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Thunder Lecture I ?????? ??????????Ingo
Rechenberg????????????? ??????10?14?(??)180020
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CNS??
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Evolution Strategy
Natures way of optimization
Ingo Rechenberg
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How Evolution Strategy works
Shanghai Institute for Advanced Studies
Technische Universität Berlin
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Search for a document
(Search)Strategies are of no use in an disordered
world
(Search)Strategies need a predictable order of
the world
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An optimization strategy
(the Evolution Strategy)
makes use of some order principles of the world
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An universal world order is
Causality
Weak Causality
Strong Causality
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Search area
Experimenter
Plumbing the depth
The search for the optimum
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Search area
Experimenter
Plumbing the depth
The search for the optimum
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Strong Causality
Strategies need a predictable order of the world
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Solve
when n1 to n6 are natural numbers
and you get famous !!!
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Edge was too small to note the proof
For m gt 2
Pierre de Fermats print of Diophants Arithmetica
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No Solution ! (Fermat, Wiles)

EULERs conjecture No solution !

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Euler has been mistaken
!
958004 2175194 4145604 4224814
(Frye, 1988)
!
275 845 1105 1335 1445
(Lander/Parkin, 1966)
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Minimize exactly
when n1 to n6 are natural numbers and you get
famous !!!
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Minimize exactly
when n1 to n5 are natural numbers
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Evolutionary Computation (1 , 4 (1 , 100) 200
-ES
676 1246 4566 8846 13276
(1346.00000000004163)6
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Weak Causality
Strategies are of no use in a disordered world
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1. Global deterministic search
2. Global stochastic search
3. Local deterministic search
4. Local stochastic search
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1. Global deterministic search
Systematic scanning of the variable space
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2. Global stochastic search
To find the target with 95 probability
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1. Global deterministic search
2. Global stochastic search
3. Local deterministic search
4. Local stochastic search
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Definition of a local convergence measure
The rate of progress
j
distance moved uphill
j
number of generations
Condition Strong Causality !!!
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d
Progress
d
Linearity radius
3. Local deterministic search
Walking following the steepest ascent
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2. Offspring
Parent
1. Offspring
d
Linearity radius
4. Local stochastic search
Random drifting along the steepest ascent
n gtgt 1
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Plus-offspring
Center of gravity
Minus-offspring
Parent
Linearity radius
Determiation of the linear rate of progress
Statistical mean of the progress
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Center of gravity
s
s
s
n Dim.
2 Dim.
3 Dim.
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2. Offspring
Parent
1. Offspring
d
Linearity radius
4. Local stochastic search
Random drifting along the steepest ascent
n gtgt 1
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Gradient Strategie contra Evolution Strategy
For n gtgt 1
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Gradient Strategy
contra
Evolution Strategy
Linear local climbing theory within a strong
causal optimization landscape
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Algorithm of the (1 1) - ES

Arbitrarily large ?
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Where is the optimum ???
End of the linearity
Global stochastic search
Search for the maximum rate of progress
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Nonlinear models
Near to the optimum
Far from the optimum
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Two solutions for the (11)-ES
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DARWINs theory in maximal abstraction
More correct imitation of the Biological Evolution
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Basic-Algorithm of the (1, l ) Evolution
Strategie
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The general idea (in 1 dimension)
Set of functions
The TAYLOR series expansion in the MACLAURIN form
!
All functions have the same form
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TAYLOR series expansion in n dimensions
(MACLAURIN series)
Transformation to the principle axes
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Tabel
of the progress coefficients
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r
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D
D
2
F

-
Central law of progress
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The Evolution Strategist
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not so
but so
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Demonstration of the necessity of a step-size
regulation
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Definition of the success probability
The theory gives the result
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Step-size adaptation D using the success-rule
D ?
D ?
0.227
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1 / 5
Development of the 1/5-Success-Rule
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We gt 1/5
We lt 1/5
Cosmic ray
Mutations
Biologically impossible
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Assessment of the climbing style
Climbing alone
Climbing in a group
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Duplicator
DNA
Mutation
cator
dupli
the
made
Has
Heredity of the mutability
Crucial point of the Evolution Strategy
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Algorithm of the (1, l ) Evolution Strategy
with MSC
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Fraidycat
Columbus
Amundsen
Hothead
Four mountaineers, four climbing styles
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Four moutaineers, four climbing styles
In a compact notation
Nested Evolution Strategy
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On the way to an evolution-strategic algebra
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On the way to an evolution-strategic algebra
,
m
,
l
)
(
- ES
1

1

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On the way to an evolution-strategic algebra
,
r
m
l
)
(
- ES
/

Example r 2
,
m
l
(
)
- ES

/
2
Only half of the parental information builds up
an offspring
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On the way to an evolution-strategic algebra
g
,
m
l
)
(
- ES

Example
4
(1 6)
(1 6)
(1 6)
(1 6)
(1 6)

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On the way to an evolution-strategic algebra
g
g
?
,
,
?

?
m
m
l

)
l
(
- ES


Biological equivalent to the strategy nesting
Family ? Genus Species Variety ( Individual
) ?
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Nested Evolution Strategy
?
g
g
,
,
?

m
?
m
l

)
l
(
- ES


Adaptation of the objektive variables xk
Adaptation of the mutation size d
to operate in the Evolution Window!
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MATLAB-program of the (1, l )-ES
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MATLAB-program of the (1, l )-ES
v100 de1 xeones(v,1)
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MATLAB-program of the (1, l )-ES
v100 de1 xeones(v,1) for
g11000 end
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MATLAB-program of the (1, l )-ES
v100 de1 xeones(v,1) for g11000
qb1e20 end
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MATLAB-program of the (1, l )-ES
v100 de1 xeones(v,1) for g11000
qb1e20 for k110 end end
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MATLAB-program of the (1, l )-ES
v100 de1 xeones(v,1) for g11000
qb1e20 for k110 if rand lt 0.5
dnde1.3 else dnde/1.3
end end end
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MATLAB-program of the (1, l )-ES
v100 de1 xeones(v,1) qesum(xe.2) for
g11000 qb10000 for k110 if
rand lt 0.5 dnde1.3 else dnde/1.3
end xnxednrandn(v,1)/sqrt(v)
end end
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MATLAB-programm of the (1, l )-ES
v100 de1 xeones(v,1) for g11000
qb1e20 for k110 if rand lt 0.5
dnde1.3 else dnde/1.3 end
xnxednrandn(v,1)/sqrt(v)
qnsum(xn.2) end end
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MATLAB-programm of the (1, l )-ES
v100 de1 xeones(v,1) for g11000
qb1e20 for k110 if rand lt 0.5
dnde1.3 else dnde/1.3 end
xnxednrandn(v,1)/sqrt(v)
qnsum(xn.2) if qn lt qb
qbqn dbdn xbxn end end
end
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MATLAB-programm of the (1, l )-ES
v100 de1 xeones(v,1) for g11000
qb1e20 for k110 if rand lt 0.5
dnde1.3 else dnde/1.3 end
xnxednrandn(v,1)/sqrt(v)
qnsum(xn.2) if qn lt qb
qbqn dbdn xbxn end end
qeqb dedb xexb end
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MATLAB-programm of the (1, l )-ES
v100 de1 xeones(v,1) for g11000
qb1e20 for k110 if rand lt 0.5
dnde1.3 else dnde/1.3 end
xnxednrandn(v,1)/sqrt(v)
qnsum(xn.2) if qn lt qb
qbqn dbdn xbxn end end
qeqb dedb xexb semilogy(g,qe,'b.')
hold on drawnow end
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Thank you for your attention