No free lunch in search and optimization
Some computational problems are solved by searching for good solutions in a space of candidate solutions. A description of how to repeatedly select candidate solutions for evaluation is called a search algorithm. On a particular problem, different search algorithms may obtain different results, but over all problems, they are indistinguishable. It follows that if an algorithm achieves superior results on some problems, it must pay with inferiority on other problems. In this sense there is no free lunch in search. Alternatively, following Schaffer, search performance is conserved. Usually search is interpreted as optimization, and this leads to the observation that there is no free lunch in optimization. "The 'no free lunch' theorem of Wolpert and Macready," as stated in plain language by Wolpert and Macready themselves, is that "any two algorithms are equivalent when their performance is averaged across all possible problems." The "no free lunch" results indicate that matching algorithms to problems gives higher average performance than does applying a fixed algorithm to all. Igel and Toussaint and English have established a general condition under which there is no free lunch. While it is physically possible, it does not hold precisely. Droste, Jansen, and Wegener have proved a theorem they interpret as indicating that there is "(almost) no free lunch" in practice. To make matters more concrete, consider an optimization practitioner confronted with a problem. Given some knowledge of how the problem arose, the practitioner may be able to exploit the knowledge in selection of an algorithm that will perform well in solving the problem. If the practitioner does not understand how to exploit the knowledge, or simply has no knowledge, then he or she faces the question of whether some algorithm generally outperforms others on real-world problems. The authors of the "(almost) no free lunch" theorem say that the answer is essentially no, but admit some reservations as to whether the theorem addresses practice. A "problem" is, more formally, an objective function that associates candidate solutions with goodness values. A search algorithm takes an objective function as input and evaluates candidate solutions one-by-one. The output of the algorithm is the sequence of observed goodness values. Wolpert and Macready stipulate that an algorithm never reevaluates a candidate solution, and that algorithm performance is measured on outputs. For simplicity, we disallow randomness in algorithms. Under these conditions, when a search algorithm is run on every possible input, it generates each possible output exactly once. Because performance is measured on the outputs, the algorithms are indistinguishable in how often they achieve particular levels of performance. Some measures of performance indicate how well search algorithms do at optimization of the objective function. Indeed, there seems to be no interesting application of search algorithms in the class under consideration but to optimization problems. A common performance measure is the least index of the least value in the output sequence. This is the number of evaluations required to minimize the objective function. For some algorithms, the time required to find the minimum is proportional to the number of evaluations. The original no free lunch (NFL) theorems assume that all objective functions are equally likely to be input to search algorithms. It has since been established that there is NFL if and only if, loosely speaking, "shuffling" objective functions has no impact on their probabilities. Although this condition for NFL is physically possible, it has been argued that it certainly does not hold precisely. The obvious interpretation of "not NFL" is "free lunch," but this is misleading. NFL is a matter of degree, not an all-or-nothing proposition. If the condition for NFL holds approximately, then all algorithms yield approximately the same results over all objective functions. "Not NFL" implies only that algorithms are inequivalent overall by "some" measure of performance. For a performance measure of interest, algorithms may remain equivalent, or nearly so. Almost all elements of the set of all possible functions (in the set-theoretic sense of "function") are Kolmogorov random, and hence the NFL theorems apply to a set of functions almost all of which cannot be expressed more compactly than as a lookup table that contains a distinct (and random) entry for each point in the search space. Functions that can be expressed more compactly (for example, by a mathematical expression of reasonable size) are by definition not Kolmogorov random. Further, within the set of all possible objective functions, levels of goodness are equally represented among candidate solutions, hence good solutions are scattered throughout the space of candidates. Accordingly, a search algorithm will rarely evaluate more than a small fraction of the candidates before locating a very good solution. Almost all objective functions are of such high Kolmogorov complexity that they cannot be stored in a particular computer. More precisely, if we model a given physical computer as a register machine with a given size memory on the order of the memories of modern computers, then most objective functions cannot be stored in their memories. There is more information in the typical objective function or algorithm than Seth Lloyd estimates the observable universe is capable of registering. For instance, if each candidate solution is encoded as a sequence of 300 0's and 1's, and the goodness values are 0 and 1, then most objective functions have Kolmogorov complexity of at least 2300 bits, and this is greater than Lloyd's bound of 1090 ≈ 2299 bits. It follows that the original "no free lunch" theorem does not apply to what can be stored in a physical computer; instead the so-called "tightened" no free lunch theorems need to be applied. It has also been shown that NFL results apply to incomputable functions. formula_1 is the set of all objective functions "f":"X"→"Y", where formula_2 is a finite solution space and formula_3 is a finite poset. The set of all permutations of "X" is "J". A random variable "F" is distributed on formula_1. For all "j" in "J", "F" o "j" is a random variable distributed on formula_1, with P("F" o "j" = "f") = P("F" = "f" o "j"−1) for all "f" in formula_1.
Let "a"("f") denote the output of search algorithm "a" on input "f". If "a"("F") and "b"("F") are identically distributed for all search algorithms "a" and "b", then "F" has an "NFL distribution". This condition holds if and only if "F" and "F" o "j" are identically distributed for all "j" in "J". In other words, there is no free lunch for search algorithms if and only if the distribution of objective functions is invariant under permutation of the solution space. Set-theoretic NFL theorems have recently been generalized to arbitrary cardinality formula_2 and formula_3. Wolpert and Macready give two principal NFL theorems, the first regarding objective functions that do not change while search is in progress, and the second regarding objective functions that may change. "Theorem 1": For any pair of algorithms "a"1 and "a"2
formula_9
where formula_10 denotes the ordered set of size formula_11 of the cost values formula_12 associated to input values formula_13, formula_14 is the function being optimized and formula_15 is the conditional probability of obtaining a given sequence of cost values from algorithm formula_16 run formula_11 times on function formula_18. In essence, this says that when all functions "f" are equally likely, the probability of observing an arbitrary sequence of "m" values in the course of search does not depend upon the search algorithm. Theorem 1 establishes a "more subtle" NFL result for time-varying objective functions. A conventional, but not entirely accurate, interpretation of the NFL results is that "a general-purpose universal optimization strategy is theoretically impossible, and the only way one strategy can outperform another is if it is specialized to the specific problem under consideration". Several comments are in order:


In practice, only highly compressible (far from random) objective functions fit in the storage of computers, and it is not the case that each algorithm performs well on almost all compressible functions. There is generally a performance advantage in incorporating prior knowledge of the problem into the algorithm. While the NFL results constitute, in a strict sense, full employment theorems for optimization professionals, it is important to bear the larger context in mind. For one thing, humans often have little prior knowledge to work with. For another, incorporating prior knowledge does not give much of a performance gain on some problems. Finally, human time is very expensive relative to computer time. There are many cases in which a company would choose to optimize a function slowly with an unmodified computer program rather than rapidly with a human-modified program. The NFL results do not indicate that it is futile to take "pot shots" at problems with unspecialized algorithms. No one has determined the fraction of practical problems for which an algorithm yields good results rapidly. And there is a practical free lunch, not at all in conflict with theory. Running an implementation of an algorithm on a computer costs very little relative to the cost of human time and the benefit of a good solution. If an algorithm succeeds in finding a satisfactory solution in an acceptable amount of time, a small investment has yielded a big payoff. If the algorithm fails, then little is lost. Wolpert and Macready have proved that there are free lunches in coevolutionary optimization. Their analysis "covers 'self-play' problems. In these problems, the set of players work together to produce a champion, who then engages one or more antagonists in a subsequent multiplayer game." That is, the objective is to obtain a good player, but without an objective function. The goodness of each player (candidate solution) is assessed by observing how well it plays against others. An algorithm attempts to use players and their quality of play to obtain better players. The player deemed best of all by the algorithm is the champion. Wolpert and Macready have demonstrated that some coevolutionary algorithms are generally superior to other algorithms in quality of champions obtained. Generating a champion through self-play is of interest in evolutionary computation and game theory. The results are inapplicable to coevolution of biological species, which does not yield champions.
Formal synopsis
There is no free lunch in search if and only if the distribution on objective functions is invariant under permutation of the space of candidate solutions.