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Distributed regression estimation with incomplete data in multi-agent networks

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In this paper, distributed regression estimation problem with incomplete data in a time-varying multi-agent network is investigated. Regression estimation is carried out based on local agent information with incomplete in the non-ignorable mechanism. By virtue of gradient-based design and adaptive filter, a distributed algorithm is proposed to deal with a regression estimation problem with incomplete data. With the help of convex analysis and stochastic approximation techniques, the exact convergence is obtained for the proposed algorithm with incomplete data and a jointly-connected multi-agent topology. Moreover, online regret analysis is also given for real-time learning. Then, simulations for the proposed algorithm are also given to demonstrate how it can solve the estimation problem in a distributed way, even when the network configuration is time-varying.

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  1. 1

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

  2. 2

    Shi G, Johansson K. Robust consensus for continuous-time multiagent dynamics. SIAM J Control Optim, 2013, 51: 3673–3691

  3. 3

    Zhang Y Q, Lou Y C, Hong Y G, et al. Distributed projection-based algorithms for source localization in wireless sensor networks. IEEE Trans Wirel Commun, 2015, 43: 3131–3142

  4. 4

    Feng H, Jiang Z D, Hu B, et al. The incremental subgradient methods on distributed estimations in-network. Sci China Inf Sci, 2014, 57: 092103

  5. 5

    Lou Y C, Hong Y G, Wang S Y. Distributed continuous-time approximate projection protocols for shortest distance optimization problems. Automatica, 2016, 69: 289–297

  6. 6

    Yi P, Hong Y G, Liu F. Initialization-free distributed algorithms for optimal resource allocation with feasibility con-straints and application to economic dispatch of power systems. Automatica, 2016, 74: 259–269

  7. 7

    Kokaram A C. On missing data treatment for degraded video and film archives: a survey and a new Bayesian approach. IEEE Trans Image Process, 2004, 13: 397–415

  8. 8

    Molenberghs G, Kenward M G. Missing Data in Clinical Studies. New York: Wiley, 2007

  9. 9

    Ibrahim J G, Chen M H, Lipsitz S R, et al. Missing data methods for generalized linear models: a comparative review. J Am Stat Assoc, 2005, 100: 332–346

  10. 10

    Gholami M R, Jansson M, Strom E G, et al. Diffusion estimation over cooperative multi-agent networks with missing data. IEEE Trans Signal Inf Process Netw, 2016, 2: 276–289

  11. 11

    Davey A, Savla J. Statistical Power Analysis with Missing Data: A Structural Equation Modeling Approach. Oxford, UK: Routledge Academic, 2009

  12. 12

    Ram S S, Nedić A, Veeravalli V V. Distributed stochastic subgradient projection algorithms for convex optimization. J Optim Theory Appl, 2010, 147: 516–545

  13. 13

    Graybill F, Iyer H K. Regression Analysis: Concepts and Applications. California: Duxbury Press Belmont, 1994

  14. 14

    Feng Y, Sundaram S, Vishwanathan S V N, et al. Distributed autonomous online learning: regrets and intrinsic privacy-preserving properties. IEEE Trans Knowl Data Eng, 2013, 25: 2483–2493

  15. 15

    Hazan E, Kale S. Beyond the regret minimization barrier: optimal algorithms for stochastic strongly-convex optimiza-tion. J Mach Learn Res, 2014, 15: 2489–2512

  16. 16

    Shamir O, Zhang T. Stochastic gradient descent for non-smooth optimization: convergence results and optimal aver-aging schemes. In: Proceedings of International Conference on Machine Learning, Edinburgh, 2012. 71–79

  17. 17

    Towfic Z J, Chen J S, Sayed A H. On distributed online classification in the midst of concept drifts. Neurocomputing, 2013, 112: 138–152

  18. 18

    Widrow B, Stearns S D. Adaptive Signal Processing. Cliffs: Prentice-Hall, 1985. 1–32

  19. 19

    Sayed A H. Adaptation, learning, and optimization over networks. Found Trends Mach Learn, 2014, 7: 311–801

  20. 20

    Sayed A H, Tu S Y, Chen J S, et al. Diffusion strategies for adaptation and learning. IEEE Signal Proc Mag, 2013, 30: 155–171

  21. 21

    Polyak B T. Introduction to Optimization. New York: Optimization Software Inc., 1983. 2–8

  22. 22

    Godsil C, Royle G. Algebraic Graph Theory. New York: Springer-Verlag, 2001. 1–18

  23. 23

    Ferguson T S. A Course in Large Sample Theory. London: Chapman and Hall Ltd., 1996. 3–4

  24. 24

    Durrett R. Probability Theory and Examples. Camberidge, UK: Camberidge Press, 2010. 328–347

  25. 25

    Enders C K. Applied Missing Data Analysis. New York: The Guilford Press, 2010

  26. 26

    Kushner H J, Yin G. Stochastic Approximation and Recursive Algorithms and Applications. New York: Springer-Verlag, 1997. 117–157

  27. 27

    Widrow B, Mccool J, Larimore M G, et al. Stationary and nonstationary learning characteristics of the LMS adaptive filter. Proc IEEE, 1976, 64: 1151–1162

  28. 28

    Yi P, Hong Y G. Stochastic sub-gradient algorithm for distributed optimization with random sleep scheme. Control Theory Technol, 2015, 13: 333–347

  29. 29

    Larsen R J, Max M L. An Introduction to Mathematical Statistics and Its Applications. 4th ed. New York: Pearson, 2006. 221–280

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This work was supported by National Key Research and Development Program of China (Grant No. 2016YFB0901902) and National Natural Science Foundation of China (Grant Nos. 61573344, 61333001, 61374168).

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Correspondence to Yinghui Wang.

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Wang, Y., Lin, P. & Hong, Y. Distributed regression estimation with incomplete data in multi-agent networks. Sci. China Inf. Sci. 61, 092202 (2018). https://doi.org/10.1007/s11432-016-9173-8

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  • multi-agent systems
  • time-varying network
  • estimation with incomplete data
  • online learning
  • stochastic approximation