Abstract
Randomization is an efficient tool for global optimization. We here define a method which keeps :
– the order 0 of evolutionary algorithms (no gradient) ;
– the stochastic aspect of evolutionary algorithms ;
– the efficiency of so-called ”low-dispersion” points ;
and which ensures under mild assumptions global convergence with linear convergence rate. We use i) sampling on a ball instead of Gaussian sampling (in a way inspired by trust regions), ii) an original rule for step-size adaptation ; iii) quasi-monte-carlo sampling (low dispersion points) instead of Monte-Carlo sampling. We prove in this framework linear convergence rates i) for global optimization and not only local optimization ; ii) under very mild assumptions on the regularity of the function (existence of derivatives is not required). Though the main scope of this paper is theoretical, numerical experiments are made to backup the mathematical results.
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References
Auger, A.: Convergence results for (1,λ)-SA-ES using the theory of ϕ-irreducible markov chains. Theoretical Computer Science 334(1-3), 35–69 (2005)
Beyer, H.-G.: The Theory of Evolution Strategies. Springer, Heidelberg (2001)
Cerf, R.: An asymptotic theory of genetic algorithms. In: Alliot, J.-M., Ronald, E., Lutton, E., Schoenauer, M., Snyers, D. (eds.) AE 1995. LNCS, vol. 1063, pp. 37–53. Springer, Heidelberg (1996)
Droste, S., Jansen, T., Wegener, I.: On the analysis of the (1+1) evolutionary algorithm. Theoretical Computer Science 276, 51–81 (2002)
Fang, K., Wang, Y.: Number-Theoretic Methods in Statistics. Chapman and Hall, London (1994)
Garnier, J., Kallel, L., Schoenauer, M.: Rigorous hitting times for binary mutations. Evolutionary Computation 7(2), 167–203 (1999)
Goldfeld, S., Quandt, R., Trotter, H.: Maximization by quadratic hill climbing. Econometrica 34(3), 541 (1966)
Hansen, N., Ostermeier, A.: Completely Derandomized Self-Adaptation in Evolution Strategies. Evolutionary Computation 9(2), 159–195 (2001)
Landrin-Schweitzer, Y., Lutton, E.: Perturbation theory for eas: towards an estimation of convergence speed. In: Schoenauer, M., Deb, K., Rudolph, G., Yao, X., Lutton, E., Merelo, J., Schwefel, H.-P. (eds.) PPSN VI, Springer, Heidelberg (2000)
Meyer, Y.: Wavelets, Vibrations and Scaling. CRM Monograph Series. American Mathematical Society (1997)
Niedereiter, H.: Random Number Generation and Quasi-Monte Carlo Methods. SIAM, Philadelphia (1992)
Rechenberg, I.: Evolutionstrategie: Optimierung Technisher Systeme nach Prinzipien des Biologischen Evolution. Fromman-Hozlboog Verlag, Stuttgart (1973)
Rudolph, G.: Convergence analysis of canonical genetic algorithm. IEEE Transactions on Neural Networks 5(1), 96–101 (1994)
Rudolph, G.: Convergence of non-elitist strategies. In: Michalewicz, Z., Schaffer, J.D., Schwefel, H.-P., Fogel, D.B., Kitano, H. (eds.) Proceedings of the First IEEE International Conference on Evolutionary Computation, pp. 63–66. IEEE Press, Los Alamitos (1994)
Rudolph, G.: How mutation and selection solve long path problems in polynomial expected time. Evolutionary Computation 4, 195–205 (1996)
Rudolph, G.: Convergence rates of evolutionary algorithms for a class of convex objective functions. Control and Cybernetics 26(3), 375–390 (1997)
Schwefel, H.-P.: Numerical Optimization of Computer Models. John Wiley & Sons, New-York (1981), 2nd edn. (1995)
Tricot, C.: Curves and Fractal Dimension. Springer, Heidelberg (1995)
Vehel, J.L., Lutton, E.: Holder functions and deception of genetic algorithms. IEEE transactions on Evolutionary computing 2(2) (1998)
Yakowitz, S., L’Ecuyer, P., Vazquez-Abad, F.: Global stochastic optimization with low-dispersion point sets (2000)
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Auger, A., Jebalia, M., Teytaud, O. (2006). Algorithms (X, sigma, eta): Quasi-random Mutations for Evolution Strategies. In: Talbi, EG., Liardet, P., Collet, P., Lutton, E., Schoenauer, M. (eds) Artificial Evolution. EA 2005. Lecture Notes in Computer Science, vol 3871. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11740698_26
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DOI: https://doi.org/10.1007/11740698_26
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-33589-4
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