RAJ ASHAR CAS/GRS COMPUTER SCIENCE CS 591: ALGORITHMS FOR THE NEW AGE DECEMBER 9 , 2002

1. RAJ ASHAR CAS/GRS COMPUTER SCIENCECS 591: ALGORITHMS FOR THE NEW AGEDECEMBER 9 , 2002 Instructor: Prof. Shang-Hua Teng

2. Evolutionary Algorithms • Opaque means to clearly-good solutions

3. Agenda • Background • General Evolutionary Algorithm (EA) • Evolutionary Operators • EA classes • Ants Demo

4. Evolution from a CS Perspective: Part I • Population’s individuals compete for an environment’s finite resources • Genetic composition determines success • Goal: Be a • Survival: black-box objective function • Environment dictates black-box’s inner workings • Surprise: Evolution itself requires diversity

5. Evolution from a CS Perspective: Part II • NOT a purely-stochastic event • 1017 seconds to winnow through ~1019,500,000 possible genotypes • Extremely unlikely that random search would optimize survival functions • Nature adapts, not optimizes, but … • Optima do exist • Nature’s success piques CS curiosity

6. Why Evolution? • Process complements traditional search and optimization techniques • Copes well with noisy, inaccurate, incomplete data • Highly-parallelized • Potentially multiple solutions to same problem • Does not require in-depth problem knowledge • Awesome proof of concept

7. Computation, Meet Evolution • Evolutionary computation • Evolutionary Algorithms (EAs)

8. EA Basics • Seek high-”fitness” structures • Each structure encoded as a chromosome • Genes make up chromosome • Each gene represents value for some parameter

9. EA Flowchart

10. General EA Pseudocode • Generate [P(0)]t = 0WHILE NOTTermination_Criterion [P(t)] DOEvaluate [P(t)]P' (t) = Select [P(t)]P''(t) = ApplyReproductionOperators [P'(t)]P(t+1) = Replace [P(t), P''(t)]t= t + 1 ENDRETURNBest_Solution

11. Evolutionary Operators • Selection • Recombination • Mutation • Reinsertion

12. Selection Operation • Selection Idea: • Compute each individual’s fitness level • Rank-based methods only: weight each individual’s reproduction probability by rank • Select which individuals shall mate • Several selection methods exist • Concerns • Maintain Population Diversity • Improve Overall Fitness • Spread

13. Tournament Selection • For Nind iterations: • Randomly choose Tour number of individuals from the population for a group G. • Select the objectively-fittest individual from G for reproduction. • Tour may range from [2, Nind] • Generates “uniform at random offspring,” • Attempts to mimic “stags rut to vie for the privilege of mating with a herd of hinds.”

14. Truncation Selection • Pick top Trunc percent of population to reproduce • Produces uniform at random offspring • Artificial selection method

15. Ranking Methods • Compare individuals’ fitness • Proportional fitness assignment • Assigns each individual I ’s reproduction probability proportionally to I ’s fitness, and normalizes all probabilities to the unity • Scales poorly • Rank-based fitness assignment • Sorts population according to individual fitness • More robust than proportional fitness assignment

16. Rank-Based Selection • Pos = individual’s ranked position • SP = selective pressure • Two weighting formulae: • Linear ranking: Fitness(Pos) = 2 - SP + 2·(SP - 1)·(Pos - 1) / (Nind - 1) • SP may assume a value in [1.0, 2.0]. • non-linear ranking: Fitness(Pos) = Nind·X(Pos - 1) / Σ(X(i - 1)); i = 1:Nind • X stands for the root of the polynomial 0 = (SP - 1)·X(Nind - 1) + SP·X(Nind - 2) + ... + SP·X + SP

17. Comparing Ranking Formulae • Read right to left (position vs. fitness assignment) • Nonlinear increases weight more quickly with position • Better for smaller populations

18. Roulette wheel selection • Roulette wheel selection • Map each individual to segments of a continuous line “such that each individual's segment is equal in size to its fitness” by rank • Highest-ranked individual enjoys the largest line segment • Lowest-ranked individual occupies no line segment. • MatingPopulation, the number of individuals to reproduce: • For MatingPopulation iterations: • Generate a random number from independent sampling and a uniform distribution. • Select for reproduction “individual whose segment spans the random number”.

19. Stochastic universal sampling • Like Roulette wheel sampling, but differs slightly • NPointer = number of individuals to select • Place NPointer equally-spaced pointers over the line • Each pointer at 1/NPointer distance from another pointer. • Randomly generate number r within the range [0, 1/NPointer], • Place first pointer at r and every pointer thereafter at 1/NPointer distance from the previous pointer • Select for reproduction each individual whose line segment receives a pointer. • Zero bias and minimum spread

21. Recombination • Two categories • Real-valued recombination • Binary recombination • Discrete recombination algorithm • Applies to both categories • Exchanges gene values between individuals parent 1 12 25 5 parent 2 123 4 34 offspring 1 123 4 5 offspring 2 12 4 5 mask 1 2 2 1 mask 2 1 2 1

22. Real-valued recombination • Intermediate-value recombination • Set offspring value according to offspring = parent 1 + α(parent 2 - parent 1) • α = scaling factor chosen uniformly at random over an interval [-d, 1 + d] • α differs for each gene • d = 0.25 represents a good choice. parent 1 12 25 5 parent 2 123 4 34 offspring 1 67.5 1.9 2.1 offspring 2 23.1 8.2 19.5 sample 1 0.5 1.1 -0.1 sample 2 0.1 0.8 0.5

23. Discrete recombination • Single-point crossover • Randomly choose a gene at which to juxtapose two chromosomes parent 1 0 1 1 1 0 0 1 1 0 1 0 parent 2 1 0 1 0 1 1 0 0 1 0 1 offspring 1 0 1 1 1 0| 1 0 0 1 0 1 offspring 2 1 0 1 0 1| 0 1 1 0 1 0 crossover position 5

24. Mutation • Adds “small random values,” bounded by a mutation step value, to randomly chosen genes • Probability of chromosomal mutation inversely proportional to the number of genes • Again, two categories • Real-valued recombination • Binary recombination

25. Reinsertion • Introducing offspring to population • Questions • Add all offspring, or just fittest offspring • Replace all parents, least-fit parents, randomly-chosen parents • Two categories based on selection methods • Local reinsertion • Global reinsertion

26. Genetic Algorithms • Fixed-size chromosome encodes parameter values • Genetic operations and fitness measures improve population • At each iteration, • GA evaluates fitness of each individual • Creates new population by performing operations such as crossover, fitness-proportionate reproduction and mutation measured • Replaces entirely the previous population with the offspring. • Useful in solving multidimensional optimization problems • Maximize low-orbit satellite Earth coverage

27. Genetic Programming • Extends genetic learning to programming • Population consists of variable-length programs • Represented as parse trees • When executed, solve the given problem • Crossover involves exchanging random subtrees • Mutation generally does not take place • Already, “human competitive” results • Among others, “Creation of a cellular automata rule for the majority classification problem that is better than the Gacs-Kurdyumov-Levin (GKL) rule and all other known rules written by humans”

28. Evolutionary Strategies • Capable of solving high dimensional, multimodal, nonlinear problems subject to linear and/or nonlinear constraints. • Objective function can also be the result of a simulation, and does not have to be given in a closed form. • Two strategies: • plus strategy: reinserts parents and children based on fitness. Plus strategy departs from reality in theoretically allowing an individual to remain within the population for perpetuity. • comma strategy: perform only selection on the offspring, and replace all parents with the selected offspring. • Individual • Object variables • Strategy variables • Requires knowledge of probability theory and applied statistics

29. Evolutionary Programming • Strategy for stochastic optimization • No constraint on representation • Perturbs offspring chromosomes during reproduction • Does not use crossover • Degree of mutation depends on degree of functional change imposed by parents • Stochastic tournament to determine reinsertion

30. Open questions? • Complexity • Why do these methods work?

31. Ants! • Genetic algorithm • Can alter parameters • Proof of concept for systems?