  Distributed stochastic gradient descent~(DSGD) has been widely used for
optimizing large-scale machine learning models, including both convex and
non-convex models. With the rapid growth of model size, huge communication cost
has been the bottleneck of traditional DSGD. Recently, many communication
compression methods have been proposed. Memory-based distributed stochastic
gradient descent~(M-DSGD) is one of the efficient methods since each worker
communicates a sparse vector in each iteration so that the communication cost
is small. Recent works propose the convergence rate of M-DSGD when it adopts
vanilla SGD. However, there is still a lack of convergence theory for M-DSGD
when it adopts momentum SGD. In this paper, we propose a universal convergence
analysis for M-DSGD by introducing \emph{transformation equation}. The
transformation equation describes the relation between traditional DSGD and
M-DSGD so that we can transform M-DSGD to its corresponding DSGD. Hence we get
the convergence rate of M-DSGD with momentum for both convex and non-convex
problems. Furthermore, we combine M-DSGD and stagewise learning that the
learning rate of M-DSGD in each stage is a constant and is decreased by stage,
instead of iteration. Using the transformation equation, we propose the
convergence rate of stagewise M-DSGD which bridges the gap between theory and
practice.
