Erik van Nimwegen, James P. Crutchfield, and Melanie Mitchel. Statistical dynamics of the Royal Road genetic algorithm. Theoretical Computer Science, To Appear.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....hx(t 1)i 2 1 is approximated by using the sampling process to obtain x(t 1) 2 m . The deviations thereby possible become larger with decreasing m and distort the nite population dynamics as compared to the in nite population case. This results in uctuations and epoch formation as shown in [10, 11, 12]. In the following, we will consider the in nite population limit, because it re ects the exact ow of probabilities for a particular tness landscape. In a second step, the uctuations and epoch formation introduced by the niteness of a real population can be studied on the basis of that ....

E. van Nimwegen, J. P. Crutch eld and M. Mitchell. Statistical Dynamics of the Royal-Road genetic algorithms. Theoretical Computer Science, special issue on Evolutionary Computation, A. Eiben, G. Rudolph, editors, in press, 1998. 23

....the evolutionary dynamics of populations on neutral landscapes follow a characteristic pattern. Long periods of stasis occur in which the population explores the current neutral layer alternating with rapid fitness increases when an individual discovers a transition point to a fitter neutral layer [11]. This pattern of evolution reflects the phenomenon of punctuated equilibria observed in several biological populations [3] The NK landscape model was initially developed to model the fitness landscapes resulting from systems with various levels of interaction between the components, for example ....

van Nimwegen, E., Crutchfield, J. P., & Mitchell, M., Statistical dynamics of the Royal Road genetic algorithm. In A. Eiben and G. Rudolph, (Eds.), Theoretical Computer Science, Special issue on Evolutionary Computation, 1998.

....algorithms can be challenging because one must physically separate DNA strands according to their fitness. 3 The Royal Road Problem The Royal Road family of problems are of particular interest because it is one of very few families of problems for which theoretical predictions are available [38]. A fixed length target is specified consisting of N blocks, each block consisting of K bits. Each block of a candidate bitstring makes no contribution to fitness unless it is a perfect match to the corresponding block on the target. Conventionally, the fitness of a candidate is taken to be ....

....target. This family of problems got its name from the fact that it was intended to be especially suitable for genetic algorithms using crossover [25] Distressingly, computer trials for Royal road problems exhibit a wide variety of unpleasant convergence behaviors (see Figure 1, reproduced from [38]) 0 1 2 3 4 5 6 7 8 9 10 2500 5000 7500 10000 12500 15000 10 0 1000 2000 3000 4000 5000 10 100 200 300 400 500 600 700 0 2 4 6 8 10 12 14 16 18 20 0 500 1000 1500 2000 2500 0 1 2 3 4 0 25 50 75 100 125 150 (x 1000) f f f ttt (a) b) c) d) e) f) g) h) ....

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Erik van Nimwegen, James P. Crutchfield, and Melanie Mitchell. Statistical dynamics of the Royal Road genetic algorithm. Theoretical Computer Science, to Appear.

....on Royal Road functions which turned out to induce a meta stable evolution for which the magnitude and frequency of improvements in tness were predicted by a theoretical model. Since the Royal Road functions have been used for the meta stable nature of the evolutionary dynamics they involve ([NCM97]) the unexpected results obtained previously make sense. The DGA , by enforcing presence of duals, eases a drift between dual spaces which favors, in our context, the discovery of better tted individuals just as neutral drift does in natural evolution. V. Conclusion This paper examinated the ....

....disruptive in only 50 of cases while acting like an Implicit Mirroring Rate the rest of time. In natural evolutionary systems, preserving multiple solutions featuring the same tness eases the discovery of better individuals. This is particularly true when evolution is meta stable ([NCM97]) since, once stuck in a step, the GA converges toward homogeneous populations from whitch it will not be able to escape later on. In order to reach a new tness level, it is therefore usefull to encourage the preservation of multiple equally tted individuals featuring di erent genotypes. Their ....

E.van Nimwegen, J.P. Crutcheld, and M. Mitchell. Statistical dynamics of the royal road genetic algorithm. Working Paper 97-04-035, Santa Fe Institute, 1997.

....and a temperature for the simple genetic algorithm [5] Suzuki [26, 27] analyzed Markov chain models of this algorithm using the analogy with simulated annealing. Rudolph [21] and Mahfoud and Goldberg [18] compared massively parallel simulated annealing and genetic algorithms. Van Nimwegen et al. [19] studied meta stability in the royal road genetic algorithm deeply with low mutation probabilities. The connection has been investigated by Cerf in a series of papers [3, 4] for a genetic algorithm with Boltzmann roulette selection. Trying to associate energy functionals or random fields with ....

E. Van Nimwegen, J.P. Crutchfield and M. Mitchell. Statistical dynamics of the royal road genetic algorithm. Theor. Comput. Sci.,22, 9 (1999), 41-102.

....in average time O(n log n) for the ONE MAX problem. Droste et al. 11] carried out a rigorous complexity analysis of (1 1) EAs for linear functions with Boolean inputs. However, all of these results were based on EAs with a population size of 1 and without any crossover operators. Nimwegen et al. [12, 13] developed a theory which predicts the total number of tness function evaluations needed to reach a global Accepted by Arti cial Intelligence journal in 2000. To appear in 2001. y Corresponding author. 1 optimum by epochal dynamics as a function of mutation rate and population size. ....

E. van Nimwegen, J. P. Crutcheld, and M. Mitchell, \ Statistical dynamics of the royal road genetic algorithm". Theoretical Computer Science, vol.229, no.1, pp.41-102, 1999.

....and to introduce the basic principles and results of its abstract framework. While some of this material has appeared elsewhere, this paper brings those scattered results together into a unified theory. 2 The particular example considered has been previously analyzed by van Nimwegen et al. [17,18]. 2 2 Random Heuristic Search This section introduces random heuristic search as an abstract search method. Whereas the emphasis here is on generality, RHS has been instantiated to particular search methods with remarkable success. The interested reader is referred to [25] for a concrete ....

....lattice points (i.e. alternative populations) This phenomenon is made precise by theorem 1 and the characterization, given in section 2.2, of the finite population state space as 1 r X r n . This same phenomenon was later rediscovered for a particular instance of RHS by van Nimwegen et al. [17,18]) According to theorem 2, the expected next generation from population p is known, but what about the variance It decreases like 1=r (see [22] and depends upon the distance of G(p) from a vertex of (see [23,25] 16 Theorem 3 Let E denote the expectation operator. E(k(p) Gamma G(p)k 2 ) ....

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E. van Nimwegen,J. Crutchfield, and M. Mitchell, Statistical Dynamics of the Royal Road Genetic Algorithm, Theoretical Computer Science, this issue.

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Erik van Nimwegen, James P. Crutchfield, and Melanie Mitchel. Statistical dynamics of the Royal Road genetic algorithm. Theoretical Computer Science, To Appear.

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E. van Nimwegen, J. P. Crutchfield, and M. Mitchell. Statistical dynamics of the Royal Road genetic algorithm. Theoretical Computer Science, 229:41--102, 1999.

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