Evolvability (computer science)

The term evolvability is used for a recent framework of computational learning introduced by Leslie Valiant in his paper of the same name and described below. The aim of this theory is to model biological evolution and categorize which types of mechanisms are evolvable. Evolution is an extension of PAC learning and learning from statistical queries.

General framework

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Let   and   be collections of functions on   variables. Given an ideal function  , the goal is to find by local search a representation   that closely approximates  . This closeness is measured by the performance   of   with respect to  .

As is the case in the biological world, there is a difference between genotype and phenotype. In general, there can be multiple representations (genotypes) that correspond to the same function (phenotype). That is, for some  , with  , still   for all  . However, this need not be the case. The goal then, is to find a representation that closely matches the phenotype of the ideal function, and the spirit of the local search is to allow only small changes in the genotype. Let the neighborhood   of a representation   be the set of possible mutations of  .

For simplicity, consider Boolean functions on  , and let   be a probability distribution on  . Define the performance in terms of this. Specifically,

 

Note that   In general, for non-Boolean functions, the performance will not correspond directly to the probability that the functions agree, although it will have some relationship.

Throughout an organism's life, it will only experience a limited number of environments, so its performance cannot be determined exactly. The empirical performance is defined by   where   is a multiset of   independent selections from   according to  . If   is large enough, evidently   will be close to the actual performance  .

Given an ideal function  , initial representation  , sample size  , and tolerance  , the mutator   is a random variable defined as follows. Each   is classified as beneficial, neutral, or deleterious, depending on its empirical performance. Specifically,

  •   is a beneficial mutation if  ;
  •   is a neutral mutation if  ;
  •   is a deleterious mutation if  .

If there are any beneficial mutations, then   is equal to one of these at random. If there are no beneficial mutations, then   is equal to a random neutral mutation. In light of the similarity to biology,   itself is required to be available as a mutation, so there will always be at least one neutral mutation.

The intention of this definition is that at each stage of evolution, all possible mutations of the current genome are tested in the environment. Out of the ones who thrive, or at least survive, one is chosen to be the candidate for the next stage. Given  , we define the sequence   by  . Thus   is a random variable representing what   has evolved to after   generations.

Let   be a class of functions,   be a class of representations, and   a class of distributions on  . We say that   is evolvable by   over   if there exists polynomials  ,  ,  , and   such that for all   and all  , for all ideal functions   and representations  , with probability at least  ,

 

where the sizes of neighborhoods   for   are at most  , the sample size is  , the tolerance is  , and the generation size is  .

  is evolvable over   if it is evolvable by some   over  .

  is evolvable if it is evolvable over all distributions  .

Results

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The class of conjunctions and the class of disjunctions are evolvable over the uniform distribution for short conjunctions and disjunctions, respectively.

The class of parity functions (which evaluate to the parity of the number of true literals in a given subset of literals) are not evolvable, even for the uniform distribution.

Evolvability implies PAC learnability.

References

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  1. Valiant, L. G. (2006), Evolvability, ECCC TR06-120.