Parameter Control

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Chapter 8
Parameter Control
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Motivation 1
An EA has many strategy parameters, e.g.
 mutation operator and mutation rate
 crossover operator and crossover rate
 selection mechanism and selective pressure (e.g.
tournament size)
 population size
Good parameter values facilitate good performance
Q1 How to find good parameter values ?
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Motivation 2
EA parameters are rigid (constant during a run)
BUT
an EA is a dynamic, adaptive process
THUS
optimal parameter values may vary during a run
Q2: How to vary parameter values?
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Parameter tuning
Parameter tuning: the traditional way of testing and
comparing different values before the “real” run
Problems:
 users mistakes in settings can be sources of errors or suboptimal performance
 costs much time
 parameters interact: exhaustive search is not practicable
 good values may become bad during the run
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Parameter control
Parameter control: setting values on-line, during the
actual run, e.g.
 predetermined time-varying schedule p = p(t)
 using feedback from the search process
 encoding parameters in chromosomes and rely on natural selection
Problems:
 finding optimal p is hard, finding optimal p(t) is harder
 still user-defined feedback mechanism, how to ``optimize"?
 when would natural selection work for strategy parameters?
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Example
Task to solve:
 min f(x1,…,xn)
 Li  xi  Ui for i = 1,…,n
 gi (x)  0
 hi (x) = 0
bounds
for i = 1,…,q
for i = q+1,…,m
inequality constraints
equality constraints
Algorithm:
 EA with real-valued representation (x1,…,xn)
 arithmetic averaging crossover
 Gaussian mutation: x‟ i = xi + N(0, )
standard deviation  is called mutation step size
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Varying mutation step size: option1
Replace the constant  by a function (t)
 (t )  1 - 0.9  Tt
0  t  T is the current generation number
 Features:
 changes in  are independent from the search progress
 strong user control of  by the above formula
  is fully predictable
 a given  acts on all individuals of the population
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Varying mutation step size: option2
Replace the constant  by a function (t) updated after every
n steps by the 1/5 success rule (cf. ES chapter):
 (t  n ) / c if ps  1/5

 (t )   (t  n )  c if ps  1/5
 (t  n )
otherwise

 Features:
 changes in  are based on feedback from the search progress
 some user control of  by the above formula
  is not predictable
 a given  acts on all individuals of the population
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Varying mutation step size: option3
 Assign a personal  to each individual
 Incorporate this  into the chromosome: (x1, …, xn, )
 Apply variation operators to xi„s and 
     e N ( 0, )
xi  xi  N (0, )
 Features:
 changes in  are results of natural selection
 (almost) no user control of 
  is not predictable
 a given  acts on one individual
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Varying mutation step size: option4
Assign a personal  to each variable in each individual
Incorporate ‟s into the chromosomes: (x1, …, xn, 1, …,  n)
Apply variation operators to xi„s and i„s
N ( 0, )

 i  i  e
xi  xi  N (0, i )
 Features:
 changes in i are results of natural selection
 (almost) no user control of i
 i is not predictable
 a given i acts on 1 gene of one individual
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Example cont‟d
Constraints
 gi (x)  0
 hi (x) = 0
for i = 1,…,q
for i = q+1,…,m
inequality constraints
equality constraints
are handled by penalties:
eval(x) = f(x) + W × penalty(x)
where
1 for violated constraint
penalty ( x )   
j 1 0 for satisfied constraint
m
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Varying penalty: option 1
Replace the constant W by a function W(t)
W (t )  (C  t )
α
0  t  T is the current generation number
 Features:
 changes in W independent from the search progress
 strong user control of W by the above formula
 W is fully predictable
 a given W acts on all individuals of the population
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Varying penalty: option 2
Replace the constant W by W(t) updated in each generation
 β  W(t) if last k champions all feasible

W (t  1)  γ  W(t) if last k champions all infeasible
W(t)
otherwise

 < 1,  > 1,     1 champion: best of its generation
 Features:
 changes in W are based on feedback from the search progress
 some user control of W by the above formula
 W is not predictable
 a given W acts on all individuals of the population
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Varying penalty: option 3
Assign a personal W to each individual
Incorporate this W into the chromosome: (x1, …, xn, W)
Apply variation operators to xi„s and W
Alert:
eval ((x, W)) = f (x) + W × penalty(x)
while for mutation step sizes we had
eval ((x, )) = f (x)
this option is thus sensitive “cheating”  makes no sense
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Lessons learned from examples
Various forms of parameter control can be distinguished by:
 primary features:
 what component of the EA is changed
 how the change is made
 secondary features:
 evidence/data backing up changes
 level/scope of change
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What
Practically any EA component can be parameterized and
thus controlled on-the-fly:
 representation
 evaluation function
 variation operators
 selection operator (parent or mating selection)
 replacement operator (survival or environmental selection)
 population (size, topology)
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How
Three major types of parameter control:
 deterministic: some rule modifies strategy parameter
without feedback from the search (based on some
counter)
 adaptive: feedback rule based on some measure
monitoring search progress
 self-adaptative: parameter values evolve along with
solutions; encoded onto chromosomes they undergo
variation and selection
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Global taxonomy
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Evidence informing the change
The parameter changes may be based on:
 time or nr. of evaluations (deterministic control)
 population statistics (adaptive control)
 progress made
 population diversity
 gene distribution, etc.
 relative fitness of individuals creeated with given values
(adaptive or self-adaptive control)
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Evidence informing the change
 Absolute evidence: predefined event triggers change, e.g.
increase pm by 10% if population diversity falls under
threshold x
 Direction and magnitude of change is fixed
 Relative evidence: compare values through solutions
created with them, e.g. increase pm if top quality offspring
came by high mut. Rates
 Direction and magnitude of change is not fixed
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Scope/level
The parameter may take effect on different levels:
 environment (fitness function)
 population
 individual
 sub-individual
Note: given component (parameter) determines possibilities
Thus: scope/level is a derived or secondary feature in the
classification scheme
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Refined taxonomy
 Combinations
of types and evidences
 Possible:
+
 Impossible: -
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Evaluation / Summary
 Parameter control offers the possibility to use appropriate
values in various stages of the search
 Adaptive and self-adaptive parameter control
 offer users “liberation” from parameter tuning
 delegate parameter setting task to the evolutionary process
 the latter implies a double task for an EA: problem solving + self-
calibrating (overhead)
 Robustness, insensivity of EA for variations assumed
 If no. of parameters is increased by using (self)adaptation
 For the “meta-parameters” introduced in methods
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