Lecture 8: Genetic Algorithms

1. Lecture 8: Genetic Algorithms Contents: Miming nature The steps of the algorithm Coosing parents Reproduction Mutation Deeper in GA Stochastic Universal Selection Elitarism Steady State

2. Miming Nature Some considerations on evolution of organisms: • The organisms that are ill-suited for an environment have little chances to reproduce (natural selection) • Conversely, the best fitting have more chances to survive and reproduce • Offspring are similar to their parents • Random mutations occur and they can bring to better (or worse) fitting individuals “The Origin of the Species on the Basis of Natural Selection” C. Darwin (1859) • An organism is fully represented by its DNA string, that is a string over a finite alphabet (4 symbols) • Each element of this string is called gene • As for Neural Networks, the idea is that what woks in nature should be good also for artificial systems “All above in the assumption you are not Jehovah’s witnesses”

3. Intro • An individual (an organisms) is intended to be a possible solution for the problem you want to solve • An individual is represented by a binary string. Such a string is intended to be the complete description of the individual • A fitness function is a function that says how good is a solution, i.e. how well an individual fit the environment • Example: Suppose you have to find a number between 0 and 255, which binary representation contains the same number of 1s and 0s. A individual is a string of 8 bits, ex: The fitness function is: note that the fitness function gets the minimum value (i.e. 0) when or and the maximum value (i.e. 8) when h = = 126

4. Genetic Algorithm • A genetic algorithm is a way to obtain an individual that maximizes the fitness function Scheme of the (canonical) algorithm • Start with a population of N individuals • Apply the fitness function to all the individuals • Select the pairs of individuals for reproduction (repetition allowed). • Each pair generates two children (reproduction) • Apply a random mutation to the children. The children become the next generation • Apply steps 1,2,3 until some termination criteria applies In the following we explain the details of these steps using the previous example. Suppose to have the initial population:

5. Selection • For each element, compute the its contribution to the global fitness as • The choice of the pairs for reproduction consists of randomly choosing the individuals (with replacement) with distribution given by P (roulette-wheel selection ) • How to do it? • Split the interval [0,1] in N parts of length • Generate a random value r in [0,1] and choose individual associated with the interval containing r

6. Crossover • Suppose the chosen pairs are: and • The parents are combined by means of the crossover operator. This operator apply to a pair and return two individual. It consists of the following steps: • Randomly choose a cross over point “c”, i.e. a number between 1 and n • return two children: one composed by the first c bits of the first parent and the last n-c bits of the second parent, the other composed by the first c bits of the second parent and the n-c bits of the first parents

7. Mutation • After all the children have been created, some of theirs bits are changed (with a small, independent probability), so implementing a mutation on the individuals • The following new generation is the result of the first loop • In this simple case we have a solution for which the fitness function is at its maximum, but in general it is not so easy: when to stop evolution?

8. Stopping Criteria • Convergence: • A population is said to converge when all the genes have converged, I.e. when the value of every bit is the same at least in the 95% of the individuals in the population • Since convergence is not guaranteed, we must consider other stopping criteria: • Number of generations • Almost constant value of the best fitting individual • Almost constant value of the average fitness of the population

9. Deeper in GAs • Have a deeper look to what we need to implement a GA and which are the degrees of freedom • Encoding: an individual (a solution) is encoded by a string of binary values that you have to provide. The encoding was the ‘identity function’ in the previous example. Suppose you have the following problem: Given a set of four numbers, find their position in an increasing ordered list. Ex: input = [ 41,11,23,25] solution = [3,0,1,2] • Fitness Function: can be useful to distinguish between Fitness Function and Evaluation Function (or Objective Function) • The evaluation function measures how a solution is good with respect to the parameters of the problem • The Fitness Function assigns the probability of being chosen for reproduction on the base of the results of the evaluation function.

10. Applying Gas (cntd) • Suppose to use the following evaluation function Where a wrong pair is a pair of values in the string such that What is the evaluation of the following individual? Not legal Solutions • While to every solution corresponds a string, the converse is, in general, not true. • In the example above, there should not be two identical pair of bits (same position given to two numbers) • How to avoid it: • Strongly penalize the “illegal” individuals, so they have little chances to transmit they bad genes • Convert an illegal solution in a legal one, inverting the appropriate bits • Apply a crossover operator that does not produce illegal solution

11. Applying Gas (cntd) • Selection: other techniques than roulette-wheel exist for selection Eg: Remainder Stochastic Sampling decompose the pie proportionally to the values After a spin of the pie, pick an individual for each pointer In this example we would select twice, and note: the pointers are equally spaced and their number is the size of the population

12. Applying GAs Creating the new population: • Elitism: it consists in copying the best k individuals directly in the new population without combine them with any other, where k is some percentage of the population. The effect is to preserve the best solution found • Steady State: • choose two individuals from the population with roulette–wheel strategy and perform crossover to obtain their two children • “kill” two “low fitting” individuals from the population and put the children instead • Note that the concept of generation is not involved in this case

13. Exercise • Suppose you want to solve the problem of the eight queens by means of a genetic algorithm. You have 8 queens to use, therefore you cannot have solutions where less or more than 8 queens are used: such a solutions are illegal. • Give an encoding that cannot represent these illegal solutions • Suppose to have the following encoding: a solution is a string of 64 bits (a bit for each square) where 0 indicates an empty square and 1 indicates a queen in the square. Furthermore, suppose your starting population is made of only legal solutions. Define a crossover operator that produces only legal children • Can your operator be applied to any pair of legal solutions? • Is the algorithm guaranteed to produce always populations of legal solutions?