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We consider the task of minimizing the sum of smooth and strongly convexfunctions stored in a decentralized manner across the nodes of a communicationnetwork whose links are allowed to change in time. We solve two fundamentalproblems for this task. First, we establish the first lower bounds on thenumber of decentralized communication rounds and the number of localcomputations required to find an $\epsilon$-accurate solution. Second, wedesign two optimal algorithms that attain these lower bounds: (i) a variant ofthe recently proposed algorithm ADOM (Kovalev et al., 2021) enhanced via amulti-consensus subroutine, which is optimal in the case when access to thedual gradients is assumed, and (ii) a novel algorithm, called ADOM+, which isoptimal in the case when access to the primal gradients is assumed. Wecorroborate the theoretical efficiency of these algorithms by performing anexperimental comparison with existing state-of-the-art methods.

Lower Bounds and Optimal Algorithms for Smooth and Strongly Convex Decentralized Optimization Over Time-Varying Networks