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Stochastic sub-gradient algorithm for distributed optimization with random sleep scheme

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Abstract

In this paper, we consider a distributed convex optimization problem of a multi-agent system with the global objective function as the sum of agents’ individual objective functions. To solve such an optimization problem, we propose a distributed stochastic sub-gradient algorithm with random sleep scheme. In the random sleep scheme, each agent independently and randomly decides whether to inquire the sub-gradient information of its local objective function at each iteration. The algorithm not only generalizes distributed algorithms with variable working nodes and multi-step consensus-based algorithms, but also extends some existing randomized convex set intersection results. We investigate the algorithm convergence properties under two types of stepsizes: the randomized diminishing stepsize that is heterogeneous and calculated by individual agent, and the fixed stepsize that is homogeneous. Then we prove that the estimates of the agents reach consensus almost surely and in mean, and the consensus point is the optimal solution with probability 1, both under randomized stepsize. Moreover, we analyze the algorithm error bound under fixed homogeneous stepsize, and also show how the errors depend on the fixed stepsize and update rates.

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Authors and Affiliations

  1. Key Lab of Systems and Control, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Peng Yi & Yiguang Hong

Authors
  1. Peng Yi
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  2. Yiguang Hong
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Corresponding author

Correspondence to Yiguang Hong.

Additional information

This work was supported by the Beijing Natural Science Foundation (No. 4152057), the National Natural Science Foundation of China (No. 61333001) and the Program 973 (No. 2014CB845301/2/3).

Peng YI is a Ph.D. candidate at the Academy of Mathematics and Systems Science, Chinese Academy of Sciences. He received his B.Sc. degree of Automation from University of Science and Technology of China in 2011. His research interest covers multi-agents system, distributed optimization, hybrid system and smart grid.

Yiguang HONG received the B.Sc. and M.Sc. degrees from Peking University, Beijing, China, and the Ph.D. degree from the Chinese Academy of Sciences (CAS), Beijing. He is currently a professor in Academy of Mathematics and Systems Science, CAS, and serves as the Director of Key Lab of Systems and Control, CAS and the Director of the Information Technology Division, National Center for Mathematics and Interdisciplinary Sciences, CAS. His research interests include nonlinear dynamics and control, multi-agent systems, distributed optimization, social networks, and software reliability.

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Yi, P., Hong, Y. Stochastic sub-gradient algorithm for distributed optimization with random sleep scheme. Control Theory Technol. 13, 333–347 (2015). https://doi.org/10.1007/s11768-015-5100-8

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  • DOI: https://doi.org/10.1007/s11768-015-5100-8

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