  Distributed stochastic gradient descent (SGD) algorithms are widely deployed
in training large-scale deep learning models, while the communication overhead
among workers becomes the new system bottleneck. Recently proposed gradient
sparsification techniques, especially Top-$k$ sparsification with error
compensation (TopK-SGD), can significantly reduce the communication traffic
without an obvious impact on the model accuracy. Some theoretical studies have
been carried out to analyze the convergence property of TopK-SGD. However,
existing studies do not dive into the details of Top-$k$ operator in gradient
sparsification and use relaxed bounds (e.g., exact bound of Random-$k$) for
analysis; hence the derived results cannot well describe the real convergence
performance of TopK-SGD. To this end, we first study the gradient distributions
of TopK-SGD during the training process through extensive experiments. We then
theoretically derive a tighter bound for the Top-$k$ operator. Finally, we
exploit the property of gradient distribution to propose an approximate top-$k$
selection algorithm, which is computing-efficient for GPUs, to improve the
scaling efficiency of TopK-SGD by significantly reducing the computing
overhead. Codes are available at:
\url{https://github.com/hclhkbu/GaussianK-SGD}.
