A No-Free-Lunch Theorem for Non-Uniform Distributions of Target Functions

Christian Igel, Marc Toussaint
2004 Journal of Mathematical Modelling and Algorithms  
The sharpened No-Free-Lunch-theorem (NFL-theorem) states that, regardless of the performance measure, the performance of all optimization algorithms averaged uniformly over any finite set F of functions is equal if and only if F is closed under permutation (c.u.p.). In this paper, we first summarize some consequences of this theorem, which have been proven recently: The number of subsets c.u.p. can be neglected compared to the total number of possible subsets. In particular, problem classes
more » ... vant in practice are not likely to be c.u.p. The average number of evaluations needed to find a desirable (e.g., optimal) solution can be calculated independent of the optimization algorithm in certain scenarios. Second, as the main result, the NFL-theorem is extended. Necessary and sufficient conditions for NFL-results to hold are given for arbitrary distributions of target functions. This yields the most general NFL-theorem for optimization presented so far. (2000) : 90C27, 68T20. Mathematics Subject Classifications
doi:10.1023/b:jmma.0000049381.24625.f7 fatcat:zt5wtywjq5ds3ipxqqzoq5ztga