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Genetic programming convergence

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Abstract

We study both genotypic and phenotypic convergence in GP floating point continuous domain symbolic regression over thousands of generations. Subtree fitness variation across the population is measured and shown in many cases to fall. In an expanding region about the root node, both genetic opcodes and function evaluation values are identical or nearly identical. Bottom up (leaf to root) analysis shows both syntactic and semantic (including entropy) similarity expand from the outermost node. Despite large regions of zero variation, fitness continues to evolve and near zero crossover disruption suggests improved GP systems within existing memory use.

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Notes

  1. We define protected division so that division by zero yields a defined result, i.e. 1.0

  2. In [3] Koza uses a 90% bias in favour of internal nodes to reduce the fraction of crossovers which simply move leafs within the trees.

  3. Schema theory has proved quite popular with GP authors [32,33,34,35,36,37].

  4. See http://www.cs.ucl.ac.uk/staff/W.Langdon/gggp/#Langdon:2017:GECCO for animations of the evolution of convergence in binary 6-multiplexor populations. (Also YouTube video: https://youtu.be/gwCwvwJcWbQ.) C++ code available from http://www.cs.ucl.ac.uk/staff/W.Langdon/ftp/gp-code/GPbmux6.tar.gz.

  5. See http://www.cs.ucl.ac.uk/staff/W.Langdon/seminars/aigp3/ for an animation of [67, Figure 8.5].

  6. Koza [3] uses 50 fitness cases. However, to run conveniently on AVX-512 hardware we replaced 50 by the closest multiple of 16, i.e. \(3 \times 16 = 48\) test cases. Therefore we actually bank together sets of three AVX-512 instructions to evaluate 48 fitness cases together  [69, 70].

  7. Run aborted due to external power failure. We used GPavx, which on publication [69, 70], was the world’s fastest general GP system http://www.cs.ucl.ac.uk/staff/W.Langdon/ftp/gp-code/GPavx.tar.gz, on a 3.00GHz Intel Xeon Gold 6136 server with 48 cores and 3 TBytes. See also RN/20/01 [71].

  8. Figures 8, 15 and 17 use Daida’s [40] lattice in which trees are shown with their root at the origin and branches splayed out from the centre using the full 360°C. http://www.cs.ucl.ac.uk/staff/W.Langdon/gp2lattice/gp2lattice.html This circular display allows populations, indeed multiple generations, of trees to be displayed together, as if plotted on top of each other. It also highlights the asymmetry of the highly evolved trees.

  9. In pretty much any GP system the huge growth with tree size in the number of possible tree shapes (the Catalan number), and the exponential rise in the number of ways of labeling each tree with functions from the function set and leafs from the terminal set, means there must be multiple GP trees with the same fitness value. For example, in our sextic polynomial representation, the number of trees with up to 7 nodes is \(1.25\ 10^{12}\), whilst the number of fitness values cannot exceed \(2^{32} = 4.29\ 10^{9}\).

  10. We could even imagine entropy as being a secondary fitness objective, e.g. to penalise initial random trees which calculate fixed values near the average of the target function, rather than trying to match its variation.

  11. Where recursion is not well supported, interpreting trees as reverse polish expressions can be efficient [82, 83].

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Acknowledgements

I would like to thank Simon Tatham for PuTTY. This work was inspired by conversations at Dagstuhl Seminars 17191 on the theory of randomized heuristics and 18052 on Genetic Improvement of Software [90].

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Funded by EPSRC GGGP and InfoTestSS grants EP/M025853/1, EP/P005888/1.

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Langdon, W.B. Genetic programming convergence. Genet Program Evolvable Mach 23, 71–104 (2022). https://doi.org/10.1007/s10710-021-09405-9

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