  We consider decentralized time-varying stochastic optimization problems where
each of the functions held by the nodes has a finite sum structure. Such
problems can be efficiently solved using variance reduction techniques. Our aim
is to explore the lower complexity bounds (for communication and number of
stochastic oracle calls) and find optimal algorithms. The paper studies
strongly convex and nonconvex scenarios. To the best of our knowledge, variance
reduced schemes and lower bounds for time-varying graphs have not been studied
in the literature. For nonconvex objectives, we obtain lower bounds and develop
an optimal method GT-PAGE. For strongly convex objectives, we propose the first
decentralized time-varying variance-reduction method ADOM+VR and establish
lower bound in this scenario, highlighting the open question of matching the
algorithms complexity and lower bounds even in static network case.
