Abstract
We study evolutionary algorithms in a dynamic setting, where for each generation a different fitness function is chosen, and selection is performed with respect to the current fitness function. Specifically, we consider Dynamic BinVal, in which the fitness functions for each generation is given by the linear function BinVal, but in each generation the order of bits is randomly permuted. For the \((1 + 1)\)-EA it was known that there is an efficiency threshold \(c_0\) for the mutation parameter, at which the runtime switches from quasilinear to exponential. Previous empirical evidence suggested that for larger population size \(\mu \), the threshold may increase. We prove rigorously that this is at least the case in an \(\varepsilon \)-neighborhood around the optimum: the threshold of the \((\mu + 1)\)-EA becomes arbitrarily large if the \(\mu \) is chosen large enough.
However, the most surprising result is obtained by a second order analysis for \(\mu =2\): the threshold increases with increasing proximity to the optimum. In particular, the hardest region for optimization is not around the optimum. (Extended Abstract. A full version is available on arxiv at [17].)
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.
- 2.
They argued that it might either be called noisy linear functions or dynamic linear functions, but we prefer the term dynamic.
- 3.
The o(1) term is needed since we do not make assumptions about \(\varepsilon \). If we assumed that \(\varepsilon =\varepsilon (n)\) goes to zero sufficiently slowly, we could swallow the o(1) into \(O(\varepsilon ^2)\).
- 4.
Here we use the convention that if an offspring is identical to the parent, and they have lowest fitness in the population, then the offspring is rejected. Since the outcome of ejecting offspring or parent is the same, this convention does not change the course of the algorithm.
References
Colin, S., Doerr, B., Férey, G.: Monotonic functions in EC: anything but monotone! In: Genetic and Evolutionary Computation Conference (GECCO), pp. 753–760. ACM (2014)
Dang-Nhu, R., Dardinier, T., Doerr, B., Izacard, G., Nogneng, D.: A new analysis method for evolutionary optimization of dynamic and noisy objective functions. In: Genetic and Evolutionary Computation Conference (GECCO), pp. 1467–1474. ACM (2018)
Doerr, B., Goldberg, L.A.: Adaptive drift analysis. Algorithmica 65(1), 224–250 (2013). https://doi.org/10.1007/s00453-011-9585-3
Doerr, B., Hota, A., Kötzing, T.: Ants easily solve stochastic shortest path problems. In: Genetic and Evolutionary Computation Conference (GECCO), pp. 17–24. ACM (2012)
Doerr, B., Jansen, T., Sudholt, D., Winzen, C., Zarges, C.: Mutation rate matters even when optimizing monotonic functions. Evol. Comput. 21(1), 1–27 (2013)
Doerr, B., Johannsen, D., Winzen, C.: Multiplicative drift analysis. Algorithmica 64(4), 673–697 (2012). https://doi.org/10.1007/s00453-012-9622-x
Droste, S.: Analysis of the \((1+ 1)\) EA for a dynamically changing OneMax-variant. In: Congress on Evolutionary Computation (CEC), vol. 1, pp. 55–60. IEEE (2002)
Horoba, C., Sudholt, D.: Ant colony optimization for stochastic shortest path problems. In: Genetic and Evolutionary Computation Conference (GECCO), pp. 1465–1472. ACM (2010)
Jansen, T.: On the brittleness of evolutionary algorithms. In: Stephens, C.R., Toussaint, M., Whitley, D., Stadler, P.F. (eds.) FOGA 2007. LNCS, vol. 4436, pp. 54–69. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-73482-6_4
Kötzing, T., Lissovoi, A., Witt, C.: (1+1) EA on generalized dynamic OneMax. In: Foundations of Genetic Algorithms (FOGA), pp. 40–51. Springer (2015). https://dblp.org/rec/conf/foga/KotzingLW15.html?view=bibtex
Kötzing, T., Molter, H.: ACO beats EA on a dynamic pseudo-Boolean function. In: Coello, C.A.C., Cutello, V., Deb, K., Forrest, S., Nicosia, G., Pavone, M. (eds.) PPSN 2012. LNCS, vol. 7491, pp. 113–122. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-32937-1_12
Lengler, J., Steger, A.: Drift analysis and evolutionary algorithms revisited. Comb. Probab. Comput. 27(4), 643–666 (2018)
Lengler, J.: A general dichotomy of evolutionary algorithms on monotone functions. IEEE Trans. Evol. Comput. 24(6), 995–1009 (2019)
Lengler, J.: Drift analysis. Theory of Evolutionary Computation. NCS, pp. 89–131. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-29414-4_2
Lengler, J., Meier, J.: Large population sizes and crossover help in dynamic environments. In: Bäck, T., et al. (eds.) PPSN 2020. LNCS, vol. 12269, pp. 610–622. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-58112-1_42
Lengler, J., Meier, J.: Large population sizes and crossover help in dynamic environments, full version. arXiv preprint. http://arxiv.org/abs/2004.09949 (2020)
Lengler, J., Riedi, S.: Runtime analysis of the \((\mu +1)\)-EA on the dynamic BinVal function, full version. arXiv preprint. http://arxiv.org/abs/2010.13428 (2020)
Lengler, J., Schaller, U.: The (1+1)-EA on noisy linear functions with random positive weights. In: Symposium Series on Computational Intelligence (SSCI), pp. 712–719. IEEE (2018)
Lengler, J., Zou, X.: Exponential slowdown for larger populations: the (\(\mu \)+1)-EA on monotone functions. In: Foundations of Genetic Algorithms (FOGA), pp. 87–101. ACM (2019)
Lissovoi, A., Witt, C.: MMAS versus population-based EA on a family of dynamic fitness functions. Algorithmica 75(3), 554–576 (2016). https://doi.org/10.1007/s00453-015-9975-z
Lissovoi, A., Witt, C.: A runtime analysis of parallel evolutionary algorithms in dynamic optimization. Algorithmica 78(2), 641–659 (2016). https://doi.org/10.1007/s00453-016-0262-4
Lissovoi, A., Witt, C.: The impact of a sparse migration topology on the runtime of island models in dynamic optimization. Algorithmica 80(5), 1634–1657 (2018). https://doi.org/10.1007/s00453-017-0377-2
Neumann, F., Pourhassan, M., Roostapour, V.: Analysis of evolutionary algorithms in dynamic and stochastic environments. Theory of Evolutionary Computation. NCS, pp. 323–357. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-29414-4_7
Neumann, F., Witt, C.: On the runtime of randomized local search and simple evolutionary algorithms for dynamic makespan scheduling. In: International Joint Conference on Artificial Intelligence (IJCAI), pp. 3742–3748. AAAI Press (2015)
Pourhassan, M., Gao, W., Neumann, F.: Maintaining 2-approximations for the dynamic vertex cover problem using evolutionary algorithms. In: Genetic and Evolutionary Computation Conference (GECCO), pp. 903–910. ACM (2015)
Shi, F., Schirneck, M., Friedrich, T., Kötzing, T., Neumann, F.: Reoptimization time analysis of evolutionary algorithms on linear functions under dynamic uniform constraints. Algorithmica 81(2), 828–857 (2019). https://doi.org/10.1007/s00453-018-0451-4
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this paper
Cite this paper
Lengler, J., Riedi, S. (2021). Runtime Analysis of the \((\mu + 1)\)-EA on the Dynamic BinVal Function. In: Zarges, C., Verel, S. (eds) Evolutionary Computation in Combinatorial Optimization. EvoCOP 2021. Lecture Notes in Computer Science(), vol 12692. Springer, Cham. https://doi.org/10.1007/978-3-030-72904-2_6
Download citation
DOI: https://doi.org/10.1007/978-3-030-72904-2_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-72903-5
Online ISBN: 978-3-030-72904-2
eBook Packages: Computer ScienceComputer Science (R0)