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Data perturbations in stochastic generalized equations: statistical robustness in static and sample average approximated models

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Abstract

Sample average approximation which is also known as Monte Carlo method has been widely used for solving stochastic programming and equilibrium problems. In a data-driven environment, samples are often drawn from empirical data and hence may be potentially contaminated. Consequently it is legitimate to ask whether statistical estimators obtained from solving the sample average approximated problems are statistically robust, that is, the difference between the laws of the statistical estimators based on contaminated data and real data is controllable under some metrics. In Guo and Xu (Math Program 190:679–720, 2021), we address the issue for the estimators of the optimal values of a wide range of stochastic programming problems. In this paper, we complement the research by investigating the optimal solution estimators and we do so by considering stochastic generalized equations (SGE) as a unified framework. Specifically, we look into the impact of a single data perturbation on the solutions of the SGE using the notion of influence function in robust statistics. Since the SGE may have multiple solutions, we use the proto-derivative of a set-valued mapping to introduce the notion of generalized influence function (GIF) and derive sufficient conditions under which the GIF is well defined, bounded and uniformly bounded. We then move on to quantitative statistical analysis of the SGE when all of sample data are potentially contaminated and demonstrate under moderate conditions quantitative statistical robustness of the solutions obtained from solving sample average approximated SGE.

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References

  1. Balasubramanian, K., Yuan, M.: Discussion of “estimating structured high-dimensional covariance and precision matrices: optimal rates and adaptive estimation’’. Electron. J. Stat. 10, 71–73 (2016)

    MathSciNet  MATH  Google Scholar 

  2. Basar, T., Olsder, G.: Dynamic Noncooperative Game Theory. SIAM, Philadelphia (1999)

    MATH  Google Scholar 

  3. Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problem. Springer, New York (2000)

    MATH  Google Scholar 

  4. Chen, X., Shapiro, A., Sun, H.: Convergence analysis of sample average approximation of two-stage stochastic generalized equations. SIAM J. Optim. 29, 135–161 (2019)

    MathSciNet  MATH  Google Scholar 

  5. Chen, Y., Lan, G., Ouyang, Y.: Accelerated schemes for a class of variational inequalities. Math. Program. 165, 113–149 (2017)

    MathSciNet  MATH  Google Scholar 

  6. Claus, M.: Advancing stability analysis of mean-risk stochastic programs: bilevel and two-stage models. Ph.D. thesis, Universität Dusburg-Essen (2016)

  7. Claus, M., Krätschmer, V., Schultz, R.: Weak continuity of risk functionals with applications to stochastic programming. SIAM J. Optim. 27, 91–109 (2017)

    MathSciNet  MATH  Google Scholar 

  8. Cont, R., Deguest, R., Scandolo, G.: Robustness and sensitivity analysis of risk measurement procedures. Quant. Finance 10, 593–606 (2010)

    MathSciNet  MATH  Google Scholar 

  9. Dontchev, A.L.: A local selection theorem for metrically regular mappings. J. Convex Anal. 11, 81–94 (2004)

    MathSciNet  MATH  Google Scholar 

  10. Dontchev, A.L., Rockafellar, R.T.: Characterizations of strong regularity for variational inequalities over polyhedral convex sets. SIAM J. Optim. 6, 1087–1105 (1996)

    MathSciNet  MATH  Google Scholar 

  11. Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York (2007)

    MATH  Google Scholar 

  12. Guo, S., Xu, H.: Statistical robustness in utility preference robust optimization models. Math. Program. 190, 679–720 (2021)

    MathSciNet  MATH  Google Scholar 

  13. Guo, S., Xu, H., Zhang, L.: Statistical robustness of empirical risks in machine learning. arXiv:2005.08458v (2020)

  14. Gürkan, G., Ozge, A.Y., Robinson, S.M.: Sample path solution of stochastic variational inequalities, with applications to option pricing. In: Charnes, J.M., Morrice, D.M., Brunner, D.T., Swai, J.J. (eds.) Proceedings of the 1996 Winter Simulation Conference, pp. 337–344 (1996)

  15. Gürkan, G., Ozge, A.Y., Robinson, S.M.: Sample-path solution of stochastic variational inequalities. Math. Program. 84, 313–333 (1999)

    MathSciNet  MATH  Google Scholar 

  16. Hampel, F.R.: A general statistical definition of robustness. Ann. Math. Stat. 42, 1887–1896 (1971)

    MATH  Google Scholar 

  17. Hampel, F. R.: Contribution to the theory of robust estimation. Ph.D. thesis, University of California, Berkeley (1968)

  18. Henrion, R., Jourani, A., Outrata, J.: On the calmness of a class of multifunctions. SIAM J. Optim. 13, 603–618 (2002)

    MathSciNet  MATH  Google Scholar 

  19. Huber, P.J.: Robust Statistics. Wiley, New York (1981)

    MATH  Google Scholar 

  20. Huber, P.J., Ronchetti, E.M.: Robust Statistics, 2nd edn. Wiley, Hoboken (2009)

    MATH  Google Scholar 

  21. Iusem, A.N., Jofré, A., Oliveira, R.I., Thompson, P.: Extragradient method with variance reduction for stochastic variational inequalities. SIAM J. Optim. 27, 686–724 (2017)

    MathSciNet  MATH  Google Scholar 

  22. Jiang, J., Li, S.: Statistical robustness of two-stage stochastic variational inequalities. Optim. Lett. 16, 2591–2605 (2022)

    MathSciNet  MATH  Google Scholar 

  23. Jiang, H., Xu, H.: Stochastic approximation approaches to the stochastic variational inequality problem. IEEE Trans. Autom. Control 53, 1462–1475 (2008)

    MathSciNet  MATH  Google Scholar 

  24. Kakutani, S.: A generalization of Brouwer’s fixed point theorem. Duke Math. J. 8, 457–459 (1941)

    MathSciNet  MATH  Google Scholar 

  25. King, A.J., Rockafellar, R.T.: Asymptotic theory for solutions in statistical estimation and stochastic programming. Math. Oper. Res. 18, 148–162 (1993)

    MathSciNet  MATH  Google Scholar 

  26. King, A.J., Rockafellar, R.T.: Sensitivity analysis for nonsmooth generalized equations. Math. Program. 55, 191–212 (1992)

    MathSciNet  MATH  Google Scholar 

  27. Krätschmer, V., Schied, A., Zähle, H.: Comparative and qualitative robustness for law-invariant risk measures. Finance Stoch. 18, 271–295 (2014)

    MathSciNet  MATH  Google Scholar 

  28. Krätschmer, V., Schied, A., Zähle, H.: Qualitative and infinitesimal robustness of tail-dependent statistical functionals. J. Multivar. Anal. 103, 35–47 (2012)

    MathSciNet  MATH  Google Scholar 

  29. Krätschmer, V., Schied, A., Zähle, H.: Domains of weak continuity of statistical functionals with a view toward robust statistics. J. Multivar. Anal. 158, 1–19 (2017)

    MathSciNet  MATH  Google Scholar 

  30. Lamm, M., Lu, S., Budhiraja, A.: Individual confidence intervals for solutions to expected value formulations of stochastic variational inequalities. Math. Program. 165, 151–196 (2017)

    MathSciNet  MATH  Google Scholar 

  31. Lecué, G., Lerasle, M.: Robust machine learning by median-of-means: theory and practice. arXiv:1711.10306 (2017)

  32. Levy, A.B.: Implicit multifunction theorems for the sensitivity analysis of variational conditions. Math. Program. 74, 215–223 (1996)

    MathSciNet  MATH  Google Scholar 

  33. Levy, A.B., Rockafellar, T.: Sensitivity analysis of solutions to generalized equations. Trans. Am. Math. Soc. 345, 661–671 (1994)

    MathSciNet  MATH  Google Scholar 

  34. Liu, Y., Römisch, W., Xu, H.: Quantitative stability analysis of stochastic generalized equations. SIAM J. Optim. 24, 467–497 (2014)

    MathSciNet  MATH  Google Scholar 

  35. Lu, S., Budhiraja, A.: Confidence regions for stochastic variational inequalities. Math. Oper. Res. 38, 545–568 (2013)

    MathSciNet  MATH  Google Scholar 

  36. Mahajan, S., van Ryzin, G.: Inventory competition under dynamic consumer choice. Oper. Res. 49, 646–657 (2001)

    MathSciNet  MATH  Google Scholar 

  37. Ngo, M.H., Krishnamurthy, V.K.: Game theoretic cross-layer transmission policies in multipacket reception wireless networks. IEEE Trans. Signal Process. 55, 1911–1926 (2007)

    MathSciNet  MATH  Google Scholar 

  38. Prokhorov, Y.V.: Convergence of random processes and limit theorems in probability theory. Theory Probab. Appl. 1, 157–214 (1956)

    MathSciNet  Google Scholar 

  39. Rachev, S.T.: Probability Metrics and the Stability of Stochastic Models, vol. 269. Wiley, London (1991)

    MATH  Google Scholar 

  40. Rachev, S.T., Römisch, W.: Quantitative stability in stochastic programming: the method of probability metrics. Math. Oper. Res. 27, 792–818 (2002)

    MathSciNet  MATH  Google Scholar 

  41. Ravat, U., Shanbhag, U.V.: On the existence of solutions to stochastic variational inequality and complementarity problems. Math. Program. 165, 291–330 (2017)

    MathSciNet  MATH  Google Scholar 

  42. Robinson, S.M.: Regularity and stability for convex multivalued functions. Math. Oper. Res. 1, 131–143 (1976)

    MathSciNet  Google Scholar 

  43. Rockafellar, R.T.: Proto-differentiability of set-valued mappings and its applications in optimization. In: Attouch, H., et al. (eds.) Analyse Non Linéaire, pp. 449–482. Gauthier-Villars, Paris (1989)

    Google Scholar 

  44. Rockafellar, R.T., Sun, J.: Solving monotone stochastic variational inequalities and complementarity problems by progressive hedging. Math. Program. 174, 453–471 (2019)

    MathSciNet  MATH  Google Scholar 

  45. Rockafellar, R.T., Wets, J.B.: Variational Analysis. Springer, Berlin (1998)

    MATH  Google Scholar 

  46. Römisch, W.: Stability of stochastic programming problems. In: Ruszczyński, A., Shapiro, A. (eds.) Stochastic Programming, Handbooks in Operations Research and Management Science, vol. 10. Elsevier, Amsterdam (2003)

    Google Scholar 

  47. Shapiro, A.: Monte Carlo sampling methods. In: Rusczyński, A., Shapiro, A. (eds.) Stochastic Programming, Handbooks in OR & MS, vol. 10. North-Holland Publishing Company, Amsterdam (2003)

    Google Scholar 

  48. Shapiro, A., Dentcheva, D., Ruszczyński, A.: Lectures on Stochastic Programming: Modeling and Theory, 2nd edn. SIAM, Philadelphia (2014)

    MATH  Google Scholar 

  49. Steinwart, I., Christmann, A.: Support Vector Machines. Springer, New York (2008)

    MATH  Google Scholar 

  50. Tukey, J.W.: A survey of sampling from contaminated distributions. Contributions to Probability and Statistics 2, 448–485 (1960)

    MathSciNet  MATH  Google Scholar 

  51. Tukey, J.W.: The future of data analysis. Ann. Math. Stat. 33(1), 1–67 (1962)

    MathSciNet  MATH  Google Scholar 

  52. Wang, M., Bertsekas, D.: Incremental constraint projection methods for variational inequalities. Math. Program. 150, 321–363 (2015)

    MathSciNet  MATH  Google Scholar 

  53. Wang, W., Xu, H., Ma, T.: Quantitative statistical robustness for tail-dependent law invariant risk measures. Quant. Finance 21, 1669–1685 (2021)

    MathSciNet  Google Scholar 

  54. Watling, D.: User equilibrium traffic network assignment with stochastic travel times and late arrival penalty. Eur. J. Oper. Res. 175, 1539–1556 (2006)

    MATH  Google Scholar 

  55. Xu, H.: Sample average approximation methods for a class of stochastic variational inequality problems. Asia-Pac. J. Oper. Res. 27, 103–119 (2010)

    MathSciNet  MATH  Google Scholar 

  56. Xu, H.: Uniform exponential convergence of sample average random functions under general sampling with applications in stochastic programming. J. Math. Anal. Appl. 368, 692–710 (2010)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank two anonymous referees for insightful comments which help us significantly strengthen the paper, they would also like to thank the associate editor for effective handling of the review. Finally the second author would like to thank Alex Shapiro and Henryk Zähale for valuable discussions about topology of weak convergence and Sainan Zhang for careful reading of the manuscript.

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Authors and Affiliations

  1. School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China

    Shaoyan Guo

  2. Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, Hong Kong

    Huifu Xu

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  1. Shaoyan Guo
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  2. Huifu Xu
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Correspondence to Huifu Xu.

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This paper is a part of a project supported by GRC Grant (14204821). The research of the first author is supported by the National Natural Science Foundation of China (12271077, 11801057).

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Guo, S., Xu, H. Data perturbations in stochastic generalized equations: statistical robustness in static and sample average approximated models. Math. Program. 202, 135–168 (2023). https://doi.org/10.1007/s10107-023-01925-9

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