  We consider the distributed learning problem where a network of $n$ agents
seeks to minimize a global function $F$. Agents have access to $F$ through
noisy gradients, and they can locally communicate with their neighbors a
network. We study the Decentralized Local SDG method, where agents perform a
number of local gradient steps and occasionally exchange information with their
neighbors. Previous algorithmic analysis efforts have focused on the specific
network topology (star topology) where a leader node aggregates all agents'
information. We generalize that setting to an arbitrary network by analyzing
the trade-off between the number of communication rounds and the computational
effort of each agent. We bound the expected optimality gap in terms of the
number of iterates $T$, the number of workers $n$, and the spectral gap of the
underlying network. Our main results show that by using only $R=\Omega(n)$
communication rounds, one can achieve an error that scales as $O({1}/{nT})$,
where the number of communication rounds is independent of $T$ and only depends
on the number of agents. Finally, we provide numerical evidence of our
theoretical results through experiments on real and synthetic data.
