The Dynamics of Collapse and Recovery in Clustered Federated Learning

Clustered federated learning (CFL) can suffer from cluster collapse. In particular, under unfavorable initialization, early client assignments concentrate heterogeneous groups on one cluster model, which is then trained toward a pooled compromise, while another model receives no updates. Since the current models determine client routing and routing determines subsequent model updates, we formulate clustered federated learning as control of an unknown dynamical system, with cluster models as the evolving state, client routing patterns as the control input, and routing interventions as state dependent controllers. In a two group linear model, we identify an initialization region in which greedy routing makes collapse self reinforcing. We then derive finite horizon error dynamics showing how routing interventions can reverse collapse by repeatedly training the starved model, primarily with updates from its compatible group. Motivated by this condition, each client uses blocked EXP3 to choose between greedy routing and interventions that redirect updates toward unused models. Experiments show that with simpler predictors and faster specialist formation, autonomous controller selection recovers usable specialists from unfavorable starts that trap standard CFL baselines. In harder neural settings, recovery instead requires coordinated and persistent commitment from group of clients, allowing compatible updates to accumulate before resuming lowest loss routing as in standard CFL baselines.