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Lower Bounds for Comparison Based Evolution Strategies Using VC-dimension and Sign Patterns

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Abstract

We derive lower bounds on the convergence rate of comparison based or selection based algorithms, improving existing results in the continuous setting, and extending them to non-trivial results in the discrete case. This is achieved by considering the VC-dimension of the level sets of the fitness functions; results are then obtained through the use of the shatter function lemma. In the special case of optimization of the sphere function, improved lower bounds are obtained by an argument based on the number of sign patterns.

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References

  1. Arnold, D.V.: Optimal weighted recombination. In: Foundations of Genetic Algorithms 8. Lecture Notes in Computer Science, vol. 3469, pp. 215–237. Springer, Berlin (2005)

    Chapter  Google Scholar 

  2. Auger, A.: Convergence results for (1,λ)-SA-ES using the theory of φ-irreducible Markov chains. Theor. Comput. Sci. 334(1–3), 35–69 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  3. Auger, A., Schoenauer, M., Teytaud, O.: Local and global order 3/2 convergence of a surrogate evolutionary algorithm. In: GECCO’05: Proceedings of the 2005 Conference on Genetic and Evolutionary Computation, pp. 857–864. ACM, New York (2005)

    Chapter  Google Scholar 

  4. Bäck, T., Hoffmeister, F., Schwefel, H.-P.: Extended selection mechanisms in genetic algorithms. In: Belew, R.K., Booker, L.B. (eds.) Proceedings of the Fourth International Conference on Genetic Algorithms, pp. 92–99. Morgan Kaufmann, San Mateo (1991)

    Google Scholar 

  5. Baker, J.E.: Reducing bias and inefficiency in the selection algorithm. In: Proceedings of the Second International Conference on Genetic Algorithms on Genetic Algorithms and Their Application, pp. 14–21. Lawrence Erlbaum Associates, Mahwah (1987)

    Google Scholar 

  6. Beyer, H.-G.: Toward a theory of evolution strategies: On the benefit of sex—the (μ/μ,λ)-theory. Evol. Comput. 3(1), 81–111 (1995)

    Article  MathSciNet  Google Scholar 

  7. Beyer, H.-G., Schwefel, H.-P.: Evolution strategies: a comprehensive introduction. Nat. Comput. 1(1), 3–52 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  8. Devroye, L., Györfi, L., Lugosi, G.: A probabilistic Theory of Pattern Recognition. Springer, Berlin (1997)

    Google Scholar 

  9. Droste, S.: Not all linear functions are equally difficult for the compact genetic algorithm. In: Proc. of the Genetic and Evolutionary Computation COnference (GECCO 2005), pp. 679–686 (2005)

  10. Droste, S., Jansen, T., Wegener, I.: A rigorous complexity analysis of the (1+1) evolutionary algorithm for separable functions with boolean inputs. Evol. Comput. 6(2), 185–196 (1998)

    Article  Google Scholar 

  11. Gelly, S., Ruette, S., Teytaud, O.: Comparison-based algorithms are robust and randomized algorithms are anytime. Evol. Comput. J. 15(4), 411–434 (2007). Special issue on bridging Theory and Practice

    Article  Google Scholar 

  12. Halperin, D.: Arrangements. In: Goodman, J.E., O’Rourke, J. (eds.) Handbook of Discrete and Computational Geometry, pp. 529–562. CRC Press, Boca Raton (2004). Chap. 24

    Google Scholar 

  13. Hansen, N., Ostermeier, A.: Completely derandomized self-adaptation in evolution strategies. Evol. Comput. 9(2), 159–195 (2001)

    Article  Google Scholar 

  14. Hooke, R., Jeeves, T.A.: “Direct search” solution of numerical and statistical problems. J. ACM 8(2), 212–229 (1961)

    Article  MATH  Google Scholar 

  15. Jägersküpper, J.: Analysis of a simple evolutionary algorithm for minimization in Euclidean spaces. In: 30th International Colloquium on Automata, Languages, and Programming (ICALP 2003). LNCS, vol. 2719, pp. 1068–1079. Springer, Berlin (2003)

    Chapter  Google Scholar 

  16. Jägersküpper, J.: Analysis of a simple evolutionary algorithm for minimization in Euclidean spaces. Theor. Comput. Sci. 379(3), 329–347 (2007). Special issue on ICALP 2003

    Article  MATH  Google Scholar 

  17. Jägersküpper, J., Witt, C.: Rigorous runtime analysis of a (μ+1)-ES for the sphere function. In: GECCO, pp. 849–856 (2005)

  18. Matoušek, J.: Lectures on Discrete Geometry. Graduate Texts in Mathematics, vol. 212. Springer, Berlin (2002)

    MATH  Google Scholar 

  19. Moraglio, A., Poli, R.: Topological crossover for the permutation representation. In: GECCO’05: Proceedings of the 2005 Workshops on Genetic and Evolutionary Computation, pp. 332–338. ACM, New York (2005)

    Chapter  Google Scholar 

  20. Neumann, F.: Expected runtimes of evolutionary algorithms for the Eulerian cycle problem. Computers & OR 35(9), 2750–2759 (2008)

    Article  MATH  Google Scholar 

  21. Rechenberg, I.: Evolutionstrategie: Optimierung Technischer Systeme nach Prinzipien des Biologischen Evolution. Fromman-Holzboog Verlag, Stuttgart (1973)

    Google Scholar 

  22. Rónyai, L., Babai, L., Ganapathy, M.K.: On the number of zero-patterns of a sequence of polynomials. J. Am. Math. Soc. 14(3), 717–735 (2001)

    Article  MATH  Google Scholar 

  23. Rudolph, G.: Convergence rates of evolutionary algorithms for a class of convex objective functions. Control Cybern. 26(3), 375–390 (1997)

    MATH  MathSciNet  Google Scholar 

  24. Sauer, N.: On the density of families of sets. J. Comb. Theory, Ser. A 13(1), 145–147 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  25. Sudholt, D., Witt, C.: Runtime analysis of binary PSO. In: GECCO’08: Proceedings of the 10th Annual Conference on Genetic and Evolutionary Computation, pp. 135–142. ACM, New York (2008)

    Chapter  Google Scholar 

  26. Teytaud, O., Gelly, S.: General lower bounds for evolutionary algorithms. In: Proceedings of PPSN, pp. 21–31 (2006)

  27. Teytaud, O., Fournier, H.: Lower bounds for evolution strategies using vc-dimension. In: PPSN, pp. 102–111 (2008)

  28. Vapnik, V.N., Chervonenkis, A.Ya.: On the uniform convergence of relative frequencies of events to their probabilities. Theory Probab. Appl. XVI(2), 264–280 (1971)

    Article  MathSciNet  Google Scholar 

  29. Whitley, D.: The GENITOR algorithm and selection pressure: Why rank-based allocation of reproductive trials is best. In: Schaffer, J.D. (ed.) Proceedings of the Third International Conference on Genetic Algorithms, pp. 116–121. Morgan Kaufman, San Mateo (1989)

    Google Scholar 

  30. Witt, C.: Theory of randomised search heuristics in combinatorial optimisation: an algorithmic point of view. In: GECCO’09: Proceedings of the 11th Annual Conference Companion on Genetic and Evolutionary Computation Conference, pp. 3551–3592. ACM, New York (2009)

    Chapter  Google Scholar 

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Correspondence to Hervé Fournier.

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A preliminary version of this paper appeared in PPSN 2008 [27].

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Fournier, H., Teytaud, O. Lower Bounds for Comparison Based Evolution Strategies Using VC-dimension and Sign Patterns. Algorithmica 59, 387–408 (2011). https://doi.org/10.1007/s00453-010-9391-3

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