The Dynamics of Cumulative Step Size Adaptation on the Ellipsoid Model

The behavior of the -Evolution Strategy (ES) with cumulative step size adaptation (CSA) on the ellipsoid model is investigated using dynamic systems analysis. At first a nonlinear system of difference equations is derived that describes the mean value evolution of the ES. This system is successively simplified to finally allow for deriving closed-form solutions of the steady state behavior in the asymptotic limit case of large search space dimensions. It is shown that the system exhibits linear convergence order. The steady state mutation strength is calculated, and it is shown that compared to standard settings in self-adaptive ESs, the CSA control rule allows for an approximately -fold larger mutation strength. This explains the superior performance of the CSA in non-noisy environments. The results are used to derive a formula for the expected running time. Conclusions regarding the choice of the cumulation parameter c and the damping constant D are drawn.

[1]  Hans-Georg Beyer,et al.  Self-adaptation on the Ridge Function Class: First Results for the Sharp Ridge , 2006, PPSN.

[2]  Nikolaus Hansen,et al.  Verallgemeinerte individuelle Schrittweitenregelung in der Evolutionsstrategie , 1998 .

[3]  Dirk V. Arnold,et al.  On the use of evolution strategies for optimising certain positive definite quadratic forms , 2007, GECCO '07.

[4]  Alpha Microprocessor,et al.  Performance of an , 1998 .

[5]  H. Beyer,et al.  Noisy Local Optimization with Evolution Strategies , 2002 .

[6]  Hans-Georg Beyer,et al.  Performance analysis of evolutionary optimization with cumulative step length adaptation , 2004, IEEE Transactions on Automatic Control.

[7]  Hans-Georg Beyer,et al.  On the analysis of self-adaptive evolution strategies on elliptic model: first results , 2010, GECCO '10.

[8]  Hans-Georg Beyer,et al.  The Dynamical Systems Approach - Progress Measures and Convergence Properties , 2012, Handbook of Natural Computing.

[9]  N. Hansen,et al.  Step-Size Adaptation Based on Non-Local Use Selection Information , 1994 .

[10]  Hans-Georg Beyer,et al.  On the Behaviour of Evolution Strategies Optimising Cigar Functions , 2010, Evolutionary Computation.

[11]  Dipl. Ing. Karl Heinz Kellermayer.  NUMERISCHE OPTIMIERUNG VON COMPUTER-MODELLEN MITTELS DER EVOLUTIONSSTRATEGIE Hans-Paul Schwefel Birkhäuser, Basel and Stuttgart, 1977 370 pages Hardback SF/48 ISBN 3-7643-0876-1 , 1977 .

[12]  Hans-Georg Beyer,et al.  The Dynamics of Self-Adaptive Multirecombinant Evolution Strategies on the General Ellipsoid Model , 2014, IEEE Transactions on Evolutionary Computation.

[13]  Dirk V. Arnold,et al.  Noisy Optimization With Evolution Strategies , 2002, Genetic Algorithms and Evolutionary Computation.

[14]  Hans-Georg Beyer,et al.  The Theory of Evolution Strategies , 2001, Natural Computing Series.

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

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

[17]  H. Beyer,et al.  Performance of the -ES on a Class of PDQFs , 2010 .

[18]  Hans-Georg Beyer,et al.  On the analysis of self-adaptive recombination strategies: first results , 2005, 2005 IEEE Congress on Evolutionary Computation.

[19]  Hans-Georg Beyer,et al.  Toward a Theory of Evolution Strategies: Self-Adaptation , 1995, Evolutionary Computation.

[20]  Nikolaus Hansen,et al.  Step-Size Adaption Based on Non-Local Use of Selection Information , 1994, PPSN.

[21]  Ingo Rechenberg,et al.  Evolutionsstrategie '94 , 1994, Werkstatt Bionik und Evolutionstechnik.

[22]  Jens Jägersküpper,et al.  How the (1+1) ES using isotropic mutations minimizes positive definite quadratic forms , 2006, Theor. Comput. Sci..

[23]  Nikolaus Hansen,et al.  Verallgemeinerte individuelle Schrittweitenregelung in der Evolutionsstrategie - eine Untersuchung zur entstochastisierten, koordinatensystemunabhängigen Adaptation der Mutationsverteilung , 1998 .

[24]  Michael Herdy,et al.  Reproductive Isolation as Strategy Parameter in Hierarichally Organized Evolution Strategies , 1992, PPSN.