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Genetic algorithms
Optimization in DM
f’(x) = 0
f’’(x) > 0 … minimum
f’’(x) < 0 … maximum
Optimization in DM
• traditional methods (exact)
– e.g. gradient based methods
• heuristics (approximate)
– deterministic
– stochastic (chance)
• e.g. genetic algorithms, simulated annealing, ant colony
optimization, particle swarm optimization, memetic
algorithms
Optimization in DM
• Applications of optimization techniques in DM
are numerous.
• Optimize parameters to obtain the best
performance.
• Optimize weights in NN
• From many features, find the best (small)
subset giving the best performance (feature
selection).
• …
http://biology.unm.edu/ccouncil/Biology_124/Images/chromosome.gif
Biology Inspiration
• Every organism has a set of rules describing how that
organism is built up from the tiny building blocks of life. These
rules are encoded in genes.
• Genes are connected together into long strings called
chromosomes.
• Genes + alleles = genotype.
• Physical expression of the
genotype = phenotype.
locus
• gene for color
of teeth
• allele for blue
teeth
• When two organisms
mate they share their
genes. The resultant
offspring may end up
having half the genes
from one parent and half
from the other. This
process is called
recombination
(crossover).
• Very occasionally a gene
may be mutated.
http://members.cox.net/amgough/Chromosome_recombination-01_05_04.jpg
• Life on earth has evolved through the
processes of natural selection, recombination
and mutation.
• The individuals with better traits will survive
longer and produce more offsprings.
– Their survivability is given by their fitness.
• This continues to happen, with the individuals
becoming more suited to their environment
every generation.
• It was this continuous improvement that
inspired John Holland in 1970’s, to create
genetic algorithms.
GA step by step
• Objective: find the maximum of the function
O(x1, x2) = x12 + x22
– This function is called objective function.
– And it will be use to evaluate the fitness.
Adopted from Genetic Algorithms – A step by step tutorial, Max Moorkap, Barcelona, 29th November 2005
Encoding
• A model parameters (x1, x2) are encoded into a
binary strings.
– Strings with 0 and 1.
• How to encode (and decode back) real
number as a binary string?
– For each real valued variable x we need to know:
• the domain of the variable x ϵ [xL,xU]
• length of the gene k
x1 ϵ [-1, 1] x2 ϵ [0, 3.1]
5-bit
x1
x2
xL
-1
0
xU
1
3.1
Stepsize 0.0645 0.1
𝑥𝑈 − 𝑥𝐿
stepsize = 𝑛
2 −1
number = 𝑥 𝐿 + stepsize × (decoded value of string)
gene
c1 = (0101110011) → (01011) = -1 + 11 * 0.0645 = -0.29
(10011) = 0 + 19 * 0.1 = 1.9
chromosome
• At the start a population of N random models is generated
• c1 = (0101110011) → (01011) = -1 + 11 * 0.0645 = -0.29
(10011) = 0 + 19 * 0.1 = 1.9
• c2 = (1111010110) → (11110) = -1 + 30 * 0.0645 = 0.935
(10110) = 0 + 22 * 0.1 = 2.2
• c3 = (1001010001) → (10010) = -1 + 18 * 0.0645 = 0.161
(10001) = 0 + 17 * 0.1 = 1.7
• c4 = (0110100001) → (01101) = -1 + 13 * 0.0645 = -0.161
(00001) = 0 + 1 * 0.1 = 0.1
• For each member of the population calculate
the value of the objective function O(x1, x2) =
x12 + x22
O1 = O(-0.29, 1.9) = 3.69
O2 = O(0.935, 2.2) = 5.71
O3 = O(0.161, 1.7) = 2.92
O4 = O(-0.161, 0.1) = 0.04
genotype
phenotype
• Chromosome with bigger fitness has higher
probability to be selected for breeding.
• We will use the following formula
𝑃𝑖 =
𝑂𝑖
𝑛
𝑗=1 𝑂𝑗
O1 = 3.69
O2 = 5.71
O3 = 2.92
O4 = 0.04
∑Oj = 12.36
P1 = 0.30
P2 = 0.46
P3 = 0.24
P4 = 0.003
Roulette wheel
p1(30%)
p2 (46%)
p3 (24%)
P4 (0.3%)
• Now select two chromosomes according to
roulette wheel.
– Allow the same chromosome to be selected more
than once for breeding.
• These two chromosomes will:
1. cross over
2. mutate
• Let’s say c2 = (1111010110) and c3 =
(1001010001) chromosomes were selected.
• With probability Pc these two chromosomes
will exchange their parts at the randomly
selected locus (crossover point).
1
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• Crossover point is selected randomly.
• Pc generally should be high, about 80%-95%
– If the crossover is not performed, just clone two parents
into new generation.
• Pm should be very low, about 0.5%-1%
– Perform mutation on each of the two offsprings at each
locus.
• Very big population size usually does not improve
performance of GA.
– Good size: 20-30, sometimes 50-100 reported as
best
– Depends on size of encoded string
• Repeat the previous steps till the size of new
population reaches N.
– The new population replaces the old one.
• Each cycle throught this algorithm is called
generation.
• Check whether termination criteria have been
met.
– Change in the mean fitness from generation to
generation.
– Preset the number of generations.
1.
2.
3.
4.
5.
6.
[Start] Generate random population of N chromosomes (suitable solutions
for the problem)
[Fitness] Evaluate the fitness f(x) of each chromosome x in the population
[New population] Create a new population by repeating following steps
until the new population is complete
1. [Selection] Select two parent chromosomes from a population
according to their fitness (the better fitness, the bigger chance to be
selected)
2. [Crossover] With a crossover probability cross over the parents to
form a new offspring (children). If no crossover was performed,
offspring is an exact copy of parents.
3. [Mutation] With a mutation probability mutate new offspring at each
locus (position in chromosome).
4. [Accepting] Place new offspring in a new population
[Replace] Use new generated population for a further run of algorithm
[Test] If the end condition is satisfied, stop, and return the best solution in
current population
[Loop] Go to step 2.
• Check the following applets
– http://www.obitko.com/tutorials/geneticalgorithms/example-function-minimum.php
– http://userweb.elec.gla.ac.uk/y/yunli/ga_demo/
maximum
x = 6.092
y = 7.799
f(x,y)max = 100
More difficult problem
O(x) = (x - 6)(x - 2)(x + 4)(x + 6)
We search x ϵ [-10,20]
15-bit
N = 20
6 generations
O(-5.11) = -78.03
O(4.41) = -335.476
from Genetic Algorithms – A step by step tutorial, Max Moorkap, Barcelona, 29th November 2005
from Genetic Algorithms – A step by step tutorial, Max Moorkap, Barcelona, 29th November 2005
• The GA described so far is similar to Holland’s
original GA.
• It is now known as the simple genetic
algorithm (SGA).
• Other GAs use different:
– Representations
– Mutations
– Crossovers
– Selection mechanisms
Selection enhancements
• Balance fitness with diversity
– fitness is favored over variability: a set of highly fit but
suboptimal chromosomes will dominate the population,
reducing the ability of the GA to find the global optimum
– diversity is favored over fitness: model convergence will be
too slow
– e.g., one chromosome is extremely fit leading to high
selection probability. It begins to reproduce and dominates
the population (this is an example of selection pressure).
• By reducing the diversity of the population, the ability of the GA to
continue to explore new regions of the search space is impaired.
• fitness-sharing function – decrease the fitness
by the presence of similar population member
– similar = small Hamming distance
• do not allow matings between similar
chromosomes
• elitism – protect best chromosomes against
destruction by crossover, mutation
– retain certain number of chromosomes from one
generation to another
– greatly improves GA performance
• rank selection
– ranks the chromosomes according to fitness
– avoids the selection pressure exerted by the
proportional fitness method
– but it also ignores the absolute differences among
the chromosome fitnesses
– Ranking does not take variability into account and
provides a moderate adjusted fitness measure,
since the probability of selection between
chromosomes ranked k and k + 1 is the same
regardless of the absolute differences in fitness.
• tournament selection
– Run several "tournaments" among a few (k)
individuals chosen at random from the
population.
– For crossover, select best of these (the one with
the best fitness).
– Probability of selecting xi will depend on:
• Rank of xi
• Size of sample k
– Higher k increases selection pressure
• Whether contestants are picked with replacement
– Picking without replacement increases selection pressure
• Whether a fittest contestant always wins
(deterministic) or this happens with probability p
Crossover enhancements
• Multipoint crossover
• Uniform crossover
Crossover OR mutation?
• Decade long debate: which one is better
• Answer (at least, rather wide
agreement):
– it depends on the problem, but
– in general, it is good to have both
– these two have different roles
– a mutation-only-GA is possible, an xoveronly-GA would not work
• Crossover is explorative
– Discovering promising areas in the search space.
– It makes a big jump to an area somewhere “in
between” two (parent) areas.
• Mutation is exploitative
– Optimizing present information within an already
discovered promising region.
– Creating small random deviations and thereby not
wandering far from the parents.
• They complement each other.
– Only crossover can bring together information
from both parents.
– Only mutation can introduce completely new
information.
Feature Selection, Feature
Extraction
Need for reduction
• Classification of leukemia tumors from microarray
gene expression data1
– 72 patients (data points)
– 7130 features (expression levels of different genes)
• Text mining, document classification
– features are words
• Quantitative Structure-Activity Relationship
(QSAR)
– features are molecular descriptors, there exist plenty
of them
1
Xing, Jordan, Karp, Feature Selection for High-Dimensional Genomic Microarray Data, 2001
QSAR
• biological activity
– an expression describing the beneficial or adverse
effects of a drug on living matter
• Structure-Activity Relationship (SAR)
– hypotheses that similar molecules have similar
activities
• molecular descriptor
– mathematical procedure transforms chemical
information encoded within a symbolic
representation of a molecule into a useful number
Molecular descriptor
adjacency (connectivity) matrix
total adj. index AV – sum all aij
measure of the graph connectedness
Randic connectivity indices
measure of the molecular branching
2.183
QSAR
• Form a mathematical/statistical relationship (model) between structural
(physiochemical) properties and activity.
• The mathematical expression can then be used to predict the biological
response of other chemical structures.
Selection vs. Extraction
• In feature selection we try to find the best subset
of the input feature set.
• In feature extraction we create new features
based on transformation or combination of the
original feature set.
• Both selection and extraction lead to the
dimensionality reduction.
• No clear cut evidence that one of them is
superior to the other on all types of task.
Why to do it?
1. We’re interested in features – we want to
know which are relevant. If we fit a model, it
should be interpretable.
•
•
facilitate data visualization and data understanding
reduce experimental costs (measurements)
2. We’re interested in prediction – features are
not interesting in themselves, we just want to
build a good predictor.
•
•
faster training
defy the curse of dimensionality