Dining with GAs: Operator Lunch Theorems

There has been considerable discussion of the pros cons of recombination and mutation operators in the context of Holland s schema theory In this paper we de ne a common framework for extending and relating previous disruption and construction analyses for both recombination and mutation This results in several insights into the properties of recombination and mutation including a No Free Lunch theorem for recombination operators as well as the lack of such a theorem for mutation Introduction The motivation for the original schema analysis of Holland was to compute the expected number of instances of hyperplanes at time t given their number at time t To use the notation of Goldberg let mt H be the number of individuals in hyperplane H at time t Then let ft H be the observed average tness of the hyperplane at time t and let ft be the observed average tness of the population at time t Then the expected number of individuals in H at time t is given by the schema theorem E mt H mt H ft H ft Psurvival H where Psurvival H is the probability that the hyperplane will not be disrupted by either mutation or recombination i e it survives The inequality refers to the fact that not only may a hyperplane H survive it also may be constructed from other hyperplanes See Holland or Goldberg for further discussions of this equation In order to improve this result to a precise equality detailed aspects of the makeup of the entire population must be modeled Goldberg Whitley Vose resulting in exact but di cult to analyze characterizations of GA behavior In this paper we extend earlier work in which a less precise but useful characterization of population makeup is adopted resulting in more tractable models of GA behavior Spears and De Jong De Jong and Spears This allows us to de ne a framework for extending and relating previous disruption and construction analyses for both recombination and mutation This results in several insights into the properties of recombination and mutation including a No Free Lunch theorem Wolpert Sha er Rao Gordon and Spears Culberson English Wolpert and Macready for recombination operators as well as the lack of such a theorem for mutation Framework The schema theorem articulates how the expected makeup of the next generation E mt is a function of the current generationmt selection and the reproductive operators If cloning is the only reproductive operator E mt is completely determined bymt and selection Of interest then is characterizing the perturbation e ects due to reproductive operators such as recombination and mutation Following the standard schema analysis we focus on a particular kth order hyperplane Hk and ask how recombination and mutation perturb the expected number of o spring in Hk in the next generation We do so in the following manner First to allow a direct comparison of the e ects of recombination and mutation we treat them both as reproductive operators that take two parents as input and produce two children as output In the case of mutation this is equivalent to two independent applications of the standard one parent mutation operator but has the advantage that it allows us to compare the expected number of o spring residing in Hk as a result of a single application of either operator More formally let Bk be a random variable describing the number of o spring residing in Hk as a result of a single application of any two parent reproductive operator Then the expected number of o spring in Hk can be computed as follows E Bk X b f g b P Bk b P Bk P Bk Survival analysis Historically the schema theorem has emphasized the disruptive aspects of reproductive operators by assuming that at least one of the parents is a member of Hk and calculating the likelihood that neither of the children will be instances of Hk The complement of this is survival in which at least one of the o spring are in Hk More formally let Ak be a random variable which describes the number of parents that are instances of Hk For survival analysis then Ak can take on values and Again let Bk be the random variable describing the number of o spring that are instances of Hk Bk can take on values and We can then express the probability of survival as Ps Hk P Bk j Ak P Bk j Ak P Bk j Ak For survivability the general formula given by equation for E Bk the expected number of o spring in Hk specializes to Es Bk X