  Federated learning (FL) has emerged as an appealing machine learning approach
to deal with massive raw data generated at multiple mobile devices, {which
needs to aggregate the training model parameter of every mobile device at one
base station (BS) iteratively}. For parameter aggregating in FL, over-the-air
computation is a spectrum-efficient solution, which allows all mobile devices
to transmit their parameter-mapped signals concurrently to a BS. Due to
heterogeneous channel fading and noise, there exists difference between the
BS's received signal and its desired signal, measured as the mean-squared error
(MSE). To minimize the MSE, we propose to jointly optimize the signal
amplification factors at the BS and the mobile devices as well as the data size
(the number of data samples involved in local training) at every mobile device.
The formulated problem is challenging to solve due to its non-convexity. To
find the optimal solution, with some simplification on cost function and
variable replacement, which still preserves equivalence, we transform the
changed problem to be a bi-level problem equivalently. For the lower-level
problem, optimal solution is found by enumerating every candidate solution from
the Karush-Kuhn-Tucker (KKT) condition. For the upper-level problem, the
optimal solution is found by exploring its piecewise convexity. Numerical
results show that our proposed method can greatly reduce the MSE and can help
to improve the training performance of FL compared with benchmark methods.
