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In federated learning (FL), a global model is trained at a Parameter Server (PS) by aggregating model updates obtained from multiple remote learners. Generally, the communication between the remote users and the PS is rate-limited, while the transmission from the PS to the remote users are unconstrained. The FL setting gives rise to the distributed learning scenario in which the updates from the remote learners have to be compressed so as to meet communication rate constraints in the uplink transmission toward the PS. For this problem, one wishes to compress the model updates so as to minimize the loss in accuracy resulting from the compression error. In this paper, we take a rate-distortion approach to address the compressor design problem for the distributed training of deep neural networks (DNNs). In particular, we define a measure of the compression performance under communication-rate constraints -- the \emph{per-bit accuracy} -- which addresses the ultimate improvement of accuracy that a bit of communication brings to the centralized model. In order to maximize the per-bit accuracy, we consider modeling the DNN gradient updates at remote learners as a generalized normal distribution. Under this assumption on the DNN gradient distribution, we propose a class of distortion measures to aid the design of quantizers for the compression of the model updates. We argue that this family of distortion measures, which we refer to as "$M$-magnitude weighted $L_2$" norm, captures the practitioner's intuition in the choice of gradient compressor. Numerical simulations are provided to validate the proposed approach for the CIFAR-10 dataset.