  The optimal implementation of federated learning (FL) in practical edge
computing systems has been an outstanding problem. In this paper, we propose an
optimization-based quantized FL algorithm, which can appropriately fit a
general edge computing system with uniform or nonuniform computing and
communication resources at the workers. Specifically, we first present a new
random quantization scheme and analyze its properties. Then, we propose a
general quantized FL algorithm, namely GQFedWAvg. Specifically, GQFedWAvg
applies the proposed quantization scheme to quantize wisely chosen model
update-related vectors and adopts a generalized mini-batch stochastic gradient
descent (SGD) method with the weighted average local model updates in global
model aggregation. Besides, GQFedWAvg has several adjustable algorithm
parameters to flexibly adapt to the computing and communication resources at
the server and workers. We also analyze the convergence of GQFedWAvg. Next, we
optimize the algorithm parameters of GQFedWAvg to minimize the convergence
error under the time and energy constraints. We successfully tackle the
challenging non-convex problem using general inner approximation (GIA) and
multiple delicate tricks. Finally, we interpret GQFedWAvg's function principle
and show its considerable gains over existing FL algorithms using numerical
results.
