Best approximations of fitness functions of binary strings

Authors:
Hang Zhang;Jonathan E. Rowe
Affiliations:
School of Computer Science, University of Birmingham, Birmingham B15 2TT, UK (E-mails: H.Zhang@cs.bham.ac.uk);School of Computer Science, University of Birmingham, Birmingham B15 2TT, UK (E-mails: J.E.Rowe@cs.bham.ac.uk)
Venue:
Natural Computing: an international journal
Year:
2004

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Abstract

Fitness functions of binary strings (pseudo-boolean functions) canbe represented as polynomials over a set of boolean variables. Weshow that any such function has a unique best approximation in thelinear span of any subset of polynomials. For example, there is aunique best linear approximation and a unique best quadraticapproximation. The error of an approximation here isroot-mean-squared error. If all the details of the function to beapproximated are known, then the approximation can be calculateddirectly. Of more practical importance, we give a method for usingsampling to estimate the coefficients of the approximation, anddescribe its limitations.