  Does Federated Learning (FL) work when both uplink and downlink
communications have errors? How much communication noise can FL handle and what
is its impact to the learning performance? This work is devoted to answering
these practically important questions by explicitly incorporating both uplink
and downlink noisy channels in the FL pipeline. We present several novel
convergence analyses of FL over simultaneous uplink and downlink noisy
communication channels, which encompass full and partial clients participation,
direct model and model differential transmissions, and non-independent and
identically distributed (IID) local datasets. These analyses characterize the
sufficient conditions for FL over noisy channels to have the same convergence
behavior as the ideal case of no communication error. More specifically, in
order to maintain the O(1/T) convergence rate of FedAvg with perfect
communications, the uplink and downlink signal-to-noise ratio (SNR) for direct
model transmissions should be controlled such that they scale as O(t^2) where t
is the index of communication rounds, but can stay constant for model
differential transmissions. The key insight of these theoretical results is a
"flying under the radar" principle - stochastic gradient descent (SGD) is an
inherent noisy process and uplink/downlink communication noises can be
tolerated as long as they do not dominate the time-varying SGD noise. We
exemplify these theoretical findings with two widely adopted communication
techniques - transmit power control and diversity combining - and further
validating their performance advantages over the standard methods via extensive
numerical experiments using several real-world FL tasks.
