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.

翻译：空中联邦学习（AirFL）部署的关键挑战之一，在于设计既能满足共享信道的功率与带宽约束，又能将对学习性能的损害降至最低（与无噪声基准实现相比）的机制。针对具有瞬时每设备功率约束的加性高斯白噪声（AWGN）信道，先前的研究已证明基于范数裁剪的功率控制机制具有最优性——该结论通过最小化光滑学习目标（满足Polyak-Łojasiewicz (PL)条件）的最优性差距上界得出。本文为基于范数裁剪的AirFL（我们称之为AirFL-Clip）的发展做出两项贡献：首先，我们为AirFL-Clip提供了适用于一般光滑非凸学习目标的收敛界。与现有结果不同，推导出的界限不依赖于运行特定参数，从而支持离线评估。其次，我们将AirFL-Clip扩展至包含Top-k稀疏化与线性压缩。针对这一广义协议（称为AirFL-Clip-Comp），我们推导了适用于一般光滑非凸学习目标的收敛界。我们论证并通过实验表明，界限中仅有的时变量可通过利用稀疏恢复算法已充分研究的性质进行高效离线估计。