Diversity in Multipopulation Genetic Programming

In the past few years, we have done a systematic experimental investigation of the behavior of multipopulation GP [2] and we have empirically observed that distributing the individuals among several loosely connected islands allows not only to save computation time, due to the fact that the system runs on multiple machines, but also to find better solution quality. These results have often been attributed to better diversity maintenance due to the periodic migration of groups of “good” individuals among the subpopulations. We also believe that this might be the case and we study the evolution of diversity in multi-island GP. All the diversity measures that we use in this paper are based on the concept of entropy of a population P , defined as H(P ) =−∑j=1Fj log(Fj). If we are considering phenotypic diversity, we define Fj as the fraction nj/N of individuals in P having a certain fitness j, where N is the total number of fitness values in P . In this case, the entropy measure will be indicated as Hp(P ) or simply Hp. To define genotypic diversity, we use two different techniques. The first one consists in partitioning individuals in such a way that only identical individuals belong to the same group. In this case, we have considered Fj as the fraction of trees in the population P having a certain genotype j, where N is the total number of genotypes in P and the entropy measure will be indicated as HG(P ) or simply HG. The second technique consists in defining a distance measure, able to quantify the genotypic diversity between two trees. In this case, Fj is the fraction of individuals having a given distance j from a fixed tree (called origin), where N is the total number of distance values from the origin appearing in P and the entropy measure will be indicated as Hg(P ) or simply Hg . The tree distance used is Ekárt’s and Németh’s definition [1]. Figure 1 depicts the behavior of HG, Hg and Hp during evolution for the symbolic regression problem, with the classic ploynomial equation f(x) = x4+x3+x2+x, an input set composed of 1000 fitness cases and a set of functions equal to F={*,//,+,-}, where // is like / but returns 0 instead of error when the divisor is equal to 0. Fitness is the sum of the square errors at each test point. Curves are averages over 100 independent runs for generational GP, crossover rate: 95%, mutation rate: 0.1%, tournament selection of size: 10, ramped half and half initialization, maximum depth of individuals for the creation phase: 6, maximum depth of individuals for crossover: 17, elitism. Genotypic diversity for the panmictic case tends to remain constant over time, and to have higher values than the distributed case. On the contrary, the average phenotypic entropy for the multipopulation case tends to remain higher than in the panmictic case. The oscillating behavior of the multipopulation curves when groups of individuals are sent and received is not surprising: it is due to the sudden change in diversity when new individuals enter a subpopulation. Finally, we remark that the behavior of the two measures used to calculate genotypic diversity (HG and Hg) is qualitatively equivalent. Analogous results