An Extension of Geiringer's Theorem for a Wide Class of Evolutionary Search Algorithms.
Evolutionary Computation
An extension of geiringer's theorem for a wide class of evolutionary search algorithms
Evolutionary Computation
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In this thesis a general mathematical framework to describe evolutionary algorithms is developed. The following two applications of the framework are presented in detail: (1) This framework allows us to use category theory to compare various evolutionary algorithms via their representation. In fact, we shall introduce a certain category, the morphisms of which are intuitively thought of as re-encoding functions. It turns out that these morphisms are related to the family of the so-called recombination-invariant subsets in a similar way as continuous maps are related to the family of open sets and measurable functions are related to the σ-algebra of measurable sets. This allows us to classify all possible re-encodings of a given evolutionary search algorithm in terms of a classical genetic algorithm. (2) The classical theorem of Geiringer from theoretical biology can be generalized to a great extent in this new framework. Moreover, the proof of the generalized version depends on a classical fact about random walks associated to group actions and allows one to use the elegant techniques developed by Diaconis and his colleagues to estimate the rate of convergence towards the uniform stationary distribution for many evolutionary algorithms including various kinds of genetic programming techniques. The notion of a Geiringer algorithm (the type of algorithm to which the generalized Geiringer theorem applies) also has some ties to the representation (the topic of the first part).