  In this paper, a communication-efficient federated learning (FL) framework is
proposed for improving the convergence rate of FL under a limited uplink
capacity. The central idea of the proposed framework is to transmit the values
and positions of the top-$S$ entries of a local model update for uplink
transmission. A lossless encoding technique is considered for transmitting the
positions of these entries, while a linear transformation followed by the
Lloyd-Max scalar quantization is considered for transmitting their values. For
an accurate reconstruction of the top-$S$ values, a linear minimum mean squared
error method is developed based on the Bussgang decomposition. Moreover, an
error feedback strategy is introduced to compensate for both compression and
reconstruction errors. The convergence rate of the proposed framework is
analyzed for a non-convex loss function with consideration of the compression
and reconstruction errors. From the analytical result, the key parameters of
the proposed framework are optimized for maximizing the convergence rate for
the given capacity. Simulation results on the MNIST and CIFAR-10 datasets
demonstrate that the proposed framework outperforms state-of-the-art FL
frameworks in terms of classification accuracy under the limited uplink
capacity.
