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CSM6120 Introduction to Intelligent Systems Evolutionary and Genetic Algorithms [email protected] Informal biological terminology Genes Chromosomes Encoding rules that describe how an organism is built up from the tiny building blocks of life Long strings formed by connecting genes together Recombination Process of two organisms mating, producing offspring that may end up sharing genes of their parents Basic ideas of EAs An EA is an iterative procedure which evolves a population of individuals Each individual is a candidate solution to a given problem Each individual is evaluated by a fitness function, which measures the quality of its candidate solution At each iteration (generation): The best individuals are selected Genetic operators are applied to selected individuals in order to produce new individuals (offspring) New individuals are evaluated by fitness function Taxonomy Search Techniques Informed Uninformed DFS A* Evolutionary Strategies Hill Climbing Swarm Intelligence Evolutionary Algorithms Genetic Programming BFS Simulated Annealing Genetic Algorithms The Genetic Algorithm Directed search algorithms based on the mechanics of biological evolution Developed by John Holland, University of Michigan (1970s) To understand the adaptive processes of natural systems To design artificial systems software that retains the robustness of natural systems Provide efficient, effective techniques for optimization and machine learning applications Some GA applications Domain Application Types Control gas pipeline, pole balancing, missile evasion, pursuit Design semiconductor layout, aircraft design, keyboard configuration, communication networks Scheduling manufacturing, facility scheduling, resource allocation Robotics trajectory planning Machine Learning designing neural networks, improving classification algorithms, classifier systems Signal Processing filter design Game Playing poker, checkers, prisoner’s dilemma Combinatorial Optimization set covering, travelling salesman, routing, bin packing, graph colouring and partitioning Application: function optimisation (1) 4 1 3 .5 0.8 3 2 .5 0.6 2 0.4 1 .5 1 0.2 0 .5 0 0 -1 -0.8 -0.6 f(x) = -0.4 x2 -0.2 0 0.2 0.4 0.6 0.8 1 -1 0 -5 0 5 10 g(x) = sin(x) - 0.1 x + 2 h(x,y) = x.sin(4x) - y.sin(4y+ ) + 1 Application: function optimisation (2) Conventional approaches: Often requires knowledge of derivatives or other specific mathematical technique Evolutionary algorithm approach: Requires only a measure of solution quality (fitness function) Components of a GA A problem to solve, and ... Encoding technique Initialization procedure Evaluation function Selection of parents Genetic operators Parameter settings (gene, chromosome) (creation) (environment) (reproduction) (mutation, recombination) (practice and art) GA terminology Population Parents/Children The collection of potential solutions (i.e. all the chromosomes) Both are chromosomes Children are generated from the parent chromosomes Generations Number of iterations/cycles through the GA process Simple GA initialize population; evaluate population; while TerminationCriteriaNotSatisfied { select parents for reproduction; perform recombination and mutation; evaluate population; } The GA cycle chosen parents recombination selection children modification modified children parents evaluation population evaluated children deleted members discard Population Chromosomes could be: Bit strings Real numbers Permutations of element Lists of rules Program elements ... any data structure ... (0101 ... 1100) (43.2 -33.1 ... 0.0 89.2) (E11 E3 E7 ... E1 E15) (R1 R2 R3 ... R22 R23) (genetic programming) Example: Discrete representation Representation of an individual can be using discrete values (binary, integer, or any other system with a discrete set of values) The following is an example of binary representation: CHROMOSOME 1 0 1 0 GENE 0 0 1 1 Example: Discrete representation Phenotype: • Integer 8 bits Genotype 1 0 1 0 0 0 1 1 • Real Number • Schedule • ... • Anything? Example: Discrete representation Phenotype could be integer numbers Genotype: 1 0 1 0 Phenotype: 0 0 1 1 = 163 1*27 + 0*26 + 1*25 + 0*24 + 0*23 + 0*22 + 1*21 + 1*20 = 128 + 32 + 2 + 1 = 163 Example: Discrete representation Phenotype could be real numbers e.g. a number between 2.5 and 20.5 using 8 binary digits Genotype: 1 0 1 0 x 2 .5 0 Phenotype: 0 163 256 1 1 = 13.9609 20 . 5 2 . 5 13 . 9609 Example: Discrete representation Phenotype could be a schedule e.g. 8 jobs, 2 time steps Phenotype Job Genotype: 1 0 1 0 0 0 1 1 = 1 2 3 4 5 6 7 8 Time Step 2 1 2 1 1 1 2 2 Example: Real-valued representation A very natural encoding if the solution we are looking for is a list of real-valued numbers, then encode it as a list of real-valued numbers! (i.e., not as a string of 1s and 0s) Lots of applications, e.g. parameter optimisation Representation Task – how to represent the travelling salesman problem (TSP)? Find a tour of a given set of cities so that Each city is visited only once The total distance travelled is minimised Representation One possibility - an ordered list of city numbers (this is known as an order-based GA) 1) London 2) Venice 3) Dunedin 4) Singapore Chromosome 1 Chromosome 2 5) Beijing 7) Tokyo 6) Phoenix 8) Victoria (3 5 7 2 1 6 4 8) (2 5 7 6 8 1 3 4) Selection selection parents population Selection Need to choose which chromosomes to use based on their ‘fitness’ Why not choose the best chromosomes? We want a balance between exploration and exploitation Roulette wheel selection Rank-based selection 1st step 2nd step Sort (rank) individuals according to fitness Ascending or descending order (minimization or maximization) Select individuals with probability proportional to their rank only (ignoring the fitness value) The better the rank, the higher the probability of being selected It avoids most of the problems associated with roulette-wheel selection, but still requires global sorting of individuals, reducing potential for parallel processing Tournament selection A number of “tournaments” are run Several chromosomes chosen at random The chromosome with the highest fitness is selected each time Larger tournament size means that weak chromosomes are less likely to be selected Advantages It is efficient to code It works on parallel architectures The GA cycle chosen parents recombination selection children modification modified children parents evaluation population evaluated children deleted members discard Crossover: recombination (0 1 1 0 1 0 1 1) (1 1 0 1 1 0 0 1) P1 P2 (1 1 0 1 1 0 1 1) (0 1 1 0 1 0 0 1) C1 C2 Crossover is a critical feature of GAs: It greatly accelerates search early in evolution of a population It leads to effective combination of sub-solutions on different chromosomes Several methods for crossover exist… Crossover How would we implement crossover for TSPs? Parent 1 Parent 2 (3 5 7 2 1 6 4 8) (2 5 7 6 8 1 3 4) Crossover Parent 1 Parent 2 Child 1 Child 2 (3 5 7 2 1 6 4 8) (2 5 7 6 8 1 3 4) (3 5 7 6 8 1 3 4) (2 5 7 2 1 6 4 8) Mutation: local modification Before: (1 0 1 1 0 1 1 0) After: (0 1 1 0 0 1 1 0) Before: (1.38 -69.4 326.44 0.1) After: (1.38 -67.5 326.44 0.1) Causes movement in the search space (local or global) Restores lost information to the population Mutation Given the representation for TSPs, how could we achieve mutation? Mutation Mutation involves reordering of the list: * Before: After: * (5 8 7 2 1 6 3 4) (5 8 6 2 1 7 3 4) Note Both mutation and crossover are applied based on usersupplied probabilities We usually use a fairly high crossover rate and fairly low mutation rate Why do you think this is? Evaluation of fitness modified children evaluation evaluated children The evaluator decodes a chromosome and assigns it a fitness measure The evaluator is the only link between a classical GA and the problem it is solving Fitness functions Evaluate the ‘goodness’ of chromosomes (How well they solve the problem) Critical to the success of the GA Often difficult to define well Must be fairly fast, as each chromosome must be evaluated each generation (iteration) Fitness functions Fitness function for the TSP? (3 5 7 2 1 6 4 8) As we’re minimizing the distance travelled, the fitness is the total distance travelled in the journey defined by the chromosome Deletion population deleted members discard Generational GA: entire populations replaced with each iteration Steady-state GA: a few members replaced each generation The GA cycle chosen parents recombination selection children modification modified children parents evaluation population evaluated children deleted members discard Stopping! The GA cycle continues until The system has ‘converged’; or A specified number of iterations (‘generations’) has been performed An abstract example Distribution of Individuals in Generation 0 Distribution of Individuals in Generation N Good demo of the GA components http://www.obitko.com/tutorials/genetic-algorithms/examplefunction-minimum.php TSP example: 30 cities 120 100 y 80 60 40 20 0 0 10 20 30 40 50 x 60 70 80 90 100 TSP30 (Performance = 941) 120 100 y 80 60 40 20 0 0 10 20 30 40 50 x 60 70 80 90 100 TSP30 (Performance = 800) 120 100 80 y 44 62 69 67 78 64 62 54 42 50 40 40 38 21 35 67 60 60 40 42 50 99 60 40 20 0 0 10 20 30 40 50 x 60 70 80 90 100 TSP30 (Performance = 652) 120 100 y 80 60 40 20 0 0 10 20 30 40 50 x 60 70 80 90 100 TSP30 Solution (Performance = 420) 120 100 80 y 42 38 35 26 21 35 32 7 38 46 44 58 60 69 76 78 71 69 67 62 84 94 60 40 20 0 0 10 20 30 40 50 x 60 70 80 90 100 Overview of performance TSP30 - Overview of Performance 1800 1600 1400 Distance 1200 1000 800 600 400 200 0 Best 1 3 5 7 9 11 13 15 17 19 21 Generations (1000) 23 25 27 29 31 Worst Average Example: n-queens Put n queens on an n × n board with no two queens on the same row, column, or diagonal Examples Eaters http://math.hws.edu/xJava/GA/ TSP http://www.heatonresearch.com/articles/65/page1.html http://www.ads.tuwien.ac.at/raidl/tspga/TSPGA.html Exercise: The Card Problem You have 10 cards numbered from 1 to 10.You have to choose a way of dividing them into 2 piles, so that the cards in Pile0 *sum* to a number as close as possible to 36, and the remaining cards in Pile1 *multiply* to a number as close as possible to 360 Encoding Each card can be in Pile0 or Pile1, there are 1024 possible ways of sorting them into 2 piles, and you have to find the best. Think of a sensible way of encoding any possible solution. Fitness Some of these chromosomes will be closer to the target than others. Think of a sensible way of evaluating any chromosome and scoring it with a fitness measure. Issues for GA practitioners Choosing basic implementation issues: Representation Population size, mutation rate, ... Selection, deletion policies Crossover, mutation operators Termination criteria Performance, scalability Solution is only as good as the fitness function (often hardest part) Your assignment will be to code a GA for a given task! Be aware of the above issues… Benefits of GAs Concept is easy to understand Supports multi-objective optimization Good for “noisy” environments Always an answer; answer gets better with time Inherently parallel; easily distributed