A Brief Introduction to GA Theory

Principles of adaptation in complex systems John Holland proposed a general principle for adaptation in complex systems: –Adaptation requires the correct balance between “exploration” and “exploitation”

“Exploration” –Adaptation consists of “searching” for new useful traits “Exploitation” –Adaptation also consists of spreading useful traits once they are discovered

Holland developed a mathematical theory of the correct balance between exploitation and exploration in adaptive systems The theory used an idealized example: a “two-armed bandit”. (From D. E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning. Addison-Wesley 1989.

You are given n quarters to play with, and don’t know the probabilities of payoffs of the respective arms. What is the optimal way to allocate your quarters between the two arms so as to maximize your earnings (or minimize your losses) over the n arm-pulls ? (From D. E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning. Addison-Wesley 1989.

In Holland’s theory, each arm roughly corresponds with a possible “strategy” to test. The question is, if one strategy (arm) seems good, how much time should you spend exploiting it, and how much time should you spend exploring other, possibly better, strategies?

Holland developed a mathematical model to answer this question, meant to apply to all complex adaptive systems. Basic result: you should continue to play both arms (exploration), but, over time, exponentially increase the number of quarters (samples) you give to the observed better arm. This led to the development of “genetic algorithms” and the notion of “schema processing” and the “Schema Theorem”.

Schema Theory General idea (“traditional” theory): GAs work by discovering, emphasizing, and recombining good “building blocks” of solutions in a highly parallel fashion. Implies that good solutions are made up of good “building blocks”--- pieces of solutions that confer higher fitness on many strings they are part of.

Schemas Formalizations of “building blocks.” Here, assume individuals are bit strings. (There are many generalizations of schemas to other representations.) Subsets of bit strings expressed as templates. E.g., A = B = H 1 = ****11 H 2 = **0*** H 3 = **0**1 H 4 = **0*11 A and B are instances of H 1 - H 4. **0*11 is order 3 (has 3 defined bits). **0*11 has defining length 3.

Schema Processing in GAs General ideas: Any string is an instance of 2 l schemas, so its fitness gives some information about those schemas. The population of N individuals contains between 2 l and N2 l schemas (“implicit parallelism”). A GA explicitly samples strings, it implicitly samples schemas. The sampling procedure is gradually biased towards above-average fitness schemas.