Applied Intelligence

, Volume 33, Issue 1, pp 3–20 | Cite as

Optimal sampling for estimation with constrained resources using a learning automaton-based solution for the nonlinear fractional knapsack problem

Article

Abstract

While training and estimation for Pattern Recognition (PR) have been extensively studied, the question of achieving these when the resources are both limited and constrained is relatively open. This is the focus of this paper. We consider the problem of allocating limited sampling resources in a “real-time” manner, with the explicit purpose of estimating multiple binomial proportions (the extension of these results to non-binomial proportions is, in our opinion, rather straightforward). More specifically, the user is presented with ‘ntraining sets of data points, S1,S2,…,Sn, where the set Si has Ni points drawn from two classes {ω1,ω2}. A random sample in set Si belongs to ω1 with probability ui and to ω2 with probability 1−ui, with {ui}, i=1,2,…n, being the quantities to be learnt. The problem is both interesting and non-trivial because while both n and each Ni are large, the number of samples that can be drawn is bounded by a constant, c. A web-related problem which is based on this model (Snaprud et al., The Accessibility for All Conference, 2003) is intriguing because the sampling resources can only be allocated optimally if the binomial proportions are already known. Further, no non-automaton solution has ever been reported if these proportions are unknown and must be sampled.

Using the general LA philosophy as a paradigm to tackle this real-life problem, our scheme improves a current solution in an online manner, through a series of informed guesses which move towards the optimal solution. We solve the problem by first modelling it as a Stochastic Non-linear Fractional Knapsack Problem. We then present a completely new on-line Learning Automata (LA) system, namely, the Hierarchy of Twofold Resource Allocation Automata (H-TRAA), whose primitive component is a Twofold Resource Allocation Automaton (TRAA), both of which are asymptotically optimal. Furthermore, we demonstrate empirically that the H-TRAA provides orders of magnitude faster convergence compared to the Learning Automata Knapsack Game (LAKG) which represents the state-of-the-art. Finally, in contrast to the LAKG, the H-TRAA scales sub-linearly. Based on these results, we believe that the H-TRAA has also tremendous potential to handle demanding real-world applications, particularly those dealing with the world wide web.

Keywords

Sample size determination Training with constrained resources Constrained estimation for pattern recognition Nonlinear knapsack problems Hierarchical learning Learning automata Stochastic optimization 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bhattacharyya GK, Johnson RA (1977) Statistical concepts and methods. Wiley, New York Google Scholar
  2. 2.
    Bickel P, Doksum K (2000) Mathematical statistics: basic ideas and selected topics, vol 1, 2nd edn. Prentice Hall, New York Google Scholar
  3. 3.
    Black PE (2004) Fractional knapsack problem. In: Dictionary of algorithms and data structures Google Scholar
  4. 4.
    Bretthauer KM, Shetty B (2002) The nonlinear knapsack problem—algorithms and applications. Eur J Oper Res 138:459–472 MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Casella G, Berger R (2001) Statistical inference, 2nd edn. Brooks/Cole, Pacific Grove Google Scholar
  6. 6.
    Dean BC, Goemans MX, Vondrdk J (2004) Approximating the stochastic knapsack problem: the benefit of adaptivity. In: 45th annual IEEE symposium on foundations of computer science. IEEE, Los Alamitos, pp 208–217 CrossRefGoogle Scholar
  7. 7.
    Duda R, Hart P, Stork D (2000) Pattern classification, 2nd edn. Wiley, New York Google Scholar
  8. 8.
    Fox B (1966) Discrete optimization via marginal analysis. Manag Sci 13(3):211–216 CrossRefGoogle Scholar
  9. 9.
    Fukunaga K (1990) Introduction to statistical pattern recognition. Academic Press, San Diego MATHGoogle Scholar
  10. 10.
    Granmo O-C, Oommen BJ (2006) On allocating limited sampling resources using a learning automata-based solution to the fractional knapsack problem. In: Proceedings of the 2006 international intelligent information processing and web mining conference (IIS:IIPW’06). Advances in soft computing. Springer, Berlin, pp 263–272 Google Scholar
  11. 11.
    Granmo O-C, Oommen BJ, Myrer SA, Olsen MG (2006) Determining optimal polling frequency using a learning automata-based solution to the fractional knapsack problem. In: Proceedings of the 2006 IEEE international conferences on cybernetics & intelligent systems (CIS) and robotics, automation & mechatronics (RAM). IEEE, Los Alamitos, pp 73–79 Google Scholar
  12. 12.
    Granmo O-C, Oommen BJ, Myrer SA, Olsen MG (2007) Learning automata-based solutions to the nonlinear fractional knapsack problem with applications to optimal resource allocation. IEEE Trans Syst Man Cybern, Part B 37(1):166–175 CrossRefGoogle Scholar
  13. 13.
    Herbrich R (2001) Learning kernel classifiers: theory and algorithms. MIT Press, Cambridge Google Scholar
  14. 14.
    Jones B, Garthwaite P, Jolliffe I (2002) Statistical inference, 2nd edn. Oxford University Press, London MATHGoogle Scholar
  15. 15.
    Kellerer H, Pferschy U, Pisinger D (2004) Knapsack problems. Springer, Berlin MATHGoogle Scholar
  16. 16.
    Narendra KS, Thathachar MAL (1989) Learning automata: an introduction. Prentice Hall, New York Google Scholar
  17. 17.
    Oommen BJ (1986) Absorbing and ergodic discretized two action learning automata. IEEE Trans Syst Man Cybern 16:282–293 MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Oommen BJ (1997) Stochastic searching on the line and its applications to parameter learning in nonlinear optimization. IEEE Trans Syst Man Cybern, Part B 27(4):733–739 CrossRefMathSciNetGoogle Scholar
  19. 19.
    Oommen BJ, Rueda L (2006) Stochastic learning-based weak estimation of multinomial random variables and its applications to pattern recognition in non-stationary environments. Pattern Recogn 39:328–341 MATHCrossRefGoogle Scholar
  20. 20.
    Pandey S, Ramamritham K, Chakrabarti S (2003) Monitoring the dynamic web to respond to continuous queries. In: 12th international world wide web conference. ACM, New York, pp 659–668 Google Scholar
  21. 21.
    Ross KW, Tsang DHK (1989) The stochastic knapsack problem. IEEE Trans Commun 37(7):740–747 MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Ross S (2002) Introduction to probability models, 2nd edn. Academic Press, San Diego Google Scholar
  23. 23.
    Shao J (2003) Mathematical statistics, 2nd edn. Springer, Berlin MATHGoogle Scholar
  24. 24.
    Snaprud M, Ulltveit-Moe N, Granmo O-C, Rafoshei-Klev M, Wiklund A, Sawicka A (2003) Quantitative assessment of public web sites accessibility—some early results. In: The accessibility for all conference Google Scholar
  25. 25.
    Sprinthall J (2002) Basic statistical analysis, 2nd edn. Allyn and Bacon, Needhom Heights Google Scholar
  26. 26.
    Steinberg E, Parks MS (1979) A preference order dynamic program for a knapsack problem with stochastic rewards. J Oper Res Soc 30(2):141–147 MATHGoogle Scholar
  27. 27.
    Thathachar MAL, Sastry PS (2004) Networks of learning automata: techniques for online stochastic optimization. Kluwer Academic, Dordrecht Google Scholar
  28. 28.
    Tsetlin ML (1973) Automaton theory and modeling of biological systems. Academic Press, San Diego Google Scholar
  29. 29.
    Webb A (2000) Statistical pattern recognition, 2nd edn. Wiley, New York Google Scholar
  30. 30.
    Wolf JL, Squillante MS, Sethuraman J, Ozsen K (2002) Optimal crawling strategies for web search engines. In: 11th international world wide web conference. ACM, New York, pp 136–147 Google Scholar
  31. 31.
    Xiaoming Z (2004) Evaluation and enhancement of web content accessibility for persons with disabilities. PhD thesis, University of Pittsburgh Google Scholar
  32. 32.
    Zhu Q, Oommen BJ (2009) Estimation of distributions involving unobservable events: The case of optimal search with unknown target distributions. Pattern Anal Appl J 12(1):37–53 CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of Information and Communication TechnologyUniversity of AgderGrimstadNorway
  2. 2.School of Computer ScienceCarleton UniversityOttawaCanada
  3. 3.University of AgderGrimstadNorway

Personalised recommendations