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Quotient Models and Graphs:

Using Quotient Graphs to Model Neutrality in Evolutionary Search Dominic Wilson Devinder Kaur University of Toledo. Quotient Models and Graphs:. Are widely applicable. Binary and non-binary Genetic Algorithms Grammatical Evolution Cartesian Genetic Programming. Performance. Generations.

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Quotient Models and Graphs:

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  1. Using Quotient Graphs to Model Neutrality in Evolutionary SearchDominic WilsonDevinder KaurUniversity of Toledo

  2. Quotient Models and Graphs: • Are widely applicable. • Binary and non-binary Genetic Algorithms • Grammatical Evolution • Cartesian Genetic Programming

  3. Performance Generations Quotient Models and Graphs: • Can explain why performance improvements are usually smaller for later generations of evolution (e.g. ONEMAX);

  4. Quotient Models and Graphs: • Can explain the change of the location of a steady state population with mutation rate; J. Richter, A. Wright and J. Paxton. "Exploration of Population Fixed Points Versus Mutation Rates for Functions of Unitation", GECCO-2004.

  5. Quotient Models and Graphs: • Exact Markov models; • Reduce the degrees of freedom needed for modeling; • Show aspects of evolutionary search that are not obvious (e.g. correlated mutational drives). • Can track population movements on complex landscapes;

  6. Why Models? • To understand and explain the complex dynamics of Evolutionary Computing systems; • Examples of models: • Schema. (J. H. Holland. Adaptation in Natural and Artificial Systems. University of Michigan Press, 1975.) • Predicates. (M.D. Vose, “Generalizing the notion of schema in genetic algorithms. “,Artificial Intelligence, 50 1991.) • Formae. (N. J. Radcliffe. “Equivalence class analysis of genetic algorithms.” Complex Systems, 5(2),1991.) • Unitation Functions. (J. E. Rowe, “Population fixed-points for functions of unitation,” FOGA 5, 1999.)

  7. Model Similarities • Schemata, Predicates, Formae and Unitation Functions are defined based on subsets of the genotype space. • They are oblivious of the genotype-to phenotype map.

  8. Quotient Models and Graphs • Quotient models are formed by grouping subsets of the genotype space that have the same fitness and search behavior. They are therefore aware of the structure of the genotype-to-phenotype map. • Quotient graphs visually portray quotient models. They consist of nodes that have the same fitness and search behavior, connected by directed arcs.

  9. Content • Create an example quotient model. • Show how quotient models can be used to explain evolutionary search behavior.

  10. Example Genotype to Fitness Map F is like ONEMAX except for string “111”

  11. 110 (2) 100 (1) 111 (0) 101 (2) 010 (1) 000 (0) 011 (2) 001 (1) Example Map on a Cube

  12. 110 (2) 100 (1) 111 (0) 101 (2) 010 (1) 000 (0) 011 (2) 001 (1) Fitness Distribution on Mutation Each string with only one bit set to “1” has the same neighborhood! They also have the same fitness.

  13. 110 (2) 100 (1) 111 (0) 101 (2) 010 (1) 000 (0) 011 (2) 001 (1) Fitness Distribution on Mutation

  14. 110 (2) 100 (1) 111 (0) 101 (2) 010 (1) 000 (0) 011 (2) 001 (1) Fitness Distribution on Mutation

  15. 110 (2) 100 (1) 111 (0) 101 (2) 010 (1) 000 (0) 011 (2) 001 (1) Fitness Distribution on Mutation String with fitness “0” do not have the same neighborhood!

  16. 110 (2) 100 (1) 111 (0) 101 (2) 010 (1) 000 (0) 011 (2) 001 (1) Quotient Graph Quotient Graph

  17. 110 (2) 100 (1) 111 (0) 101 (2) 010 (1) 000 (0) 011 (2) 001 (1) Quotient Graph Represents the same neighborhood information as the cube

  18. 110 (2) 100 (1) 111 (0) 101 (2) 010 (1) 000 (0) 011 (2) 001 (1) Quotient Graph Correlated mutational drives

  19. 110 (2) 100 (1) 111 (0) 101 (2) 010 (1) 000 (0) 011 (2) 001 (1) Quotient Graph 4 nodes 8 nodes

  20. Larger Quotient Graphs 8 bit ONEMAX n bit ONEMAX

  21. StringFitness Map as Linear Map and , and F: Fitness X: String A: String to fitness map (linear operator)

  22. Mapping

  23. Bit mutation probability: Mutation rate matrix: Mutation

  24. Probability distribution of fitness on mutation X: Current String; MX: Probability distribution of string after mutation; AMX: Probability distribution of string fitness after mutation

  25. Search distribution

  26. Search distribution Probability distribution of string fitness after mutation Rows 1, 2 and 4 are identical; Rows 3, 5 and 6 are identical;

  27. 110 (2) 100 (1) 111 (0) 101 (2) 010 (1) 000 (0) 011 (2) 001 (1) Example Map on a Cube

  28. 110 (2) 100 (1) 111 (0) 101 (2) 010 (1) 000 (0) 011 (2) 001 (1) Quotient Graph Quotient Graph

  29. Quotient sets One set for each color. quotient set assignment matrix:

  30. [0b] [2] [1] [0a] Quotient model

  31. Quotient Mutation Rate Matrix . Mutation rate matrix: Quotient mutation rate matrix: Quotient assignment matrix:

  32. Quotient Mutation Rate Matrix Quotient mutation rate matrix:

  33. [3f] [3f] [4] [4] [3e] [3e] [3a] [3a] [3d] [3d] [2] [2] [3c] [3c] [1] [1] [3b] [3b] [0] [0] Quotient Graph of 4 bit ONEMAX with neutral layer of fitness 3 Fitness Drives Correlated mutational Drives E. Galvan-Lopez , R. Poli, “An Empirical Investigation of How and Why Neutrality Affects Evolutionary Search” GECCO’06.

  34. Example Quotient Graphs

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