Is Consensus Acceleration Possible in Decentralized Optimization over Slowly Time-Varying Networks?

We consider decentralized optimization problems where one aims to minimize a sum of convex smooth objective functions distributed between nodes in the network. The links in the network can change from time to time. For the setting when the amount of changes is arbitrary, lower complexity bounds and corresponding optimal algorithms are known, and the consensus acceleration is not possible. However, in practice the magnitude of network changes may be limited. We derive lower communication complexity bounds for several regimes of velocity of networks changes. Moreover, we show how to obtain accelerated communication rates for a certain class of time-varying graphs using a specific consensus algorithm. 1/2 g χ log(1/ε)) (Kovalev et al., 2021a). The corresponding optimal algorithms with primal oracle are ADOM+ (Kovalev et al., 2021a) and Acc-GT (Li & Lin, 2021) with multi-step communication. An optimal dual algorithm is ADOM (Kovalev et al., 2021b). Prior to optimal algorithms, several non-accelerated schemes like DIGing (Nedic et al., 2017) and sub-optimal methods with