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, memory-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.