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On the Linear Convergence of Two Decentralized Algorithms

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Abstract

Decentralized algorithms solve multi-agent problems over a connected network, where the information can only be exchanged with the accessible neighbors. Though there exist several decentralized optimization algorithms, there are still gaps in convergence conditions and rates between decentralized and centralized algorithms. In this paper, we fill some gaps by considering two decentralized algorithms: EXTRA and NIDS. They both converge linearly with strongly convex objective functions. We will answer two questions regarding them. What are the optimal upper bounds for their stepsizes? Do decentralized algorithms require more properties on the functions for linear convergence than centralized ones? More specifically, we relax the required conditions for linear convergence of both algorithms. For EXTRA, we show that the stepsize is comparable to that of centralized algorithms. For NIDS, the upper bound of the stepsize is shown to be exactly the same as the centralized ones. In addition, we relax the requirement for the objective functions and the mixing matrices. We provide the linear convergence results for both algorithms under the weakest conditions.

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Notes

  1. The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

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Acknowledgements

This work is partially supported by NSF grants DMS-1621798 and DMS-2012439.

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Authors and Affiliations

  1. Department of Mathematics, Michigan State University, East Lansing, USA

    Yao Li

  2. Department of Computational Mathematics, Science and Engineering and Department of Mathematics, Michigan State University, East Lansing, USA

    Ming Yan

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  1. Yao Li
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  2. Ming Yan
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Correspondence to Ming Yan.

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Communicated by Zenon Mróz.

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Li, Y., Yan, M. On the Linear Convergence of Two Decentralized Algorithms. J Optim Theory Appl 189, 271–290 (2021). https://doi.org/10.1007/s10957-021-01833-y

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  • DOI: https://doi.org/10.1007/s10957-021-01833-y

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