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

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

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.

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;

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.)

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.

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.

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

Example Genotype to Fitness Map 3 bit Strings X Fitness F 000 001 1 010 011 2 100 101 110 111 F is like ONEMAX except for string “111”

Example Map on a Cube F 3 bit Strings X Fitness 000 001 1 010 011 2 001 1 010 011 2 100 101 110 111 000 (0) 001 (1) 100 011 (2) 110 111 101 010

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

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

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

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

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

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

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

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

Larger Quotient Graphs 8 bit ONEMAX n bit ONEMAX

StringFitness Map as Linear Map Strings X Fitness F 000 001 1 010 011 2 100 101 110 111 and , and F: Fitness X: String A: String to fitness map (linear operator)

Mapping F 3 bit Strings (X) Fitness 000 001 1 010 011 2 100 101 110 001 1 010 011 2 100 101 110 111

Mutation Bit mutation probability: Mutation rate matrix:

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

Search distribution

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

Example Map on a Cube F 3 bit Strings X Fitness 000 001 1 010 011 2 001 1 010 011 2 100 101 110 111 000 (0) 001 (1) 100 011 (2) 110 111 101 010

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

Quotient sets F 3 bit Strings X Fitness 000 001 1 010 011 2 100 101 001 1 010 011 2 100 101 110 111 One set for each color. quotient set assignment matrix:

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

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

Quotient Mutation Rate Matrix

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

Example Quotient Graphs