1. RAJ ASHAR CAS/GRS COMPUTER SCIENCE CS 591: ALGORITHMS FOR THE NEW AGE DECEMBER 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  10 17 seconds to winnow through ~10 19,500,000 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 = 0 WHILE NOT Termination_Criterion [P(t)] DO Evaluate [P(t)] P' (t) = Select [P(t)] P''(t) = ApplyReproductionOperators [P'(t)] P(t+1) = Replace [P(t), P''(t)] t = t + 1 END RETURN Best_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 mask 1 2 2 1 mask 2 1 2 1 offspring 1 123 4 5 offspring 2 12 4 5

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 sample 1 0.5 1.1 -0.1 sample 2 0.1 0.8 0.5 offspring 1 67.5 1.9 2.1 offspring 2 23.1 8.2 19.5

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 crossover position 5 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

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?