  One of the main focuses in distributed learning is communication efficiency,
since model aggregation at each round of training can consist of millions to
billions of parameters. Several model compression methods, such as gradient
quantization and sparsification, have been proposed to improve the
communication efficiency of model aggregation. However, the
information-theoretic minimum communication cost for a given distortion of
gradient estimators is still unknown. In this paper, we study the fundamental
limit of communication cost of model aggregation in distributed learning from a
rate-distortion perspective. By formulating the model aggregation as a vector
Gaussian CEO problem, we derive the rate region bound and sum-rate-distortion
function for the model aggregation problem, which reveals the minimum
communication rate at a particular gradient distortion upper bound. We also
analyze the communication cost at each iteration and total communication cost
based on the sum-rate-distortion function with the gradient statistics of
real-world datasets. It is found that the communication gain by exploiting the
correlation between worker nodes is significant for SignSGD, and a high
distortion of gradient estimator can achieve low total communication cost in
gradient compression.
