research-article

Open access

Lower bounds from fitness levels made easy

Authors:

Benjamin Doerr,

Timo KötzingAuthors Info & Claims

GECCO '21: Proceedings of the Genetic and Evolutionary Computation Conference

Pages 1142 - 1150

https://doi.org/10.1145/3449639.3459352

Published: 26 June 2021 Publication History

Abstract

One of the first and easy to use techniques for proving run time bounds for evolutionary algorithms is the so-called method of fitness levels by Wegener. It uses a partition of the search space into a sequence of levels which are traversed by the algorithm in increasing order, possibly skipping levels. An easy, but often strong upper bound for the run time can then be derived by adding the reciprocals of the probabilities to leave the levels (or upper bounds for these). Unfortunately, a similarly effective method for proving lower bounds has not yet been established. The strongest such method, proposed by Sudholt (2013), requires a careful choice of the viscosity parameters γ_i,j, 0 ≤ i ≤ j ≤ n.

In this paper we present two new variants of the method, one for upper and one for lower bounds. Besides the level leaving probabilities, they only rely on the probabilities that levels are visited at all. We show that these can be computed or estimated without greater difficulties and apply our method to reprove the following known results in an easy and natural way. (i) The precise run time of the (1 + 1) EA on LeadingOnes. (ii) A lower bound for the run time of the (1 + 1) EA on OneMax, tight apart from an O(n) term. (iii) A lower bound for the run time of the (1 + 1) EA on long k-paths.

References

[1]

Denis Antipov, Maxim Buzdalov, and Benjamin Doerr. 2020. First steps towards a runtime analysis when starting with a good solution. In Parallel Problem Solving From Nature, PPSN 2020, Part II. Springer, 560--573.

[2]

Süntje Böttcher, Benjamin Doerr, and Frank Neumann. 2010. Optimal fixed and adaptive mutation rates for the LeadingOnes problem. In Parallel Problem Solving from Nature, PPSN 2010. Springer, 1--10.

Digital Library

[3]

Maxim Buzdalov, Benjamin Doerr, Carola Doerr, and Dmitry Vinokurov. 2020. Fixed-target runtime analysis. In Genetic and Evolutionary Computation Conference, GECCO 2020. ACM, 1295--1303.

Digital Library

[4]

Dogan Corus, Duc-Cuong Dang, Anton V. Eremeev, and Per Kristian Lehre. 2018. Level-based analysis of genetic algorithms and other search processes. IEEE Transactions on Evolutionary Computation 22 (2018), 707--719.

Digital Library

[5]

Duc-Cuong Dang and Per Kristian Lehre. 2016. Runtime analysis of non-elitist populations: from classical optimisation to partial information. Algorithmica 75 (2016), 428--461.

Digital Library

[6]

Benjamin Doerr. 2019. Analyzing randomized search heuristics via stochastic domination. Theoretical Computer Science 773 (2019), 115--137.

Digital Library

[7]

Benjamin Doerr. 2020. Probabilistic tools for the analysis of randomized optimization heuristics. In Theory of Evolutionary Computation: Recent Developments in Discrete Optimization, Benjamin Doerr and Frank Neumann (Eds.). Springer, 1--87. Also available at https://arxiv.org/abs/1801.06733.

[8]

Benjamin Doerr, Carola Doerr, and Timo Kötzing. 2018. Static and self-adjusting mutation strengths for multi-valued decision variables. Algorithmica 80 (2018), 1732--1768.

Digital Library

[9]

Benjamin Doerr, Carola Doerr, and Jing Yang. 2020. Optimal parameter choices via precise black-box analysis. Theoretical Computer Science 801 (2020), 1--34.

[10]

Benjamin Doerr, Mahmoud Fouz, and Carsten Witt. 2010. Quasirandom evolutionary algorithms. In Genetic and Evolutionary Computation Conference, GECCO 2010. ACM, 1457--1464.

Digital Library

[11]

Benjamin Doerr, Mahmoud Fouz, and Carsten Witt. 2011. Sharp bounds by probability-generating functions and variable drift. In Genetic and Evolutionary Computation Conference, GECCO 2011. ACM, 2083--2090.

Digital Library

[12]

Benjamin Doerr, Thomas Jansen, Carsten Witt, and Christine Zarges. 2013. A method to derive fixed budget results from expected optimisation times. In Genetic and Evolutionary Computation Conference, GECCO 2013. ACM, 1581--1588.

Digital Library

[13]

Benjamin Doerr, Daniel Johannsen, and Carola Winzen. 2012. Multiplicative drift analysis. Algorithmica 64 (2012), 673--697.

Digital Library

[14]

Benjamin Doerr and Timo Kötzing. 2019. Multiplicative up-drift. In Genetic and Evolutionary Computation Conference, GECCO 2019. ACM, 1470--1478.

Digital Library

[15]

Benjamin Doerr and Timo Kötzing. 2021. Lower Bounds from Fitness Levels Made Easy. CoRR abs/2104.03372 (2021). arXiv:2104.03372

[16]

Benjamin Doerr, Timo Kötzing, J. A. Gregor Lagodzinski, and Johannes Lengler. 2020. The impact of lexicographic parsimony pressure for ORDER/MAJORITY on the run time. Theoretical Computer Science 816 (2020), 144--168.

Digital Library

[17]

Stefan Droste, Thomas Jansen, and Ingo Wegener. 1998. A Rigorous complexity analysis of the (1 + 1) evolutionary algorithm for separable functions with boolean inputs. Evolutionary Computation 6 (1998), 185--196.

Digital Library

[18]

Stefan Droste, Thomas Jansen, and Ingo Wegener. 2002. On the analysis of the (1 + 1) evolutionary algorithm. Theoretical Computer Science 276 (2002), 51--81.

Digital Library

[19]

Matthias Feldmann and Timo Kötzing. 2013. Optimizing expected path lengths with ant colony optimization using fitness proportional update. In Foundations of Genetic Algorithms, FOGA 2013. ACM, 65--74.

Digital Library

[20]

Josselin Garnier, Leila Kallel, and Marc Schoenauer. 1999. Rigorous hitting times for binary mutations. Evolutionary Computation 7 (1999), 173--203.

Digital Library

[21]

Christian Gießen and Carsten Witt. 2018. Optimal mutation rates for the (1 + λ) EA on OneMax through asymptotically tight drift analysis. Algorithmica 80 (2018), 1710--1731.

Digital Library

[22]

Jun He and Xin Yao. 2001. Drift analysis and average time complexity of evolutionary algorithms. Artificial Intelligence 127 (2001), 51--81.

Digital Library

[23]

Hsien-Kuei Hwang, Alois Panholzer, Nicolas Rolin, Tsung-Hsi Tsai, and Wei-Mei Chen. 2018. Probabilistic analysis of the (1+1)-evolutionary algorithm. Evolutionary Computation 26 (2018), 299--345.

Digital Library

[24]

Hsien-Kuei Hwang and Carsten Witt. 2019. Sharp bounds on the runtime of the (1+1) EA via drift analysis and analytic combinatorial tools. In Foundations of Genetic Algorithms, FOGA 2019. ACM, 1--12.

Digital Library

[25]

Jens Jägersküpper. 2007. Algorithmic analysis of a basic evolutionary algorithm for continuous optimization. Theoretical Computer Science 379 (2007), 329--347.

Digital Library

[26]

Thomas Jansen and Christine Zarges. 2014. Performance analysis of randomised search heuristics operating with a fixed budget. Theoretical Computer Science 545 (2014), 39--58.

Digital Library

[27]

Daniel Johannsen. 2010. Random Combinatorial Structures and Randomized Search Heuristics. Ph.D. Dissertation. Universität des Saarlandes.

[28]

Jörg Lässig and Dirk Sudholt. 2014. General upper bounds on the runtime of parallel evolutionary algorithms. Evolutionary Computation 22 (2014), 405--437.

Digital Library

[29]

Per Kristian Lehre. 2011. Fitness-levels for non-elitist populations. In Genetic and Evolutionary Computation Conference, GECCO 2011. ACM, 2075--2082.

Digital Library

[30]

Per Kristian Lehre and Carsten Witt. 2014. Concentrated hitting times of randomized search heuristics with variable drift. In International Symposium on Algorithms and Computation, ISAAC 2014. Springer, 686--697.

[31]

Johannes Lengler. 2020. Drift analysis. In Theory of Evolutionary Computation: Recent Developments in Discrete Optimization, Benjamin Doerr and Frank Neumann (Eds.). Springer, 89--131. Also available at https://arxiv.org/abs/1712.00964.

[32]

Boris Mitavskiy, Jonathan E. Rowe, and Chris Cannings. 2009. Theoretical analysis of local search strategies to optimize network communication subject to preserving the total number of links. International Journal on Intelligent Computing and Cybernetics 2 (2009), 243--284.

[33]

Günter Rudolph. 1996. How mutation and selection solve long path problems in polynomial expected time. Evolutionary Computation 4 (1996), 195--205.

Digital Library

[34]

Günter Rudolph. 1997. Convergence Properties of Evolutionary Algorithms. Verlag Dr. Kovǎc.

[35]

Dirk Sudholt. 2009. The impact of parametrization in memetic evolutionary algorithms. Theoretical Computer Science 410 (2009), 2511--2528.

Digital Library

[36]

Dirk Sudholt. 2010. General lower bounds for the running time of evolutionary algorithms. In Parallel Problem Solving from Nature, PPSN 2010, Part I. Springer, 124--133.

[37]

Dirk Sudholt. 2013. A new method for lower bounds on the running time of evolutionary algorithms. IEEE Transactions on Evolutionary Computation 17 (2013), 418--435.

Digital Library

[38]

Ingo Wegener. 2001. Theoretical aspects of evolutionary algorithms. In Automata, Languages and Programming, ICALP 2001. Springer, 64--78.

[39]

Ingo Wegener. 2002. Methods for the analysis of evolutionary algorithms on pseudo-Boolean functions. In Evolutionary Optimization, R. Sarker, X. Yao, and M. Mohammadian (Eds.). Kluwer, 349--369.

[40]

Carsten Witt. 2013. Tight bounds on the optimization time of a randomized search heuristic on linear functions. Combinatorics, Probability & Computing 22 (2013), 294--318.

[41]

Carsten Witt. 2014. Fitness levels with tail bounds for the analysis of randomized search heuristics. Inform. Process. Lett. 114 (2014), 38--41.

Digital Library

Cited By

Doerr BLi XHandl J(2024)A Gentle Introduction to Theory (for Non-Theoreticians)Proceedings of the Genetic and Evolutionary Computation Conference Companion10.1145/3638530.3648402(800-829)Online publication date: 14-Jul-2024
https://dl.acm.org/doi/10.1145/3638530.3648402
Lv ZQian CYen GSun Y(2024)Analyzing the Expected Hitting Time of Evolutionary Computation-Based Neural Architecture Search AlgorithmsIEEE Transactions on Emerging Topics in Computational Intelligence10.1109/TETCI.2024.33776838:6(3899-3911)Online publication date: Dec-2024
https://doi.org/10.1109/TETCI.2024.3377683
Doerr BKelley A(2024)Fourier Analysis Meets Runtime Analysis: Precise Runtimes on PlateausAlgorithmica10.1007/s00453-024-01232-586:8(2479-2518)Online publication date: 10-May-2024
https://doi.org/10.1007/s00453-024-01232-5
Show More Cited By

Index Terms

Lower bounds from fitness levels made easy
1. Theory of computation
  1. Theory and algorithms for application domains
    1. Theory of randomized search heuristics

Recommendations

Lower Bounds from Fitness Levels Made Easy
Abstract
One of the first and easy to use techniques for proving run time bounds for evolutionary algorithms is the so-called method of fitness levels by Wegener. It uses a partition of the search space into a sequence of levels which are traversed by the ... $_{}$ $^{}$
On Lower Bounds for the Redundancy of Optimal Codes

The problem of providing bounds on the redundancy of an optimal code for a discrete memoryless source in terms of the probability distribution of the source, has been extensively studied in the literature. The attention has mainly focused on binary ...
Sampling lower bounds via information theory
STOC '03: Proceedings of the thirty-fifth annual ACM symposium on Theory of computing

We present a novel technique, based on the Jensen-Shannon divergence from information theory, to prove lower bounds on the query complexity of sampling algorithms that approximate functions over arbitrary domain and range. Unlike previous methods, our ...

Comments

Information & Contributors

Information

Published In

cover image ACM Conferences

GECCO '21: Proceedings of the Genetic and Evolutionary Computation Conference

June 2021

1219 pages

ISBN:9781450383509

DOI:10.1145/3449639

Editor:
Francisco Chicano
University of Malaga
,
General Chair:
Krzysztof Krawiec
Poznan University of Technology

Copyright © 2021 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected].

Sponsors

SIGEVO: ACM Special Interest Group on Genetic and Evolutionary Computation

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 26 June 2021

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Research-article

Funding Sources

Conference

GECCO '21

Sponsor:

SIGEVO

GECCO '21: Genetic and Evolutionary Computation Conference

July 10 - 14, 2021

Lille, France

Acceptance Rates

Overall Acceptance Rate 1,669 of 4,410 submissions, 38%

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

14
Total Citations
View Citations
295
Total Downloads

Downloads (Last 12 months)107
Downloads (Last 6 weeks)10

Reflects downloads up to 18 Feb 2025

Other Metrics

View Author Metrics

Citations

Cited By

Doerr BLi XHandl J(2024)A Gentle Introduction to Theory (for Non-Theoreticians)Proceedings of the Genetic and Evolutionary Computation Conference Companion10.1145/3638530.3648402(800-829)Online publication date: 14-Jul-2024
https://dl.acm.org/doi/10.1145/3638530.3648402
Lv ZQian CYen GSun Y(2024)Analyzing the Expected Hitting Time of Evolutionary Computation-Based Neural Architecture Search AlgorithmsIEEE Transactions on Emerging Topics in Computational Intelligence10.1109/TETCI.2024.33776838:6(3899-3911)Online publication date: Dec-2024
https://doi.org/10.1109/TETCI.2024.3377683
Doerr BKelley A(2024)Fourier Analysis Meets Runtime Analysis: Precise Runtimes on PlateausAlgorithmica10.1007/s00453-024-01232-586:8(2479-2518)Online publication date: 10-May-2024
https://doi.org/10.1007/s00453-024-01232-5
Friedrich TKötzing TRadhakrishnan ASchiller LSchirneck MTennigkeit GWietheger S(2023)Crossover for Cardinality Constrained OptimizationACM Transactions on Evolutionary Learning and Optimization10.1145/36036293:2(1-32)Online publication date: 28-Jun-2023
https://dl.acm.org/doi/10.1145/3603629
Doerr BSilva SPaquete L(2023)A Gentle Introduction to Theory (for Non-Theoreticians)Proceedings of the Companion Conference on Genetic and Evolutionary Computation10.1145/3583133.3595042(946-975)Online publication date: 15-Jul-2023
https://dl.acm.org/doi/10.1145/3583133.3595042
Friedrich TKötzing TNeumann ANeumann FRadhakrishnan ASilva SPaquete L(2023)Analysis of (1+1) EA on LeadingOnes with ConstraintsProceedings of the Genetic and Evolutionary Computation Conference10.1145/3583131.3590453(1584-1592)Online publication date: 15-Jul-2023
https://dl.acm.org/doi/10.1145/3583131.3590453
Doerr BKelley ASilva SPaquete L(2023)Fourier Analysis Meets Runtime Analysis: Precise Runtimes on PlateausProceedings of the Genetic and Evolutionary Computation Conference10.1145/3583131.3590393(1555-1564)Online publication date: 15-Jul-2023
https://dl.acm.org/doi/10.1145/3583131.3590393
Hevia Fajardo MSudholt D(2023)Theoretical and Empirical Analysis of Parameter Control Mechanisms in the (1 + (λ, λ)) Genetic AlgorithmACM Transactions on Evolutionary Learning and Optimization10.1145/35647552:4(1-39)Online publication date: 14-Jan-2023
https://dl.acm.org/doi/10.1145/3564755
Doerr BGhannane YIbn Brahim M(2023)Runtime Analysis for Permutation-based Evolutionary AlgorithmsAlgorithmica10.1007/s00453-023-01146-886:1(90-129)Online publication date: 26-Jul-2023
https://dl.acm.org/doi/10.1007/s00453-023-01146-8
Doerr BWagner M(2022)A gentle introduction to theory (for non-theoreticians)Proceedings of the Genetic and Evolutionary Computation Conference Companion10.1145/3520304.3533628(890-921)Online publication date: 9-Jul-2022
https://dl.acm.org/doi/10.1145/3520304.3533628
Show More Cited By

View Options

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Publication

Figures

Tables

Media

View Table of Conten