  We consider the distributed stochastic optimization problem where $n$ agents
want to minimize a global function given by the sum of agents' local functions,
and focus on the heterogeneous setting when agents' local functions are defined
over non-i.i.d. data sets. We study the Local SGD method, where agents perform
a number of local stochastic gradient steps and occasionally communicate with a
central node to improve their local optimization tasks. We analyze the effect
of local steps on the convergence rate and the communication complexity of
Local SGD. In particular, instead of assuming a fixed number of local steps
across all communication rounds, we allow the number of local steps during the
$i$-th communication round, $H_i$, to be different and arbitrary numbers. Our
main contribution is to characterize the convergence rate of Local SGD as a
function of $\{H_i\}_{i=1}^R$ under various settings of strongly convex,
convex, and nonconvex local functions, where $R$ is the total number of
communication rounds. Based on this characterization, we provide sufficient
conditions on the sequence $\{H_i\}_{i=1}^R$ such that Local SGD can achieve
linear speed-up with respect to the number of workers. Furthermore, we propose
a new communication strategy with increasing local steps superior to existing
communication strategies for strongly convex local functions. On the other
hand, for convex and nonconvex local functions, we argue that fixed local steps
are the best communication strategy for Local SGD and recover state-of-the-art
convergence rate results. Finally, we justify our theoretical results through
extensive numerical experiments.
