Mutation rates of the (1+1)-EA on pseudo-boolean functions of bounded epistasis

When the epistasis of the fitness function is bounded by a constant, we show that the expected fitness of an offspring of the (1+1)-EA can be efficiently computed for any point. Moreover, we show that, for any point, it is always possible to efficiently retrieve the "best" mutation rate at that point in the sense that the expected fitness of the resulting offspring is maximized. On linear functions, it has been shown that a mutation rate of 1/n is provably optimal. On functions where epistasis is bounded by a constant k, we show that for sufficiently high fitness, the commonly used mutation rate of 1/n is also best, at least in terms of maximizing the expected fitness of the offspring. However, we find for certain ranges of the fitness function, a better mutation rate can be considerably higher, and can be found by solving for the real roots of a degree-k polynomial whose coefficients contain the nonzero Walsh coefficients of the fitness function. Simulation results on maximum k-satisfiability problems and NK-landscapes show that this expectation-maximized mutation rate can cause significant gains early in search.

[1]  Thomas Jansen,et al.  Design and Management of Complex Technical Processes and Systems by means of Computational Intelligence Methods On the Choice of the Mutation Probability for the ( 1 + 1 ) EA , 2006 .

[2]  N. Obrechkoff Sur une généralisation du théorème de Poulain et Hermite pour les zéros réels des polynômes réels , 1964 .

[3]  Thomas Bäck,et al.  The Interaction of Mutation Rate, Selection, and Self-Adaptation Within a Genetic Algorithm , 1992, PPSN.

[4]  K. Dejong,et al.  An analysis of the behavior of a class of genetic adaptive systems , 1975 .

[5]  Rajarshi Das,et al.  A Study of Control Parameters Affecting Online Performance of Genetic Algorithms for Function Optimization , 1989, ICGA.

[6]  Alden H. Wright,et al.  Efficient Linkage Discovery by Limited Probing , 2003, Evolutionary Computation.

[7]  L. Darrell Whitley,et al.  Test Function Generators as Embedded Landscapes , 1998, FOGA.

[8]  Kyomin Jung,et al.  Almost Tight Upper Bound for Finding Fourier Coefficients of Bounded Pseudo- Boolean Functions , 2008, COLT.

[9]  Alden H. Wright,et al.  The computational complexity of N-K fitness functions , 2000, IEEE Trans. Evol. Comput..

[10]  Thomas Jansen,et al.  Optimizing Monotone Functions Can Be Difficult , 2010, PPSN.

[11]  Thomas Bck,et al.  Self-adaptation in genetic algorithms , 1991 .

[12]  S. Kauffman,et al.  Towards a general theory of adaptive walks on rugged landscapes. , 1987, Journal of theoretical biology.

[13]  Robert B. Heckendorn,et al.  Walsh analysis, epistasis, and optimization problem difficulty for evolutionary algorithms , 1999 .

[14]  Andrew M. Sutton,et al.  Computing the moments of k-bounded pseudo-Boolean functions over Hamming spheres of arbitrary radius in polynomial time , 2012, Theor. Comput. Sci..

[15]  Marc Schoenauer,et al.  Rigorous Hitting Times for Binary Mutations , 1999, Evolutionary Computation.

[16]  Reinhard Männer,et al.  Towards an Optimal Mutation Probability for Genetic Algorithms , 1990, PPSN.

[17]  John H. Holland,et al.  Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence , 1992 .

[18]  Schloss Birlinghoven,et al.  How Genetic Algorithms Really Work I.mutation and Hillclimbing , 2022 .

[19]  Hillol Kargupta,et al.  Gene Expression and Fast Construction of Distributed Evolutionary Representation , 2001, Evolutionary Computation.

[20]  Thomas Bäck,et al.  Optimal Mutation Rates in Genetic Search , 1993, ICGA.

[21]  Heinz Mühlenbein,et al.  How Genetic Algorithms Really Work: Mutation and Hillclimbing , 1992, PPSN.

[22]  Ingo Wegener,et al.  A Rigorous Complexity Analysis of the (1 + 1) Evolutionary Algorithm for Separable Functions with Boolean Inputs , 1998, Evolutionary Computation.

[23]  Thomas Bäck,et al.  Intelligent Mutation Rate Control in Canonical Genetic Algorithms , 1996, ISMIS.

[24]  Frank Neumann,et al.  Optimal Fixed and Adaptive Mutation Rates for the LeadingOnes Problem , 2010, PPSN.

[25]  Stephen R. Marsland,et al.  Convergence Properties of (μ + λ) Evolutionary Algorithms , 2011, AAAI.

[26]  John J. Grefenstette,et al.  Optimization of Control Parameters for Genetic Algorithms , 1986, IEEE Transactions on Systems, Man, and Cybernetics.