Transcript Genetic Algorithms

Genetic Algorithms
In part from: https://www.cs.wmich.edu/~elise/courses/cs6800/GeneticAlgorithms.ppt
Introduction
• After scientists became disillusioned with
classical and neo-classical attempts at modeling
intelligence, they looked in other directions.
• Two prominent fields arose, connectionism
(neural networking, parallel processing) and
evolutionary computing.
• genetic algorithms are a form of evolutionary
computing
• The belong to the category of untrained
reinforcement-based problem solving
algorithms
What is GA
• A genetic algorithm (or GA) is a search technique
used in computing to find true or approximate
solutions to optimization and search problems.
• Genetic algorithms are categorized as global search
heuristics.
• Genetic algorithms are a particular class of
evolutionary algorithms that use techniques inspired
by evolutionary biology such as inheritance,
mutation, selection, and crossover (also called
recombination).
What is GA
• Genetic algorithms are implemented as a computer
simulation in which a population of abstract
representations (called chromosomes or the genotype or
the genome) of candidate solutions (called individuals,
creatures, or phenotypes) to an optimization problem
evolves toward better solutions.
• In other words: An iterative process during which an
optimization problem is “encouraged” to produce better
solutions at each iteration
• Traditionally, solutions are represented as binary strings
of 0s and 1s, but other encodings are also possible.
What is GA
• The evolution usually starts from a population of
randomly generated individuals and happens in
generations.
• In each generation, the fitness (=evaluation function)
of every individual (=solution) in the population is
evaluated, multiple individuals are selected from the
current population (based on their fitness), and
modified (recombined and possibly mutated) to form
a new population.
What is GA
• The new population is then used in the next
iteration of the algorithm.
• Commonly, the algorithm terminates when
either a maximum number of generations has
been produced, or a satisfactory fitness level
has been reached for the population.
• If the algorithm has terminated due to a
maximum number of generations, a
satisfactory solution may or may not have
been reached.
GA terminology summary
• Individual - Any possible solution (e.g. best strategy
for a robot to perform a given task)
• Population - Group of all individuals (e.g. alternative
strategies)
• Search Space - All possible solutions to the problem
• Chromosome/genotype/genome – “Blueprint” for
an individual (abstract representation of a solution,
e.g. a graph representing a plan) NOTE: in biology
these are different things..
• Trait - Possible aspect (features) of an individual
(e.g. number of states in a plan)
In Biology
• All living organisms consist of cells, and each cell contains the
same set of one or more chromosomes—strings of DNA—that
serve as a "blueprint" for the organism.
• A chromosome can be conceptually divided into genes— each
of which encodes a particular protein. Very roughly, one can
think of a gene as encoding a trait, such as eye color.
• The different possible "settings" for a trait (e.g., eyecolor=
blue, brown, hazel) are called alleles. Each gene is located at a
particular locus (position) on the chromosome.
• The complete collection of genetic material (all chromosomes
taken together) is called the organism's genome.
• The term genotype refers to the particular set of genes
contained in a genome.
Chromosome, Genes and
Genomes
Genotype and Phenotype
• Genotype:
– Particular set of genes in a genome
• Phenotype:
– Physical characteristic of the genotype
(smart, beautiful, healthy, etc.)
Genotype and Phenotype
With reference to our previous ML
terminology
• First, we do no represent objects that we need to
classify, but rather solutions to a problem that we want
to solve
• The representation of a solution is called chromosome,
and its elements are genes
• A chromosome is a bit like a feature vector, features
are a bit like genes, traits are like feature values
• But here features (genes) can be related to each other
– more often representations are not “flat” vectors of
features (as for in supervised classification), rather,
they are structured (e.g. a graph or a set of rules)
GA Requirements: representation
• A typical genetic algorithm requires two things to be defined:
– a genetic representation of the solution domain, and
– a fitness function to evaluate the solution domain.
• A common representation of the solution is as an array of bits
(mapping to a structure, e.g. a plan, an architecture..) . Arrays
of other types and structures can be used in essentially the
same way.
• The main property that makes these binary “genetic”
representations convenient is that their parts are easily
aligned due to their fixed size, that facilitates simple crossover
operation.
• Variable length representations may also be used, but
crossover implementation is more complex in this case.
Representation examples
Chromosomes could be:
– Bit strings
(0101 ... 1100)
– Real numbers
(43.2 -33.1 ... 0.0 89.2)
– Permutations of element (E11 E3 E7 ... E1 E15)
– Lists of rules
(R1 R2 R3 ... R22 R23)
– Program elements
(genetic programming)
– ... any data structure ...
Representation example
(chromosomes= rules)
• R: If (input1=Low OR High)AND(input2=High)
THEN Output=Low
• Rules of this type can be encoded in 9 bits
where the first 6 represent the antecedent
(alternative values of Input 1 and 2) and the
last 3 bits the value(s) of the consequent
• E.g. for the first 3 bits: 000 i1=no contraints
;001i1=high; 011i1=low or high; etc.
More examples of chromosome types
• Chromosomes can represent
different architectures of neural
networks (e.g. different number of
layers, different number of nodes
per layers)
• Chromosomes can represent
alternative “body” structures for
creatures
• Chromosomes can be alternative
strategies for a prey to escape the
predator (or for a predator to
capture a prey)
• In all these cases a chromosome
encodes a graph structure
Evolving swimming creatures
Chromosome ≠ feature vector
• In previous example the “architecture” of creatures is
a set of joints (nodes) and elastic edges: elements of a
representation are structurally connected!
• Note also that a solution is made of two aspects in the
swimming creatures examples: the body of the
creature, and the strategy to swim (which also is to be
mapped into an abstract “chromosome”
representation)
• Multiple problems (e.g. best body for best swimming
stategy) can be solved jointly or separately
GA Requirements :Fitness function
• The fitness function is defined over the genetic representation
and measures the quality of the represented solution.
• The fitness function is always problem dependent.
• Usually the fitness function is defined with relation to the
objective task: for example, for flying creatures, the time they
are able to fly before the fall down
• If we evolvea neural network architectures, the fitness will be
the classification precision of that particular architecture and
the convergence time
• If the evolve a driving agent, the fitness can be the number of
times it correctly stops when recognizing and obstacle, or the
time to reach a target location, etc.
A fitness function produces a
ranking of individuals
Basics of GA: algorithm steps
• The most common type of genetic algorithm works like this:
– a population is created with a group of individuals created
randomly.
– The individuals in the population are then evaluated.
– The evaluation function (fitness) is provided by the
programmer and gives the individuals a score based on
how well they perform at the given task.
– A subset of individuals are then selected based on their
fitness, the higher the fitness, the higher the chance of
being selected (sample probability is fitness-dependent).
– These individuals are paired and then "reproduce" to
create one or more “offspring”, after which the offspring
are “mutated” randomly.
– This continues until a suitable solution has been found or a
certain number of generations have passed, depending on
the needs of the programmer.
Let’s take a problem: motion control
• Motion control applications can be found in
almost every sector of industry, from factory
automation and robotics to high-tech computer
hard-disk drives.
• They are used to regulate mechanical motions in
terms of position, velocity, and acceleration,
and/or to coordinate the motions of multiple
axes or machine parts.
• How to select the most appropriate control law
and its parameters?
Encoding the parameters (1)
• This implies creating a representation for each
parameter setting (=a solution) and a stepsize
(the increment/decrement of parameter values)
1. Model parameters to a binary string
(chromosome):
Each value is a setting of a parameter (e.g. velocity,
acceleration, position of a mechanical arm, joint rotation..
Encoding parameters (2)
• For each parameter, a minimum value,
stepsize and encoding length has to be prespecified :
Generating a population
• Initialization
• Initially many individual solutions are randomly
generated to form an initial population P.
• The population size depends on the nature of the
problem, but typically contains several hundreds or
thousands of possible solutions.
• Traditionally, the population is generated randomly,
covering the entire range of possible solutions (the
search space).
• Occasionally, the solutions may be "seeded" in areas
where optimal solutions are likely to be found (if some
initial knowledge is available).
Initial population P
Rating the individuals of a population
P
• A fitness function is computed (as we said
fitness depends on the application domain)
for every individual of P
• For example, let’s say that (un)fitness function
is: O(x1,x2)=x12+x22
• f(x)=1/O(x)
Creating successive generations
• Selection
• During each successive generation, a
proportion p of the existing population P:
p1,p2..pn is selected to breed a new
generation.
• Individual solutions are selected through a
fitness-based process, where “fitter” solutions
(as measured by a fitness function) are
typically more likely to be selected.
Creating successive generations (2)
• Certain selection methods rate the fitness of each
solution and preferentially select the best
solutions. Other methods rate only a random
sample of the population, as this process may be
very time-consuming.
• Most functions are stochastic and designed so
that also a small proportion of less fit solutions
are selected. This helps keep the diversity of the
population large, preventing premature
convergence on poor solutions.
• Popular and well-studied selection methods
include roulette wheel selection and tournament
selection.
Creating successive generations (3)
• In roulette wheel selection, individuals are
given a probability of being selected that is
directly proportionate to their fitness.
• Two individuals are then chosen randomly
based on these probabilities and produce
offspring.
Example (roulette wheel)
• Based on the objective function value, a
probability is assigned to make it into the next
generation P(pj ) = f (1/O(xi )). In our example:
NOTE: individuals with a higher fitness are more likely to be selected
in the new generation. Also note than an individual can be sampled
more than once.
Modify selected members of
population (4)
• Reproduction
• The next step is to generate a second generation population
of solutions from those selected through genetic operators:
crossover (also called recombination), and/or mutation.
• For each new solution to be produced, a pair of "parent"
solutions is selected for breeding from the pool selected
previously.
• By producing a "child" solution using the above methods of
crossover and mutation, a new solution is created which
typically shares many of the characteristics of its "parents".
New parents are selected for each child, and the process
continues until a new population of solutions of appropriate
size is generated.
Crossover
Here the “random” point is 3, e.g. parents exchange their chromosome after bit 3,
generating two new individuals
Mutuation
Example
With probability (1-Pc) no
crossover
With probability Pc
crossover, and random
crossover point=4
With probability Pm
permutation
Iterate
• These processes ultimately result in the next
generation population of chromosomes that is
different from the initial generation.
• Generally the average fitness will have increased
by this procedure for the population, since only
the best organisms from the first generation are
selected for breeding, along with a small
proportion of less fit solutions, for reasons
already mentioned above.
Crossover (more on)
• The most common type is single point crossover. In single
point crossover, you choose a locus at which you swap the
remaining alleles from on parent to the other (e.g. you swap
eye color values of eye color gene). This is complex and is best
understood visually.
• The point at which the chromosome is broken depends on the
randomly selected crossover point (but should be at the
beginning –locus- of a gene).
• This particular method is called single point crossover because
only one crossover point exists. Sometimes only child 1 or
child 2 is created, but oftentimes both offspring are created
and put into the new population.
• Crossover does not always occur, however. Sometimes, based
on a probability (Pc), no crossover occurs and the parents are
copied directly to the new population. The probability of
crossover occurring is usually 60% to 70%.
Single point crossover
Mutation (more on)
• After selection and crossover, you now have a new population full
of individuals.
• Some are directly copied, and others are produced by crossover.
• In order to ensure that the individuals are not all exactly the same,
you allow for a small chance of mutation.
• You loop through all the alleles of all the individuals, and if that
allele is selected for mutation, you can either change it by a small
amount or replace it with a new value. The probability of mutation
is usually between 1-2%.
• Mutation is fairly simple. You just change the selected alleles based
on what you feel is necessary and move on. Mutation is, however,
vital to ensuring genetic diversity within the population.
• In other terms it avoids that at some point all the members of a
generation are too similar to each other (like a “local” maximum)
Mutation
Termination condition (5)
• This generational process is repeated until a
termination condition has been reached.
• Common terminating conditions are:
–
–
–
–
A solution is found that satisfies minimum criteria
Fixed number of generations reached
Allocated budget (computation time/money) reached
The highest ranking solution's fitness is reaching or has
reached a plateau such that successive iterations no longer
produce better results
– Manual inspection
– Any combinations of the above
GA Pseudo-code (summary)
Choose initial population
Evaluate the fitness of each individual in the population
Repeat
Select best-ranking individuals to reproduce
Breed new generation through crossover and mutation (genetic
operations) and give birth to offspring
Evaluate the individual fitnesses of the offspring
Replace worst ranked part of population with offspring
Until <terminating condition>
An example of application: Evolving
Neural Networks
• Evolving the architecture of neural network is
slightly more complicated, and there have
been several ways of doing it.
• For small nets, a simple matrix represents
which neuron connects which, and then this
matrix is, in turn, converted into the necessary
'genes', and various combinations of these are
evolved.
Other Areas
• Genetic Algorithms can be applied to virtually any problem
that has a large search space.
• Genetic algorithms used to filter out 'good' and 'bad' riffs for
jazz improvisation.
• The military uses GAs to evolve equations to differentiate
between different radar returns.
• Stock companies use GA-powered programs to predict the
stock market.
Checkboard example
– We are given an n by n checkboard in which every field can
have a different colour from a set of four colors.
– Goal is to achieve a checkboard in a way that there are no
neighbours with the same color (not diagonal)
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
Checkboard example Cont’d
– Chromosomes represent the way the checkboard is
colored.
– Chromosomes are not represented by bitstrings but by
bitmatrices
– The bits in the bitmatrix can have one of the four values 0,
1, 2 or 3, depending on the color.
– Crossing-over involves matrix manipulation instead of
point wise operating.
– Crossing-over can be combining the parential matrices in a
horizontal, vertical, triangular or square way.
– Mutation remains bitwise changing bits in either one of
the other numbers.
Checkboard example Cont’d
• This problem can be seen as a graph with n nodes
and (n-1) edges, so the fitness f(x) is defined as:
f(x) = 2 · (n-1) ·n
Checkboard example Cont’d
• Fitnesscurves for different cross-over rules:
Fitness
Lower-Triangular Crossing Over
Square Crossing Over
180
180
170
170
160
160
150
150
140
140
130
0
100
200
300
400
500
130
0
Fitness
Horizontal Cutting Crossing Over
180
170
170
160
160
150
150
140
140
0
200
400
Generations
600
400
600
800
Verical Cutting Crossing Over
180
130
200
800
130
0
500
1000
Generations
1500
Evolving body shapes
• embedded graphs to
represent body shape
and controller
• different environments
and/or fitness functions
to evolve various
behaviours, e.g.
– Swimming
– Walking
– Jumping
– Following
We end with something you saw at the
beginning..