  Federated learning (FL) is a promising learning paradigm that can tackle the
increasingly prominent isolated data islands problem while keeping users' data
locally with privacy and security guarantees. However, FL could result in
task-oriented data traffic flows over wireless networks with limited radio
resources. To design communication-efficient FL, most of the existing studies
employ the first-order federated optimization approach that has a slow
convergence rate. This however results in excessive communication rounds for
local model updates between the edge devices and edge server. To address this
issue, in this paper, we instead propose a novel over-the-air second-order
federated optimization algorithm to simultaneously reduce the communication
rounds and enable low-latency global model aggregation. This is achieved by
exploiting the waveform superposition property of a multi-access channel to
implement the distributed second-order optimization algorithm over wireless
networks. The convergence behavior of the proposed algorithm is further
characterized, which reveals a linear-quadratic convergence rate with an
accumulative error term in each iteration. We thus propose a system
optimization approach to minimize the accumulated error gap by joint device
selection and beamforming design. Numerical results demonstrate the system and
communication efficiency compared with the state-of-the-art approaches.
