  This paper investigates the role of dimensionality reduction in efficient
communication and differential privacy (DP) of the local datasets at the remote
users for over-the-air computation (AirComp)-based federated learning (FL)
model. More precisely, we consider the FL setting in which clients are prompted
to train a machine learning model by simultaneous channel-aware and limited
communications with a parameter server (PS) over a Gaussian multiple-access
channel (GMAC), so that transmissions sum coherently at the PS globally aware
of the channel coefficients. For this setting, an algorithm is proposed based
on applying federated stochastic gradient descent (FedSGD) for training the
minimum of a given loss function based on the local gradients,
Johnson-Lindenstrauss (JL) random projection for reducing the dimension of the
local updates, and artificial noise to further aid user's privacy. For this
scheme, our results show that the local DP performance is mainly improved due
to injecting noise of greater variance on each dimension while keeping the
sensitivity of the projected vectors unchanged. This is while the convergence
rate is slowed down compared to the case without dimensionality reduction. As
the performance outweighs for the slower convergence, the trade-off between
privacy and convergence is higher but is shown to lessen in high-dimensional
regime yielding almost the same trade-off with much less communication cost.
