Novel Results on Random Walk-Jump Chains That Possess Tree-Based Transitions

Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 578)

Abstract

The most difficult task in analyzing and appraising algorithms in Artificial Intelligence (AI) involves their formal mathematical analysis. In general, such an analysis is intractable because of the size of the search space and the fact that the transitions between the states within this space can be very intricate. That is why AI algorithms are, for the most part, evaluated empirically and experimentally, i.e., by simulations. However, whenever such an analysis is undertaken, it usually involves an analysis of the underlying stochastic process. In this connection, the most common tools used involve Random Walks (RWs), which is a field that has been extensively studied for more than a century [6]. These walks have traditionally been on a line, and the generalizations for two and three dimensions, have been by extending the random steps to the corresponding neighboring positions in one or many of the dimensions. The analysis of RWs on a tree have received little attention, even though it is an important topic since a tree is a counter-part space representation of a line whenever there is some ordering on the nodes on the line.

Nevertheless, RWs on a tree entail moving to non-neighbor states in the space, which makes the analysis involved, and in many cases, impossible. This is precisely what we achieve in this rather pioneering paper. The applications of this paper are numerous. Indeed, the RW on the tree that this paper models, is a type of generalization of dichotomous search with faulty feedback about the direction of the search, rendering the real-life application of the model to be pertinent. To resolve this, we advocate the concept of “backtracking” transitions in order to efficiently explore the search space. Interestingly, it is precisely these “backtracking” transitions that naturally render the chain to be “time reversible”. By doing this, we are able to bridge the gap between deterministic dichotomous search and its faulty version, explained, in detail, in [21].

Keywords

Time reversibility Controlled random walk Random walk with jumps Dichotomous search Learning systems 

References

  1. 1.
    Altman, A., Tennenholtz, M.: Ranking systems: the pagerank axioms. In: EC 2005: Proceedings of the 6th ACM Conference on Electronic Commerce, pp. 1–8. ACM, New York (2005)Google Scholar
  2. 2.
    Ben Or, M., Hassidim, A.: The bayesian learner is optimal for noisy binary search (and pretty good for quantum as well). In: IEEE 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008, pp. 221–230. IEEE (2008)Google Scholar
  3. 3.
    Berg, H.C.: Random Walks in Biology, Revised edn. Princeton University Press, Princeton (1993)Google Scholar
  4. 4.
    Bishop, P.G., Pullen, F.D.: A random walk through software reliability theory. Math. Struct. Softw. Eng., 83–111 (1991)Google Scholar
  5. 5.
    Bower, G.H.: A turning point in mathematical learning theory. Psychol. Rev. 101(2), 290–300 (1994)CrossRefGoogle Scholar
  6. 6.
    Feller, W.: An Introduction to Probability Theory and Its Applications, vol. 1, 3rd edn. Wiley, New York (1968)MATHGoogle Scholar
  7. 7.
    Fouss, F., Pirotte, A., Renders, J.M., Saerens, M.: Random-walk computation of similarities between nodes of a graph with application to collaborative recommendation. IEEE Trans. Knowl. Data Eng. 19(3), 355–369 (2007)CrossRefGoogle Scholar
  8. 8.
    Granmo, O.C., Oommen, B.J.: Solving stochastic nonlinear resource allocation problems using a hierarchy of twofold resource allocation automata. IEEE Trans. Comput. 59, 545–560 (2009)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Gross, D., Harris, C.M.: Fundamentals of Queueing Theory (Wiley Series in Probability and Statistics). Wiley-Interscience, New York (1998)Google Scholar
  10. 10.
    Karlin, S., Taylot, H.: A First Course in Stochastic Processes. Academic Press, New York (1975)Google Scholar
  11. 11.
    Kelly, F.: Reversibility and Stochastic Networks. Wiley Series in Probability and Mathematical Statistics. Tracts on Probability and Statistics, Wiley, Chichester (1987)Google Scholar
  12. 12.
    Nowak, M.A.: Evolutionary Dynamics: Exploring the Equations of Life. Belknap Press of Harvard University Press, Cambridge (2006)MATHGoogle Scholar
  13. 13.
    Oommen, B.J.: Stochastic searching on the line and its applications to parameter learning in nonlinear optimization. IEEE Trans. Syst. Man Cybern. Part B: Cybern. 27(4), 733–739 (1997)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Oommen, J., Dong, J.: Generalized swap-with-parent schemes for self-organizing sequential linear lists. In: Leong, H.W., Imai, H., Jain, S. (eds.) ISAAC 1997. LNCS, vol. 1350, pp. 414–423. Springer, Heidelberg (1997). doi:10.1007/3-540-63890-3_44 CrossRefGoogle Scholar
  15. 15.
    Paulsen, J.: Ruin theory with compounding assets - a survey. Insur. Math. Econ. 22(1), 3–16 (1998). Special issue on the interplay between insurance, finance and controlMathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Pearson, K.: The problem of the random walk. Nature 72(1867), 342 (1905). http://dx.doi.org/10.1038/072342a0 CrossRefMATHGoogle Scholar
  17. 17.
    Ross, S.: Introduction to Probability Models. Academic Press, New York (1980)Google Scholar
  18. 18.
    Takacs, L.: On the classical ruin problems. J. Am. Stat. Assoc. 64(327), 889–906 (1969)MathSciNetMATHGoogle Scholar
  19. 19.
    Yazidi, A., Granmo, O.C., Oommen, B.J.: On the analysis of a random interleaving walk-jump process with applications to testing. Sequential Anal. 30(4), 457–478 (2011)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Yazidi, A., Granmo, O.C., Oommen, B.J., Goodwin, M.: A novel strategy for solving the stochastic point location problem using a hierarchical searching scheme. IEEE Trans. Cybern. 44(11), 2202–2220 (2014)CrossRefGoogle Scholar
  21. 21.
    Yazidi, A., Oommen, B.J.: On the analysis of a random walk-jump chain with tree-based transitions, and its applications to faulty dichotomous search. Unabridged version of this paper (2016). To be submitted for publicationGoogle Scholar
  22. 22.
    Zhang, J., Wang, Y., Wang, C., Zhou, M.: Symmetrical hierarchical stochastic searching on the line in informative and deceptive environments. To appear in IEEE Trans. Cybern. (2016)Google Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Department of Computer ScienceOslo and Akershus University College of Applied SciencesOsloNorway
  2. 2.School of Computer ScienceCarleton UniversityOttawaCanada

Personalised recommendations