An Introduction to Genetic Algorithms - Boente

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experiments with grammatical encoding. It is possible that the advantage of grammatical encoding in these experiments was simply due to the GA's finding network topologies that make the problem trivial; the comparison is thus unfair, since this route was not available to the particular direct encoding approach being compared. Kitano's idea of evolving grammars is intriguing, and his informal arguments are plausible reasons to believe that the grammatical encoding method (or extensions of it) will work well on the kinds of problems on which complex neural networks could be needed. However, the particular experiments used to support the arguments are not convincing, since the problems may have been too simple. An extension of Kitano's initial work, in which the evolution of network architecture and the setting of weights are integrated, is reported in Kitano 1994. More ambitious approaches to grammatical encoding have been tried by Gruau (1992) and Belew (1993). Evolving a Learning Rule David Chalmers (1990) took the idea of applying genetic algorithms to neural networks in a different direction: he used GAs to evolve a good learning rule for neural networks. Chalmers limited his initial study to fully connected feedforward networks with input and output layers only, no hidden layers. In general a learning rule is used during the training procedure for modifying network weights in response to the network's performance on the training data. At each training cycle, one training pair is given to the network, which then produces an output. At this point the learning rule is invoked to modify weights. A learning rule for a single−layer, fully connected feedforward network might use the following local information for a given training cycle to modify the weight on the link from input unit i to output unit j: ai: the activation of input unit i oj: the activation of output unit j tj: the training signal (i.e., correct activation, provided by a teacher) on output unit j wij: the current weight on the link from i to j. Chapter 2: Genetic Algorithms in Problem Solving The change to make in weight wij, ”wij, is a function of these values: The chromosomes in the GA population encoded such functions. Chalmers made the assumption that the learning rule should be a linear function of these variables and all their pairwise products. That is, the general form of the learning rule was The km (1dmd10) are constant coefficients, and k0 is a scale parameter that affects how much the weights can change on any one cycle. (k0 is called the "learning rate.") Chalmers's assumption about the form of the learning rule came in part from the fact that a known good learning rule for such networks—the "Widrow−Hoff" or "delta" rule—has the form 58
Chapter 2: Genetic Algorithms in Problem Solving (Rumelhart et al. 1986), where n is a constant representing the learning rate. One goal of Chalmers's work was to see if the GA could evolve a rule that performs as well as the delta rule. The task of the GA was to evolve values for the km's. The chromosome encoding for the set of km's is illustrated in figure 2.25. The scale parameter k 0 is encoded as five bits, with the zeroth bit encoding the sign (1 encoding + and 0 encoding ) and the first through fourth bits encoding an integer n: k 0 = 0 if n = 0; otherwise |k 0 |=2 n–9 . Thus k0 can take on the values 0, ±1/256, ±1/128,…, ±32, ±64. The other coefficients km are encoded by three bits each, with the zeroth bit encoding the sign and the Figure 2.25: Illustration of the method for encoding the kms in Chalmers's system. first and second bits encoding an integer n. For i = 1…10, km = 0 if n = 0; otherwise |km | = 2 n–1 . It is known that single−layer networks can learn only those classes of input−output mappings that are "linearly separable" (Rumelhart et al. 1986). As an "environment" for the evolving learning rules, Chalmers used 30 different linearly separable mappings to be learned via the learning rules. The mappings always had a single output unit and between two and seven input units. The fitness of each chromosome (learning rule) was determined as follows. A subset of 20 mappings was selected from the full set of 30 mappings. For each mapping, 12 training examples were selected. For each of these mappings, a network was created with the appropriate number of input units for the given mapping (each network had one output unit). The network's weights were initialized randomly. The network was run on the training set for some number of epochs (typically 10), using the learning rule specified by the chromosome. The performance of the learning rule on a given mapping was a function of the network's error on the training set, with low error meaning high performance. The overall fitness of the learning rule was a 59
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An Introduction to Genetic Algorith
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Table of Contents Chapter 4: Theore
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Chapter 1: Genetic Algorithms: An O
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Chapter 1: Genetic Algorithms: An O
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1.4 SEARCH SPACES AND FITNESS LANDS
Page 12 and 13: potential energy is a measure of ho
Page 14 and 15: A simple method of implementing fit
Page 16 and 17: from an initial state to a goal. Fo
Page 18 and 19: Robert Axelrod of the University of
Page 20 and 21: Chapter 1: Genetic Algorithms: An O
Page 26 and 27: instances of H at time t, and let
Page 28 and 29: What is the total payoff after 10 g
Page 30 and 31: 6. * c. d. a. b. Chapter 1: Genetic
Page 32 and 33: As a simple example, consider a pro
Page 34 and 35: Chapter 2: Genetic Algorithms in Pr
Page 36 and 37: it to get one fitness case correct:
Page 38 and 39: Evolving Cellular Automata Chapter
Page 42 and 43: up most of the computation time. Ch
Page 46 and 47: the prospect of using GAs to automa
Page 48 and 49: where Ã is the standard deviation
Page 54 and 55: the brain. In a feedforward network
Page 58 and 59: Figure 2.21: An illustration of Mil
Page 64 and 65: function of the average error of th
Page 66 and 67: COMPUTER EXERCISES 1. 2. 3. 4. 5. 6
Page 70 and 71: simulated generations, and such sim
Page 72 and 73: environments, they can fairly quick
Page 74 and 75: Chapter 3: Genetic Algorithms in Sc
Page 76 and 77: Figure 3.6: A schematic illustratio
Page 82 and 83: an initial population in which the
Page 88 and 89: these random effects, Bedau and Pac
Page 90 and 91: 3. * 4. * the female child in the n
Page 92 and 93: calculating the observed average fi
Page 94 and 95: (4.1) The goal is to find n = n* th
Page 96 and 97: competition in the d *···* parti
Page 98 and 99: schemas with the best observed fitn
Page 100 and 101: Steepest−ascent hill climbing (SA
Page 102 and 103: (If the algorithm spends only 1/m o
Page 104 and 105: strings—then in principle crossov
Page 106 and 107: (4.7) Chapter 4: Theoretical Founda
Page 108 and 109: 2. 3. Calculate the fitness f(x) of
Page 110 and 111: Defining ri,j(k) and is somewhat tr
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Results of the Formalization How ca
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Chapter 4: Theoretical Foundations
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Figure 4.5: Predicted and observed
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COMPUTER EXERCISES 1. 2. 3. 4. 5. 6
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prone to rather arbitrary orderings
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Inversion works by choosing two poi
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The one hitch is that, as was seen
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used in the work of Tanese (1989),
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Two individuals are chosen at rando
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strategies community, in which para
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* 10. * 11. * 12. * Chapter 4: Theo
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Chapter 6: Conclusions and Future D
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Chapter 6: Conclusions and Future D
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Appendix A: Selected General Refere
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Evolution Artificielle Foundations
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Kaufmann. Appendix B: Other Resourc
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Morgan Kaufmann. Appendix B: Other
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Appendix B: Other Resources Forrest
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Appendix B: Other Resources Grefens
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Kirkpatrick, S., Gelatt, C.D., Jr.,
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Appendix B: Other Resources Mitchel
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Appendix B: Other Resources Roughga
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Appendix B: Other Resources Vose, M
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