Mini-batch stochastic gradient descent (SGD) is state of the art in large scale distributed training. The scheme can reach a linear speed-up 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 speed-up 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.
We prove that parallel local SGD achieves linear speedup with much lesser communication than parallel mini-batch SGD. Provides a convergence proof for local SGD, and proves that local SGD can provide the same speedup gains as minibatch, but may be able to communicate significantly less. This paper presents an analysis of local SGD and bounds on how frequent the estimators obtained by running SGD required to be averaged in order to yield linear parallelization speedups. The authors analyze the local SGD algorithm, where $K$ parallel chains of SGD are run, and the iterates are occasionally synchronized across machines by averaging