Analyzing the impact of mirrored sampling and sequential selection in elitist evolution strategies

This paper presents a refined single parent evolution strategy that is derandomized with mirrored sampling and/or uses sequential selection. The paper analyzes some of the elitist variants of this algorithm. We prove, on spherical functions with finite dimension, linear convergence of different strategies with scale-invariant step-size and provide expressions for the convergence rates as the expectation of some known random variables. In addition, we derive explicit asymptotic formulae for the convergence rate when the dimension of the search space goes to infinity. Convergence rates on the sphere reveal lower bounds for the convergence rate of the respective step-size adaptive strategies. We prove the surprising result that the (1+2)-ES with mirrored sampling converges at the same rate as the (1+1)-ES without and show that the tight lower bound for the (1+λ-ES with mirrored sampling and sequential selection improves by 16% over the (1+1)-ES reaching an asymptotic value of about -0.235.

[1]  Nikolaus Hansen,et al.  Completely Derandomized Self-Adaptation in Evolution Strategies , 2001, Evolutionary Computation.

[2]  Anne Auger,et al.  Benchmarking the (1,4)-CMA-ES with mirrored sampling and sequential selection on the noiseless BBOB-2010 testbed , 2010, GECCO '10.

[3]  Anne Auger,et al.  Mirrored variants of the (1,2)-CMA-ES compared on the noisy BBOB-2010 testbed , 2010, GECCO '10.

[4]  Olivier Teytaud,et al.  On the Ultimate Convergence Rates for Isotropic Algorithms and the Best Choices Among Various Forms of Isotropy , 2006, PPSN.

[5]  J. Hammersley,et al.  Monte Carlo Methods , 1965 .

[6]  Anne Auger,et al.  Investigating the impact of sequential selection in the (1,2)-CMA-ES on the noiseless BBOB-2010 testbed , 2010, GECCO '10.

[7]  Anne Auger,et al.  Convergence results for the (1, lambda)-SA-ES using the theory of phi-irreducible Markov chains , 2005, Theor. Comput. Sci..

[8]  Anne Auger,et al.  Mirrored variants of the (1,4)-CMA-ES compared on the noiseless BBOB-2010 testbed , 2010, GECCO '10.

[9]  Anne Auger,et al.  Mirrored variants of the (1,2)-CMA-ES compared on the noiseless BBOB-2010 testbed , 2010, GECCO '10.

[10]  Ingo Rechenberg,et al.  Evolutionsstrategie : Optimierung technischer Systeme nach Prinzipien der biologischen Evolution , 1973 .

[11]  Nikolaus Hansen,et al.  An Analysis of Mutative -Self-Adaptation on Linear Fitness Functions , 2006, Evolutionary Computation.

[12]  Anne Auger,et al.  Log-Linear Convergence and Optimal Bounds for the (1+1)-ES , 2007, Artificial Evolution.

[13]  Anne Auger,et al.  Mirrored variants of the (1,4)-CMA-ES compared on the noisy BBOB-2010 testbed , 2010, GECCO '10.

[14]  A. Auger.  Convergence results for the ( 1 , )-SA-ES using the theory of-irreducible Markov chains , 2005 .

[15]  Anne Auger,et al.  Mirrored Sampling and Sequential Selection for Evolution Strategies , 2010, PPSN.

[16]  Anne Auger,et al.  Comparing the (1+1)-CMA-ES with a mirrored (1+2)-CMA-ES with sequential selection on the noiseless BBOB-2010 testbed , 2010, GECCO '10.

[17]  Anne Auger,et al.  Investigating the impact of sequential selection in the (1,2)-CMA-ES on the noisy BBOB-2010 testbed , 2010, GECCO '10.

[18]  Anne Auger,et al.  Theory of Evolution Strategies: A New Perspective , 2011, Theory of Randomized Search Heuristics.

[19]  Anne Auger,et al.  Benchmarking the (1,4)-CMA-ES with mirrored sampling and sequential selection on the noisy BBOB-2010 testbed , 2010, GECCO '10.

[20]  Anne Auger,et al.  Real-Parameter Black-Box Optimization Benchmarking 2009: Noiseless Functions Definitions , 2009 .

[21]  Dirk V. Arnold,et al.  Optimal Weighted Recombination , 2005, FOGA.

[22]  Olivier Teytaud,et al.  DCMA: yet another derandomization in covariance-matrix-adaptation , 2007, GECCO '07.

[23]  Olivier Teytaud,et al.  General Lower Bounds for Evolutionary Algorithms , 2006, PPSN.

[24]  Dirk V. Arnold,et al.  Evolutionary Gradient Search Revisited , 2007, IEEE Transactions on Evolutionary Computation.