  Distributed parallel stochastic gradient descent algorithms are workhorses
for large scale machine learning tasks. Among them, local stochastic gradient
descent (Local SGD) has attracted significant attention due to its low
communication complexity. Previous studies prove that the communication
complexity of Local SGD with a fixed or an adaptive communication period is in
the order of $O (N^{\frac{3}{2}} T^{\frac{1}{2}})$ and $O (N^{\frac{3}{4}}
T^{\frac{3}{4}})$ when the data distributions on clients are identical (IID) or
otherwise (Non-IID), where $N$ is the number of clients and $T$ is the number
of iterations. In this paper, to accelerate the convergence by reducing the
communication complexity, we propose \textit{ST}agewise \textit{L}ocal
\textit{SGD} (STL-SGD), which increases the communication period gradually
along with decreasing learning rate. We prove that STL-SGD can keep the same
convergence rate and linear speedup as mini-batch SGD. In addition, as the
benefit of increasing the communication period, when the objective is strongly
convex or satisfies the Polyak-\L ojasiewicz condition, the communication
complexity of STL-SGD is $O (N \log{T})$ and $O (N^{\frac{1}{2}}
T^{\frac{1}{2}})$ for the IID case and the Non-IID case respectively, achieving
significant improvements over Local SGD. Experiments on both convex and
non-convex problems demonstrate the superior performance of STL-SGD.
