  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.
