Wireless connectivity is instrumental in enabling scalable federated learning\n(FL), yet wireless channels bring challenges for model training, in which\nchannel randomness perturbs each worker's model update while multiple workers'\nupdates incur significant interference under limited bandwidth. To address\nthese challenges, in this work we formulate a novel constrained optimization\nproblem, and propose an FL framework harnessing wireless channel perturbations\nand interference for improving privacy, bandwidth-efficiency, and scalability.\nThe resultant algorithm is coined analog federated ADMM (A-FADMM) based on\nanalog transmissions and the alternating direction method of multipliers\n(ADMM). In A-FADMM, all workers upload their model updates to the parameter\nserver (PS) using a single channel via analog transmissions, during which all\nmodels are perturbed and aggregated over-the-air. This not only saves\ncommunication bandwidth, but also hides each worker's exact model update\ntrajectory from any eavesdropper including the honest-but-curious PS, thereby\npreserving data privacy against model inversion attacks. We formally prove the\nconvergence and privacy guarantees of A-FADMM for convex functions under\ntime-varying channels, and numerically show the effectiveness of A-FADMM under\nnoisy channels and stochastic non-convex functions, in terms of convergence\nspeed and scalability, as well as communication bandwidth and energy\nefficiency.\n