This paper investigates the transmission power control in over-the-air
federated edge learning (Air-FEEL) system. Different from conventional power
control designs (e.g., to minimize the individual mean squared error (MSE) of
the over-the-air aggregation at each round), we consider a new power control
design aiming at directly maximizing the convergence speed. Towards this end,
we first analyze the convergence behavior of Air-FEEL (in terms of the
optimality gap) subject to aggregation errors at different communication
rounds. It is revealed that if the aggregation estimates are unbiased, then the
training algorithm would converge exactly to the optimal point with mild
conditions; while if they are biased, then the algorithm would converge with an
error floor determined by the accumulated estimate bias over communication
rounds. Next, building upon the convergence results, we optimize the power
control to directly minimize the derived optimality gaps under both biased and
unbiased aggregations, subject to a set of average and maximum power
constraints at individual edge devices. We transform both problems into convex
forms, and obtain their structured optimal solutions, both appearing in a form
of regularized channel inversion, by using the Lagrangian duality method.
Finally, numerical results show that the proposed power control policies
achieve significantly faster convergence for Air-FEEL, as compared with
benchmark policies with fixed power transmission or conventional MSE
minimization.

研究了 Air-FEEL 系统中的传输功率控制，通过最大化收敛速度的方法优化功率控制策略，并使用拉格朗日对偶方法得到了优化解，结果表明该策略比传统策略实现了更快的收敛速度。