Server-Proximal Aggregation for Federated Domain-Incremental Learning under Partial Participation: Task-Uniform Convergence and Backward Transfer

Real-world federated systems seldom operate on static data: input distributions drift while privacy rules forbid raw data sharing. We study Federated Domain-Incremental Learning (FDIL), where (i) clients are heterogeneous, (ii) tasks arrive sequentially with shifting domains, and (iii) the label space remains fixed. Two theoretical pillars remain missing for FDIL under partial participation: a guarantee of backward knowledge transfer (BKT) and a convergence rate that holds *uniformly across the task sequence*. We introduce SPECIAL (Server-Proximal Efficient Continual Aggregation for Learning), a simple, replay-free FDIL algorithm that adds a single server-side ``anchor'' to FedAvg: in each round, the server aggregates updates from a uniformly sampled subset of clients and then blends the result with the previous global model via a lightweight proximal step. This anchor curbs cumulative drift without replay buffers, synthetic data, or task-specific heads, leaving communication cost and model size unchanged. Our theory shows that SPECIAL (i) *preserves earlier tasks*: a BKT bound caps any increase in earlier-task loss by a drift-controlled term that shrinks with more rounds, local epochs, and participating clients; and (ii) *achieves task-uniform, communication-efficient convergence* for non-convex FDIL with partial participation: $\mathcal{O}\big(\sqrt{E/(NT)}\big)$ in expected gradient norm, with $E$ local epochs, $T$ rounds, and $N$ participating clients, while explicitly separating optimization variance from inter-task drift. Experiments on standard FDIL benchmarks corroborate the theory.

Machine learning systems are often trained using data from many organizations or devices, such as hospitals, phones, or sensors. These participants usually cannot share their raw data because of privacy or policy restrictions. At the same time, the data they see can change over time. A medical model may encounter images from new scanners, or a mobile application may see changing user behavior. The challenge is to help the model learn from new situations without forgetting what it learned before. This paper presents SPECIAL, a simple method for making shared, privacy-preserving learning systems more stable over time. Each participant continues to learn from its current data in the usual way, while the central coordinator gently anchors the updated model to knowledge learned from earlier data. This helps the model adapt to new information while reducing forgetting. The method does not store old raw data, generate artificial past data, or require extra communication. We show mathematically that SPECIAL can protect earlier knowledge and keep training efficient, even when only some participants are available at a time. Experiments on image and language tasks show that SPECIAL performs well compared with existing methods. These results suggest that a simple anchoring strategy can make privacy-preserving machine learning more reliable in changing real-world environments.