Influence of History in Federated Unlearning

Client deletion in federated learning must address influence propagated through later retained-client updates. We study when a public bound on this historical influence is tight enough to support useful certified repair. Building on historical sensitivity propagation, we formulate a finite-horizon public influence ledger that accounts for numerical local-solver error, and combine its initialization-distance certificate with noisy retained-data repair to obtain an $(\varepsilon,\delta)$ unlearning guarantee relative to clean retraining followed by the same repair procedure. In controlled convex experiments, the realized recursion closely tracks the true leave-one-client-out separation, but public round-level bounds introduce substantial slack. A stability-controlled proximal training family reduces this additional slack from $67.7$ to $6.99$. Under standard training, history-based repair does not reliably improve over tuned generic clipping. Under the modified protocol, direct retained-reference repair outperforms generic clipping at every tested privacy budget and across all five held-out training seeds. It also outperforms projected ascent, while zero coupling remains competitive. These results suggest that the usefulness of historical certification depends on how tightly the training dynamics can be bounded.