  We propose an improved convergence analysis technique that characterizes the
distributed learning paradigm of federated learning (FL) with imperfect/noisy
uplink and downlink communications. Such imperfect communication scenarios
arise in the practical deployment of FL in emerging communication systems and
protocols. The analysis developed in this paper demonstrates, for the first
time, that there is an asymmetry in the detrimental effects of uplink and
downlink communications in FL. In particular, the adverse effect of the
downlink noise is more severe on the convergence of FL algorithms. Using this
insight, we propose improved Signal-to-Noise (SNR) control strategies that,
discarding the negligible higher-order terms, lead to a similar convergence
rate for FL as in the case of a perfect, noise-free communication channel while
incurring significantly less power resources compared to existing solutions. In
particular, we establish that to maintain the $O(\frac{1}{\sqrt{K}})$ rate of
convergence like in the case of noise-free FL, we need to scale down the uplink
and downlink noise by $\Omega({\sqrt{k}})$ and $\Omega({k})$ respectively,
where $k$ denotes the communication round, $k=1,\dots, K$. Our theoretical
result is further characterized by two major benefits: firstly, it does not
assume the somewhat unrealistic assumption of bounded client dissimilarity, and
secondly, it only requires smooth non-convex loss functions, a function class
better suited for modern machine learning and deep learning models. We also
perform extensive empirical analysis to verify the validity of our theoretical
findings.
