Browsing by Author "Nguyen, Edward Duc Hien"

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    On Graphs with Finite-Time Consensus and Their Use in Gradient Tracking
    (2024-05-20) Nguyen, Edward Duc Hien; Uribe, César
    A crucial design decision when employing distributed or decentralized optimization algorithms in practice is the choice of topology. A topology should be sufficiently well connected such that when agents communicate, agents reach a consensus faster. However, more densely connected topologies come with a price of higher bandwidth cost or latency. To address this issue, we study sequences of graphs satisfying the finite-time consensus property (i.e., iterating through such a finite sequence is equivalent to performing global or exact averaging) and their use in the decentralized optimization algorithm Gradient Tracking. We provide an explicit weight matrix representation of the studied sequences and prove their finite-time consensus property. Moreover, we incorporate the studied finite-time consensus topologies into Gradient Tracking and present a new algorithmic scheme called Gradient Tracking for Finite-Time Consensus Topologies (GT-FT). We analyze the new scheme for nonconvex problems with stochastic gradient estimates. Our analysis shows that the convergence rate of GT-FT does not depend on the heterogeneity of the agents' functions or the connectivity of any individual graph in the topology sequence. Furthermore, owing to the sparsity of the graphs, GT-FT requires lower communication costs than Gradient Tracking using the static counterpart of the topology sequence.