Genetic%20Algorithms

1
Genetic Algorithms

• José Galaviz Casas
• Facultad de Ciencias
• UNAM

2
Contents

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

3
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.)

4
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.

5
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.

6
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.

7
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.

8
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?

9
Selection

• Proportional to fitness (simple genetic algorithm
  SGA). But there are other alternatives.
• Survival of the fittest.

10
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.

11
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.

12
The general procedure

1. Define the problem domain encoding.
2. Generate an initial population of codes
   (genotypes). This will be called current
   generation.
3. Evaluate the fitness of every individual in the
   current generation.
4. Perform the selection of two individuals in
   current generation.
5. Determine if the selected individuals must be
   crossed. Random event pc .

13

1. If selected individuals must be crossed, then
   perform crossover, generate two offspring called
   new individuals.
2. If selected individuals must not be crossed the
   selected individuals are called new individuals.
3. For every new individual determine if mutation
   must be performed for every element in its code.
   Random event pm .
4. Add the two new individuals in the new
   generation.
5. If new generation has N individuals, call it
   current generation, return to step 3 until some
   convergence criteria has been accomplished.
6. Else return to step 4.

14
Proportional selection
15
1-point crossover

• Choose a random cut point in the genetic code of
  every individual.
• Mix the complementary parts.

16
Example
 
Maximum in xm 7/11 0.6363... Not in genotypes
space!
17
 
 
18
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.

19
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

20
The schema theorem

• The expected number of instances of schema H in
  generation at time t1

21
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.

22
Other kinds of crossover
2-point crossover
Uniform crossover
23
Some crossover operators
24
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).

25
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.

26
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.

27
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.

28
GAs and the bandit

• If we consider proportional selection only.
• Let H be a schema with above average fitness
  
• The schema theorem says

29
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).

30
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.

31
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.

32
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.

33
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.