Fitness Clouds and Problem Hardness in Genetic Programming

Abstract

This paper presents an investigation of genetic programming fitness landscapes. We propose a new indicator of problem hardness for tree-based genetic programming, called negative slope coefficient, based on the concept of fitness cloud. The negative slope coefficient is a predictive measure, i.e. it can be calculated without prior knowledge of the global optima. The fitness cloud is generated via a sampling of individuals obtained with the Metropolis-Hastings method. The reliability of the negative slope coefficient is tested on a set of well known and representative genetic programming benchmarks, comprising the binomial-3 problem, the even parity problem and the artificial ant on the Santa Fe trail.

Negative Slope Coefficient: A Measure to Characterize Genetic Programming Fitness Landscapes

“…Since selection used by GP is likely to eliminate bad individuals from the population, importance sampling techniques must to be used. As in [17,18], also in this work the well-known MetropolisHastings technique [10] is used to sample the search space and the k-tournament selection algorithm [8] (with k = 10) is used to sample neighborhoods (see [17] for a detailed motivation of these choices). Using random samples would assume that the space is relatively uniform, e.g.…”

“…But the mere observation of the scatterplot is not sufficient to quantify these features. In [17,18] …”

“…The hypothesis proposed in [18] is that ncs should classify problems in the following way: if nsc= 0, the problem is easy; if nsc< 0 the problem is difficult and the value of nsc quantifies this difficulty: the smaller its value, the more difficult the problem. The idea behind this hypothesis is that the presence of a segment with negative slope indicates a bad evolvability for individuals having fitness values contained in that segment (see [17] for a detailed discussion on this issue).…”

“…Thus, it is important to investigate other approaches based on quantities that can be measured without any explicit knowledge of the genotype of optimal solutions. Preliminary results of this enquiry can be found in [18], where a new measure called negative slope coefficient (nsc) has been introduced.…”

“…It is structured as follows: section 2 introduces the concept of fitness cloud on which nsc is based. Section 3 takes up the original definition of the nsc as it has been presented in [18] and section 4 points out its main limitations. Section 5 proposes a new method for calculating the nsc and shows some experimental results pointing out that this method enables to overcome some drawbacks of the original definition of the nsc.…”