Skip to main content
Log in

Stochastic gradient processes: A survey of convergence theory using lyapunov second method

  • Systems Analysis
  • Published:
Cybernetics and Systems Analysis Aims and scope

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

References

  1. R. S. Bucy, “Stability and positive supermartingales,” J. Diff. Eq.,1, No. 2, 151–155 (1965).

    MATH  MathSciNet  Google Scholar 

  2. J. Doob, Stochastic Processes [Russian translation], IL, Moscow (1956).

    Google Scholar 

  3. V. Lakshmikantam, S. Lila, and A. A. Martynyuk, Stability of Motion: Comparison Method [in Russian], Naukova Dumka, Kiev (1991).

    Google Scholar 

  4. A. A. Lyapunov, General Problem of Stability of Motion [in Russian], Gostekhizdat, Moscow (1950).

    Google Scholar 

  5. A. N. Nakonechnyi, “Probability-theoretical generalization of Lyapunov second method,” Kibern. Sistem. Anal., No. 1, 68–81 (1993).

    MathSciNet  Google Scholar 

  6. A. N. Nakonechnyi, “Probability-theoretical analog of the method of vector Lyapunov functions,” Kibern. Sistem. Anal., No. 3, 110–119 (1994).

    MATH  MathSciNet  Google Scholar 

  7. A. N. Nakonechnyi, “Iterative processes: a survey of convergence theory using the Lyapunov second method,” Kibern. Sistem. Anal., No. 4, 66–85 (1994).

    MATH  MathSciNet  Google Scholar 

  8. J. B. Blum, “Multidimensional stochastic approximation method,” Ann. Math. Stat.,25, No. 4, 737–744 (1954).

    MATH  MathSciNet  Google Scholar 

  9. T. Morozan, “Sur l'approximation stochastique,” C. R. Acad. Sci., Ser. A,264, No. 14, 633–635 (1967).

    MathSciNet  Google Scholar 

  10. E. M. Braverman and L. I. Rozonoer, “Convergence of stochastic processes in machine learning theory, I,” Avtomat. Telemekh., No. 1, 57–77 (1969).

    MathSciNet  Google Scholar 

  11. B. T. Polyak, “Convergence and rate of convergence of iterative stochastic algorithms. I. The general case,” Avtomat. Telemekh., No. 12, 83–94 (1976).

    MATH  MathSciNet  Google Scholar 

  12. B. T. Polyak, An Introduction to Optimization [in Russian], Nauka, Moscow (1983).

    Google Scholar 

  13. V. Ya. Katkovnik, Linear Estimators and Stochastic Optimization Problems [in Russian], Nauka, Moscow (1976).

    Google Scholar 

  14. G. Ladde and D. Siljak, “Convergence and stability of distributed stochastic iterative processes,” IEEE Trans. Autom. Contr.,35, No. 6, 655–672 (1990).

    MathSciNet  Google Scholar 

  15. M. Bazara and C. M. Shetty, Nonlinear Programming: Theory and Algorithms [Russian translation], Mir, Moscow (1982).

    Google Scholar 

  16. H. Robins and S. Monro, “A stochastic approximation method,” Ann. Math. Stat.,22, No. 3, 400–407 (1951).

    Google Scholar 

  17. J. Kiefer and J. Wolfowitz, “Stochastic estimation of the maximum of a regression function,” Ann. Math. Stat.,23, No. 3, 426–466 (1952).

    MathSciNet  Google Scholar 

  18. M. Wasan, Stochastic Approximation [Russian translation], Mir, Moscow (1972).

    Google Scholar 

  19. H. Kesten, “Accelerated stochastic approximation method,” Ann. Math. Stat.,29, No. 1, 41–59 (1958).

    MATH  MathSciNet  Google Scholar 

  20. G. N. Saridis, “Stochastic approximation methods for identification and control,” IEEE Trans. Autom. Contr.,19, No. 6, 798–809 (1974).

    MATH  MathSciNet  Google Scholar 

  21. B. T. Polyak, “A new stochastic approximation method,” Avtomat. Telemekh., No. 7, 98–107 (1990).

    MATH  MathSciNet  Google Scholar 

  22. Yu. M. Ermol'ev, Stochastic Programming Methods [in Russian], Nauka, Moscow (1976).

    Google Scholar 

  23. A. Ruszczynski and W. Syski, “On convergence of the stochastic subgradient method with one-line stopsize rules,” J. Math. Anal. Appl.,114, No. 2, 512–527 (1986).

    MathSciNet  Google Scholar 

  24. S. P. Uryas'ev, Adaptive Algorithms for Stochastic Optimization and Game Theory [in Russian], Nauka, Moscow (1990).

    Google Scholar 

  25. V. F. Dem'yanov and A. B. Pevnyi, “Numerical method to find saddle points,” Zh. Vychisl. Matem. i Mat. Fiz.,12, No. 5, 1099–1127 (1972).

    Google Scholar 

  26. A. Auslender, “Recherche des points de sell d'une function,” Cahiers Cent. Etudes Rech. Oper.,12, No. 2, 57–75 (1970).

    MATH  MathSciNet  Google Scholar 

  27. K. J. Arrow, L. Hurwicz, and H. Uzawa, Studies in Linear and Nonlinear Programming [Russian translation], IL, Moscow (1962).

    Google Scholar 

  28. I. N. Kovalenko and A. N. Nakonechnyi, Approximate Calculation and Optimization of Reliability [in Russian], Naukova Dumka, Kiev (1989).

    Google Scholar 

  29. A. N. Nakonechnyi, “Extremal problems with rare events, I,” Kibernetika, No. 5, 55–58 (1990).

    MathSciNet  Google Scholar 

  30. A. N. Nakonechnyi, “Extremal problems with rare events, II,” Kibern. Sistem. Anal., No. 2, 33–39 (1992).

    MathSciNet  Google Scholar 

  31. S. M. Ermakov, Monte-Carlo Method and Related Topics [in Russian], Nauka, Moscow (1975).

    Google Scholar 

  32. N. Yu. Kuznetsov, “Analytical-statistical method for construction of quantitative estimates of continuity of characteristics of queueing systems and redundant systems,” in: Problems of Stability of Stochastic Models [in Russian], VNIISI, Moscow (1986), pp. 54–62.

    Google Scholar 

  33. I. N. Kovalenko, N. Yu. Kuznetsov, and A. N. Nakonechnyi, “Optimization of system reliability characteristics using quantitative continuity estimates and accelerated modeling methods,” in: Problems of Stability of Stochastic Models [in Russian], VNIISI, Moscow (1986), pp. 42–48.

    Google Scholar 

Download references

Authors

Additional information

The research described in this publication (and in publications [6, 7] in References) was made possible in part by Grant No. UAL000 from the International Science foundation.

Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 46–62, January–February, 1995.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Nakonechnyi, A.N. Stochastic gradient processes: A survey of convergence theory using lyapunov second method. Cybern Syst Anal 31, 37–51 (1995). https://doi.org/10.1007/BF02366794

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02366794

Keywords

Navigation