  Decentralized stochastic optimization methods have gained a lot of attention
recently, mainly because of their cheap per iteration cost, data locality, and
their communication-efficiency. In this paper we introduce a unified
convergence analysis that covers a large variety of decentralized SGD methods
which so far have required different intuitions, have different applications,
and which have been developed separately in various communities.
  Our algorithmic framework covers local SGD updates and synchronous and
pairwise gossip updates on adaptive network topology. We derive universal
convergence rates for smooth (convex and non-convex) problems and the rates
interpolate between the heterogeneous (non-identically distributed data) and
iid-data settings, recovering linear convergence rates in many special cases,
for instance for over-parametrized models. Our proofs rely on weak assumptions
(typically improving over prior work in several aspects) and recover (and
improve) the best known complexity results for a host of important scenarios,
such as for instance coorperative SGD and federated averaging (local SGD).
