Local Epochs as Optimal Stopping: Theory and Evidence in Federated Learning

How long should each client train before communicating? Federated learning usually answers with a fixed, hand-tuned number of local epochs, yet an extra epoch helps only while the loss it removes outweighs the client drift it adds. We cast local training as finite-horizon optimal stopping: a client trains while the next epoch's expected loss reduction exceeds its priced squared drift. We prove that this one-stage look-ahead threshold is optimal whenever stopping decisions are nested, derive closed forms for geometric and power-law returns, and show that an estimated rule loses utility only through wrong-signed decisions. Epoch-level evidence supports the model's structure (geometric diminishing returns and additive drift) and reveals a simple regularity: the stage of training, not the client, sets the budget. The resulting rule, FedStop, is never more than one point below tuned FedAvg in mean accuracy. Where its budgets vary, it raises worst-decile accuracy by 4-8 points over tuned FedAvg and an adaptive-interval baseline, and every one of these gains remains significant after a Holm correction. Against the client-side stopping rules ALT and FLASH, it has the highest CIFAR-10 worst decile on every seed and is level elsewhere. It stays stable where per-client early stopping collapses, and each stopping decision takes constant time.