Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-23T13:50:16.542Z Has data issue: false hasContentIssue false
  • English
  • Français

Tail bounds on hitting times of randomized search heuristics using variable drift analysis

Part of: Algorithms - Computer Science Theory of computing

Published online by Cambridge University Press:  05 November 2020

P. K. Lehre  and
C. Witt
P. K. Lehre
Affiliation:
School of Computer Science, University of Birmingham, BirminghamB15 2TT, UK
C. Witt*
Affiliation:
DTU Compute, Technical University of Denmark, Kongens Lyngby, Denmark
*
*Corresponding author. Email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Drift analysis is one of the state-of-the-art techniques for the runtime analysis of randomized search heuristics (RSHs) such as evolutionary algorithms (EAs), simulated annealing, etc. The vast majority of existing drift theorems yield bounds on the expected value of the hitting time for a target state, for example the set of optimal solutions, without making additional statements on the distribution of this time. We address this lack by providing a general drift theorem that includes bounds on the upper and lower tail of the hitting time distribution. The new tail bounds are applied to prove very precise sharp-concentration results on the running time of a simple EA on standard benchmark problems, including the class of general linear functions. On all these problems, the probability of deviating by an r-factor in lower-order terms of the expected time decreases exponentially with r. The usefulness of the theorem outside the theory of RSHs is demonstrated by deriving tail bounds on the number of cycles in random permutations. All these results handle a position-dependent (variable) drift that was not covered by previous drift theorems with tail bounds. Finally, user-friendly specializations of the general drift theorem are given.

MSC classification

Secondary: 68W20: Randomized algorithms 68Q87: Probability in computer science (algorithm analysis, random structures, phase transitions, etc.)
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 30 , Issue 4 , July 2021 , pp. 550 - 569
Check for updates
Copyright
© The Author(s), 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

†

A preliminary version of this paper appeared in the proceedings of ISAAC 2014 [28].

References

Arratia, R. and Tavaré, S. (1992) The cycle structure of random permutations. Ann. Probab. 20 1567–1591.CrossRefGoogle Scholar
Auger, A. and Doerr, B., eds, (2011) Theory of Randomized Search Heuristics: Foundations and Recent Developments. World Scientific.CrossRefGoogle Scholar
Coffman, E. G., Feldmann, A., Kahale, N. and Poonen, B. (1999) Computing call admission capacities in linear networks. Probab. Engrg Inform. Sci. 13 387–406.CrossRefGoogle Scholar
Doerr, B., Doerr, C. and Kötzing, T. (2016) The right mutation strength for multi-valued decision variables. In Proceedings of the Genetic and Evolutionary Computation Conference (GECCO 2016), pp. 1115–1122. ACM Press.CrossRefGoogle Scholar
Doerr, B., Doerr, C. and Yang, J. (2020) Optimal parameter choices via precise black-box analysis. Theoret. Comput. Sci. 801 1–34.CrossRefGoogle Scholar
Doerr, B., Fouz, M. and Witt, C. (2011) Sharp bounds by probability-generating functions and variable drift. In Proceedings of the Genetic and Evolutionary Computation Conference (GECCO 2011), pp. 2083–2090. ACM Press.Google Scholar
Doerr, B. and Goldberg, L. A. (2013) Adaptive drift analysis. Algorithmica 65 224–250.CrossRefGoogle Scholar
Doerr, B., Jansen, T., Witt, C. and Zarges, C. (2013) A method to derive fixed budget results from expected optimisation times. In Proceedings of the Genetic and Evolutionary Computation Conference (GECCO 2013), pp. 1581–1588. ACM Press.CrossRefGoogle Scholar
Doerr, B., Johannsen, D. and Winzen, C. (2012) Multiplicative drift analysis. Algorithmica 64 673–697.CrossRefGoogle Scholar
Doerr, B. and Neumann, F., eds, (2020) Theory of Evolutionary Computation: Recent Developments in Discrete Optimization. Springer.CrossRefGoogle Scholar
Droste, S., Jansen, T. and Wegener, I. (2002) On the analysis of the (1+1) evolutionary algorithm. Theoret. Comput. Sci. 276 51–81.CrossRefGoogle Scholar
Eryilmaz, A. and Srikant, R. (2012) Asymptotically tight steady-state queue length bounds implied by drift conditions. Queueing Syst. Theory Appl. 72 311–359.CrossRefGoogle Scholar
Feldmann, M. and Kötzing, T. (2013) Optimizing expected path lengths with ant colony optimization using fitness proportional update. In Proceedings of Foundations of Genetic Algorithms (FOGA 2013), pp. 65–74. ACM Press.Google Scholar
Gießen, C. and Witt, C. (2018) Optimal mutation rates for the (1 + λ) EA on OneMax through asymptotically tight drift analysis. Algorithmica 80 1710–1731.CrossRefGoogle Scholar
Hajek, B. (1982) Hitting and occupation time bounds implied by drift analysis with applications. Adv. Appl. Probab. 14 502–525.CrossRefGoogle Scholar
He, J. and Yao, X. (2001) Drift analysis and average time complexity of evolutionary algorithms. Artif. Intell. 127 57–85. Erratum in Artif. Intell. 140 (2002) 245–248.CrossRefGoogle Scholar
Hwang, H., Panholzer, A., Rolin, N., Tsai, T. and Chen, W. (2018) Probabilistic analysis of the (1+1)-evolutionary algorithm. Evol. Comput. 26 299–345.CrossRefGoogle ScholarPubMed
Jägersküpper, J. (2011) Combining Markov-chain analysis and drift analysis: the (1+1) evolutionary algorithm on linear functions reloaded. Algorithmica 59 409–424.CrossRefGoogle Scholar
Jansen, T. (2013) Analyzing Evolutionary Algorithms: The Computer Science Perspective, Natural Computing Series. Springer.CrossRefGoogle Scholar
Jansen, T. (2020) Analysing stochastic search heuristics operating on a fixed budget. In Theory of Evolutionary Computation: Recent Developments in Discrete Optimization (Doerr, B. and Neumann, F., eds), pp. 249–270. Springer.CrossRefGoogle Scholar
Johannsen, D. (2010) Random combinatorial structures and randomized search heuristics. PhD thesis, Universität des Saarlandes, Germany.Google Scholar
Karp, R. M. (1994) Probabilistic recurrence relations. J. Assoc. Comput. Mach. 41 1136–1150.CrossRefGoogle Scholar
Kötzing, T. (2016) Concentration of first hitting times under additive drift. Algorithmica 75 490–506.CrossRefGoogle Scholar
Kötzing, T. and Krejca, M. S. (2019) First-hitting times under drift. Theoret. Comput. Sci. 796 51–69.CrossRefGoogle Scholar
Kötzing, T. and Witt, C. (2020) Improved fixed-budget results via drift analysis. In Proceedings of Parallel Problem Solving from Nature (PPSN 2020), Vol. 12270 of Lecture Notes in Computer Science, pp. 648–660. Springer.CrossRefGoogle Scholar
Lehre, P. K. (2011) Negative drift in populations. In Proceedings of Parallel Problem Solving from Nature (PPSN 2010), Vol. 6238 of Lecture Notes in Computer Science, pp. 244–253. Springer.Google Scholar
Lehre, P. K. (2012) Drift analysis (tutorial). In Companion to GECCO 2012, pp. 1239–1258. ACM Press.Google Scholar
Lehre, P. K. and Witt, C. (2014) Concentrated hitting times of randomized search heuristics with variable drift. In Proceedings of the 25th International Symposium on Algorithms and Computation (ISAAC 2014), Vol. 8889 of Lecture Notes in Computer Science, pp. 686–697. Springer.Google Scholar
Lehre, P. K. and Witt, C. (2018) General drift analysis with tail bounds. Preprint of this paper, including supplementary material. arXiv:1211.7184Google Scholar
Lengler, J. (2020) Drift analysis. In Theory of Evolutionary Computation: Recent Developments in Discrete Optimization (Doerr, B. and Neumann, F., eds), pp. 89–131. Springer.CrossRefGoogle Scholar
Lengler, J. and Steger, A. (2018) Drift analysis and evolutionary algorithms revisited. Combin. Probab. Comput. 27 643–666.CrossRefGoogle Scholar
Mitavskiy, B., Rowe, J. E. and Cannings, C. (2009) Theoretical analysis of local search strategies to optimize network communication subject to preserving the total number of links. Int. J. Intelligent Computing Cybernetics 2 243–284.CrossRefGoogle Scholar
Neumann, F. and Witt, C. (2010) Bioinspired Computation in Combinatorial Optimization: Algorithms and Their Computational Complexity, Natural Computing Series. Springer.CrossRefGoogle Scholar
Oliveto, P. S. and Witt, C. (2011) Simplified drift analysis for proving lower bounds in evolutionary computation. Algorithmica 59 369–386.CrossRefGoogle Scholar
Oliveto, P. S. and Witt, C. (2012) Erratum: Simplified drift analysis for proving lower bounds in evolutionary computation. arXiv:1211.7184Google Scholar
Rowe, J. E. and Sudholt, D. (2012) The choice of the offspring population size in the (1, λ) EA. In Proceedings of the Genetic and Evolutionary Computation Conference (GECCO 2012), pp. 1349–1356. ACM Press.Google Scholar
Sasak, G. H. and Hajek, B. (1988) The time complexity of maximum matching by simulated annealing. J. Assoc. Comput. Mach. 35 387–403.CrossRefGoogle Scholar
Sudholt, D. (2013) A new method for lower bounds on the running time of evolutionary algorithms. IEEE Trans. Evol. Comput. 17 418–435.Google Scholar
Wegener, I. (2001) Theoretical aspects of evolutionary algorithms. In Proceedings of the 28th International Colloquium on Automata, Languages and Programming (ICALP 2001), Vol. 2076 of Lecture Notes in Computer Science, pp. 64–78. Springer.Google Scholar
Witt, C. (2013) Tight bounds on the optimization time of a randomized search heuristic on linear functions. Combin. Probab. Comput. 22 294–318. Preliminary version in STACS 2012.CrossRefGoogle Scholar