  Federated learning (FL), as an emerging edge artificial intelligence
paradigm, enables many edge devices to collaboratively train a global model
without sharing their private data. To enhance the training efficiency of FL,
various algorithms have been proposed, ranging from first-order to second-order
methods. However, these algorithms cannot be applied in scenarios where the
gradient information is not available, e.g., federated black-box attack and
federated hyperparameter tuning. To address this issue, in this paper we
propose a derivative-free federated zeroth-order optimization (FedZO) algorithm
featured by performing multiple local updates based on stochastic gradient
estimators in each communication round and enabling partial device
participation. Under non-convex settings, we derive the convergence performance
of the FedZO algorithm on non-independent and identically distributed data and
characterize the impact of the numbers of local iterates and participating edge
devices on the convergence. To enable communication-efficient FedZO over
wireless networks, we further propose an over-the-air computation (AirComp)
assisted FedZO algorithm. With an appropriate transceiver design, we show that
the convergence of AirComp-assisted FedZO can still be preserved under certain
signal-to-noise ratio conditions. Simulation results demonstrate the
effectiveness of the FedZO algorithm and validate the theoretical observations.
