An Introduction to Genetic Algorithms - Boente

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strategies community, in which parameters such as mutation rate are encoded as part of the chromosome. Here I will describe Lawrence Davis's approach to self−adaptation of operator rates (Davis 1989,1991). Davis assigns to each operator a "fitness" which is a function of how many highly fit individuals that operator has contributed to creating over the last several generations. Operators gain high fitness both for directly creating good individuals and for "setting the stage" for good individuals to be created (that is, creating the ancestors of good individuals). Davis tested this method in the context of a steady−state GA. Each operator (e.g., crossover, mutation) starts out with the same initial fitness. At each time step a single operator is chosen probabilistically (on the basis of its current fitness) to create a new individual, which replaces a low−fitness member of the population. Each individual i keeps a record of which operator created it. If i has fitness higher than the current best fitness, then i receives some credit for the operator that created it, as do i' parents, grandparents, and so on, back to a prespecified level of ancestor. The fitness of each operator over a given time interval is a function of its previous fitness and the sum of the credits received by all the individuals created by that operator during that time period. (The frequency with which operator fitnesses are updated is a parameter of the method.) In principle, the dynamically changing fitnesses of operators should keep up with their actual usefulness at different stages of the search, causing the GA to use them at appropriate rates at different times. As far as I know, this ability for the operator fitnesses to keep up with the actual usefulness of the operators has not been tested directly in any way, though Davis showed that this method improved the performance of a GA on some problems (including, it turns out, Montana and Davis's project on evolving weights for neural networks). A big question, then, for any adaptive approach to setting parameters— including Davis's—is this: How well does the rate of adaptation of parameter settings match the rate of adaptation in the GA population? The feedback for setting parameters comes from the population's success or failure on the fitness function, but it might be difficult for this information to travel fast enough for the parameter settings to stay up to date with the population's current state. Very little work has been done on measuring these different rates of adaptation and how well they match in different parameter−adaptation experiments. This seems to me to be the most important research to be done in order to get self−adaptation methods to work well. THOUGHT EXERCISES 1. 2. Formulate an appropriate definition of "schema" in the context of tree encodings (á la genetic programming). Give an example of a schema in a tree encoding, and calculate the probability of disruption of that schema by crossover and by mutation. Using your definition of schema in thought exercise 1, can a version of the Schema Theorem be stated for tree encodings? What (if anything) might make this difficult? 3. Derive the formula 4. Chapter 4: Theoretical Foundations of Genetic Algorithms where n is the number of schemas of order k in a search space of length l bit strings. 132
5. 6. Derive the requirements for rank selection given in the subsection on rank selection: 1 dMaxd2 and Min = 2Max. Derive the expressions Exp Val[i] and Exp Val[i] for the minimum and the maximum number of times an individual will reproduce under SUS. In the discussion on messy GAs, it was noted that Goldberg et al. explored a "probabilistically complete initialization" scheme in which they calculate what pairs of l' and ng will ensure that, on average, each schema of order k will be present in the initial population. Give examples of l' and ng that will guarantee this for k = 5. COMPUTER EXERCISES 1. Implement SUS and use it on the fitness function described in computer exercise 1 in chapter 1. How does this GA differ in behavior from the original one with roulette−wheel selection? Measure the "spread" (the range of possible actual number of offspring, given an expected number of offspring) of both sampling methods. 2. Implement a GA with inversion and test it on Royal Road function R1. Is the performance improved? 3. 4. 5. 6. Design a fitness function on which you think inversion will be helpful, and compare the performance of the GA with and without inversion on that fitness function. Implement Schaffer and Morishima's crossover template method and see if it improves the GA's performance on R1. Where do the exclamation points end up? Design a fitness function on which you think the crossover template method should help, and compare the performance of the GA with and without crossover templates on that fitness function. Design a fitness function on which you think uniform crossover should perform better than one−point or two−point crossover, and test your hypothesis. 7. Compare the performance of GAs using one−point, two−point, and uniform crossover on R1. 8. 9. Chapter 4: Theoretical Foundations of Genetic Algorithms Compare the performance of GAs using the various selection methods described in this chapter, using R1 as the fitness function. Which results in the best performance? 133
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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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1.4 SEARCH SPACES AND FITNESS LANDS
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potential energy is a measure of ho
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A simple method of implementing fit
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from an initial state to a goal. Fo
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Robert Axelrod of the University of
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instances of H at time t, and let
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What is the total payoff after 10 g
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6. * c. d. a. b. Chapter 1: Genetic
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As a simple example, consider a pro
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Chapter 2: Genetic Algorithms in Pr
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it to get one fitness case correct:
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Evolving Cellular Automata Chapter
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up most of the computation time. Ch
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the prospect of using GAs to automa
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where Ã is the standard deviation
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the brain. In a feedforward network
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Figure 2.21: An illustration of Mil
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experiments with grammatical encodi
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function of the average error of th
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COMPUTER EXERCISES 1. 2. 3. 4. 5. 6
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simulated generations, and such sim
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environments, they can fairly quick
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Chapter 3: Genetic Algorithms in Sc
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Figure 3.6: A schematic illustratio
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an initial population in which the
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An Introduction to Genetic Algorithms - Boente

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