Population Size vs . Runtime of a Simple Evolutionary Algorithm

Evolutionary algorithms (EAs) find numerous applications, and practical knowledge on EAs is immense. In practice, sophisticated population-based EAs employing selection, mutation and crossover are applied. In contrast, theoretical analysis of EAs often concentrates on very simple algorithms like the (1+1) EA, where the population size equals 1. In this paper, the question is addressed whether the use of a population by itself can be advantageous. A population-based EA that does neither make use of crossover nor any diversity-maintaining operator is investigated on an example function. It is shown that an increase of the population size by a constant factor decreases the expected runtime from exponential to polynomial. Thereby, the so far best known gap is improved from superpolynomial vs. polynomial to exponential vs. polynomial. Moreover, it is proved that the exponential and polynomial runtime bounds occur with a probability exponentially close to one if the population size is a constant resp. a small polynomial. Finally, a second example function, where only a small population leads to a polynomial runtime, and a hierarchy result on the appropriate population size are presented.

[1]  Feller William,et al.  An Introduction To Probability Theory And Its Applications , 1950 .

[2]  David E. Goldberg,et al.  Genetic Algorithms in Search Optimization and Machine Learning , 1988 .

[3]  Melanie Mitchell,et al.  The royal road for genetic algorithms: Fitness landscapes and GA performance , 1991 .

[4]  John H. Holland,et al.  When will a Genetic Algorithm Outperform Hill Climbing , 1993, NIPS.

[5]  Boris G. Pittel,et al.  Note on the Heights of Random Recursive Trees and Random m-ary Search Trees , 1994, Random Struct. Algorithms.

[6]  Günter Rudolph,et al.  Theory of Evolutionary Algorithms: A Bird's Eye View , 1999, Theor. Comput. Sci..

[7]  K. Mehlhorn,et al.  On the Expected Depth of Random Circuits , 1999, Combinatorics, Probability and Computing.

[8]  M. Vose.  The Simple Genetic Algorithm , 1999 .

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

[10]  Russ Bubley,et al.  Randomized algorithms , 1995, CSUR.

[11]  Thomas Bäck,et al.  Theory of Genetic Algorithms , 2001, Current Trends in Theoretical Computer Science.

[12]  Thomas Jansen,et al.  A New Framework for the Valuation of Algorithms for Black-Box Optimization , 2002, FOGA.

[13]  Xin Yao,et al.  From an individual to a population: an analysis of the first hitting time of population-based evolutionary algorithms , 2002, IEEE Trans. Evol. Comput..

[14]  Thomas Jansen,et al.  On the analysis of the (1+1) evolutionary algorithm , 2002, Theor. Comput. Sci..

[15]  Thomas Jansen,et al.  An Analysis Of The Role Of Offspring Population Size In EAs , 2002, GECCO.

[16]  Hans-Paul Schwefel,et al.  How to analyse evolutionary algorithms , 2002, Theor. Comput. Sci..

[17]  C. Witt.  Population size vs. runtime of a simple EA , 2003, The 2003 Congress on Evolutionary Computation, 2003. CEC '03..

[18]  Martin Dietzfelbinger,et al.  The analysis of a recombinative hill-climber on H-IFF , 2003, IEEE Trans. Evol. Comput..

[19]  I. Wegener,et al.  On the Optimization of Monotone Polynomials by the (1+1) EA and Randomized Local Search , 2003, GECCO.

[20]  Thomas Jansen,et al.  Exploring the Explorative Advantage of the Cooperative Coevolutionary (1+1) EA , 2003, GECCO.

[21]  Jens Jägersküpper,et al.  Analysis of a Simple Evolutionary Algorithm for Minimization in Euclidean Spaces , 2003, ICALP.

[22]  Xin Yao,et al.  Towards an analytic framework for analysing the computation time of evolutionary algorithms , 2003, Artif. Intell..

[23]  Ingo Wegener,et al.  Real royal road functions for constant population size , 2003, Theor. Comput. Sci..

[24]  Thomas Jansen,et al.  UNIVERSITY OF DORTMUND REIHE COMPUTATIONAL INTELLIGENCE COLLABORATIVE RESEARCH CENTER 531 Design and Management of Complex Technical Processes and Systems by means of Computational Intelligence Methods Upper and Lower Bounds for Randomized Search Heuristics in Black-Box Optimization , 2004 .

[25]  Thomas Jansen.  On the utility of populations , 2004 .

[26]  Kenneth A. De Jong,et al.  Design and Management of Complex Technical Processes and Systems by Means of Computational Intelligence Methods on the Choice of the Offspring Population Size in Evolutionary Algorithms on the Choice of the Offspring Population Size in Evolutionary Algorithms , 2004 .

[27]  Ingo Wegener,et al.  Real royal road functions--where crossover provably is essential , 2001, Discret. Appl. Math..

[28]  Ingo Wegener,et al.  On the Optimization of Monotone Polynomials by Simple Randomized Search Heuristics , 2005, Combinatorics, Probability and Computing.