Cumulative Step-Size Adaptation on Linear Functions

The CSA-ES is an Evolution Strategy with Cumulative Step size Adaptation, where the step size is adapted measuring the length of a so-called cumulative path. The cumulative path is a combination of the previous steps realized by the algorithm, where the importance of each step decreases with time. This article studies the CSA-ES on composites of strictly increasing functions with affine linear functions through the investigation of its underlying Markov chains. Rigorous results on the change and the variation of the step size are derived with and without cumulation. The step-size diverges geometrically fast in most cases. Furthermore, the influence of the cumulation parameter is studied.

[1]  Nikolaus Hansen,et al.  Adapting arbitrary normal mutation distributions in evolution strategies: the covariance matrix adaptation , 1996, Proceedings of IEEE International Conference on Evolutionary Computation.

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

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

[4]  Dirk V. Arnold,et al.  On the Behaviour of the (1+1)-ES for a Simple Constrained Problem , 2008, PPSN.

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

[6]  Hans-Georg Beyer,et al.  Optimum Tracking with Evolution Strategies , 2006, Evolutionary Computation.

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

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

[9]  Dirk V. Arnold,et al.  Cumulative Step Length Adaptation on Ridge Functions , 2006, PPSN.

[10]  Anne Auger,et al.  Cumulative Step-size Adaptation on Linear Functions: Technical Report , 2012, PPSN 2012.

[11]  Richard L. Tweedie,et al.  Markov Chains and Stochastic Stability , 1993, Communications and Control Engineering Series.

[12]  Hans-Georg Beyer,et al.  Evolution strategies with cumulative step length adaptation on the noisy parabolic ridge , 2008, Natural Computing.

[13]  Dirk V. Arnold,et al.  On the behaviour of the (1,λ)-es for a simple constrained problem , 2011, FOGA '11.

[14]  Hans-Georg Beyer,et al.  Random Dynamics Optimum Tracking with Evolution Strategies , 2002, PPSN.