Advertisement

Target-sensitive control of Markov and semi-Markov processes

  • Abhijit Gosavi
Technical Notes and Correspondence

Abstract

We develop the theory for Markov and semi-Markov control using dynamic programming and reinforcement learning in which a form of semi-variance which computes the variability of rewards below a pre-specified target is penalized. The objective is to optimize a function of the rewards and risk where risk is penalized. Penalizing variance, which is popular in the literature, has some drawbacks that can be avoided with semi-variance.

Keywords

Relative value iteration semi-Markov control semi-variance stochastic shortest path problem target-sensitive 

References

  1. [1]
    J. Abounadi, D. Bertsekas, and V. Borkar, “Learning algorithms for Markov decision processes with average cost,” SIAM Journal of Control and Optimization, vol. 40, pp. 681–698, 2001.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    E. Altman, Constrained Markov Decision Processes, CRC Press, Boca Raton, 1998.Google Scholar
  3. [3]
    J. Baxter and P. Bartlett, “Infinite-horizon policygradient estimation,” Journal of Artificial Intelligence, vol. 15, pp. 319–350, 2001.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    D. P. Bertsekas and J. Tsitsiklis, Neuro-Dynamic Programming, Athena, Belmont, 1996.zbMATHGoogle Scholar
  5. [5]
    D. P. Bertsekas, Dynamic Programming and Optimal Control, 2nd edition, Athena, Belmont, 2000.Google Scholar
  6. [6]
    T. Bielecki, D. Hernandez-Hernandez, and S. Pliska, “Risk-sensitive control of finite state Markov chains in discrete time,” Math. Methods of Opns. Research, vol. 50, pp. 167–188, 1999.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    K. Boda and J. Filar, “Time consistent dynamic risk measures,” Mathematical Methods of Operations Research, vol. 63, pp. 169–186, 2005.MathSciNetCrossRefGoogle Scholar
  8. [8]
    V. Borkar and S. Meyn, “Risk-sensitive optimal control for Markov decision processes with monotone cost,” Mathematics of Operations Research, vol. 27, pp. 192–209, 2002.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    V. S. Borkar, “Stochastic approximation with two-time scales,” Systems and Control Letters, vol. 29, pp. 291–294, 1997.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    V. S. Borkar, “Asynchronous stochastic approximation,” SIAM Journal of Control and Optimization, vol. 36, no. 3, pp. 840–851, 1998.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    V. S. Borkar and S. P. Meyn, “The ODE method for convergence of stochastic approximation and reinforcement learning,” SIAM Journal of Control and Optimization, vol. 38, no. 2, pp. 447–469, 2000.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    V. S. Borkar and K. Soumyanath, “A new analog parallel scheme for fixed point computation, part I: Theory,” IEEE Trans. on Circuits and Systems I: Theory and Applications, vol. 44, pp. 351–355, 1997.MathSciNetCrossRefGoogle Scholar
  13. [13]
    M. Bouakiz and Y. Kebir, “Target-level criterion in Markov decision processes,” Journal of Optimization Theory and Applications, vol. 86, pp. 1–15, 1995.MathSciNetzbMATHCrossRefGoogle Scholar
  14. [14]
    S. J. Bradtke and M. Duff, “Reinforcement learning methods for continuous-time MDPs,” In Advances in Neural Information Processing Systems 7. MIT Press, Cambridge, MA, USA, 1995.Google Scholar
  15. [15]
    F. Brauer and J. Nohel, The Qualitative Theory of Ordinary Differential Equations: An Introduction, Dover Publishers, New York, 1989.Google Scholar
  16. [16]
    X.-R. Cao, “From perturbation analysis to Markov decision processes and reinforcement learning,” Discrete-Event Dynamic Systems: Theory and Applications, vol. 13, pp. 9–39, 2003.MathSciNetzbMATHCrossRefGoogle Scholar
  17. [17]
    X.-R. Cao, “Semi-Markov decision problems and performance sensitivity analysis,” IEEE Trans. on Automatic Control, vol. 48, no. 5, pp. 758–768, 2003.CrossRefGoogle Scholar
  18. [18]
    R. Cavazos-Cadena, “Solution to risk-sensitive average cost optimality equation in a class of MDPs with finite state space,” Math. Methods of Opns. Research, vol. 57, pp. 253–285, 2003.MathSciNetGoogle Scholar
  19. [19]
    R. Cavazos-Cadena and E. Fernandez-Gaucherand, “Controlled Markov chains with risk-sensitive criteria,” Mathematical Models of Operations Research, vol. 43, pp. 121–139, 1999.Google Scholar
  20. [20]
    R.-R. Chen and S. Meyn, “Value iteration and optimization of multiclass queueing networks,” Queueing Systems, vol. 32, pp. 65–97, 1999.MathSciNetzbMATHCrossRefGoogle Scholar
  21. [21]
    K. Chung and M. Sobel, “Discounted MDPs: distribution functions and exponential utility maximization,” SIAM Journal of Control and Optimization, vol. 25, pp. 49–62, 1987.MathSciNetzbMATHCrossRefGoogle Scholar
  22. [22]
    G. Di Masi and L. Stettner, “Risk-sensitive control of discrete-time Markov processes with infinite horizon,” SIAM Journal of Control and Optimization, vol. 38, no. 1, pp. 61–78, 1999.zbMATHCrossRefGoogle Scholar
  23. [23]
    J. Estrada, “Mean-semivariance behavior: Downside risk and capital asset pricing,” International Review of Economics and Finance, vol. 16, pp. 169–185, 2007.CrossRefGoogle Scholar
  24. [24]
    J. Filar, L. Kallenberg, and H. Lee, “Variancepenalized Markov decision processes,” Mathematics of Operations Research, vol. 14, no 1, pp. 147–161, 1989.MathSciNetzbMATHCrossRefGoogle Scholar
  25. [25]
    J. Filar, D. Krass, and K. Ross, “Percentile perfor mance criteria for limiting average Markov decision processes,” IEEE Trans. on Automatic Control, vol. 40, pp. 2–10, 1995.MathSciNetzbMATHCrossRefGoogle Scholar
  26. [26]
    W. Fleming and D. Hernandez-Hernandez, “Risksensitive control of finite state machines on an infinte horizon,” SIAM Journal of Control and Optimization, vol. 35, pp. 1790–1810, 1997.MathSciNetzbMATHCrossRefGoogle Scholar
  27. [27]
    A. Gosavi, “Reinforcement learning for long-run average cost,” European Journal of Operational Research, vol. 155, pp. 654–674, 2004.MathSciNetzbMATHCrossRefGoogle Scholar
  28. [28]
    A. Gosavi, “A risk-sensitive approach to total productive maintenance,” Automatica, vol. 42, pp. 1321–1330, 2006.MathSciNetzbMATHCrossRefGoogle Scholar
  29. [29]
    A. Gosavi, S. L. Murray, V. M. Tirumalasetty, and S. Shewade, “A budget-sensitive approach to scheduling maintenance in a total productive maintenance (TPM),” Engineering Management Journal, vol. 23, no. 3, pp. 46–56, 2011.Google Scholar
  30. [30]
    D. Hernandez-Hernandez and S. Marcus, “Risksensitive control of Markov processes in countable state space,” Systems and Control Letters, vol. 29, pp. 147–155, 1996.MathSciNetzbMATHCrossRefGoogle Scholar
  31. [31]
    R. Howard and J. Matheson, “Risk-sensitive MDPs,” Management Science, vol. 18, no. 7, pp. 356–369, 1972.MathSciNetzbMATHCrossRefGoogle Scholar
  32. [32]
    G. Hübner, “Improved procedures for eliminating sub-optimal actions in Markov programming by the use of contraction properties,” Transactions of 7th Prague Conference on Information Theory, Statistical Decision Functions, Random Processes, pp. 257–263, Dordrecht, 1978.Google Scholar
  33. [33]
    Q. Jiang, H.-S. Xi, and B.-Q. Yin, “Dynamic file grouping for load balancing in streaming media clustered server systems,” International Journal of Control, Automation, and Systems, vol. 7, no. 4, pp. 630–637, 2009.CrossRefGoogle Scholar
  34. [34]
    W. Y. Kwon, H. Suh, and S. Lee, “SSPQL: stochastic shortest path-based Q-learning,” International Journal of Control, Automation, and Systems, vol. 9, no. 2, pp. 328–338, 2011.CrossRefGoogle Scholar
  35. [35]
    A. E. B. Lim and X. Y. Zhou, “Risk-sensitive control with HARA utility,” IEEE Trans. on Automatic Control, vol. 46, no. 4, pp. 563–578, 2001.MathSciNetzbMATHCrossRefGoogle Scholar
  36. [36]
    R. Porter, “Semivariance and stochastic dominance,” American Economic Review, vol. 64, pp. 200–204, 1974.Google Scholar
  37. [37]
    M. L. Puterman, Markov Decision Processes, Wiley Interscience, New York, 1994.zbMATHCrossRefGoogle Scholar
  38. [38]
    S. Ross, Applied Probability Models with Optimization Applications, Dover, New York, 1992.zbMATHGoogle Scholar
  39. [39]
    E. Seneta, Non-Negative Matrices and Markov Chains, Springer-Verlag, NY, 1981.zbMATHGoogle Scholar
  40. [40]
    S. Singh, V. Tadic, and A. Doucet, “A policygradient method for semi-Markov decision processes with application to call admission control,” European Journal of Operational Research, vol. 178, no. 3, pp. 808–818, 2007.MathSciNetzbMATHCrossRefGoogle Scholar
  41. [41]
    M. Sobel, “The variance of discounted Markov decision processes,” Journal of Applied Probability, vol. 19, pp. 794–802, 1982.MathSciNetzbMATHCrossRefGoogle Scholar
  42. [42]
    H. C. Tijms, A First Course in Stochastic Models, 2nd edition, Wiley, 2003.Google Scholar
  43. [43]
    C. G. Turvey and G. Nayak, “The semi-varianceminimizing hedge ratios,” Journal of Agricultural and Resource Economics, vol. 28, no. 1, pp. 100–115, 2003.Google Scholar
  44. [44]
    D. White, “Minimizing a threshold probability in discounted Markov decision processes,” Journal of Mathematical Analysis and Applications, vol. 173, pp. 634–646, 1993.MathSciNetzbMATHCrossRefGoogle Scholar
  45. [45]
    C. Wu and Y. Lin, “Minimizing risk models in Markov decision processes with policies depending on target values,” Journal of Mathematical Analysis and Applications, vol. 231, pp. 47–67, 1999.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Institute of Control, Robotics and Systems and The Korean Institute of Electrical Engineers and Springer-Verlag Berlin Heidelberg  2011

Authors and Affiliations

  1. 1.Department of Engineering ManagementMissouri University of Science and Technology, 219 Engineering ManagementRollaUSA

Personalised recommendations