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Distributed Stochastic Subgradient Projection Algorithms for Convex Optimization

Author

Listed:
  • S. Sundhar Ram

    (University of Illinois at Urbana-Champaign)

  • A. Nedić

    (University of Illinois at Urbana-Champaign)

  • V. V. Veeravalli

    (University of Illinois at Urbana-Champaign)

Abstract

We consider a distributed multi-agent network system where the goal is to minimize a sum of convex objective functions of the agents subject to a common convex constraint set. Each agent maintains an iterate sequence and communicates the iterates to its neighbors. Then, each agent combines weighted averages of the received iterates with its own iterate, and adjusts the iterate by using subgradient information (known with stochastic errors) of its own function and by projecting onto the constraint set. The goal of this paper is to explore the effects of stochastic subgradient errors on the convergence of the algorithm. We first consider the behavior of the algorithm in mean, and then the convergence with probability 1 and in mean square. We consider general stochastic errors that have uniformly bounded second moments and obtain bounds on the limiting performance of the algorithm in mean for diminishing and non-diminishing stepsizes. When the means of the errors diminish, we prove that there is mean consensus between the agents and mean convergence to the optimum function value for diminishing stepsizes. When the mean errors diminish sufficiently fast, we strengthen the results to consensus and convergence of the iterates to an optimal solution with probability 1 and in mean square.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:147:y:2010:i:3:d:10.1007_s10957-010-9737-7
    DOI: 10.1007/s10957-010-9737-7
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    References listed on IDEAS

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    1. M. V. Solodov & S. K. Zavriev, 1998. "Error Stability Properties of Generalized Gradient-Type Algorithms," Journal of Optimization Theory and Applications, Springer, vol. 98(3), pages 663-680, September.
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    Cited by:

    1. Bin Hu & Zhi-Hong Guan & Rui-Quan Liao & Ding-Xue Zhang & Gui-Lin Zheng, 2015. "Consensus-based distributed optimisation of multi-agent networks via a two level subgradient-proximal algorithm," International Journal of Systems Science, Taylor & Francis Journals, vol. 46(7), pages 1307-1318, May.
    2. 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.
    3. Jueyou Li & Chuanye Gu & Zhiyou Wu & Changzhi Wu, 2017. "Distributed Optimization Methods for Nonconvex Problems with Inequality Constraints over Time-Varying Networks," Complexity, Hindawi, vol. 2017, pages 1-10, December.
    4. Maude J. Blondin & Matthew Hale, 2021. "A Decentralized Multi-objective Optimization Algorithm," Journal of Optimization Theory and Applications, Springer, vol. 189(2), pages 458-485, May.
    5. Junlong Zhu & Ping Xie & Mingchuan Zhang & Ruijuan Zheng & Ling Xing & Qingtao Wu, 2019. "Distributed Stochastic Subgradient Projection Algorithms Based on Weight-Balancing over Time-Varying Directed Graphs," Complexity, Hindawi, vol. 2019, pages 1-16, August.
    6. Haimonti Dutta, 2022. "A Consensus Algorithm for Linear Support Vector Machines," Management Science, INFORMS, vol. 68(5), pages 3703-3725, May.
    7. Wei Ni & Xiaoli Wang, 2022. "A Multi-Scale Method for Distributed Convex Optimization with Constraints," Journal of Optimization Theory and Applications, Springer, vol. 192(1), pages 379-400, January.

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