  One of the key challenges towards the deployment of over-the-air federated
learning (AirFL) is the design of mechanisms that can comply with the power and
bandwidth constraints of the shared channel, while causing minimum
deterioration to the learning performance as compared to baseline noiseless
implementations. For additive white Gaussian noise (AWGN) channels with
instantaneous per-device power constraints, prior work has demonstrated the
optimality of a power control mechanism based on norm clipping. This was done
through the minimization of an upper bound on the optimality gap for smooth
learning objectives satisfying the Polyak-{\L}ojasiewicz (PL) condition. In
this paper, we make two contributions to the development of AirFL based on norm
clipping, which we refer to as AirFL-Clip. First, we provide a convergence
bound for AirFLClip that applies to general smooth and non-convex learning
objectives. Unlike existing results, the derived bound is free from
run-specific parameters, thus supporting an offline evaluation. Second, we
extend AirFL-Clip to include Top-k sparsification and linear compression. For
this generalized protocol, referred to as AirFL-Clip-Comp, we derive a
convergence bound for general smooth and non-convex learning objectives. We
argue, and demonstrate via experiments, that the only time-varying quantities
present in the bound can be efficiently estimated offline by leveraging the
well-studied properties of sparse recovery algorithms.
