  This paper presents an approximate wireless communication scheme for
federated learning (FL) model aggregation in the uplink transmission. We
consider a realistic channel that reveals bit errors during FL model exchange
in wireless networks. Our study demonstrates that random bit errors during
model transmission can significantly affect FL performance. To overcome this
challenge, we propose an approximate communication scheme based on the
mathematical and statistical proof that machine learning (ML) model gradients
are bounded under certain constraints. This bound enables us to introduce a
novel encoding scheme for float-to-binary representation of gradient values and
their QAM constellation mapping. Besides, since FL gradients are
error-resilient, the proposed scheme simply delivers gradients with errors when
the channel quality is satisfactory, eliminating extensive error-correcting
codes and/or retransmission. The direct benefits include less overhead and
lower latency. The proposed scheme is well-suited for resource-constrained
devices in wireless networks. Through simulations, we show that the proposed
scheme is effective in reducing the impact of bit errors on FL performance and
saves at least half the time than transmission with error correction and
retransmission to achieve the same learning performance. In addition, we
investigated the effectiveness of bit protection mechanisms in high-order
modulation when gray coding is employed and found that this approach
considerably enhances learning performance.
