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No Free Lunch Theorems: Limitations and Perspectives of Metaheuristics

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Theory and Principled Methods for the Design of Metaheuristics

Part of the book series: Natural Computing Series ((NCS))

Abstract

The No Free Lunch (NFL) theorems for search and optimization are reviewed and their implications for the design of metaheuristics are discussed. The theorems state that any two search or optimization algorithms are equivalent when their performance is averaged across all possible problems and even over subsets of problems fulfilling certain constraints. The NFL results show that if there is no assumption regarding the relation between visited and unseen search points, efficient search and optimization is impossible. There is no well-performing universal metaheuristic, but the heuristics must be tailored to the problem class at hand using prior knowledge. In practice, it is not likely that the preconditions of the NFL theorems are fulfilled for a problem class and thus differences between algorithms exist. Therefore, tailored algorithms can exploit structure underlying the optimization problem. Given full knowledge about the problem class, it is, in theory, possible to construct an optimal algorithm.

Thus not only our reason fails us in the discovery of the ultimate connexion of causes and effects, but even after experience has informed us of their constant conjunction, it is impossible for us to satisfy ourselves by our reason, why we should extend that experience beyond those particular instances, which have fallen under our observation. We suppose, but are never able to prove, that there must be a resemblance betwixt those objects, of which we have had experience, and those which lie beyond the reach of our discovery. (David Hume, 1739 [10, 14])

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Notes

  1. 1.

    In general, this cannot be done in the theorems because \(\sum _{f\in F}\delta (k,\sum _{a\prime\in A}p_{a}(a\prime)c(Y (f,m,a\prime)))\neq \sum _{f\in F}\sum _{a\prime\in A}p_{a}(a\prime)\delta (k,c(Y (f,m,a\prime)))\).

  2. 2.

    It is also possible to map the optimization to an infinite horizon problem with appropriate absorbing states.

  3. 3.

    In a S-MDP no state is ever revisited. Hence, for any policy, the transition probability graph is acyclic and thus value iteration finds the optimal policy after at most m steps, where each step needs \(O{(\vert \mathcal{A}\vert \vert \mathcal{S}\vert )}^{2}\) computations (see [3, Sect. 2.2.2] or [2]).

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Acknowledgements

Parts of this text are based on former joint work with Marc Toussaint. The author acknowledges support from the German Federal Ministry of Education and Research within the National Network Computational Neuroscience – Bernstein Fokus: “Learning behavioral models: From human experiment to technical assistance”, grant FKZ 01GQ0951.

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  1. Department of Computer Science, University of Copenhagen, Universitetsparken 5, 2100, Copenhagen, Denmark

    Christian Igel

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Correspondence to Christian Igel .

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  1. VisualDNA, London, United Kingdom

    Yossi Borenstein

  2. University of Birmingham School of Computer Science, Birmingham, United Kingdom

    Alberto Moraglio

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Igel, C. (2014). No Free Lunch Theorems: Limitations and Perspectives of Metaheuristics. In: Borenstein, Y., Moraglio, A. (eds) Theory and Principled Methods for the Design of Metaheuristics. Natural Computing Series. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33206-7_1

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  • DOI: https://doi.org/10.1007/978-3-642-33206-7_1

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