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Kernel representations for evolving continuous functions

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Abstract

To parameterize continuous functions for evolutionary learning, we use kernel expansions in nested sequences of function spaces of growing complexity. This approach is particularly powerful when dealing with non-convex constraints and discontinuous objective functions. Kernel methods offer a number of beneficial properties for parameterizing continuous functions, such as smoothness and locality, which make them attractive as a basis for mutation operators. Beyond such practical considerations, kernel methods make heavy use of inner products in function space and offer a well established regularization framework. We show how evolutionary computation can profit from these properties. Searching function spaces of iteratively increasing complexity allows the solution to evolve from a simple first guess to a complex and highly refined function. At transition points where the evolution strategy is confronted with the next level of functional complexity, the kernel framework can be used to project the search distribution into the extended search space. The feasibility of the method is demonstrated on challenging trajectory planning problems where redundant robots have to avoid obstacles.

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Notes

  1. Care has to be taken that the function space is restricted to L 2 functions, that is, functions of finite norm.

  2. For any (not necessarily square) matrix M we write M −1 for the Moore–Penrose pseudo-inverse.

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Acknowledgments

This work was funded through the 7th Framework Programme of the EU under grant number 231576 (STIFF project) and SNF grant 200020-125038/1.

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Correspondence to Tobias Glasmachers.

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Glasmachers, T., Koutník, J. & Schmidhuber, J. Kernel representations for evolving continuous functions. Evol. Intel. 5, 171–187 (2012). https://doi.org/10.1007/s12065-012-0070-y

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  • DOI: https://doi.org/10.1007/s12065-012-0070-y

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