An Introduction to Genetic Algorithms - Boente

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function of the average error of the 20 networks over the chosen subset of 20 mappings—low average error translated to high fitness. This fitness was then transformed to be a percentage, where a high percentage meant high fitness. Using this fitness measure, the GA was run on a population of 40 learning rules, with two−point crossover and standard mutation. The crossover rate was 0.8 and the mutation rate was 0.01. Typically, over 1000 generations, the fitness of the best learning rules in the population rose from between 40% and 60% in the initial generation (indicating no significant learning ability) to between 80% and 98%, with a mean (over several runs) of about 92%. The fitness of the delta rule is around 98%, and on one out of a total of ten runs the GA discovered this rule. On three of the ten runs, the GA discovered slight variations of this rule with lower fitness. These results show that, given a somewhat constrained representation, the GA was able to evolve a successful learning rule for simple single−layer networks. The extent to which this method can find learning rules for more complex networks (including networks with hidden units) remains an open question, but these results are a first step in that direction. Chalmers suggested that it is unlikely that evolutionary methods will discover learning methods that are more powerful than back−propagation, but he speculated that the GA might be a powerful method for discovering learning rules for unsupervised learning paradigms (e.g., reinforcement learning) or for new classes of network architectures (e.g., recurrent networks). Chalmers also performed a study of the generality of the evolved learning rules. He tested each of the best evolved rules on the ten mappings that had not been used in the fitness calculation for that rule (the "test set"). The mean fitness of the best rules on the original mappings was 92%, and Chalmers found that the mean fitness of these rules on the test set was 91.9%. In short, the evolved rules were quite general. Chalmers then looked at the question of how diverse the environment has to be to produce general rules. He repeated the original experiment, varying the number of mappings in each original environment between 1 and 20. A rule's evolutionary fitness is the fitness obtained by testing a rule on its original environment. A rule's test fitness is the fitness obtained by testing a rule on ten additional tasks not in the original environment. Chalmers then measured these two quantities as a function of the number of tasks in the original environment. The results are shown in figure 2.26. The two curves are the mean evolutionary fitness and the mean test fitness for rules that were tested in an environment with the given number of tasks. This plot shows that while the evolutionary fitness stays roughly constant for different numbers of environmental tasks, the test fitness increases sharply with the number of tasks, leveling off somewhere between 10 and 20 tasks. The conclusion is that the evolution of a general learning rule requires a diverse environment of tasks. (In this case of simple single−layer networks, the necessary degree of diversity is fairly small.) THOUGHT EXERCISES 1. Chapter 2: Genetic Algorithms in Problem Solving Using the function set {AND, OR, NOT} and the terminal set {s –1 , s 0 , s +1 }, construct a parse tree (or Lisp expression) that encodes the r = 1 majority−rule CA, where s i denotes the state of the neighborhood site i sites away from the central cell (with indicating distance to the left and + indicating distance to the right). AND and OR each take two arguments, and NOT takes one argument. 60
2. Figure 2.26: Results of Chalmers's experiments testing the effect of diversity of environment on generalization ability. The plot gives the evolutionary fitness (squares) and test fitness (diamonds) as a function of the number of tasks in the environment. (Reprinted from D. S. Touretzky et al. (eds.), Proceedings of the 1990 Connectionist Models Summer School Reprinted by permission of the publisher. © 1990 Morgan Kaufmann.) Assume that "MUTATE−TREE(TREE)" is a function that replaces a subtree in TREE by a randomly generated subtree. Using this function, write pseudo code for a steepest−ascent hill climbing algorithm that searches the space of GP parse trees, starting from a randomly chosen parse tree. Do the same for random−mutation hill climbing. 3. Write a formula for the number of CA rules of radius r. 4. Follow the same procedure as in figure 2.23 to construct the network given by the grammar displayed in figure 2.27. Figure 2.27: Grammar for thought exercise 4 5. Design a grammar that will produce the network architecture given in figure 2.28. Figure 2.28: Network for thought exercise 5. Chapter 2: Genetic Algorithms in Problem Solving 61
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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
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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 62 and 63: experiments with grammatical encodi
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
Page 112 and 113: 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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