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Genetic Algorithms

- José Galaviz Casas
- Facultad de Ciencias
- UNAM

Contents

- Introduction, motivation, fundamental concepts.
- How genetic algorithms work.
- Operators.
- Theoretical framework.
- Variations around the same theme

Nature as optimizer

- Although the human mind can create a lot of

inventions, it cannot create better, more simple

and direct inventions than nature, since in its

creations nothing is missing and nothing is

superfluous. (Leonardo da Vinci, Notebook). - Optimal individuals live in very complicated

environment. Lots of variables (atmospheric

pressure, temperature, predators, resources,

chemical substances, etc.)

How (we think that) nature works

- Evolutionary process.
- Selection of well adapted individuals. The better

the fitness is, the larger the reproduction

chance. - Searching phenotypes is hard (How can we

determine the shape, color, physiological and

functional features of an optimal marine predator

if we dont know a white shark?) - Easiest way searching genotypes. Encoding the

problem domain. - Mutation is evolutions engine.

Genetic algorithms

- Searching methods inspired by natural evolution.

Let the nature be your guide. - John Holland (60s).
- Originally A model for natural evolution.
- Now Method for optimization and machine learning.

How GA works?

- Domain encoding.
- Creation of initial population of codes. Proposed

solutions for the optimization problem. - Evaluation of fitness.
- Selection of individuals as a function of their

fitness. - Creation of new proposed solutions based on the

individuals selected. - Creation of new proposed solutions based on a

random alterations of genetic codes. - Iteration of whole process.

Domain encoding

- We must know the entire domain of problem

(phenotypes space). - We define an encoding procedure that maps

phenotypes to codes (genetic code, genotype). - Typically this is not an injective function

(several phenotypes may be mapped to the same

genotype). - We are interested in the inverse mapping.

Formally this is not a function, but we impose

restrictions. - We hope that actual solution can be obtained from

some code in the genotypes space. At least we

want some code(s) to be mapped close enough to

such solution.

Evaluation

- Fitness function. Maps every possible genotype to

an aptitude level. - Formally a non-negative function. However this is

violated in practice. Greatest value for better

individuals. - Population relative. How fast an impala must be

in order to survive a cheetah hunting?

Selection

- Proportional to fitness (simple genetic algorithm

SGA). But there are other alternatives. - Survival of the fittest.

New individuals(from selected codes)

- Given two (or more) selected individuals their

codes are mixed in order to generate offspring. - We manipulate only codes. The genotypes obtained

correspond to some phenotypes in the problems

domain, but generally we dont care about that. - Sometimes we need to guarantee that hybrid

individuals are valid phenotypes.

New individuals(from random alterations)

- Some elements in the code of new individuals are

randomly changed. - Generally we dont care about phenotypes.
- Sometimes we need to restrict the changes in

order to obtain codes for valid phenotypes.

The general procedure

- Define the problem domain encoding.
- Generate an initial population of codes

(genotypes). This will be called current

generation. - Evaluate the fitness of every individual in the

current generation. - Perform the selection of two individuals in

current generation. - Determine if the selected individuals must be

crossed. Random event pc .

- If selected individuals must be crossed, then

perform crossover, generate two offspring called

new individuals. - If selected individuals must not be crossed the

selected individuals are called new individuals. - For every new individual determine if mutation

must be performed for every element in its code.

Random event pm . - Add the two new individuals in the new

generation. - If new generation has N individuals, call it

current generation, return to step 3 until some

convergence criteria has been accomplished. - Else return to step 4.

Proportional selection

1-point crossover

- Choose a random cut point in the genetic code of

every individual. - Mix the complementary parts.

Example

Maximum in xm 7/11 0.6363... Not in genotypes

space!

Why GA works?

- We suppose individuals are encoded as binary

strings. - Schema pattern, template accomplished by several

codewords. - Example 010010110 and 111010010 are instances of

schema 101010, also are instances of

1, 100, etc. - Defining length ?(H) distance between first and

last defined position in the schema. 7 in the

example. - Order o(H) number of defined positions in the

schema. 6 in the example.

The first model

- Let m(H, t) be the number of instances of schema

H in the t-th generation of a simple genetic

algorithm. - We assume
- proportional selection
- 1-point crossover P(breaking H) ?(H)/(l-1)

where l is the string length. - uniform mutation P(survival H) ? 1- o(H) pm

The schema theorem

- The expected number of instances of schema H in

generation at time t1

Not very useful

- Only disruptive effects of genetic operators are

considered. - Only a lower bound, not very tight. Long time

behavior is not accurately predicted. - Very particular.

Other kinds of crossover

2-point crossover

Uniform crossover

Some crossover operators

The building block hypothesis

- Since the schemas survive easily if
- They are short
- They are good (high fitness)
- Therefore the solutions obtained by the GA must

be constructed using schemas with these

characteristics, building blocks. - Contradictory evidence (hitchhicking).

Implicit parallelism

- Every binary string of length l is instance of 2l

schemas. - Evaluation of one string is the implicit

evaluation of a sample of exponentially many

schemas.

Exploitation Vs. exploration

- There are two opposite forces working in a GA.
- Selection pressure. Exploitation of acquired

knowledge. - Mutation. Random exploration of search space.
- Selection causes convergence, even to a

sub-optimal solution. Gravity. - Mutation favors finding of optimal solution, but

causes divergence. Expansive pressure. - Trade-off emphasizing one of them diminishes the

other. Impact in performance and/or robustness.

Two armed bandit

- Bandit machine, two arms, the payoff of each arm

is a normally distributed random variable. The

mean of one of the arms is higher, but we doesnt

know which one. - We have a limited amount of money for exploration

and exploitation, simultaneously. - How the sampling must be performed?
- Answer the arm with the current best observed

mean must receive exponentially many more

experiments than the other one.

GAs and the bandit

- If we consider proportional selection only.
- Let H be a schema with above average fitness

- The schema theorem says

The building blocks hypo

- Mitchell-Holland-Forrest, 1994. Royal Roads.
- Functions created ad-hoc to support the BBH.
- Genetic algorithm carefully adapted to prevent

premature convergence. - Three hillclimbers for comparison purposes. SAHC,

NAHC, RMHC. - RMHC outperforms GA. Oops!
- Spurious correlation (hitchhicking).

The idealized GA (IGA)

- Works on single strings, not population.
- Always preserve the best code found.
- Chooses new individual randomly, if such

individual is better than the current best

individual, cross them.

How can we reach the desired IGA?

- Can be approximated by a GA if
- There are no locus with fixed value in high

proportion of population. - Good schemas must be favored by strong enough

selection, but also hitchhicking must be avoided. - Crossover probability must be high enough to

guarantee that good schemas have enough diffusion

in population.

Alternative GAs

- Elitism. Important feature. It has been proved

that elitism is the sufficient condition for

convergence to the optimal. - Deterministic selection schemes. In a population

sorted decreasingly by fitness - Nietzsche. The i-th individual is crossed with

the (i1)-th. - Vasconcelos. The i-th individual is crossed with

the (N-i)-th. Good approximation to IGA. Has been

statistically proved (Kuri, 2002) that

GA(Vasconcelos)elitism achieve best performance. - Self-adaptation. The control parameters such pm

and pc, are encoded in the individuals. - AGhillclimbers or AGcatastrophic events.
- Co evolution.

Variations

- Several encoding schemes binary (Gray, weighted

positional, etc.), non-binary. - Ad-hoc operators problem-dependent. Panmictic

(orgiastic) crossover operators. - Knowledge-based biased initial population.
- Several selection schemes fitness based

(proportional, sigma truncation), tournament,

ranking (linear, non-linear). - Additional features. Imagination is the limit.

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