  Synchronous mini-batch SGD is state-of-the-art for large-scale distributed
machine learning. However, in practice, its convergence is bottlenecked by slow
communication rounds between worker nodes. A natural solution to reduce
communication is to use the \emph{`local-SGD'} model in which the workers train
their model independently and synchronize every once in a while. This algorithm
improves the computation-communication trade-off but its convergence is not
understood very well. We propose a non-asymptotic error analysis, which enables
comparison to \emph{one-shot averaging} i.e., a single communication round
among independent workers, and \emph{mini-batch averaging} i.e., communicating
at every step. We also provide adaptive lower bounds on the communication
frequency for large step-sizes ($ t^{-\alpha} $, $ \alpha\in (1/2 , 1 ) $) and
show that \emph{Local-SGD} reduces communication by a factor of
$O\Big(\frac{\sqrt{T}}{P^{3/2}}\Big)$, with $T$ the total number of gradients
and $P$ machines.
