  We consider speeding up stochastic gradient descent (SGD) by parallelizing it
across multiple workers. We assume the same data set is shared among $n$
workers, who can take SGD steps and coordinate with a central server.
Unfortunately, this could require a lot of communication between the workers
and the server, which can dramatically reduce the gains from parallelism. The
Local SGD method, proposed and analyzed in the earlier literature, suggests
machines should make many local steps between such communications. While the
initial analysis of Local SGD showed it needs $\Omega ( \sqrt{T} )$
communications for $T$ local gradient steps in order for the error to scale
proportionately to $1/(nT)$, this has been successively improved in a string of
papers, with the state-of-the-art requiring $\Omega \left( n \left( \mbox{
polynomial in log } (T) \right) \right)$ communications. In this paper, we give
a new analysis of Local SGD. A consequence of our analysis is that Local SGD
can achieve an error that scales as $1/(nT)$ with only a fixed number of
communications independent of $T$: specifically, only $\Omega(n)$
communications are required.
