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Decentralized optimization over slowly time-varying graphs: algorithms and lower bounds

Author

Listed:
  • Dmitry Metelev

    (Moscow Institute of Physics and Technology)

  • Aleksandr Beznosikov

    (Moscow Institute of Physics and Technology
    Skolkovo Institute of Science and Technology
    Mohamed bin Zayed University of Artificial Intelligence)

  • Alexander Rogozin

    (Moscow Institute of Physics and Technology
    Skolkovo Institute of Science and Technology
    HSE University)

  • Alexander Gasnikov

    (Moscow Institute of Physics and Technology
    Skolkovo Institute of Science and Technology
    Institute for Information Transmission Problems)

  • Anton Proskurnikov

    (Politecnico di Torino)

Abstract

We consider a decentralized convex unconstrained optimization problem, where the cost function can be decomposed into a sum of strongly convex and smooth functions, associated with individual agents, interacting over a static or time-varying network. Our main concern is the convergence rate of first-order optimization algorithms as a function of the network’s graph, more specifically, of the condition numbers of gossip matrices. We are interested in the case when the network is time-varying but the rate of changes is restricted. We study two cases: randomly changing network satisfying Markov property and a network changing in a deterministic manner. For the random case, we propose a decentralized optimization algorithm with accelerated consensus. For the deterministic scenario, we show that if the graph is changing in a worst-case way, accelerated consensus is not possible even if only two edges are changed at each iteration. The fact that such a low rate of network changes is sufficient to make accelerated consensus impossible is novel and improves the previous results in the literature.

Suggested Citation

  • Dmitry Metelev & Aleksandr Beznosikov & Alexander Rogozin & Alexander Gasnikov & Anton Proskurnikov, 2024. "Decentralized optimization over slowly time-varying graphs: algorithms and lower bounds," Computational Management Science, Springer, vol. 21(1), pages 1-25, June.
    • Handle: RePEc:spr:comgts:v:21:y:2024:i:1:d:10.1007_s10287-023-00489-5
    DOI: 10.1007/s10287-023-00489-5
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