Abstract
In this paper we show how to use drift analysis in the case of two random variables \(X_1, X_2\), when the drift is approximatively given by \(A\cdot (X_1,X_2)^T\) for a matrix A. The non-trivial case is that \(X_1\) and \(X_2\) impede each other’s progress, and we give a full characterization of this case. As application, we develop and analyze a minimal example TwoLinof a dynamic environment that can be hard. The environment consists of two linear function \(f_1\) and \(f_2\) with positive weights 1 and n, and in each generation selection is based on one of them at random. They only differ in the set of positions that have weight 1 and n. We show that the \((1 + 1)\)-EAwith mutation rate \(\chi /n\) is efficient for small constant \(\chi \) on TwoLin, but does not find the shared optimum in polynomial time for large constant \(\chi \).
Extended Abstract. All proofs and further details are available on arxiv [11].
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
Similar content being viewed by others
Notes
- 1.
- 2.
We follow the convention that we always call drift towards the optimum positive, and drift away from the optimum negative. Since the optimum in our case is at 0, this means that we consider the difference \(X^{t}-X^{t+1}\) for the drift, not vice versa.
- 3.
For direct comparison it is important to note that [28] works with the matrix \(I-A\) instead of A, where I is the identity matrix.
- 4.
This is because for any matrix M of norm 1 we can write \(\lambda _1(A+\beta M) = \lambda _1(A) + \beta D{\lambda _1}(A)\cdot M + O(\beta ^2)\), where the total differential \(D\lambda _1(A)\) has bounded norm, and analogously for the other eigenvalues and eigenvectors.
References
Antipov, D., Doerr, B., Yang, Q.: The efficiency threshold for the offspring population size of the \((\mu , \lambda )\) EA. In: Genetic and Evolutionary Computation Conference (GECCO), pp. 1461–1469 (2019)
Colin, S., Doerr, B., Férey, G.: Monotonic functions in EC: anything but monotone! In: Genetic and Evolutionary Computation Conference (GECCO), pp. 753–760 (2014)
Dang, D.C., Eremeev, A., Lehre, P.K.: Escaping local optima with non-elitist evolutionary algorithms. In: AAAI Conference on Artificial Intelligence, vol. 35, pp. 12275–12283 (2021)
Dang, D.C., Eremeev, A., Lehre, P.K.: Non-elitist evolutionary algorithms excel in fitness landscapes with sparse deceptive regions and dense valleys. In: Genetic and Evolutionary Computation Conference (GECCO), pp. 1133–1141 (2021)
Dang, D.-C., Lehre, P.K.: Self-adaptation of mutation rates in non-elitist populations. In: Handl, J., Hart, E., Lewis, P.R., López-Ibáñez, M., Ochoa, G., Paechter, B. (eds.) PPSN 2016. LNCS, vol. 9921, pp. 803–813. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-45823-6_75
Doerr, B.: Lower bounds for non-elitist evolutionary algorithms via negative multiplicative drift. Evol. Comput. 29(2), 305–329 (2021)
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, 673–697 (2012)
Hevia Fajardo, M.A., Sudholt, D.: Self-adjusting population sizes for non-elitist evolutionary algorithms: why success rates matter. In: Genetic and Evolutionary Computation Conference (GECCO), pp. 1151–1159 (2021)
Jägersküpper, J., Storch, T.: When the plus strategy outperforms the comma strategyand when not. In: Foundations of Computational Intelligence (FOCI), pp. 25–32. IEEE (2007)
Janett, D., Lengler, J.: Two-dimensional drift analysis: optimizing two functions simultaneously can be hard (2022). https://arxiv.org/abs/2203.14547
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
Kaufmann, M., Larcher, M., Lengler, J., Zou, X.: OneMax is not the easiest function for fitness improvements (2022). https://arxiv.org/abs/2204.07017
Kaufmann, M., Larcher, M., Lengler, J., Zou, X.: Self-adjusting population sizes for the \((1, \lambda )\)-EA on monotone functions. In: Parallel Problem Solving from Nature (PPSN). Springer (2022)
Kötzing, T.: Concentration of first hitting times under additive drift. Algorithmica 75(3), 490–506 (2016)
Lehre, P.K.: Negative drift in populations. In: Schaefer, R., Cotta, C., Kołodziej, J., Rudolph, G. (eds.) PPSN 2010. LNCS, vol. 6238, pp. 244–253. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-15844-5_25
Lengler, J.: A general dichotomy of evolutionary algorithms on monotone functions. IEEE Trans. Evol. Comput. 24(6), 995–1009 (2019)
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
Lengler, J., Martinsson, A., Steger, A.: When does hillclimbing fail on monotone functions: An entropy compression argument. In: Analytic Algorithmics and Combinatorics (ANALCO), pp. 94–102. SIAM (2019)
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., Riedi, S.: Runtime Analysis of the \((\mu + 1)\)-EA on the Dynamic BinVal Function. In: Evolutionary Computation in Combinatorial Optimization (EvoCom). pp. 84–99. Springer, Heidelberg (2021)
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., Steger, A.: Drift analysis and evolutionary algorithms revisited. Comb. Probab. Comput. 27(4), 643–666 (2018)
Lengler, J., Sudholt, D., Witt, C.: The complex parameter landscape of the compact genetic algorithm. Algorithmica 83(4), 1096–1137 (2021)
Lengler, J., Zou, X.: Exponential slowdown for larger populations: the \((\mu + 1)\)-EA on monotone functions. Theoret. Comput. Sci. 875, 28–51 (2021)
Neumann, F., Sudholt, D., Witt, C.: A few ants are enough: ACO with iteration-best update. In: Genetic and Evolutionary Computation Conference (GECCO), pp. 63–70 (2010)
Oliveto, P.S., Witt, C.: Improved time complexity analysis of the simple genetic algorithm. Theoret. Comput. Sci. 605, 21–41 (2015)
Rowe, J.E.: Linear multi-objective drift analysis. Theoret. Comput. Sci. 736, 25–40 (2018)
Rowe, J.E., Sudholt, D.: The choice of the offspring population size in the (1, \(\lambda \)) evolutionary algorithm. Theoret. Comput. Sci. 545, 20–38 (2014)
Sudholt, D., Witt, C.: On the choice of the update strength in estimation-of-distribution algorithms and ant colony optimization. Algorithmica 81(4), 1450–1489 (2019)
Witt, C.: Tight bounds on the optimization time of a randomized search heuristic on linear functions. Comb. Probab. Comput. 22(2), 294–318 (2013)
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
Janett, D., Lengler, J. (2022). Two-Dimensional Drift Analysis:. 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_43
Download citation
DOI: https://doi.org/10.1007/978-3-031-14721-0_43
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)