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Unsupervised Problem Decomposition Using Genetic Programming Ahmed Kattan, Alexandros Agapitos, and Riccardo Poli A.I. Esparcia-Alcazar et al. (Eds.): EuroGP 2010, LNCS 6021, pp. 122–133, 2010. ©Springer-Verlag Berlin Heidelberg 2010 1

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Introduction Problem decomposition aims to simplify complex real world problems in order to better cope with them. This strategy is regularly used by humans when solving problems. One way to formalise the decomposition process is to assume there exist different patterns in the problem space, each pattern has particular characteristics and therefore it needs a special solution. A.I. Esparcia-Alcazar et al. (Eds.): EuroGP 2010, LNCS 6021, pp. 122–133, 2010. ©Springer-Verlag Berlin Heidelberg 2010 2

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Why its useful ? Problem decomposition is important for two reasons: Reduces the complexity of a problem. Automated problem decomposition may help researchers to better understand a problem domain by discovering regularities in the problem space. A.I. Esparcia-Alcazar et al. (Eds.): EuroGP 2010, LNCS 6021, pp. 122–133, 2010. ©Springer-Verlag Berlin Heidelberg 2010 3

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The Problem It is not difficult to split a problem into several sub-problems to be solved in cooperation with different methods. However, using the wrong decomposition may actually increase the problems complexity. A.I. Esparcia-Alcazar et al. (Eds.): EuroGP 2010, LNCS 6021, pp. 122–133, 2010. ©Springer-Verlag Berlin Heidelberg 2010 4

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The ideal problem decomposition system A.I. Esparcia-Alcazar et al. (Eds.): EuroGP 2010, LNCS 6021, pp. 122–133, 2010. ©Springer-Verlag Berlin Heidelberg 2010 5 Problem Decomposition System An ideal problem decomposition system would be one that gets the data from the user and identifies different groups in the data; each of these groups should be simpler to solve than the original problem.

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Some of the previous works Rosca, J., Johnson, M.P., Maes, P.: Evolutionary Speciation for Problem Decomposition (1996) Proposed a system called Evolutionary Speciation Genetic Programming (ESGP) to automatically discover natural decompositions of problems. Each individual consisted of two parts: A.I. Esparcia-Alcazar et al. (Eds.): EuroGP 2010, LNCS 6021, pp. 122–133, 2010. ©Springer-Verlag Berlin Heidelberg 2010 6 Condition treeOutput tree

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Some of the previous works Some of the fitness cases may be claimed by more than one individual while others are never chosen. A fitness function was proposed which encourages individuals to fully cover the problem space and minimise the overlap of the claimed fitness cases. Experimentation: Symbolic regression problem standard GP and with GP(IF) Results: GP(IF) evolved conditions in such a way as to effectively assign different fitness cases to different pieces of code and, moreover, that GP(IF) outperformed ESGP. A.I. Esparcia-Alcazar et al. (Eds.): EuroGP 2010, LNCS 6021, pp. 122–133, 2010. ©Springer-Verlag Berlin Heidelberg 2010 7

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Other works Iba, H.: Bagging, boosting, and bloating in genetic programming. In: Proceedings of the Genetic and Evolutionary Computation Conference, Orlando, Florida, USA, July 13-17, vol. 2, pp. 1053–1060. Morgan Kaufmann, San Francisco (1999) Proposed to extend GP using two well-known re- sampling techniques known as Bagging and Boosting and presented two systems referred to as BagGP and BoostGP. A.I. Esparcia-Alcazar et al. (Eds.): EuroGP 2010, LNCS 6021, pp. 122–133, 2010. ©Springer-Verlag Berlin Heidelberg 2010 8

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Other works Jackson, D.: The performance of a selection architecture for genetic programming. In: O’Neill, M., Vanneschi, L., Gustafson, S., Esparcia Alca zar, A.I., De Falco, I., Della Cioppa, A., Tarantino, E. (eds.) EuroGP 2008. LNCS, vol. 4971, pp. 170–181. Springer, Heidelberg (2008). Proposed a hierarchical architecture for GP for problem decomposition based on partitioning the input test cases into subsets. A.I. Esparcia-Alcazar et al. (Eds.): EuroGP 2010, LNCS 6021, pp. 122–133, 2010. ©Springer-Verlag Berlin Heidelberg 2010 9

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As one can see, none of the previous methods for problem decomposition via test case subdivision is fully automated. Overcoming this limitation is one of the aims of the work presented in this paper. A.I. Esparcia-Alcazar et al. (Eds.): EuroGP 2010, LNCS 6021, pp. 122–133, 2010. ©Springer-Verlag Berlin Heidelberg 2010 10

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In this Research An intelligent decomposition of problems requires understanding the problem domain and usually can only be carried out by experts. In this paper, we propose a GP system that can evolve programs that automatically decompose a problem into a collection of simpler and smaller sub-problems while simultaneously solving the sub- problems. This is an area of GP that has not been thoroughly explored thus far. A.I. Esparcia-Alcazar et al. (Eds.): EuroGP 2010, LNCS 6021, pp. 122–133, 2010. ©Springer-Verlag Berlin Heidelberg 2010 11

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The Approach TrainingResamplingSolvingTesting A.I. Esparcia-Alcazar et al. (Eds.): EuroGP 2010, LNCS 6021, pp. 122–133, 2010. ©Springer-Verlag Berlin Heidelberg 2010 12

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Training The system starts by randomly initialising a population of individuals using the ramped half-and-half method. Each individual composed of two trees: projector x projector y A.I. Esparcia-Alcazar et al. (Eds.): EuroGP 2010, LNCS 6021, pp. 122–133, 2010. ©Springer-Verlag Berlin Heidelberg 2010 13

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Resampling A.I. Esparcia-Alcazar et al. (Eds.): EuroGP 2010, LNCS 6021, pp. 122–133, 2010. ©Springer-Verlag Berlin Heidelberg 2010 14 projector xprojector y Training cases XY

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Function set A.I. Esparcia-Alcazar et al. (Eds.): EuroGP 2010, LNCS 6021, pp. 122–133, 2010. ©Springer-Verlag Berlin Heidelberg 2010 15

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A.I. Esparcia-Alcazar et al. (Eds.): EuroGP 2010, LNCS 6021, pp. 122–133, 2010. ©Springer-Verlag Berlin Heidelberg 2010 16 GP RUN1 GP RUN2 GP RUN3

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Clustering the Training Examples We used standard K-means. (any classification can be used). Why K-mean ? Simplicity of implementation. Execution speed. A.I. Esparcia-Alcazar et al. (Eds.): EuroGP 2010, LNCS 6021, pp. 122–133, 2010. ©Springer-Verlag Berlin Heidelberg 2010 17

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Clustering the Training Examples K-means is a partitioning algorithm that normally requires the user to fix the number of clusters to be formed. In our case the optimal number of subdivisions for the problem into sub-problems is unknown. A.I. Esparcia-Alcazar et al. (Eds.): EuroGP 2010, LNCS 6021, pp. 122–133, 2010. ©Springer-Verlag Berlin Heidelberg 2010 18

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Find the number of clusters Analyze the density of the data, to select the right number of the clusters K-means is a very fast algorithm. The system repeatedly instructs K-means to divide the data set into k clusters, where k = 2,3,...Kmax (Kmax = 10, in our implementation). After each call the system computes the clusters’ quality. The k value which provided the best quality for clusters is then used to split the training set and invoke GP runs on the corresponding clusters. A.I. Esparcia-Alcazar et al. (Eds.): EuroGP 2010, LNCS 6021, pp. 122–133, 2010. ©Springer-Verlag Berlin Heidelberg 2010 19

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Clusters’ Quality The Clusters’ Quality measured as: 1.Separation. 2.Representativeness. A.I. Esparcia-Alcazar et al. (Eds.): EuroGP 2010, LNCS 6021, pp. 122–133, 2010. ©Springer-Verlag Berlin Heidelberg 2010 20

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Separation Modified Davis Bouldin Index (DBI) A.I. Esparcia-Alcazar et al. (Eds.): EuroGP 2010, LNCS 6021, pp. 122–133, 2010. ©Springer-Verlag Berlin Heidelberg 2010 21 Standard deviation Distances between cluster’s centiod and data members Distance between all cluster’s centiods

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Representativeness Check whether the formed clusters are representative enough to classify unseen data ? The system penalises the values of k that lead K- mean to form clusters where less than a minimum number of members is present. In this work, the minimum allowed number of members for each cluster was simply set to 10 samples. (Why 10 !!) A.I. Esparcia-Alcazar et al. (Eds.): EuroGP 2010, LNCS 6021, pp. 122–133, 2010. ©Springer-Verlag Berlin Heidelberg 2010 22

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Clusters’ Quality A.I. Esparcia-Alcazar et al. (Eds.): EuroGP 2010, LNCS 6021, pp. 122–133, 2010. ©Springer-Verlag Berlin Heidelberg 2010 23 For each K-man call with k value 2:k_max Quality DBI index Representativeness penalty value

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Training Outputs Projector tree x. Projector tree y. Clusters centriods. Best GP solution from each cluster (or sub-problem). A.I. Esparcia-Alcazar et al. (Eds.): EuroGP 2010, LNCS 6021, pp. 122–133, 2010. ©Springer-Verlag Berlin Heidelberg 2010 24

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Testing 1.Unseen data projected via projector trees. 2.Then, they are classified based on the closest centroid. 3.The input data are passed to the evolved program associated to the corresponding cluster. A.I. Esparcia-Alcazar et al. (Eds.): EuroGP 2010, LNCS 6021, pp. 122–133, 2010. ©Springer-Verlag Berlin Heidelberg 2010 25

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Fitness Evaluation Two levels of GP-runs Top-level GP run (represented by two projection trees). Inner-GP runs (for each cluster). We evaluate the top-level GP system’s individuals by measuring how well the whole problem is solved. A.I. Esparcia-Alcazar et al. (Eds.): EuroGP 2010, LNCS 6021, pp. 122–133, 2010. ©Springer-Verlag Berlin Heidelberg 2010 26 Fitness value Inner-GP fitness value Number of clusters This fitness function encourages the individuals to project the fitness cases in such a way that a solution for the fitness cases in each group can easily be found by the inner GP runs.

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Experimental Setup Experiments have been conducted to evaluate the proposed framework. To do this we chose a variety of symbolic regression problems, which we felt were difficult enough to demonstrate the characteristics and benefits of the method. We used discontinuous functions as symbolic regression target functions. These allow us to study the ability of the system to decompose a complex problem into simpler tasks. A.I. Esparcia-Alcazar et al. (Eds.): EuroGP 2010, LNCS 6021, pp. 122–133, 2010. ©Springer-Verlag Berlin Heidelberg 2010 27

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Experimental Setup A.I. Esparcia-Alcazar et al. (Eds.): EuroGP 2010, LNCS 6021, pp. 122–133, 2010. ©Springer-Verlag Berlin Heidelberg 2010 28 Rosca’s experiments [-5,5] [-7,7]

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Experimental Setup A.I. Esparcia-Alcazar et al. (Eds.): EuroGP 2010, LNCS 6021, pp. 122–133, 2010. ©Springer-Verlag Berlin Heidelberg 2010 29 100 independent runs for each system (20 runs for each test function). 100 samples were uniformly selected from the training interval. 400 test samples. Each evolved solution has been evaluated with two different test sets: Interpolation. Extrapolation.

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Results A.I. Esparcia-Alcazar et al. (Eds.): EuroGP 2010, LNCS 6021, pp. 122–133, 2010. ©Springer-Verlag Berlin Heidelberg 2010 30 Better than Rosca’s results

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Number of Clusters Functions 1, 2 and 5, the system decided to split the training samples into 4 to 10 clusters. Function 3, the system identified 6 to 10 clusters in the problem space. Function 4, the most complex in our test set, it identified 8 to 10 clusters. A.I. Esparcia-Alcazar et al. (Eds.): EuroGP 2010, LNCS 6021, pp. 122–133, 2010. ©Springer-Verlag Berlin Heidelberg 2010 31

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Experimental results summary A.I. Esparcia-Alcazar et al. (Eds.): EuroGP 2010, LNCS 6021, pp. 122–133, 2010. ©Springer-Verlag Berlin Heidelberg 2010 32

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Conclusions In this paper we presented a new framework to automatically decompose difficult symbolic regression tasks into smaller and simpler tasks. The proposed approach is based on the idea of first projecting the training cases onto a two-dimensional Euclidian space, and then clustering them via the K- means algorithm to better see their similarities and differences. The clustering is guaranteed to be optimal thanks to the use of an iterative process. A.I. Esparcia-Alcazar et al. (Eds.): EuroGP 2010, LNCS 6021, pp. 122–133, 2010. ©Springer-Verlag Berlin Heidelberg 2010 33

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Conclusions Note that while the projection and clustering steps may seem excessive for scalar domains, they make our problem decomposition technique applicable to much more complex domains. The major disadvantage of the system is the slow training process (measured by hours). A.I. Esparcia-Alcazar et al. (Eds.): EuroGP 2010, LNCS 6021, pp. 122–133, 2010. ©Springer-Verlag Berlin Heidelberg 2010 34

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Future work Extend the experimentation by testing the technique on multi-varied problems and non- symbolic- regression problems. The system should be able to change the settings of each inner GP run according to the difficulty of the given sub-problem. We intend to investigate the relationship between the identified number of clusters and how this affects the solutions’ accuracy. A.I. Esparcia-Alcazar et al. (Eds.): EuroGP 2010, LNCS 6021, pp. 122–133, 2010. ©Springer-Verlag Berlin Heidelberg 2010 35

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A.I. Esparcia-Alcazar et al. (Eds.): EuroGP 2010, LNCS 6021, pp. 122–133, 2010. ©Springer-Verlag Berlin Heidelberg 2010 Thank you for paying attention! 36

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