Certified Characterization of Privacy, Participation, and Convergence in Over-the-Air Federated Learning

Over-the-air federated learning (OTA-FL) exploits wireless superposition to aggregate client updates in one transmission, reducing communication overhead. In truncated channel-inversion OTA-FL, the receive scaling factor $\eta$ jointly controls normalized channel noise, privacy sensitivity, and client activation thresholds. We characterize this coupling for smooth nonconvex objectives under Rayleigh fading via certified truncation envelopes. The resulting bounds yield a computable feasible set of receive amplitudes satisfying convergence, privacy, and participation constraints. We show that apparent participation fairness can be misleading at large $\eta$, where clients may be nearly uniformly excluded, and introduce a dropped-weight constraint to rule out this degenerate regime. We further propose CARES, an adaptive receive-scaling algorithm that selects $\eta_t$ online via contextual bandits, tracks privacy with zero-concentrated differential privacy, and enforces participation-envelope constraints. Experiments on image classification show that CARES achieves competitive accuracy and strong worst-client accuracy while maintaining formal $(\varepsilon,\delta)$-DP using only implicit channel noise among the OTA baselines considered.