  We study federated machine learning (ML) at the wireless edge, where power-
and bandwidth-limited wireless devices with local datasets carry out
distributed stochastic gradient descent (DSGD) with the help of a remote
parameter server (PS). Standard approaches assume separate computation and
communication, where local gradient estimates are compressed and transmitted to
the PS over orthogonal links. Following this digital approach, we introduce
D-DSGD, in which the wireless devices employ gradient quantization and error
accumulation, and transmit their gradient estimates to the PS over a multiple
access channel (MAC). We then introduce a novel analog scheme, called A-DSGD,
which exploits the additive nature of the wireless MAC for over-the-air
gradient computation, and provide convergence analysis for this approach. In
A-DSGD, the devices first sparsify their gradient estimates, and then project
them to a lower dimensional space imposed by the available channel bandwidth.
These projections are sent directly over the MAC without employing any digital
code. Numerical results show that A-DSGD converges faster than D-DSGD thanks to
its more efficient use of the limited bandwidth and the natural alignment of
the gradient estimates over the channel. The improvement is particularly
compelling at low power and low bandwidth regimes. We also illustrate for a
classification problem that, A-DSGD is more robust to bias in data distribution
across devices, while D-DSGD significantly outperforms other digital schemes in
the literature. We also observe that both D-DSGD and A-DSGD perform better by
increasing the number of devices (while keeping the total dataset size
constant), showing their ability in harnessing the computation power of edge
devices.
