Lecture 6: Genetic Algorithm. An Introduction to Meta-Heuristics, Produced by Qiangfu Zhao (Since 2012), All rights reserved

Transcription

1 Lecture 6: Genetic Algorithm An Introduction to Meta-Heuristics, Produced by Qiangfu Zhao (Since 2012), All rights reserved Lec06/1

2 Search and optimization again Given a problem, the set of all possible solutions is called the solution space, problem space or search space. A search algorithm is the process for finding the desired solution from the solution space. In many cases, we want to get the best solution, and this is why search is often called optimization. In practice, the search results may not be optimal. An Introduction to Meta-Heuristics, Produced by Qiangfu Zhao (Since 2012), All rights reserved Lec06/ 2

3 GA is a population based multi-point search algorithm Many points are used for search. Each point corresponds to a potential solution. A solution is called an individual in GA, and the set of all individuals is called the population. At the end of search, the best individual is often used as the final result. An Introduction to Meta-Heuristics, Produced by Qiangfu Zhao (Since 2012), All rights reserved Lec06/ 3

4 GA is a generate-and-test search Step 1 : Generate new solutions Step 2 : Test the solutions Step 3: If the current best solution is good enough, stop; otherwise, return to step 1. The function may not be continuous or derivative. The problem space may not be described by any function at all. An Introduction to Meta-Heuristics, Produced by Qiangfu Zhao (Since 2012), All rights reserved Lec06/ 4

5 GA is a history preserving search Each individual keeps some history of the old individuals (its parents). Search is not single path nor simple path. Often can get better solution faster than random search. Needs less memory than Tabu search. Search history is preserved in the population. Each individual in the population is a potential point for intensification. We need some mechanism for diversification. An Introduction to Meta-Heuristics, Produced by Qiangfu Zhao (Since 2012), All rights reserved Lec06/ 5

6 Basic steps of GA Start Initialization Evaluation Terminating condition? no Selection Reproduction yes End An Introduction to Meta-Heuristics, Produced by Qiangfu Zhao (Since 2012), All rights reserved Lec06/ 6

7 1-2-3 (Do-Re-Mi) for using GA 1 condition Terminating condition 2 functions Encoding/decoding and Evaluation 3 operations Selection, Crossover and Mutation An Introduction to Meta-Heuristics, Produced by Qiangfu Zhao (Since 2012), All rights reserved Lec06/ 7

8 The terminating condition Although natural evolution is an endless process, GA must stop at a certain point to get a useful solution. Usually, we see if the best current solution is good enough or not. If yes, we can terminate the program, and use the best current solution as the final answer. Different terminating conditions should be used for solving different problems. An Introduction to Meta-Heuristics, Produced by Qiangfu Zhao (Since 2012), All rights reserved Lec06/ 8

9 2 functions (1) The first function is a mapping between the genotype and the phenotype. The genotype of a individual is the genetic code of the individual. The phenotype is the body of the individual. In natural evolution, it is known that only the genotype evolves. The phenotype can be built from the genotype. The goodness of an individual is obtained by evaluating the phenotype. An Introduction to Meta-Heuristics, Produced by Qiangfu Zhao (Since 2012), All rights reserved Lec06/ 9

10 2 functions (2) The second function is the evaluation function. This function is usually different for different problems. The point is that, even if we do not have much information about the problem space, we can use GA to get a good solution, provided that a proper evaluation function is given. This function tells how good an individual is based on the performance of the phenotype. The evaluation function just provides local information. In the whole evolution process, we do not have to know the global information of the problem space. An Introduction to Meta-Heuristics, Produced by Qiangfu Zhao (Since 2012), All rights reserved Lec06/ 10

11 3 genetic operations Selection: survival strategy Crossover: generating new solutions by recombination of two or more parents For intensification! Mutation: generating a new solutions with one parent For diversification! An Introduction to Meta-Heuristics, Produced by Qiangfu Zhao (Since 2012), All rights reserved Lec06/ 11

12 A simple example Problem: Find the maximum point of f(x)=1+x+x 2 from the domain: 5.12 < x < 5.12 An Introduction to Meta-Heuristics, Produced by Qiangfu Zhao (Since 2012), All rights reserved Lec06/ 12

13 1 terminating condition In most cases, we evaluate the individuals in each generation, and terminate the evolution process if the best individual is GOOD enough. We may never get a good enough individual through evolution. We may just terminate the process when the number of generations is large enough. An Introduction to Meta-Heuristics, Produced by Qiangfu Zhao (Since 2012), All rights reserved Lec06/ 13

14 2 functions Mapping between the genotype and the phenotype Genotype: 10 bits binary number (coding) Range of fixed point integer: y=[0,1023] Phenotype: x=(y-512)/100 (decoding) The evaluation function Fitness = f(x) Maximum= The fitness is found in two steps Reconstruct the phenotype x Substitute x into f(x) [ ] y = 216 x = fitness = An Introduction to Meta-Heuristics, Produced by Qiangfu Zhao (Since 2012), All rights reserved Lec06/ 14

15 Operation-1: truncation selection Find the fitness of all individuals Sort the individuals according to their fitness Remove m worst individuals (m<popsize) from the bottom Generate m new individuals using survived ones, and put them to the empty positions m=ps x PopSize: ps is the selection rate PopSize is the size of the population An Introduction to Meta-Heuristics, Produced by Qiangfu Zhao (Since 2012), All rights reserved Lec06/ 15

16 Operator-1: truncation selection (cont.) An Introduction to Meta-Heuristics, Produced by Qiangfu Zhao (Since 2012), All rights reserved Lec06/ 16

17 Operation-2: One-point crossover Choose one point at random This point is called the crossover point Cut each parent into two parts Recombine them to generate two children crossover point An Introduction to Meta-Heuristics, Produced by Qiangfu Zhao (Since 2012), All rights reserved Lec06/ 17

18 Operation-3: Bit-by-bit mutation For each bit of the binary code Generate a random number in [0,1] If this number is less than p m, reverse the bit value (0 1 or 1 0) p m is called the mutation rate An Introduction to Meta-Heuristics, Produced by Qiangfu Zhao (Since 2012), All rights reserved Lec06/ 18

19 Results Before evolution: I[0]: I[1]: I[2]: I[3]: I[4]: I[5]: I[6]: I[7]: I[8]: I[9]: The maximum point is The maximum value is For the 9-th generation: I[0]: I[1]: I[2]: I[3]: I[4]: I[5]: I[6]: I[7]: I[8]: I[9]: The maximum point is The maximum value is An Introduction to Meta-Heuristics, Produced by Qiangfu Zhao (Since 2012), All rights reserved Lec06/ 19

20 Evolutionary learning of a neuron A neuron is the simplest neural network The decision boundary made by one neuron is a line or a hyper-plane It can be used to classify given patterns into two groups or classes An Introduction to Meta-Heuristics, Produced by Qiangfu Zhao (Since 2012), All rights reserved Lec06/ 20

21 Evolutionary learning of a neuron A neuron is represented by its weights and the threshold The threshold is considered as a special weight (with the corresponding input being -1) The genotype of a neuron can be defined as the weight list Each weight can be represented in a binary number w n+1 w 1 w 2 output w n x 1 x 2 x n w 1,w 2,,w n,w n An Introduction to Meta-Heuristics, Produced by Qiangfu Zhao (Since 2012), All rights reserved Lec06/ 21

22 Reconstruction of the phenotype B b y ik i If and : number : k where the - B k 1 b from the genotype th bit for the a 2 and weights b ik -10, of k b bits, w are the scaling take because per weight i i - value y th weight i a[ y is i ] b from [-10,10], always and shifting a factors. 20, smaller th an 1 An Introduction to Meta-Heuristics, Produced by Qiangfu Zhao (Since 2012), All rights reserved Lec06/ 22

23 Example Given patterns: points in a 2-D space Problem: find one neuron to classify the patterns into two classes Classification method: Class 1: w1x+w2y>w3 Class 2: otherwise Genotype: (w1,w2,w3) All weights are represented in B bits Fitness=1-E/N E: number of errors N: number of patterns w1 w2 w An Introduction to Meta-Heuristics, Produced by Qiangfu Zhao (Since 2012), All rights reserved Lec06/ 23

24 Evolutionary learning of an MLP Similar to evolving one neuron, we can evolve an MLP by defining the genotype as the weight list Each weight can be represented as a binary number, or as a real number The fitness can also be defined depends on the problem to solve Pattern recognition: fitness=1-e/n Function approximation: fitness=1/(1+e) An Introduction to Meta-Heuristics, Produced by Qiangfu Zhao (Since 2012), All rights reserved Lec06/ 24

25 Example For the neural network shown in the right There are 9 weights: (w 1,w 2,w 3,v 11,v 12, v 13,v 21,v 22,v 23 ) If we represent each weight using a, say, 16 bits binary number, the genotype will be a 144 bits binary string w 1 1 v 13 v 11 w 3 v 12 v 21 w 2 v 23 v 22 An Introduction to Meta-Heuristics, Produced by Qiangfu Zhao (Since 2012), All rights reserved Lec06/ 25

26 Neural network for XOR problem Suppose that a neural network is designed to approximate the XOR function. In this example, the fitness can be defined as follows: Fitness=1/(1+E) E is the mean squared error x 1 x 2 x 3 y An Introduction to Meta-Heuristics, Produced by Qiangfu Zhao (Since 2012), All rights reserved Lec06/ 26

27 Results Fitness[0]= Error[0]= Fitness[1]= Error[1]= Fitness[2]= Error[2]= Fitness[3]= Error[3]= Fitness[4]= Error[4]= Fitness[5]= Error[5]= Fitness[6]= Error[6]= Fitness[7]= Error[7]= Fitness[8]= Error[8]= Fitness[9]= Error[9]= Fitness[10]= Error[10]= Fitness[89]= Error[89]= Fitness[90]= Error[90]= Fitness[91]= Error[91]= Fitness[92]= Error[92]= Fitness[93]= Error[93]= Fitness[94]= Error[94]= Fitness[95]= Error[95]= Fitness[96]= Error[96]= Fitness[97]= Error[97]= Fitness[98]= Error[98]= Fitness[99]= Error[99]= An Introduction to Meta-Heuristics, Produced by Qiangfu Zhao (Since 2012), All rights reserved Lec06/ 27

28 Homework Following the flowchart given in p. 6, re-write the genetic algorithm in algorithm form. To design a neural network for pattern recognition, how to define the for the genetic algorithm? An Introduction to Meta-Heuristics, Produced by Qiangfu Zhao (Since 2012), All rights reserved Lec06/28