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Convergence Results of a Nested Decentralized Gradient Method for Non-strongly Convex Problems

Author

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  • Woocheol Choi

    (Sungkyunkwan University)

  • Doheon Kim

    (Hanyang University)

  • Seok-Bae Yun

    (Sungkyunkwan University)

Abstract

We are concerned with the convergence of NEAR-DGD $$^+$$ + (Nested Exact Alternating Recursion Distributed Gradient Descent) method introduced to solve the distributed optimization problems. Under the assumption of the strong convexity of local objective functions and the Lipschitz continuity of their gradients, the linear convergence is established in Berahas et al. (IEEE Trans Autom Control 64:3141-3155, 2019). In this paper, we investigate the convergence property of NEAR-DGD $$^+$$ + in the absence of strong convexity. More precisely, we establish the convergence results in the following two cases: (1) When only the convexity is assumed on the objective function. (2) When the objective function is represented as a composite function of a strongly convex function and a rank deficient matrix, which falls into the class of convex and quasi-strongly convex functions. The numerical results are provided to support the convergence results.

Suggested Citation

  • Woocheol Choi & Doheon Kim & Seok-Bae Yun, 2022. "Convergence Results of a Nested Decentralized Gradient Method for Non-strongly Convex Problems," Journal of Optimization Theory and Applications, Springer, vol. 195(1), pages 172-204, October.
    • Handle: RePEc:spr:joptap:v:195:y:2022:i:1:d:10.1007_s10957-022-02069-0
    DOI: 10.1007/s10957-022-02069-0
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    References listed on IDEAS

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    1. Ion Necoara & Yurii Nesterov & François Glineur, 2019. "Linear convergence of first order methods for non-strongly convex optimization," LIDAM Reprints CORE 3000, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. S. Sundhar Ram & A. Nedić & V. V. Veeravalli, 2010. "Distributed Stochastic Subgradient Projection Algorithms for Convex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 147(3), pages 516-545, December.
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