Communication trade-offs for Local-SGD with large step size

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.