user behaviors while safeguarding privacy. Despite addressing heterogeneous user data distributions in collaborative model training, existing methods often face limitations balancing personalization and generalization, oversimplifying user similarities, or relying heavily on global models. In this paper, we propose FedSub, a novel federated approach designed to enhance personalization through the use of class-aware prototypes and model subnetworks. Prototypes serve as compact representations of user data, clustered on the server to identify similarities based on specific label patterns. Concurrently, subnetworks -- model components necessary to process each class -- are extracted locally and fused by the server according to these clusters, producing highly tailored model updates for each user. This fine-grained, class-specific aggregation of clients' models allows FedSub to capture the unique characteristics of individual user data patterns. The effectiveness of FedSub is validated in three real-world scenarios characterized by high data heterogeneity, derived from human activity recognition and mobile health applications. Experimental evaluations demonstrate FedSub's performance improvements with respect to the state-of-the-art and significant advancements in personalization for ubiquitous systems based on personal mobile and wearable devices.