An Introduction to Genetic Algorithms - Boente

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What is the total payoff after 10 games of TIT FOR TAT playing against (a) a strategy that always defects; (b) a strategy that always cooperates; (c) ANTI−TIT−FOR−TAT, a strategy that starts out by defecting and always does the opposite of what its opponent did on the last move? (d) What is the expected payoff of TIT FOR TAT against a strategy that makes random moves? (e) What are the total payoffs of each of these strategies in playing 10 games against TIT FOR TAT? (For the random strategy, what is its expected average payoff?) 3. How many possible sorting networks are there in the search space defined by Hillis's representation? 4. Prove that any string of length l is an instance of 2 l different schemas. 5. 6. 7. Define the fitness ƒ of bit string x with l = 4 to be the integer represented by the binary number x. (e.g., f(0011) = 3, ƒ(1111) = 15). What is the average fitness of the schema 1*** under ƒ? What is the average fitness of the schema 0*** under ƒ? Define the fitness of bit string x to be the number of ones in x. Give a formula, in terms of l (the string length) and k, for the average fitness of a schema H that has k defined bits, all set to 1. When is the union of two schemas also a schema? For example, {0*}*{1*} is a schema (**), but {01}*{10} is not. When is the intersection of two schemas also a schema? What about the difference of two schemas? 8. Are there any cases in which a population of n l−bit strings contains exactly n × 2 l different schemas? COMPUTER EXERCISES (Asterisks indicate more difficult, longer−term projects.) 1. 2. Chapter 1: Genetic Algorithms: An Overview Implement a simple GA with fitness−proportionate selection, roulettewheel sampling, population size 100, single−point crossover rate pc = 0.7, and bitwise mutation rate pm = 0.001. Try it on the following fitness function: ƒ(x) = number of ones in x, where x is a chromosome of length 20. Perform 20 runs, and measure the average generation at which the string of all ones is discovered. Perform the same experiment with crossover turned off (i.e., pc = 0). Do similar experiments, varying the mutation and crossover rates, to see how the variations affect the average time required for the GA to find the optimal string. If it turns out that mutation with crossover is better than mutation alone, why is that the case? Implement a simple GA with fitness−proportionate selection, roulettewheel sampling, population size 100, single−point crossover rate pc = 0.7, and bitwise mutation rate pm = 0.001. Try it on the fitness function ƒ(x) = the integer represented by the binary number x, where x is a chromosome of length 20. 24
3. 4. 5. * Run the GA for 100 generations and plot the fitness of the best individual found at each generation as well as the average fitness of the population at each generation. How do these plots change as you vary the population size, the crossover rate, and the mutation rate? What if you use only mutation (i.e., pc = 0)? Define ten schemas that are of particular interest for the fitness functions of computer exercises 1 and 2 (e.g., 1*···* and 0*···*). When running the GA as in computer exercises 1 and 2, record at each generation how many instances there are in the population of each of these schemas. How well do the data agree with the predictions of the Schema Theorem? Compare the GA's performance on the fitness functions of computer exercises 1 and 2 with that of steepest−ascent hill climbing (defined above) and with that of another simple hill−climbing method, "random−mutation hill climbing" (Forrest and Mitchell 1993b): 1. Start with a single randomly generated string. Calculate its fitness. 2. Randomly mutate one locus of the current string. 3. If the fitness of the mutated string is equal to or higher than the fitness of the original string, keep the mutated string. Otherwise keep the original string. 4. Go to step 2. Iterate this algorithm for 10,000 steps (fitness−function evaluations). This is equal to the number of fitness−function evaluations performed by the GA in computer exercise 2 (with population size 100 run for 100 generations). Plot the best fitness found so far at every 100 evaluation steps (equivalent to one GA generation), averaged over 10 runs. Compare this with a plot of the GA's best fitness found so far as a function of generation. Which algorithm finds higher−fitness chromosomes? Which algorithm finds them faster? Comparisons like these are important if claims are to be made that a GA is a more effective search algorithm than other stochastic methods on a given problem. Implement a GA to search for strategies to play the Iterated Prisoner's Dilemma, in which the fitness of a strategy is its average score in playin 100 games with itself and with every other member of the population. Each strategy remembers the three previous turns with a given player. Use a population of 20 strategies, fitness−proportional selection, single−point crossover with pc = 0.7, and mutation with pm = 0.001. a. b. Chapter 1: Genetic Algorithms: An Overview See if you can replicate Axelrod's qualitative results: do at least 10 runs of 50 generations each and examine the results carefully to find out how the best−performing strategies work and how they change from generation to generation. 25
Page 2 and 3: An Introduction to Genetic Algorith
Page 4 and 5: Table of Contents Chapter 4: Theore
Page 6 and 7: Chapter 1: Genetic Algorithms: An O
Page 10 and 11: 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 26 and 27: instances of H at time t, and let
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 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
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Chapter 3: Genetic Algorithms in Sc
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an initial population in which the
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these random effects, Bedau and Pac
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3. * 4. * the female child in the n
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calculating the observed average fi
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(4.1) The goal is to find n = n* th
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competition in the d *···* parti
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schemas with the best observed fitn
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Steepest−ascent hill climbing (SA
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(If the algorithm spends only 1/m o
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strings—then in principle crossov
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(4.7) Chapter 4: Theoretical Founda
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2. 3. Calculate the fitness f(x) of
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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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An Introduction to Genetic Algorithms - Boente

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