In this chapter we take a closer look at the distribution of symbolic regression models generated by genetic programming in the search space. The motivation for this work is to improve the search for well-fitting symbolic regression models by using information about the similarity of models that can be precomputed independently from the target function. For our analysis, we use a restricted grammar for uni-variate symbolic regression models and generate all possible models up to a fixed length limit. We identify unique models and cluster them based on phenotypic as well as genotypic similarity. We find that phenotypic similarity leads to well-defined clusters while genotypic similarity does not produce a clear clustering. By mapping solution candidates visited by GP to the enumerated search space we find that GP initially explores the whole search space and later converges to the subspace of highest quality expressions in a run for a simple benchmark problem.
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We actually found that this assumption is wrong. We found that the search space can be split into clusters of phenotypically and genotypically similar expressions. However, we could not show that phenotypically similar expressions also are phenotypically similar and/or vice versa. This is intuitive because two highly similar expressions become dissimilar on the phenotypic level just by a multiplication with zero. Symmetrically, many different expressions can be found which produce the same output.
We have used a uni-variate variant of the benchmark function described by Pagie and Hogeweg.
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The authors thank the participants of the Genetic Programming in Theory and Practice (GPTP XVI) workshop for their valuable feedback and ideas which helped to improve the work described in this chapter. The authors gratefully acknowledge support by the Christian Doppler Research Association and the Federal Ministry for Digital and Economic Affairs within the Josef Ressel Center for Symbolic Regression.
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Kronberger, G., Kammerer, L., Burlacu, B., Winkler, S.M., Kommenda, M., Affenzeller, M. (2019). Cluster Analysis of a Symbolic Regression Search Space. In: Banzhaf, W., Spector, L., Sheneman, L. (eds) Genetic Programming Theory and Practice XVI. Genetic and Evolutionary Computation. Springer, Cham. https://doi.org/10.1007/978-3-030-04735-1_5
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