Optimal Distributed Convex Optimization on Slowly Time-Varying Graphs

We study optimal distributed first-order optimization algorithms when the network (i.e., communication constraints between the agents) changes with time. This problem is motivated by scenarios where agents experience network malfunctions. We provide a sufficient condition that guarantees a convergence rate with optimal (up to logarithmic terms) dependencies on the network and function parameters if the network changes are constrained to a small percentage alpha of the total number of iterations. We call such networks slowly time-varying networks. Moreover, we show that Nesterov's method has an iteration complexity of Omega((<(kappa(Phi) center dot <(chi)over bar>)over bar> + alpha log(kappa(Phi) center dot (chi) over bar)) log(1/epsilon)) for decentralized algorithms, where kappa(Phi) is the condition number of the objective function, and (chi) over bar is a worst case bound on the condition number of the sequence of communication graphs. Additionally, we provide an explicit upper bound on a in terms of the condition number of the objective function and network topologies.