Abstract
In this paper we investigate why the running time of lexicase parent selection is empirically much lower than its worst-case bound of \(O(N \cdot C)\). We define a measure of population diversity and prove that high diversity leads to low running times \(O(N + C)\) of lexicase selection. We then show empirically that genetic programming populations evolved under lexicase selection are diverse for several program synthesis problems, and explore the resulting differences in running time bounds.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Our results hold for the original variant, later dubbed “static” \(\varepsilon \)-lexicase selection [12].
- 2.
This worst-case example does not hold if the losses are binary, but even that does not help much. It is possible to construct a population of N individuals without duplicates that differ only on \(\log _2 N\) binary training cases, and are identical on all other training cases. In this situation, the candidate pool does not shrink before at least one of those training cases is found, and in expectation this takes \(C/\log _2 N\) iterations. Thus the expected runtime in this situation is at least \(O(N\cdot C/\log N)\), which is not much better than \(O(N\cdot C)\).
- 3.
Experiment code: https://github.com/cavalab/lexicase_runtime.
- 4.
References
Aenugu, S., Spector, L.: Lexicase selection in learning classifier systems. In: Proceedings of the Genetic and Evolutionary Computation Conference, pp. 356–364 (2019)
Doerr, B., Johannsen, D., Winzen, C.: Multiplicative drift analysis. Algorithmica 64(4), 673–697 (2012)
Dolson, E., Ofria, C.: Ecological theory provides insights about evolutionary computation. In: Proceedings of the Genetic and Evolutionary Computation Conference Companion, GECCO 2018, pp. 105–106. Association for Computing Machinery, New York, NY, USA (2018). https://doi.org/10.1145/3205651.3205780
Helmuth, T., McPhee, N.F., Spector, L.: Effects of lexicase and tournament selection on diversity recovery and maintenance. In: Proceedings of the 2016 on Genetic and Evolutionary Computation Conference Companion, pp. 983–990. ACM (2016). http://dl.acm.org/citation.cfm?id=2931657
Helmuth, T., McPhee, N.F., Spector, L.: The impact of hyperselection on lexicase selection. In: Proceedings of the 2016 on Genetic and Evolutionary Computation Conference, pp. 717–724. ACM (2016). http://dl.acm.org/citation.cfm?id=2908851
Helmuth, T., McPhee, N.F., Spector, L.: Program synthesis using uniform mutation by addition and deletion. In: Proceedings of the Genetic and Evolutionary Computation Conference, GECCO 2018, pp. 1127–1134. ACM, Kyoto, Japan, 15–19 July 2018. https://doi.org/10.1145/3205455.3205603
Helmuth, T., Pantridge, E., Spector, L.: On the importance of specialists for lexicase selection. Genet. Program. Evolvable Mach. 21(3), 349–373 (2020). https://doi.org/10.1007/s10710-020-09377-2
Helmuth, T., Spector, L.: General program synthesis benchmark suite. In: GECCO 2015: Proceedings of the 2015 conference on Genetic and Evolutionary Computation Conference, Madrid, Spain, pp. 1039–1046. ACM, 11–15 July 2015. https://doi.org/10.1145/2739480.2754769
Helmuth, T., Spector, L.: Explaining and exploiting the advantages of down-sampled lexicase selection. In: Artificial Life Conference Proceedings, pp. 341–349. MIT Press, 13–18 July 2020. https://doi.org/10.1162/isal_a_00334, https://www.mitpressjournals.org/doi/abs/10.1162/isal_a_00334
Helmuth, T., Spector, L., Matheson, J.: Solving uncompromising problems with lexicase selection. IEEE Trans. Evol. Comput. 19(5), 630–643 (2015). https://doi.org/10.1109/TEVC.2014.2362729
Jansen, T., Zarges, C.: Theoretical analysis of lexicase selection in multi-objective optimization. In: Auger, A., Fonseca, C.M., Lourenço, N., Machado, P., Paquete, L., Whitley, D. (eds.) Parallel Problem Solving from Nature - PPSN XV, pp. 153–164. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-99259-4_13
La Cava, W., Helmuth, T., Spector, L., Moore, J.H.: A probabilistic and multi-objective analysis of lexicase selection and epsilon-lexicase selection. Evol. Comput. 27(3), 377–402 (2019). https://doi.org/10.1162/evco_a_00224, https://arxiv.org/pdf/1709.05394
La Cava, W., et al.: Contemporary symbolic regression methods and their relative performance. In: Proceedings of the Neural Information Processing Systems Track on Datasets and Benchmarks, vol. 1, December 2021
La Cava, W., Spector, L., Danai, K.: Epsilon-lexicase selection for regression. In: Proceedings of the Genetic and Evolutionary Computation Conference 2016, GECCO 2016, New York, NY, USA, pp. 741–748. ACM (2016). https://doi.org/10.1145/2908812.2908898
Lengler, J.: Drift analysis. In: Theory of Evolutionary Computation. NCS, pp. 89–131. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-29414-4_2
Liskowski, P., Krawiec, K., Helmuth, T., Spector, L.: Comparison of semantic-aware selection methods in genetic programming. In: Proceedings of the Companion Publication of the 2015 Annual Conference on Genetic and Evolutionary Computation, GECCO Companion 2015, New York, NY, USA, pp. 1301–1307. ACM (2015). https://doi.org/10.1145/2739482.2768505
Moore, J.M., Stanton, A.: Tiebreaks and diversity: isolating effects in lexicase selection. In: The 2018 Conference on Artificial Life, pp. 590–597 (2018). https://doi.org/10.1162/isal_a_00109
Orzechowski, P., La Cava, W., Moore, J.H.: Where are we now? A large benchmark study of recent symbolic regression methods. In: Proceedings of the 2018 Genetic and Evolutionary Computation Conference, GECCO 2018, April 2018. https://doi.org/10.1145/3205455.3205539, tex.ids: orzechowskiWhereAreWe2018a arXiv: 1804.09331
Spector, L.: Assessment of problem modality by differential performance of lexicase selection in genetic programming: a preliminary report. In: Proceedings of the Fourteenth International Conference on Genetic and Evolutionary Computation Conference Companion, pp. 401–408 (2012). http://dl.acm.org/citation.cfm?id=2330846
Spector, L., Klein, J., Keijzer, M.: The Push3 execution stack and the evolution of control. In: GECCO 2005: Proceedings of the 2005 conference on Genetic and Evolutionary Computation, Washington DC, USA, vol. 2, pp. 1689–1696. ACM Press, 25–29 June 2005. https://doi.org/10.1145/1068009.1068292
Spector, L., Robinson, A.: Genetic programming and autoconstructive evolution with the push programming language. Genet. Program. Evolvable Mach. 3(1), 7–40 (2002). http://hampshire.edu/lspector/pubs/push-gpem-final.pdf, https://doi.org/10.1023/A:1014538503543
Vanneschi, L., Castelli, M., Silva, S.: A survey of semantic methods in genetic programming. Genet. Program. Evolvable Mach. 15(2), 195–214 (2014). https://doi.org/10.1007/s10710-013-9210-0
Acknowledgements
William La Cava was supported by the National Library of Medicine and National Institutes of Health under award R00LM012926. We would like to thank Darren Strash for discussions that contributed to the development of this work.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Helmuth, T., Lengler, J., La Cava, W. (2022). Population Diversity Leads to Short Running Times of Lexicase Selection. In: Rudolph, G., Kononova, A.V., Aguirre, H., Kerschke, P., Ochoa, G., Tušar, T. (eds) Parallel Problem Solving from Nature – PPSN XVII. PPSN 2022. Lecture Notes in Computer Science, vol 13399. Springer, Cham. https://doi.org/10.1007/978-3-031-14721-0_34
Download citation
DOI: https://doi.org/10.1007/978-3-031-14721-0_34
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-14720-3
Online ISBN: 978-3-031-14721-0
eBook Packages: Computer ScienceComputer Science (R0)