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].
J. Oommen—Chancellor’s Professor; Fellow: IEEE and Fellow: IAPR. This author is also an Adjunct Professor with the University of Agder in Grimstad, Norway. The work of this author was partially supported by NSERC, the Natural Sciences and Engineering Research Council of Canada.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
These algorithms can be designed to yield a high probabilistic guarantee of reaching the target. To do this, they formulate a limiting assumption: In order to choose a number of tests at each level that is large enough, one needs to know the exact probability by which the environment suggests the right direction of the search. In this paper, we assume that the latter probability is unknown and that the only known clue is that this quantity has a lower bound of \(\frac{1}{2}\). The case when the probability of correct transition is less than \(\frac{1}{2}\) is referred to as the symmetric environment [22].
- 2.
This is also in line with the theoretical results where we did not impose any condition on distributing \(1-p\). Thus, for the sake of brevity, we will not report those experiments with “uneven” distributions of \(1-p\) over the incorrect alternatives, and limit the reported results to the case of “even” distributions, i.e., \(\frac{1-p}{2}\).
References
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)
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)
Berg, H.C.: Random Walks in Biology, Revised edn. Princeton University Press, Princeton (1993)
Bishop, P.G., Pullen, F.D.: A random walk through software reliability theory. Math. Struct. Softw. Eng., 83–111 (1991)
Bower, G.H.: A turning point in mathematical learning theory. Psychol. Rev. 101(2), 290–300 (1994)
Feller, W.: An Introduction to Probability Theory and Its Applications, vol. 1, 3rd edn. Wiley, New York (1968)
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)
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)
Gross, D., Harris, C.M.: Fundamentals of Queueing Theory (Wiley Series in Probability and Statistics). Wiley-Interscience, New York (1998)
Karlin, S., Taylot, H.: A First Course in Stochastic Processes. Academic Press, New York (1975)
Kelly, F.: Reversibility and Stochastic Networks. Wiley Series in Probability and Mathematical Statistics. Tracts on Probability and Statistics, Wiley, Chichester (1987)
Nowak, M.A.: Evolutionary Dynamics: Exploring the Equations of Life. Belknap Press of Harvard University Press, Cambridge (2006)
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)
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
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 control
Pearson, K.: The problem of the random walk. Nature 72(1867), 342 (1905). http://dx.doi.org/10.1038/072342a0
Ross, S.: Introduction to Probability Models. Academic Press, New York (1980)
Takacs, L.: On the classical ruin problems. J. Am. Stat. Assoc. 64(327), 889–906 (1969)
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)
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)
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 publication
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)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG
About this paper
Cite this paper
Yazidi, A., John Oommen, B. (2018). Novel Results on Random Walk-Jump Chains That Possess Tree-Based Transitions. In: Kurzynski, M., Wozniak, M., Burduk, R. (eds) Proceedings of the 10th International Conference on Computer Recognition Systems CORES 2017. CORES 2017. Advances in Intelligent Systems and Computing, vol 578. Springer, Cham. https://doi.org/10.1007/978-3-319-59162-9_5
Download citation
DOI: https://doi.org/10.1007/978-3-319-59162-9_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-59161-2
Online ISBN: 978-3-319-59162-9
eBook Packages: EngineeringEngineering (R0)