On Multiplicative Noise Models for Stochastic Search

  • Mohamed Jebalia
  • Anne Auger
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5199)

Abstract

In this paper we investigate multiplicative noise models in the context of continuous optimization. We illustrate how some intrinsic properties of the noise model imply the failure of reasonable search algorithms for locating the optimum of the noiseless part of the objective function. Those findings are rigorously investigated on the (1 + 1)-ES for the minimization of the noisy sphere function. Assuming a lower bound on the support of the noise distribution, we prove that the (1 + 1)-ES diverges when the lower bound allows to sample negative fitness with positive probability and converges in the opposite case. We provide a discussion on the practical applications and non applications of those outcomes and explain the differences with previous results obtained in the limit of infinite search-space dimensionality.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Arnold, D.V.: Noisy Optimization with Evolution Strategies. GENA. Kluwer Academic Publishers, Dordrecht (2002)CrossRefMATHGoogle Scholar
  2. 2.
    Arnold, D.V., Beyer, H.-G.: Efficiency and mutation strength adaptation of the (μ/μ i,λ)-ES in a noisy environment. In: Deb, K., Rudolph, G., Lutton, E., Merelo, J.J., Schoenauer, M., Schwefel, H.-P., Yao, X. (eds.) PPSN 2000. LNCS, vol. 1917, pp. 39–48. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  3. 3.
    Arnold, D.V., Beyer, H.-G.: Investigation of the \(\left(\mu,\lambda \right)\)-ES in the presence of noise. In: Proceedings of 2001 IEEE Congress on Evolutionary Computation, pp. 332–339. IEEE Press, Los Alamitos (2001)Google Scholar
  4. 4.
    Arnold, D.V., Beyer, H.-G.: Local performance of the (1+1)-ES in a noisy environment. IEEE Transactions on Evolutionary Computation 6(1), 30–41 (2002)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Arnold, D.V., Beyer, H.-G.: A comparison of evolution strategies with other direct search methods in the presence of noise. Computational Optimization and Applications 24, 135–159 (2003)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Auger, A., Hansen, N.: Reconsidering the progress rate theory for evolution strategies in finite dimensions. In: Press, A. (ed.) Proceedings of the Genetic and Evolutionary Computation Conference (GECCO 2006), pp. 445–452 (2006)Google Scholar
  7. 7.
    Beyer, H.-G.: Evolutionary algorithms in noisy environments: Theoretical issues and guidelines for practice. Computer Methods in Applied Mechanics and Engineering 186(2-4), 239–267 (2000)CrossRefMATHGoogle Scholar
  8. 8.
    Beyer, H.-G.: The Theory of Evolution Strategies. Natural Computing Series. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  9. 9.
    Jebalia, M., Auger, A., Liardet, P.: Log-linear convergence and optimal bounds for the (1 + 1)-ES. In: Monmarché, N., Talbi, E.-G., Collet, P., Schoenauer, M., Lutton, E. (eds.) EA 2007. LNCS, vol. 4926, pp. 207–218. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  10. 10.
    Jin, Y., Branke, J.: Evolutionary Optimization in Uncertain Environments-A Survey. IEEE Transactions on Evolutionary Computation 9(3), 303–317 (2005)CrossRefGoogle Scholar
  11. 11.
    Rechenberg, I.: Evolutionsstrategie. Friedrich Frommann Verlag (Günther Holzboog KG), Stuttgart (1973)Google Scholar
  12. 12.
    Suganthan, P., Hansen, N., Liang, J., Deb, K., Chen, Y.P., Auger, A., Tiwari, S.: Problem definitions and evaluation criteria for the CEC 2005 special session on real-parameter optimization. Technical report, Nanyang Technological University, Singapore and KanGAL Report Number 2005005 (Kanpur Genetic Algorithms Laboratory, IIT Kanpur) (May 2005)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Mohamed Jebalia
    • 1
  • Anne Auger
    • 1
    • 2
  1. 1.TAO Team, INRIA SaclayUniversité Paris Sud, LRIOrsay cedexFrance
  2. 2.Microsoft Research-INRIA Joint CentreOrsay CedexFrance

Personalised recommendations