Fitness Inheritance in BOA
Martin Pelikan Kumara Sastry
Dept. of Math and CS Illinois GA Lab
Univ. of Missouri at St. Louis Univ. of Illinois at Urbana-Champaign
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Motivation
Bayesian optimization algorithm (BOA)
Scales up on decomposable problems
O(n)-O(n2) evaluations until convergence
Expensive evaluations
Real-world evaluations can be complex
FEA, simulation, …
O(n2) is often not enough
This paper
Extend probabilistic model to include fitness info
Use model to evaluate part of the population
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Outline
BOA basics
Fitness inheritance in BOA
Extend Bayesian networks with fitness.
Use extended model for evaluation.
Experiments
Future work
Summary and conclusions
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Bayesian Optimization Alg. (BOA)
Pelikan, Goldberg, and Cantu-Paz (1998)
Similar to genetic algorithms (GAs)
Replace mutation + crossover by
Build Bayesian network to model selected solutions.
Sample Bayesian network to generate new candidate
solutions.
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BOA
Bayesian New
Current network
Selection population
population
Restricted tournament replacement
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Bayesian Networks (BNs)
2 components
Structure
directed acyclic graph
nodes = variables (string positions)
Edges = dependencies between variables
Parameters
Conditional probabilities p(X|Px), where
X is a variable
Px are parents of X (variables that X depends on)
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BN example
A B p(A|B)
0 0 0.10
0 1 0.60
A 1 0 0.90
1 1 0.40
B C C A p(C|A)
0 0 0.80
0 1 0.55
B p(B)
1 0 0.20
0 0.25
1 1 0.45
1 0.75 7

Extending BNs with fitness info
A B p(A|B) f(A|B)
Basic idea
0 0 0.10 -0.5
Don’t work only with
conditional probabilities 0 1 0.60 0.5
Add also fitness info for 1 0 0.90 0.3
fitness estimation 1 1 0.40 -0.3
Fitness info attached to p(X|Px) denoted by f(X|Px)
Contribution of X restricted by Px
f (X = x | Px = px ) = f (X = x, Px = px ) − f (Px = px )
f (X = x, Px = px )
avg. fitness of solutions with X=x and Px=px
f (Px = px )
avg. fitness of solutions with Px
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Estimating fitness
Equation
( )
n
f (X1 , X2 ,K , X n ) = favg + ∑ f Xi | PXi
i=0
In words
Fitness = avg. fitness + avg. contribution of each bit
Avg. contributions taken w.r.t. context from BN
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BNs with decision trees
Local structures in BNs
More efficient representation for p ( X | Px )
Example for p ( A | B C )
B
0 1
p( A | B = 0) C
0 1
p( A | B = 1, C = 0 ) p( A | B = 1, C = 1)
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BNs with decision trees + fitness
Same idea
Attach fitness info to
each probability
B
0 1
p( A | B = 0) C
f ( A | B = 0) 0 1
p ( A | B = 1, C = 0) p ( A | B = 1, C = 1)
f ( A | B = 1, C = 0) f ( A | B = 1, C = 1)
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Estimating fitness again
Same as before…because both BNs represent the same
Equation
( )
n
f ( X 1 , X 2 ,K , X n ) = f avg + ∑ f X i | PX i
i =0
In words
Fitness = avg. fitness + avg. contribution of each bit
Avg. contributions taken w.r.t. context from BN
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Where to learn fitness from?
Evaluate entire initial population
Choose inheritance proportion, pi
After that
Evaluate (1-pi) proportion of offspring
Use evaluated parents + evaluated offspring to learn
Estimate fitness of the remaining prop. pi
Sample for learning: N(1-pi) to N+N(1-pi)
Often, 2N(1-pi)
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Simple example: Onemax
Onemax
n
f ( X 1 , X 2 ,K , X n ) = ∑ X i
i =1
What happens?
Average fitness grows (as predicted by theory)
No context is necessary
Fitness contributions stay constant
f ( X i = 1) = +0.5
f ( X i = 0 ) = −0.5
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Experiments
Problems
50-bit onemax
10 traps of order 4
10 traps of order 5
Settings
Inheritance proportion from 0 to 0.999
Minimum population size for reliable convergence
Many runs for each setting (300 runs for each setting)
Output
Speed-up (in terms of real fitness evaluations)
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Onemax
35
Speed-up (w.r.t. no inheritance)
30
25
20
15
10
5
0
0 0.2 0.4 0.6 0.8 1
Proportion inherited
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Trap-4
35
Speed-up (w.r.t. no inheritance)
30
25
20
15
10
5
0
0 0.2 0.4 0.6 0.8 1
Proportion inherited
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Trap-5
60
Speed-up (w.r.t. no inheritance)
50
40
30
20
10
0
0 0.2 0.4 0.6 0.8 1
Proportion inherited
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Discussion
Inheritance proportion
High proportions of inheritance work great.
Speed-up
Optimal speed-up of 30-53
High speed-up for almost any setting
The tougher the problem, the better the speed-up
Why so good?
Learning probabilistic model difficult, so accurate
fitness info can be added at not much extra cost.
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Conclusions
Fitness inheritance works great in BOA
Theory now exists (Satry et al., 2004) that
explains these results
High proportions of inheritance lead to high
speed-ups
Challenging problems allow much speed-up
Useful for practitioners with computationally
complex fitness function
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Contact
Martin Pelikan
Dept. of Math and Computer Science, 320 CCB
University of Missouri at St. Louis
8001 Natural Bridge Rd.
St. Louis, MO 63121
E-mail: pelikan@cs.umsl.edu
WWW: http://www.cs.umsl.edu/~pelikan/
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