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This paper studies sequences of graphs satisfying the finite-time consensusproperty (i.e., iterating through such a finite sequence is equivalent toperforming global or exact averaging) and their use in Gradient Tracking. Weprovide an explicit weight matrix representation of the studied sequences andprove their finite-time consensus property. Moreover, we incorporate thestudied finite-time consensus topologies into Gradient Tracking and present anew algorithmic scheme called Gradient Tracking for Finite-Time ConsensusTopologies (GT-FT). We analyze the new scheme for nonconvex problems withstochastic gradient estimates. Our analysis shows that the convergence rate ofGT-FT does not depend on the heterogeneity of the agents' functions or theconnectivity of any individual graph in the topology sequence. Furthermore,owing to the sparsity of the graphs, GT-FT requires lower communication coststhan Gradient Tracking using the static counterpart of the topology sequence.

On Graphs with Finite-Time Consensus and Their Use in Gradient Tracking