  We consider a distributed learning problem over multiple access channel (MAC)
using a large wireless network. The computation is made by the network edge and
is based on received data from a large number of distributed nodes which
transmit over a noisy fading MAC. The objective function is a sum of the nodes'
local loss functions. This problem has attracted a growing interest in
distributed sensing systems, and more recently in federated learning. We
develop a novel Gradient-Based Multiple Access (GBMA) algorithm to solve the
distributed learning problem over MAC. Specifically, the nodes transmit an
analog function of the local gradient using common shaping waveforms and the
network edge receives a superposition of the analog transmitted signals used
for updating the estimate. GBMA does not require power control or beamforming
to cancel the fading effect as in other algorithms, and operates directly with
noisy distorted gradients. We analyze the performance of GBMA theoretically,
and prove that it can approach the convergence rate of the centralized gradient
descent (GD) algorithm in large networks. Specifically, we establish a
finite-sample bound of the error for both convex and strongly convex loss
functions with Lipschitz gradient. Furthermore, we provide energy scaling laws
for approaching the centralized convergence rate as the number of nodes
increases. Finally, experimental results support the theoretical findings, and
demonstrate strong performance of GBMA using synthetic and real data.
