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Distributed multi-agent optimization with state-dependent communication

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Abstract

We study distributed algorithms for solving global optimization problems in which the objective function is the sum of local objective functions of agents and the constraint set is given by the intersection of local constraint sets of agents. We assume that each agent knows only his own local objective function and constraint set, and exchanges information with the other agents over a randomly varying network topology to update his information state. We assume a state-dependent communication model over this topology: communication is Markovian with respect to the states of the agents and the probability with which the links are available depends on the states of the agents. We study a projected multi-agent subgradient algorithm under state-dependent communication. The state-dependence of the communication introduces significant challenges and couples the study of information exchange with the analysis of subgradient steps and projection errors. We first show that the multi-agent subgradient algorithm when used with a constant stepsize may result in the agent estimates to diverge with probability one. Under some assumptions on the stepsize sequence, we provide convergence rate bounds on a “disagreement metric” between the agent estimates. Our bounds are time-nonhomogeneous in the sense that they depend on the initial starting time. Despite this, we show that agent estimates reach an almost sure consensus and converge to the same optimal solution of the global optimization problem with probability one under different assumptions on the local constraint sets and the stepsize sequence.

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References

  1. Acemoglu D., Ozdaglar A., ParandehGheibi A.: Spread of (mis)information in social networks. Games Econ. Behav. 70, 194–227 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bertsekas D.P., Nedić A., Ozdaglar A.E.: Convex analysis and optimization. Athena Scientific, Cambridge, MA (2003)

    MATH  Google Scholar 

  3. Bertsekas D.P., Tsitsiklis J.N.: Parallel and Distributed Computation: Numerical Methods. Athena Scientific, Belmont, MA (1997)

    Google Scholar 

  4. Blondel V.D., Hendrickx J.M., Tsitsiklis J.N.: On Krause’s multi-agent consensus model with state-dependent connectivity. IEEE Trans. Automat. Control 54(11), 2586–2597 (2009)

    Article  MathSciNet  Google Scholar 

  5. Blondel V.D., Hendrickx J.M., Tsitsiklis J.N.: Continuous-time average-preserving opinion dynamics with opinion-dependent communications. SIAM J. Control Optim. 48(8), 5214–5240 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Blondel, V.D., Hendrickx, J.M., Olshevsky, A., Tsitsiklis, J.N.: Convergence in multiagent coordination, consensus, and flocking. In: Proceedings of IEEE CDC (2005)

  7. Boyd, S., Ghosh, A., Prabhakar, B., Shah, D.: Gossip algorithms: design, analysis, and applications. In: Proceedings of IEEE INFOCOM (2005)

  8. Cao, M., Spielman, D.A., Morse, A.S.: A lower bound on convergence of a distributed network consensus algorithm. In: Proceedings of IEEE CDC (2005)

  9. Duchi, J., Agarwal, A., Wainwright, M.: Dual averaging for distributed optimization: Convergence analysis and network scaling, Preprint (2010)

  10. Fagnani F., Zampieri S.: Randomized consensus algorithms over large scale networks. IEEE J. Sel. Areas Commun. 26, 634–649 (2008)

    Article  Google Scholar 

  11. Golub B., Jackson M.O.: Naive learning in social networks: Convergence, influence, and the wisdom of crowds. Am. Econ. J. Microecon. 2(1), 112–149 (2010)

    Article  Google Scholar 

  12. Hatano Y., Mesbahi M.: Agreement over random networks. IEEE Trans. Automat. Control 50(11), 1867–1872 (2005)

    Article  MathSciNet  Google Scholar 

  13. Jadbabaie A., Lin J., Morse S.: Coordination of groups of mobile autonomous agents using nearest neighbor rules. IEEE Trans. Automat. Control 48(6), 988–1001 (2003)

    Article  MathSciNet  Google Scholar 

  14. Lobel, I., Ozdaglar, A.: Distributed subgradient methods for convex optimization over random networks. IEEE Trans. Automat. Control (2011) (to appear)

  15. Lobel, I., Ozdaglar, A., Feijer, D.: Online appendices of distributed multi-agent optimization with state-dependent communication, http://www.pages.stern.nyu.edu/~ilobel/state-dep-online-app.pdf (2011)

  16. Matei, I., Baras, J.: Distributed subgradient method under random communication topology—the effect of the probability distribution of the random graph on the performance metrics, ISR Technical Report 2009-15 (2009)

  17. Nedić A., Olshevsky A., Ozdaglar A., Tsitsiklis J.N.: On distributed averaging algorithms and quantization effects. IEEE Trans. Automat. Control 54(11), 2506–2517 (2009)

    Article  MathSciNet  Google Scholar 

  18. Nedić A., Ozdaglar A.: Distributed subgradient methods for multi-agent optimization. IEEE Trans. Automat. Control 54(1), 48–61 (2009)

    Article  MathSciNet  Google Scholar 

  19. Nedić A., Ozdaglar A., Parrilo P.A.: Constrained consensus and optimization in multi-agent networks. IEEE Trans. Automat. Control 55, 922–938 (2010)

    Article  MathSciNet  Google Scholar 

  20. Nesterov Y.: Primal-dual subgradient methods for convex problems. Math. Program. 120(1), 261–283 (2009)

    Article  MathSciNet  Google Scholar 

  21. Olfati-Saber R., Murray R.M.: Consensus problems in networks of agents with switching topology and time-delays. IEEE Trans. Automat. Control 49(9), 1520–1533 (2004)

    Article  MathSciNet  Google Scholar 

  22. Olshevsky A., Tsitsiklis J.N.: Convergence speed in distributed consensus and averaging. SIAM J. Control Optim. 48(1), 33–55 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  23. Ram S.S., Nedić A., Veeravalli V.V.: Incremental stochastic subgradient algorithms for convex optimization. SIAM J. Optim. 20(2), 691–717 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  24. Tahbaz-Salehi A., Jadbabaie A.: A necessary and sufficient condition for consensus over random networks. IEEE Trans. Automat. Control 53(3), 791–795 (2008)

    Article  MathSciNet  Google Scholar 

  25. Tsitsiklis J.N., Bertsekas D.P., Athans M.: Distributed asynchronous deterministic and stochastic gradient optimization algorithms. IEEE Trans. Automat. Control 31(9), 803–812 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  26. Wu C.W.: Synchronization and convergence of linear dynamics in random directed networks. IEEE Trans. Automat. Control 51(7), 1207–1210 (2006)

    Article  MathSciNet  Google Scholar 

  27. Zhu, M., Martínez, S.: On distributed convex optimization under inequality and equality constraints via primal-dual subgradient methods, unpublished manuscript (2010)

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Authors and Affiliations

  1. Stern School of Business, New York University, New York, NY, USA

    Ilan Lobel

  2. Laboratory for Information and Decision Systems, Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA, USA

    Asuman Ozdaglar & Diego Feijer

Authors
  1. Ilan Lobel
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  2. Asuman Ozdaglar
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  3. Diego Feijer
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Corresponding author

Correspondence to Ilan Lobel.

Additional information

This paper is dedicated to the memory of Paul Tseng, a great researcher and friend.

This research was partially supported by the National Science Foundation under Career grant DMI-0545910, the DARPA ITMANET program, and the AFOSR MURI R6756-G2.

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Lobel, I., Ozdaglar, A. & Feijer, D. Distributed multi-agent optimization with state-dependent communication. Math. Program. 129, 255–284 (2011). https://doi.org/10.1007/s10107-011-0467-x

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  • DOI: https://doi.org/10.1007/s10107-011-0467-x

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