Abstract
We analyse the impact of the selective pressure for the global optimisation capabilities of steady-state evolutionary algorithms (EAs). For the standard bimodal benchmark function TwoMax, we rigorously prove that using uniform parent selection leads to exponential runtimes with high probability to locate both optima for the standard (\(\)+1) EA and (\(\)+1) RLS with any polynomial population sizes. However, we prove that selecting the worst individual as parent leads to efficient global optimisation with overwhelming probability for reasonable population sizes. Since always selecting the worst individual may have detrimental effects for escaping from local optima, we consider the performance of stochastic parent selection operators with low selective pressure for a function class called TruncatedTwoMax, where one slope is shorter than the other. An experimental analysis shows that the EAs equipped with inverse tournament selection, where the loser is selected for reproduction and small tournament sizes, globally optimise TwoMax efficiently and effectively escape from local optima of TruncatedTwoMax with high probability. Thus, they identify both optima efficiently while uniform (or stronger) selection fails in theory and in practice. We then show the power of inverse selection on function classes from the literature where populations are essential by providing rigorous proofs or experimental evidence that it outperforms uniform selection equipped with or without a restart strategy. We conclude the article by confirming our theoretical insights with an empirical analysis of the different selective pressures on standard benchmarks of the classical MaxSat and multidimensional knapsack problems.
- T. Bäck and F. Hoffmeister. 1991. A survey of evolution strategies. In Proceedings of the 4th International Conference on Genetic Algorithms, R. K Belew and L. B. Booker (Eds.). Morgan Kaufmann, 92–99.Google Scholar
- T. Bäck, F. Hoffmeister, and H. P. Schwefel. 1991. A survey of evolution strategies. In Proceedings of the 4th International Conference on Genetic Algorithms. Morgan Kaufmann, 2--9.Google Scholar
- J. E. Beasley. 1990. OR-Library: Distributing test problems by electronic mail. Journal of the Operational Research Society 41, 11 (1990), 1069--1072. Google ScholarCross Ref
- B. Bollobas. 1985. Random Graphs. Academic Press, London, UK.Google Scholar
- Jürgen Branke, Massimo Cutaia, and Heinrich Dold. 1999. Reducing genetic drift in steady state evolutionary algorithms. In Proceedings of the Genetic and Evolutionary Computation Conference (GECCO’99). 68--74.Google Scholar
- P. C. Chu and J. E. Beasley. 1998. A genetic algorithm for the multidimensional knapsack problem. Journal of Heuristics 4, 1 (1998), 63--86. Google ScholarDigital Library
- D. Corus, D.-C. Dang, A. V. Eremeev, and P. K. Lehre. 2018a. Level-based analysis of genetic algorithms and other search processes. IEEE Transactions on Evolutionary Computation 22, 5 (2018), 707--719.Google ScholarDigital Library
- D. Corus and P. S. Oliveto. 2018. Standard steady state genetic algorithms can hillclimb faster than mutation-only evolutionary algorithms. IEEE Transactions on Evolutionary Computation 22, 5 (2018), 720--732.Google ScholarDigital Library
- D. Corus and P. S. Oliveto. 2020. On the benefits of populations for the exploitation speed of standard steady-state genetic algorithms. arXiv:1903.10976Google Scholar
- D. Corus, P. S. Oliveto, and D. Yazdani. 2018b. Fast artificial immune systems. In Parallel Problem Solving from Nature. Lecture Notes in Computer Science, Vol. 11102. Springer, 67--78.Google Scholar
- D. Corus, P. S. Oliveto, and D. Yazdani. 2019. Artificial immune systems can find arbitrarily good approximations for the NP-hard number partitioning problem. Artificial Intelligence 247 (2019), 180--196.Google ScholarDigital Library
- D.-C. Dang, T. Friedrich, T. Kötzing, M. S. Krejca, P. K. Lehre, P. S. Oliveto, D. Sudholt, and A. M. Sutton. 2018. Escaping local optima using crossover with emergent diversity. IEEE Transactions on Evolutionary Computation 22, 3 (2018), 484--497.Google ScholarCross Ref
- C. Darwin. 1859. On the Origin of the Species. John Murray.Google Scholar
- C. Darwin. 1868. The Variation of Animals and Plants Under Domestication. John Murray.Google Scholar
- K. De Jong and J. Sarma. 1993. Generation gaps revisited. In Proceedings of Foundations of Genetic Algorithms II (FOGA’93). 19--28.Google Scholar
- A. de Perthuis de Laillevault, B. Doerr, and C. Doerr. 2015. Money for nothing: Speeding up evolutionary algorithms through better initialization. In Proceedings of the Genetic and Evolutionary Computation Conference (GECCO’15). ACM, New York, NY, 815--822.Google Scholar
- B. Doerr. 2011. Analyzing randomized search heuristics: Tools from probability theory. In Theory of Randomized Search Heuristics: Foundations and Recent Developments, B. Doerr and A. Auger (Eds.). World Scientific, 1--20.Google Scholar
- B. Doerr, H. Ph. Le, R. Makhmara, and T. D. Nguyen. 2017. Fast genetic algorithms. In Proceedings of the Genetic and Evolutionary Computation Conference (GECCO’17). 777--784.Google Scholar
- T. Friedrich, P. S. Oliveto, D. Sudholt, and C. Witt. 2009. Analysis of diversity-preserving mechanisms for global exploration. Evolutionary Computation 17, 4 (2009), 455--476.Google ScholarDigital Library
- D. E. Goldberg. 1989. Genetic Algorithms in Search, Optimization and Machine Learning. Addison Wesley Longman. Google ScholarDigital Library
- D. E. Goldberg and K. Deb. 1991. A comparative analysis of selection schemes used in genetic algorithms. In Proceedings of Foundations of Genetic Algorithms I (FOGA’91). 69--93.Google Scholar
- J. H. Holland. 1992. Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence. MIT Press, Cambridge, MA.Google Scholar
- S. Holm. 1979. A simple sequentially rejective multiple test procedure. Scandinavian Journal of Statistics 6, 2 (1979), 65--70. Google Scholar
- H. H. Hoos and T. Stützle. 2000. SATLIB: An online resource for research on SAT. SAT 2000 (2000), 283--292.Google Scholar
- T. Jansen, K. A. De Jong, and I. Wegener. 2005. On the choice of the offspring population size in evolutionary algorithms. Evolutionary Computation 13, 4 (2005), 413--440.Google ScholarDigital Library
- H. Kautz and B. Selman. 1993. Domain-independent extension to GSAT: Solving large structured satisfiability problems. In Proceedings of the 13th International Joint Conference on Artificial Intelligence (IJCAI’93). 290--295.Google Scholar
- H. Kautz and B. Selman. 1996. Pushing the envelope: Planning, propositional logic, and stochastic search. In Proceedings of the 13th National Conference on Artificial Intelligence (AAAI’96). 1194--1201.Google Scholar
- P. K. Lehre. 2010. Negative drift in populations. In Parallel Problem Solving from Nature. Lecture Notes in Computer Science, Vol. 6328. Springer, 244--253.Google Scholar
- J. Lengler. 2018. A general dichotomy of evolutionary algorithms on monotone functions. In Parallel Problem Solving from Nature. Lecture Notes in Computer Science, Vol. 11101. Springer, 3--15.Google Scholar
- P. S. Oliveto, D. Sudholt, and C. Zarges. 2018. On the benefits and risks of using fitness sharing for multimodal optimisation. Theoretical Computer Science 773 (2018), 53--70.Google ScholarDigital Library
- E. Covantes Osuna and D. Sudholt. 2017. Analysis of the clearing diversity-preserving mechanism. In Proceedings of Foundations of Genetic Algorithms (FOGA’17). ACM, New York, NY, 55--63.Google Scholar
- M. Preuss. 2015. Multimodal Optimization by Means of Evolutionary Algorithms. Springer.Google Scholar
- J. Smith. 2007. On replacement strategies in steady state evolutionary algorithms. Evolutionary Computation (2007), 29--59.Google Scholar
- H. Spencer. 1864. The Principles of Biology. Williams and Norgate.Google Scholar
- D. Sudholt. 2019. The benefits of population diversity in evolutionary algorithms: A survey of rigorous runtime analyses. In Theory of Randomized Search Heuristics in Discrete Search Spaces, B. Doerr and F. Neumann (Eds.). Springer, 359--404.Google Scholar
- A. Sutton. 2018. Crossover can simulate bounded tree search on a fixed-parameter tractable optimization problem. In Proceedings of the Genetic and Evolutionary Computation Conference (GECCO’18). ACM, New York, NY, 1531--1538.Google ScholarDigital Library
- G. Syswerda. 1989. Uniform crossover in genetic algorithms. In Proceedings of the 3rd International Conference on Genetic Algorithms. 2--9.Google ScholarDigital Library
- G. Syswerda. 1991. A study of reproduction in generational and steady-state genetic algorithms. In Proceedings of Foundations of Genetic Algorithms I (FOGA’91). 94--101.Google ScholarCross Ref
- D. Whitley. 1989. The Genitor algorithm and selection pressure: Why rank-based allocation of reproductive trials is best. In Proceedings of the 3rd International Conference on Genetic Algorithms. 116--121.Google ScholarDigital Library
- F. Wilcoxon. 1945. Individual comparisons by ranking methods. Biometrics Bulletin 1, 6 (1945), 80--83. Google ScholarCross Ref
- C. Witt. 2006. Runtime analysis of the ( + 1) EA on simple pseudo-Boolean functions. Evolutionary Computation 14, 1 (2006), 65--86.Google ScholarDigital Library
- C. Witt. 2014. Fitness levels with tail bounds for the analysis of randomized search heuristics. Information Processing Letters 114, 1--2 (2014), 38--41.Google ScholarCross Ref
Index Terms
- On Steady-State Evolutionary Algorithms and Selective Pressure: Why Inverse Rank-Based Allocation of Reproductive Trials Is Best
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