  Mini-batch stochastic gradient descent (SGD) is state of the art in large
scale distributed training. The scheme can reach a linear speedup with respect
to the number of workers, but this is rarely seen in practice as the scheme
often suffers from large network delays and bandwidth limits. To overcome this
communication bottleneck recent works propose to reduce the communication
frequency. An algorithm of this type is local SGD that runs SGD independently
in parallel on different workers and averages the sequences only once in a
while.
  This scheme shows promising results in practice, but eluded thorough
theoretical analysis. We prove concise convergence rates for local SGD on
convex problems and show that it converges at the same rate as mini-batch SGD
in terms of number of evaluated gradients, that is, the scheme achieves linear
speedup in the number of workers and mini-batch size. The number of
communication rounds can be reduced up to a factor of T^{1/2}---where T denotes
the number of total steps---compared to mini-batch SGD. This also holds for
asynchronous implementations. Local SGD can also be used for large scale
training of deep learning models.
  The results shown here aim serving as a guideline to further explore the
theoretical and practical aspects of local SGD in these applications.
