Announcements

1. Announcements • Lab 3 due Tuesday, November 6 • Homework 6 due Tuesday, November 6 • Lab 4 due Thursday, November 8 • Current Event • Chelsea - today • Beth - Tuesday, November 6 CS 484 – Artificial Intelligence

2. Genetic Algorithms Lecture 13

3. Evolutionary Computation • Computational procedures patterned after biological evolution • Search procedure that probabilistically applies search operators to a set of points in the search space • Generate successor hypothesis by repeatedly mutating and recombining parts of the best currently known hypothesis CS 484 – Artificial Intelligence

4. General Procedure • At each stage has a collection of hypotheses known as the current population • Update population by replacing some fraction of it by offspring of the most fit current hypotheses • Process forms a generate-and-test search CS 484 – Artificial Intelligence

5. Reason for Popularity • Evolution is known to be successful, robust method for adaptation within biological systems • GAs can search spaces of hypotheses containing complex interacting parts • Genetic algorithms are easily parallelized and can take advantage of the decreasing cost of powerful computer hardware CS 484 – Artificial Intelligence

6. The Parameters GA(Fitness, Fitness_threshold, p, r, m) • Fitness - function evaluates how good a hypothesis is • Fitness_threshold - minimum acceptable hypothesis • p - size of the population • r - fraction of population to be replaced • m - mutation rate CS 484 – Artificial Intelligence

7. The Algorithm GA(Fitness, Fitness_threshold, p, r, m) • Initialize: P←p random hypotheses • Evaluate: for each h in P, compute Fitness(h) • While [maxhFitness(h)] < Fitness_threshold • Select: Select (1 – r) members of P to add to PS based on fitness • Crossover: Probabilistically select pairs of hypotheses from P. For each pair, <h1, h2>, produce two offspring by applying the Crossover operator. Add all offspring to PS • Mutate: Invert a randomly selected bit in m · p random members of PS • Update: P←PS • Evaluate: for each h in P, compute Fitness(h) • Return the hypothesis from P that has the highest fitness CS 484 – Artificial Intelligence

8. Fitness • Fitness is an important concept in genetic algorithms. • The fitness of a chromosome determines how likely it is that it will reproduce. • Fitness is usually measured in terms of how well the chromosome solves some goal problem. • E.g., if the genetic algorithm is to be used to sort numbers, then the fitness of a chromosome will be determined by how close to a correct sorting it produces. • Fitness can also be subjective (aesthetic) CS 484 – Artificial Intelligence

9. Representing Hypotheses • From the EnjoySport (simplified) example • Represent (Sky = Sunny vCloudy) Λ (AirTemp = Warm) by • Represent IF AirTemp = Warm THEN EnjoySport = yes by CS 484 – Artificial Intelligence

10. Single-Point Crossover • Crossover is applied as follows: • Select a random crossover point. • Break each chromosome into two parts, splitting at the crossover point. • Recombine the broken chromosomes by combining the front of one with the back of the other, and vice versa, to produce two new chromosomes. CS 484 – Artificial Intelligence

11. Other types of Crossover • Usually, crossover is applied with one crossover point, but can be applied with more, such as in the following case which has two crossover points: • Uniform crossover involves using a probability to select which genes to use from chromosome 1, and which from chromosome 2. CS 484 – Artificial Intelligence

12. Mutation • A unary operator – applies to one chromosome. • Randomly selects some bits (genes) to be “flipped” • 1 => 0 and 0 =>1 • Mutation is usually applied with a low probability, such as 1 in 1000. CS 484 – Artificial Intelligence

13. Termination Criteria • A genetic algorithm is run over a number of generations until the termination criteria are reached. • Typical termination criteria are: • Stop after a fixed number of generations. • Stop when a chromosome reaches a specified fitness level. • Stop when a chromosome succeeds in solving the problem, within a specified tolerance. • Human judgment can also be used in some more subjective cases. CS 484 – Artificial Intelligence

14. Optimizing a mathematical function • A genetic algorithm can be used to find the highest value for f(x) = sin (x). • Each chromosome consists of 4 bits, to represent the values of x from 0 to 15. • Fitness ranges from 0 (f(x) = -1) to 100 (f(x) = 1). • f'(x) = 50 * (f(x) + 1) - determines fitness • By applying the genetic algorithm it takes just a few generations to find that the value x =8 gives the optimal solution for f(x). CS 484 – Artificial Intelligence

15. What is a Schema? • How to characterize evolution of population in GA? • As with the rules used in classifier systems, a schema is a string consisting of 1’s, 0’s and *’s. E.g.: 1011*001*0 Matches the following four strings: 1011000100 1011000110 1011100100 1011100110 • A schema with n *’s will match a total of 2n chromosomes. • Each chromosome of r bits will match 2r different schemata. • Example: 0010 is a representative of 24 distinct schemas • 00**, 0*10, ****, etc. CS 484 – Artificial Intelligence

16. Schema Theorem • Characterizes evolution in the population in terms of the number of instances representing each schema • The fitness of a schema is defined as the average fitness of the chromosomes that match the schema. • The fitness of a schema, S, in generation tth is written as follows: f(S, t) • The number of occurrences of S in the population at time t is : m(S, t) • Looking for the expected value of m(S, t+1) • Defined in terms of m(S, t) and other properties CS 484 – Artificial Intelligence

17. Selection Step • Evolution of GA depends on selection step, recombination step, and mutation step • Selection step • Cloned individuals become part of next generation • How many strings in schema S will be present in the next generation? • A particular string h, is selected with a probability • Let a(t) be the average fitness of all individuals CS 484 – Artificial Intelligence

18. Fitness of the Next Generation • Interested in the probability that a single hypothesis selected by the GA will be an instance of schema S • Probability that we select a representative of S • Given that a(S,t) is the average fitness of instances of schema S • The expected number of instances of S resulting from n independent selection steps CS 484 – Artificial Intelligence

19. Schemata Properties • The defining length dL(S) of a schema, S, is the distance between the first and last defined bits. For example, the defining length of each of the following schemata is 4: **10111* 1*0*1** • The order O(S)is number of defined bits in S. The following schemata both have order 4: **10*11* 1*0*1**1 • A schema with a high order is more specific than one with a lower order. CS 484 – Artificial Intelligence

20. The Effect of Single-Point Crossover • For a schema S to survive crossover, the crossover point must be outside the defining length of S. • Hence, the probability that S will survive crossover is: • L(S) is the length of the bit strings and pc is the probability that single-point crossover will be applied to an individual • This tells us that a short schema is more likely to survive crossover than a longer schema. CS 484 – Artificial Intelligence

21. The Effect of Mutation • The probability that mutation will be applied to a bit in a string is pm • Hence, the probability that a schemaS will survive mutation is: • We can combine this with the effects of selection, crossover, and reproduction to give: CS 484 – Artificial Intelligence

22. The Schema Theorem • Holland’s Schema Theorem, represented by the above formula, can be written as: More fit schemas will tend to grow in influence, especially schemas containing small number of defined bits, and defined bits that near one another within the bit string. • This helps to explain why genetic algorithms work. • It does not provide a complete answer. CS 484 – Artificial Intelligence

23. The Building Block Hypothesis • Genetic algorithms manipulate short, low-order, high fitness schemata in order to find optimal solutions to problems.” • These short, low-order, high fitness schemata are known as building blocks. • Hence genetic algorithms work well when small groups of genes represent useful features in the chromosomes. • This tells us that it is important to choose a correct representation. CS 484 – Artificial Intelligence

24. Messy Genetic Algorithms (1) • An alternative to standard genetic algorithms that avoid deception. • Each bit in the chromosome is represented as a (position, value) pair. For example: ((1,0), (2,1), (4,0)) • In this case, the third bit is undefined, which is allowed with MGAs. A bit can also overspecified: ((1,0), (2,1), (3,1), (3,0), (4,0)) CS 484 – Artificial Intelligence

25. Messy Genetic Algorithms (2) • Underspecified bits are filled with bits taken from a template chromosome. • The template chromosome is usually the best performing chromosome from the previous generation. • Overspecified bits are usually dealt with by working from left to right and using the first value specified for each bit. CS 484 – Artificial Intelligence

26. Messy Genetic Algorithms (3) • MGAs use splice and cut instead of crossover. • Splicing involves simply joining two chromosomes together: ((1,0), (3,0), (4,1), (6,1)) ((2,1), (3,1), (5,0), (7,0), (8,0)) ((1,0), (3,0), (4,1), (6,1), (2,1), (3,1), (5,0), (7,0), (8,0)) • Cutting involves splitting a chromosome into two: ((1,0), (3,0), (4,1)) ((6,1), (2,1), (3,1), (5,0), (7,0), (8,0)) CS 484 – Artificial Intelligence

27. Co-Evolution • In the real world, the presence of predators is responsible for many evolutionary developments. • Similarly, in many artificial life systems, introducing “predators” produces better results. • This process is known as co-evolution. • For example, Ramps, which were evolved to sort numbers: “parasites” were introduced which produced sets of numbers that were harder to sort, and the ramps produced better results. CS 484 – Artificial Intelligence