Outline
CS 6776
Evolutionary Computation
January 21, 2014
Fitness Function
•  A mathematical function that quantifies
how good a solution is.
•  A problem can be modelled as a
maximization or a minimization problem.
•  Example:
•  TSP: A tour X = {x i },i = 1…n
•  minimize
f ( X ) = ∑∑ d
n
•  Problem modeling includes representation
design and Fitness Function definition.
•  Fitness function:
–  Unconstrained optimization/modeling
–  Constrained optimization/modeling
–  Multi-objective optimization/modeling
–  Relative fitness: co-evolution
•  Population size and population models
•  Convergence and termination criteria
Numerical Optimization
•  Maximize f (x1,x 2 ) = x12 + x 22,
−5.0 ≤ x1 ,x 2 ≤ 5.0
How many
optima?
What are they?
€
n
ij
i =1 j =1
€
⎧ dist( x i , x j ), x to x is in the tour
i
i
dij = ⎨
0,
⎩
otherwise
€
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Regression Model
Example
•  Regression models, such as symbolic
regressions, where the output is a real
value, the most commonly used fitness
function is minimizing error:
N
∑( y
–  MSE: mean squared error
f (x) =
i
− yˆ i )
• The true value of the
outliner point y = 60:
2
f (x) =
–  the predicted value
ŷ=80:
–  MSE: |80-60|^2=400
–  MAE: |80-60|=20
•  Non-liner model:
i=1
N
•  Biased more weighted on outliers
•  Weights large errors more heavily than small ones
N
€
∑ y i − yˆ i
–  MAE: mean absolute error
•  Less biased
•  Liner model:
–  the predicted value
ŷ=80:
–  MSE: |80-70|^2=100
–  MAE: |80-70|=10
i=1
N
€
Classification Model
Constrained Optimization
•  One way to handle solutions that violate the
constraints is to assign penalty p in the fitness
function:
•  Classification models, where the output is
a discrete label, the most commonly used
fitness function is maximizing accuracy:
f (x)' = f (x) ± p
N
∑t
f ( x) =
ti
i
i =1
,
N
⎧1, y i = y
ˆi
= ⎨
ˆi
⎩0, y i ≠ y
•  Problematic when the data set is
€
imbalanced.
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•  For maximization problems, p is subtracted from
f(x).
•  For minimization problems, p is added to f(x)
•  There are other constraints handling methods,
other than sum-penalty.
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Multi-Objective Optimization
Relative Fitness
•  Problems require satisfying more than one objective: e.g.
maximizing profit & minimizing cost.
•  Weighted sum approach:
–  Converts all fitness function for maximization, e.g.
convert fitness by multiply -1;
λ = [w , w ], w ≥ 0, ∑ w = 1
•  The fitness of an individual is measured in
relation to that of other individuals in the
same population or in a competing
population.
•  Interaction Patterns:
m
1
m
i
i
i =1
m
∑ w f ( x)
•  Maintain a set of Pareto Front solutions:
–  Fitness is a vector F(x) = [ f1 (x), f 2 (x), f 3 (x) f m (x)]
€
–  Select Pareto-optimal solution (discussed in later
lecture)
F ( x) =
i
i
i =1
–  Single population
•  Relative (competitive) fitness vs. absolute fitness
–  Multiple populations
•  All vs. previous-best
•  Neighborhood interaction
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Standard Evolutionary Algorithm
•  Individual fitness is based on its absolute
performance without interacting with
others.
Co-evolution – Single Population
•  Individuals are evaluated by having them
interact with each other, e.g. play a game.
Play N games with randomly selected N individuals from the population
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Co-evolution – Multiple Populations
•  In asymmetric games (the strategies for
player 1 are different from that of the
player 2), each member of Pop. 1 interacts
with each member of Pop. 2.
Population Size
•  Intuitively, the population size can be
viewed as a measure of the degree of
parallel search an EA supports.
•  A larger population size provides:
–  better coverage of the search space
(diversity) which helps high fitness individuals
to be included in the initial population
–  a larger past memory, so that good individuals
do not lost so quickly during evolution.
Evolutionary Algorithms Workflow
•  To design an effective
evolutionary
algorithm, one need
to consider the
problem at hand.
Population Size - Continued
•  Depending on the complexity of the
problem fitness landscape, different
population size is needed to search a
solution.
•  Increase population size beyond
necessary would take EA longer to find a
solution.
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Generational Model
•  Non-overlapping Population:
gen-0
gen-1
gen-2
gen-n
•  Canonical Genetic Algorithms:
•  Parent selection only; all offspring are kept in
the following generation.
•  Evolution Strategies (µ, λ):
•  Random parent selection to generate a large
number of offspring;
•  Select fitter offspring to form the new
generation.
Steady-state Model
•  Overlapping Population
parents
pop
replace
offspring
•  Generation Gap:
–  The proportion of the population that is replaced
[Sarma & De Jong, 1995].
–  Generational model: pop_size/pop_size=1.0
–  Steady-state model: #_replaced/pop_size
Generational Model - Continued
•  Under stochastic selection, an individual,
regardless how fit it is, may only live for
one generation, hence has a short-term
impact on evolution.
•  During stochastic selection, best solutions
might get lost and are never carried over
to new generation.
•  These can be fixed by deterministic
selection, or elite selection.
Steady-state Model - Continued
•  Offspring and parents compete to survive
in the population.
•  Fit individual can live for a long period of
time to impact the evolution.
•  A fit offspring can have impact on the
evolution immediately after its birth,
without waiting until the next generation.
•  Impact evolutionary search?
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EA Termination Criteria
•  EA search process termination criteria:
–  When a specified number of generation is
reached.
–  When the known best solution is found.
–  When the population is “converged”: no
further changes in the population may occur.
Convergence
•  Practical ways to detect convergence:
–  Measure the degree of homogeneity of the
population using spatial dispersion or entropy:
When the homogeneity measure approaches
0, the population is converged.
–  Measure the global fitness improvement:
When the best fitness does not improve for a
certain number of generation (typically 10-20),
the population is converged.
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