Jaap Hofstede Beasly, Bull, Martin Genetic Algorithms Sources Jaap Hofstede Beasly, Bull, Martin

Introduction to Evolutionary Computation Evolutionary Computation is the field of study devoted to the design, development, and analysis is problem solvers based on natural selection (simulated evolution). Evolution has proven to be a powerful search process. Evolutionary Computation has been successfully applied to a wide range of problems including: Aircraft Design, Routing in Communications Networks, Tracking Windshear, Game Playing (Checkers [Fogel])

Introduction to Evolutionary Computation (Applications cont.) Robotics, Air Traffic Control, Design, Scheduling, Machine Learning, Pattern Recognition, Job Shop Scheduling, VLSI Circuit Layout, Strike Force Allocation,

Introduction to Evolutionary Computation (Applications cont.) Theme Park Tours (Disney Land/World) http://www.TouringPlans.com Market Forecasting, Egg Price Forecasting, Design of Filters and Barriers, Data-Mining, User-Mining, Resource Allocation, Path Planning, Etc.

Example of Evolutionary Algorithm An Example Evolutionary Computation Procedure EC{ t = 0; Initialize Pop(t); Evaluate Pop(t); While (Not Done) { Parents(t) = Select_Parents(Pop(t)); Offspring(t) = Procreate(Parents(t)); Evaluate(Offspring(t)); Pop(t+1)= Replace(Pop(t),Offspring(t)); t = t + 1; }

Candidate Solutions CS In an Evolutionary Computation, a population of candidate solutions (CSs) is randomly generated. Each of the CSs is evaluated and assigned a fitness based on a user specified evaluation function. The evaluation function is used to determine the ‘goodness’ of a CS. A number of individuals are then selected to be parents based on their fitness. The Select_Parents method must be one that balances the urge for selecting the best performing CSs with the need for population diversity.

Parents and Generations The selected parents are then allowed to create a set of offspring which are evaluated and assigned a fitness using the same evaluation function defined by the user. Finally, a decision must be made as to which individuals of the current population and the offspring population should be allowed to survive. Typically, in EC , this is done to guarantee that the population size remains constant. [The study of ECs with dynamic population sizes would make an interesting project for this course]

Selecting and Stopping Once a decision is made the survivors comprise the next generation (Pop(t+1)). This process of selecting parents based on their fitness, allowing them to create offspring, and replacing weaker members of the population is repeated for a user specified number of cycles. Stopping conditions for evolutionary search could be: The discovery of an optimal or near optimal solution Convergence on a single solution or set of similar solutions, When the EC detects the problem has no feasible solution, After a user-specified threshold has been reached, or After a maximum number of cycles.

A Brief History of Evolutionary Computation The idea of using simulated evolution to solve engineering and design problems have been around since the 1950’s (Fogel, 2000). Bremermann, 1962 Box, 1957 Friedberg, 1958 However, it wasn’t until the early 1960’s that we began to see three influential forms of EC emerge (Back et al, 1997): Evolutionary Programming (Lawrence Fogel, 1962), Genetic Algorithms (Holland, 1962) Evolution Strategies (Rechenberg, 1965 & Schwefel, 1968),

A Brief History of Evolutionary Computation (cont.) The designers of each of the EC techniques saw that their particular problems could be solved via simulated evolution. Fogel was concerned with solving prediction problems. Rechenberg & Schwefel were concerned with solving parameter optimization problems. Holland was concerned with developing robust adaptive systems.

A Brief History of Evolutionary Computation (cont.) Each of these researchers successfully developed appropriate ECs for their particular problems independently. In the US, Genetic Algorithms have become the most popular EC technique due to a book by David E. Goldberg (1989) entitled, “Genetic Algorithms in Search, Optimization & Machine Learning”. This book explained the concept of Genetic Search in such a way the a wide variety of engineers and scientist could understand and apply.

A Brief History of Evolutionary Computation (cont.) However, a number of other books helped fuel the growing interest in EC: Lawrence Davis’, “Handbook of Genetic Algorithms”, (1991), Zbigniew Michalewicz’ book (1992), “Genetic Algorithms + Data Structures = Evolution Programs. John R. Koza’s “Genetic Programming” (1992), and D. B. Fogel’s 1995 book entitled, “Evolutionary Computation: Toward a New Philosophy of Machine Intelligence. These books not only fueled interest in EC but they also were instrumental in bringing together the EP, ES, and GA concepts together in a way that fostered unity and an explosion of new and exciting forms of EC.

A Brief History of Evolutionary Computation: The Evolution of Evolutionary Computation First Generation EC EP (Fogel) GA (Holland) ES (Rechenberg, Schwefel) Second Generation EC Genetic Evolution of Data Structures (Michalewicz) Genetic Evolution of Programs (Koza) Hybrid Genetic Search (Davis) Tabu Search (Glover)

A Brief History of Evolutionary Computation: The Evolution of Evolutionary Computation (cont.) Third Generation EC Artificial Immune Systems (Forrest) Cultural Algorithms (Reynolds) DNA Computing (Adleman) Ant Colony Optimization (Dorigo) Particle Swarm Optimization (Kennedy & Eberhart) Memetic Algorithms Estimation of Distribution Algorithms Fourth Generation ????

Introduction to Evolutionary Computation: A Simple Example Let’s walk through a simple example! Let’s say you were asked to solve the following problem: Maximize: f6(x,y) = 0.5 + (sin(sqrt(x2+y2))2 – 0.5)/(1.0 + 0.001(x2+y2))2 Where x and y are take from [-100.0,100.0] You must find a solution that is greater than 0.99754, and you can only evaluate a total of 4000 candidate solutions (CSs) This seems like a difficult problem. It would be nice if we could see what it looks like! This may help us determine a good algorithm for solving it.

Introduction to Evolutionary Computation: A Simple Example A 3D view of f6(x,y):

Introduction to Evolutionary Computation: A Simple Example If we just look at only one dimension f6(x,1.0)

Introduction to Evolutionary Computation: A Simple Example Let’s develop a simple EC for solving this problem An individual (chromosome or CS) <xi,yi> fiti = f6(xi,yi)

Introduction to Evolutionary Computation: A Simple Example Procedure simpleEC{ t = 0; Initialize Pop(t); /* of P individuals */ Evaluate Pop(t); while (t <= 4000-P){ Select_Parent(<xmom,ymom>); /* Randomly */ Select_Parent(<xdad,ydad>); /* Randomly */ Create_Offspring(<xkid,ykid>): xkid = rnd(xmom, xdad) + Nx(0,); ykid = rnd(ymom, ydad) + Ny(0,); fitkid = Evaluate(<xkid,ykid>); Pop(t+1) = Replace(worst,kid);{Pop(t)-{worst}}{kid} t = t + 1; }

Introduction to Evolutionary Computation: A Simple Example To simulate this simple EC we can use the applet at: http://www.eng.auburn.edu/~gvdozier/GA.html

Introduction to Evolutionary Computation: A Simple Example To get a better understanding of some of the properties of ECs let’s do the ‘in class’ lab found at: http://www.eng.auburn.edu/~gvdozier/GA_Lab.html

Hill climbing

Introduction 1 Inspired by natural evolution Population of individuals Individual is feasible solution to problem Each individual is characterized by a Fitness function Higher fitness is better solution Based on their fitness, parents are selected to reproduce offspring for a new generation Fitter individuals have more chance to reproduce New generation has same size as old generation; old generation dies Offspring has combination of properties of two parents If well designed, population will converge to optimal solution

Algorithm BEGIN Generate initial population; Compute fitness of each individual; REPEAT /* New generation /* FOR population_size / 2 DO Select two parents from old generation; /* biased to the fitter ones */ Recombine parents for two offspring; Compute fitness of offspring; Insert offspring in new generation END FOR UNTIL population has converged END

Example of convergence

Introduction 2 Reproduction mechanism has no knowledge of the problem to be solved Link between genetic algorithm and problem: Coding Fitness function

Basic principles 1 Coding or Representation Fitness function String with all parameters Fitness function Parent selection Reproduction Crossover Mutation Convergence When to stop

Basic principles 2 An individual is characterized by a set of parameters: Genes The genes are joined into a string: Chromosome The chromosome forms the genotype The genotype contains all information to construct an organism: the phenotype Reproduction is a “dumb” process on the chromosome of the genotype Fitness is measured in the real world (‘struggle for life’) of the phenotype

Coding Parameters of the solution (genes) are concatenated to form a string (chromosome) All kind of alphabets can be used for a chromosome (numbers, characters), but generally a binary alphabet is used Order of genes on chromosome can be important Generally many different codings for the parameters of a solution are possible Good coding is probably the most important factor for the performance of a GA In many cases many possible chromosomes do not code for feasible solutions

Example of coding for TSP Travelling Salesman Problem Binary Cities are binary coded; chromosome is string of bits Most chromosomes code for illegal tour Several chromosomes code for the same tour Path Cities are numbered; chromosome is string of integers Ordinal Cities are numbered, but code is complex All possible chromosomes are legal and only one chromosome for each tour Several others

Reproduction Crossover Mutation Two parents produce two offspring There is a chance that the chromosomes of the two parents are copied unmodified as offspring There is a chance that the chromosomes of the two parents are randomly recombined (crossover) to form offspring Generally the chance of crossover is between 0.6 and 1.0 Mutation There is a chance that a gene of a child is changed randomly Generally the chance of mutation is low (e.g. 0.001)

Crossover One-point crossover Two-point crossover Uniform crossover

One-point crossover 1 Randomly one position in the chromosomes is chosen Child 1 is head of chromosome of parent 1 with tail of chromosome of parent 2 Child 2 is head of 2 with tail of 1 Randomly chosen position Parents: 1010001110 0011010010 Offspring: 0101010010 0011001110

One-point crossover 2

Two-point crossover Randomly two positions in the chromosomes are chosen Avoids that genes at the head and genes at the tail of a chromosome are always split when recombined Randomly chosen positions Parents: 1010001110 0011010010 Offspring: 0101010010 0011001110

Uniform crossover A random mask is generated The mask determines which bits are copied from one parent and which from the other parent Bit density in mask determines how much material is taken from the other parent (takeover parameter) Mask: 0110011000 (Randomly generated) Parents: 1010001110 0011010010 Offspring: 0011001010 1010010110

Problems with crossover Depending on coding, simple crossovers can have high chance to produce illegal offspring E.g. in TSP with simple binary or path coding, most offspring will be illegal because not all cities will be in the offspring and some cities will be there more than once Uniform crossover can often be modified to avoid this problem E.g. in TSP with simple path coding: Where mask is 1, copy cities from one parent Where mask is 0, choose the remaining cities in the order of the other parent

Fitness Function Purpose Parent selection Measure for convergence For Steady state: Selection of individuals to die Should reflect the value of the chromosome in some “real” way Next to coding the most critical part of a GA

Parent selection Chance to be selected as parent proportional to fitness Roulette wheel To avoid problems with fitness function Tournament Not a very important parameter

Roulette wheel Sum the fitness of all chromosomes, call it T Generate a random number N between 1 and T Return chromosome whose fitness added to the running total is equal to or larger than N Chance to be selected is exactly proportional to fitness Chromosome : 1 2 3 4 5 6 Fitness: 8 2 17 7 4 11 Running total: 8 10 27 34 38 49 N (1  N  49): 23 Selected: 3

Tournament Binary tournament Probabilistic binary tournament Two individuals are randomly chosen; the fitter of the two is selected as a parent Probabilistic binary tournament Two individuals are randomly chosen; with a chance p, 0.5<p<1, the fitter of the two is selected as a parent Larger tournaments n individuals are randomly chosen; the fittest one is selected as a parent By changing n and/or p, the GA can be adjusted dynamically

Problems with fitness range Premature convergence Fitness too large Relatively superfit individuals dominate population Population converges to a local maximum Too much exploitation; too few exploration Slow finishing Fitness too small No selection pressure After many generations, average fitness has converged, but no global maximum is found; not sufficient difference between best and average fitness Too few exploitation; too much exploration

Solutions for these problems Use tournament selection Implicit fitness remapping Adjust fitness function for roulette wheel Explicit fitness remapping Fitness scaling Fitness windowing Fitness ranking

Fitness scaling Fitness values are scaled by subtraction and division so that worst value is close to 0 and the best value is close to a certain value, typically 2 Chance for the most fit individual is 2 times the average Chance for the least fit individual is close to 0 Problems when the original maximum is very extreme (super-fit) or when the original minimum is very extreme (super-unfit) Can be solved by defining a minimum and/or a maximum value for the fitness

Example of Fitness Scaling

Fitness windowing Same as window scaling, except the amount subtracted is the minimum observed in the n previous generations, with n e.g. 10 Same problems as with scaling

Fitness ranking Individuals are numbered in order of increasing fitness The rank in this order is the adjusted fitness Starting number and increment can be chosen in several ways and influence the results No problems with super-fit or super-unfit Often superior to scaling and windowing

Other parameters of GA 1 Initialization: Reproduction: Population size Random Dedicated greedy algorithm Reproduction: Generational: as described before (insects) Generational with elitism: fixed number of most fit individuals are copied unmodified into new generation Steady state: two parents are selected to reproduce and two parents are selected to die; two offspring are immediately inserted in the pool (mammals)

Other parameters of GA 2 Stop criterion: Measure: Duplicates Number of new chromosomes Number of new and unique chromosomes Number of generations Measure: Best of population Average of population Duplicates Accept all duplicates Avoid too many duplicates, because that degenerates the population (inteelt) No duplicates at all

Example run Maxima and Averages of steady state and generational replacement

Introduction to Evolutionary Computation: Reading List Bäck, T., Hammel, U., and Schwefel, H.-P. (1997). “Evolutionary Computation: Comments on the History and Current State,” IEEE Transactions on Evolutionary Computation, VOL. 1, NO. 1, April 1997. Spears, W. M., De Jong, K. A., Bäck, T., Fogel, D. B., and de Garis, H. (1993). “An Overview of Evolutionary Computation,” The Proceedings of the European Conference on Machine Learning, v667, pp. 442-459. (http://www.cs.uwyo.edu/~wspears/papers/ecml93.pdf) De Jong, Kenneth A., and William M. Spears (1993). “On the State of Evolutionary Computation”, The Proceedings of the Int'l Conference on Genetic Algorithms, pp. 618-623. (http://www.cs.uwyo.edu/~wspears/papers/icga93.pdf)