  Communication has been known to be one of the primary bottlenecks of
federated learning (FL), and yet existing studies have not addressed the
efficient communication design, particularly in wireless FL where both uplink
and downlink communications have to be considered. In this paper, we focus on
the design and analysis of physical layer quantization and transmission methods
for wireless FL. We answer the question of what and how to communicate between
clients and the parameter server and evaluate the impact of the various
quantization and transmission options of the updated model on the learning
performance. We provide new convergence analysis of the well-known FedAvg under
non-i.i.d. dataset distributions, partial clients participation, and
finite-precision quantization in uplink and downlink communications. These
analyses reveal that, in order to achieve an O(1/T) convergence rate with
quantization, transmitting the weight requires increasing the quantization
level at a logarithmic rate, while transmitting the weight differential can
keep a constant quantization level. Comprehensive numerical evaluation on
various real-world datasets reveals that the benefit of a FL-tailored uplink
and downlink communication design is enormous - a carefully designed
quantization and transmission achieves more than 98% of the floating-point
baseline accuracy with fewer than 10% of the baseline bandwidth, for majority
of the experiments on both i.i.d. and non-i.i.d. datasets. In particular, 1-bit
quantization (3.1% of the floating-point baseline bandwidth) achieves 99.8% of
the floating-point baseline accuracy at almost the same convergence rate on
MNIST, representing the best known bandwidth-accuracy tradeoff to the best of
the authors' knowledge.
