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Robbins–Monro Algorithm with \(\boldsymbol{\psi}\)-Mixing Random Errors

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Abstract

In this work, we first establish exponential inequalities for the Robbins–Monro’s algorithm under \(\psi\)-mixing random errors. Then, we present a numerical application that uses the main result of this work to approximate the theoretical solution of the objective function.

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REFERENCES

  1. N. Aiane and A. Dahmani, ‘‘On the Rate of Convergence of the Robbins–Monro’s Algorithm in a Linear Stochastic Ill-posed Problems with alpha-mixing Data,’’ Communications in Statistics-Theory and Methods 46 (13), 6694–6703 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  2. P. Alquier, P. Doukhan, and X. Fan, ‘‘Exponential inequalities for nonstationary Markov chains,’’ Depend. Model 7, 150–168 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  3. I. Arab and A. Dahmani, ‘‘Consistency of stochastic approximation algorithm with quasi-associated random errors,’’ Communications in Statistics-Theory and Methods 45 (23), 6883–6890 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  4. I. Arab and A. Dahmani, ‘‘Complete Convergence of Stochastic Approximation Algorithm in \(R^{d}\) under Random Errors,’’ Sequential Analysis: Design Methods and Applications 35 (2), 216–225 (2016).

    MATH  Google Scholar 

  5. J. R. Blum, D. L. Hanson, and L. Koopmans, ‘‘On the strong law of large numbers for a class of stochastic process,’’ Z. Wahrscheinlichkeitstheor. Verwandte Geb. 2, 1–11 (1963).

    Article  MATH  Google Scholar 

  6. B. V. Bondarev and A. Dahmani, ‘‘Exponential Bound in stochastic approximation procedures,’’ 741–745, (1990), translated from Ukrainskii Matematicheskii Zhurnal 41 (7), 867–872 (1989).

  7. I. S. Borisov and N. V. Volodko, ‘‘A note on exponential inequalities for the distribution tails of canonical von Mises’ statistics of dependent observations,’’ Statistics and Probability Letters 96, 287–291 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  8. S. Bouzebda and Y. Slaoui, ‘‘Bandwidth selector for nonparametric recursive density estimation for spatial data defined by stochastic approximation method,’’ Communications in Statistics—Theory and Methods 49 (12), 2942–2963 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  9. R. C. Bradley, Jr., ‘‘On the \(\psi\)-Mixing Condition for Stationary Random Sequences,’’ Transactions of the American Mathematical Society 276 (1) , 55–66 (1983).

    MathSciNet  MATH  Google Scholar 

  10. H. F. Chen, Stochastic Approximation and Its Applications (Kluwer Academic Publishers—Dordrecht/Boston/London, 2003), Vol. 64.

    Google Scholar 

  11. A. Dahmani, M. H. Bengrina, and A. Ait Saidi, ‘‘Bernstein-Frechet inequalities for the Robbins Monro’s Algorithm,’’ International Journal of Applied Mathematics and Statistics 3 (D05), 17–24 (2005).

    MathSciNet  MATH  Google Scholar 

  12. A. Dahmani and A. Ait Saidi, ‘‘Consistency of Robbins Monro’s algorithm within a mixing framework,’’ Bulletin de la Société Royale des Sciences de Liège 79, 131–140 (2010).

    MathSciNet  MATH  Google Scholar 

  13. J. Dedecker, P. Doukhan, G. Lang, J. R. León, S. Louhichi, and C. Prieur, Weak Dependence: With Examples and Applications, Lecture Note in Statistics (Springer, 2007), p. 190.

    Book  MATH  Google Scholar 

  14. B. Delyon, Exponential inequalities for dependent processes (hal-01072019, 2015).

  15. S. Dereich, ‘‘General multilevel adaptations for stochastic approximation algorithms II: CLTs,’’ Stochastic Processes and Their Applications 132, 226–260 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  16. X. Fan, ‘‘Cramér type moderate deviations for self-normalized \(\psi\)-mixing sequences,’’ J. Math. Anal. Appl. 486, 123902 1–17 (2020).

  17. H. Hang and I. Steinwart, ‘‘A Bernstein-type inequality for some mixing processes and dynamical systems with an application to learning,’’ The Annals of Statistics 45 (2), 708–743 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  18. L. Heinrich, ‘‘Rates of convergence in stable limit theorems for sums of exponentially \(\psi\)-mixing random variables with an application to metric theory of continued fractions,’’ Math. Nadir. 131, 149–165 (1987).

    MATH  Google Scholar 

  19. D. Hu, P. Chen, and S. H. Sung, ‘‘Strong laws for weighted sums of \(\psi\)-mixing random variables and applications in errors-in-variables regression models,’’ TEST 600–617 (2017).

  20. W. Huang and W. B. Haskell, ‘‘Stochastic approximation for risk-aware Markov decision processes,’’ IEEE Transactions on Automatic Control. 66 (3), 1314–1320 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  21. M. Iosifescu and C. Kraaikamp, Metrical Theory of Continued Fractions, Mathematics and Its Applications (Kluwer, 2002), p. 547.

    Book  MATH  Google Scholar 

  22. R. Kannan and J. R. Luedtke, A Stochastic Approximation Method for Approximating the Efficient Frontier of Chance-Constrained Nonlinear Programs (Mathematical Programming Computation, 2021).

  23. X. Liu and X. Jin, ‘‘The Spitzer law for \(\psi\)-mixing random variables,’’ Journal of Mathematical Inequalities 12 (3), 883–888 (2018).

    MathSciNet  MATH  Google Scholar 

  24. F. Maouche, A. Dahmani, and N. Rahmania, ‘‘Stochastic procedure to solve linear ill-posed problems,’’ Communications in Statistics—Theory and Methods 46 (3), 1519–1531 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  25. P. Max, ‘‘L’approximation et l’optimisation stochastiques en économie: deux exemples,’’ Revue économique 24 (1), 176–182 (1973).

    Google Scholar 

  26. M. B. Nevelson and R. Z. Hasminskii, Stochastic Approximation and Recursive Estimation (American Mathematical Society, 1976).

    Book  Google Scholar 

  27. J. Y. Ouvrard, Probabilités 2 Master—Agrégation (Cassini, Paris, 2009).

    Google Scholar 

  28. I. F. Pinelis and A. I. Sakhanenko, ‘‘Remarks on inequalities for large deviation probabilities,’’ Theory of Probability and Its Applications 30 (1), 143–148 (1986).

    Article  MATH  Google Scholar 

  29. S. Rahmani and A. Dahmani, ‘‘Exponential inequalities for the Robbins–Monro’s algorithm under mixing condition,’’ Communications in Statistics—Theory and Methods 45 (2), 520–528 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  30. H. Robbins and S. Monro, ‘‘A stochastic approximation method,’’ Ann. Math. Stat. 22, 400–407 (1951).

    Article  MathSciNet  MATH  Google Scholar 

  31. G. G. Roussas, ‘‘Exponential probability inequalities with some applications,’’ Statistics, Probability, and Game Theory IMS, Lecture Notes 30, 303–319 (1996).

    MathSciNet  Google Scholar 

  32. A. Shen, H. Zhu, and Y. Zhang, ‘‘Exponential inequality for \(\rho\)-mixing sequences and its applications,’’ Filomat 28 (4), 859–870 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  33. Y. Shen, F. Han, and D. Witten, ‘‘Exponential inequalities for dependent \(V\)-statistics via random Fourier features,’’ Electron. J. Probab. 25 (7), 1–18 (2020).

    MathSciNet  MATH  Google Scholar 

  34. M. T. Wasan, Stochastic Approximation, Cambridge University Press (Revised Edition, 2004).

    MATH  Google Scholar 

  35. X. Wang, S. Hu, Y. Shen, and W. Yang, ‘‘Maximal inequality for \(\psi\)-mixing sequences and its applications,’’ Applied Mathematics Letters 23, 1156–1161 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  36. X. Wang, S. Wang, and R. Wang, ‘‘Exponential inequalities for sums of unbounded \(\phi\)-mixing sequence and their applications,’’ Communications in Statistics—Theory and Methods 46 (1), 457–464 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  37. H. Xu and L. Tang. ‘‘Strong convergence properties for \(\psi\)-mixing random variables,’’ Journal of Inequalities and Applications 360, 1–14 (2013).

    Google Scholar 

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ACKNOWLEDGMENTS

We thank the editor and the reviewers for several comments that helped improve the original version of this paper.

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Authors and Affiliations

  1. Laboratoire de Mathématiques Appliquées, Faculté des Sciences Exactes, Université de Bejaia, 06000, Bejaia, Algeria

    AbdelKader El Moumen

  2. University of Tamenghasset, 11000, Tamenghasset, Algeria

    AbdelKader El Moumen

  3. Laboratoire d’informatique et des Technologies Avancées du Numérique, Engineer School in Computer Science and Digital Technologies (ESTIN), Bejaia, Algeria

    Salim Benslimane

  4. Faculté de Technologie, Université de Bejaia, 06000, Bejaia, Algeria

    Samir Rahmani

Authors
  1. AbdelKader El Moumen
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  2. Salim Benslimane
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  3. Samir Rahmani
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Correspondence to AbdelKader El Moumen.

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El Moumen, A., Benslimane, S. & Rahmani, S. Robbins–Monro Algorithm with \(\boldsymbol{\psi}\)-Mixing Random Errors. Math. Meth. Stat. 31, 105–119 (2022). https://doi.org/10.3103/S1066530722030024

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  • DOI: https://doi.org/10.3103/S1066530722030024

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