How to Assess Step-Size Adaptation Mechanisms in Randomised Search

Step-size adaptation for randomised search algorithms like evolution strategies is a crucial feature for their performance. The adaptation must, depending on the situation, sustain a large diversity or entertain fast convergence to the desired optimum. The assessment of step-size adaptation mechanisms is therefore non-trivial and often done in too restricted scenarios, possibly only on the sphere function. This paper introduces a (minimal) methodology combined with a practical procedure to conduct a more thorough assessment of the overall population diversity of a randomised search algorithm in different scenarios. We illustrate the methodology on evolution strategies with σ-self-adaptation, cumulative step-size adaptation and two-point adaptation. For the latter, we introduce a variant that abstains from additional samples by constructing two particular individuals within the given population to decide on the step-size change. We find that results on the sphere function alone can be rather misleading to assess mechanisms to control overall population diversity. The most striking flaws we observe for self-adaptation: on the linear function, the step-size increments are rather small, and on a moderately conditioned ellipsoid function, the adapted step-size is 20 times smaller than optimal.

[1]  Anne Auger,et al.  On Proving Linear Convergence of Comparison-based Step-size Adaptive Randomized Search on Scaling-Invariant Functions via Stability of Markov Chains , 2013, ArXiv.

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

[3]  Anne Auger,et al.  Reconsidering the progress rate theory for evolution strategies in finite dimensions , 2006, GECCO '06.

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

[5]  Robert Schaefer Parallel Problem Solving from Nature - PPSN XI, 11th International Conference, Kraków, Poland, September 11-15, 2010. Proceedings, Part II , 2010, PPSN.

[6]  Kalyanmoy Deb,et al.  On self-adaptive features in real-parameter evolutionary algorithms , 2001, IEEE Trans. Evol. Comput..

[7]  Ralf Salomon,et al.  Evolutionary algorithms and gradient search: similarities and differences , 1998, IEEE Trans. Evol. Comput..

[8]  Nikolaus Hansen,et al.  Evaluating the CMA Evolution Strategy on Multimodal Test Functions , 2004, PPSN.

[9]  Anne Auger,et al.  Impacts of invariance in search: When CMA-ES and PSO face ill-conditioned and non-separable problems , 2011, Appl. Soft Comput..

[10]  Nikolaus Hansen,et al.  The CMA Evolution Strategy: A Tutorial , 2016, ArXiv.

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

[12]  Nikolaus Hansen,et al.  CMA-ES with Two-Point Step-Size Adaptation , 2008, ArXiv.

[13]  Tom Schaul,et al.  Exponential natural evolution strategies , 2010, GECCO '10.

[14]  Hans-Paul Schwefel,et al.  Evolution and optimum seeking , 1995, Sixth-generation computer technology series.

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