Thursday, September 01, 2005

No Free Lunch Theorems

Neural networks and genetic algorithms are general search strategies that are often used as heuristics to solve otherwise intractable problems. They provide the luxury of not having to think about the internals of a specific problem, and in certain cases can perform quite well as practical heuristics.

I dislike them greatly though (even though in my misguided youth I used to play with GAs) because they are extremely general hammers that typically don't have provably performance or running time guarantees, and don't reveal any interesting insights into the structure of a problem.

Such search strategies are also victims of their own generality, and the dagger that pierces them is called a "No Free Lunch" theorem. No Free Lunch theorems are generally of the form
...all algorithms that search for an extremum of a cost function perform exactly the same, when averaged over all possible cost functions. In particular, if algorithm A outperforms algorithm B on some cost functions, then loosely speaking there must exist exactly as many other functions where B outperforms A.
The crucial point here is the "averaged over all cost functions". The NFL theorems are not saying (and this was something I used to be confused about) that search procedures are doomed; rather, they say that blind search procedures (such as those performed by genetic algorithms) are doomed. The original NFL theorem is attributed to Wolpert and Macready; many variants have been proved as well.

Joseph Culberson has a nice perspective on such theorems from an algorithmic point of view, and attempts to frame them in the context of complexity theory.

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