Over-the-air computation is a communication-efficient solution for federated
learning (FL). In such a system, iterative procedure is performed: Local
gradient of private loss function is updated, amplified and then transmitted by
every mobile device; the server receives the aggregated gradient all-at-once,
generates and then broadcasts updated model parameters to every mobile device.
In terms of amplification factor selection, most related works suppose the
local gradient's maximal norm always happens although it actually fluctuates
over iterations, which may degrade convergence performance. To circumvent this
problem, we propose to turn local gradient to be normalized one before
amplifying it. Under our proposed method, when the loss function is smooth, we
prove our proposed method can converge to stationary point at sub-linear rate.
In case of smooth and strongly convex loss function, we prove our proposed
method can achieve minimal training loss at linear rate with any small positive
tolerance. Moreover, a tradeoff between convergence rate and the tolerance is
discovered. To speedup convergence, problems optimizing system parameters are
also formulated for above two cases. Although being non-convex, optimal
solution with polynomial complexity of the formulated problems are derived.
Experimental results show our proposed method can outperform benchmark methods
on convergence performance.

通过将本地梯度归一化，改进了过程中最相关工作对于放大因子选择的假设不准确的问题，同时在平滑和强凸损失函数的情况下，提出的方法能够以次线性或线性速率收敛到稳定点，还发现了收敛速率和容差之间的权衡，并通过优化系统参数加快了收敛速度，实验结果表明该方法在收敛性能上优于基准方法。