  In this paper, a green-quantized FL framework, which represents data with a
finite precision level in both local training and uplink transmission, is
proposed. Here, the finite precision level is captured through the use of
quantized neural networks (QNNs) that quantize weights and activations in
fixed-precision format. In the considered FL model, each device trains its QNN
and transmits a quantized training result to the base station. Energy models
for the local training and the transmission with quantization are rigorously
derived. To minimize the energy consumption and the number of communication
rounds simultaneously, a multi-objective optimization problem is formulated
with respect to the number of local iterations, the number of selected devices,
and the precision levels for both local training and transmission while
ensuring convergence under a target accuracy constraint. To solve this problem,
the convergence rate of the proposed FL system is analytically derived with
respect to the system control variables. Then, the Pareto boundary of the
problem is characterized to provide efficient solutions using the normal
boundary inspection method. Design insights on balancing the tradeoff between
the two objectives while achieving a target accuracy are drawn from using the
Nash bargaining solution and analyzing the derived convergence rate. Simulation
results show that the proposed FL framework can reduce energy consumption until
convergence by up to 70\% compared to a baseline FL algorithm that represents
data with full precision without damaging the convergence rate.
