Continuous lunches are free!

This paper investigates extensions of No Free Lunch (NFL) theorems to countably infinite and uncountable infinite domains. The original NFLdue to Wolpert and Macready states that all search heuristics have the same performance when averaged over the uniform distribution over all possible functions. For infinite domains the extension of the concept of distribution over all possible functions involves measurability issues and stochastic process theory. For countably infinite domains, we prove that the natural extension of NFL theorems does not hold, but that a weaker form of NFL does hold, by stating the existence of non-trivial distributions of fitness leading to equal performance forall search heuristics. Our main result is that for continuous domains, NFL does not hold.

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