In mathematical folklore, the "no free lunch" (NFL) theorem (sometimes pluralized) of David Wolpert and William Macready appears in the 1997 "No Free Lunch Theorems for Optimization". Wolpert had previously derived no free lunch theorems for machine learning (statistical inference). The name alludes to the saying "there ain't no such thing as a free lunch", that is, there are no easy shortcuts to success. In 2005, Wolpert and Macready themselves indicated that the first theorem in their paper "state[s] that any two optimization algorithms are equivalent when their performance is averaged across all possible problems". The "no free lunch" (NFL) theorem is an easily stated and easily understood consequence of theorems Wolpert and Macready actually prove. It is weaker than the proven theorems, and thus does not encapsulate them. Various investigators have extended the work of Wolpert and Macready substantively. In terms of how the NFL theorem is used in the context of the research area, the no free lunch in search and optimization is a field that is dedicated for purposes of mathematically analyzing data for statistical identity, particularly search and optimization. While some scholars argue that NFL conveys important insight, others argue that NFL is of little relevance to machine learning research. Example Posit a toy universe that exists for exactly two days and on each day contains exactly one object, a square or a triangle. The universe has exactly four possible histories: (square, triangle): the universe contains a square on day 1, and a triangle on day 2 (square, square) (triangle, triangle) (triangle, square) Any prediction strategy that succeeds for history #2, by predicting a square on day 2 if there is a square on day 1, will fail on history #1, and vice versa. If all histories are equally likely, then any prediction strategy will score the same, with the same accuracy rate of 0.5. Origin Wolpert and Macready give two NFL theorems that are closely related to the folkloric theorem. In their paper, they state: The first theorem hypothesizes objective functions that do not change while optimization is in progress, and the second hypothesizes objective functions that may change. where denotes the ordered set of size of the cost values associated to input values , is the function being optimized and is the conditional probability of obtaining a given sequence of cost values from algorithm run times on function . The theorem can be equivalently