Don’t Evaluate, Inherit
Kumara Sastry
David E. Goldberg
Martin Pelikan
Illinois Genetic Algorithms Laboratory
University of Illinois at Urbana-Champaign
Urbana, IL 61801
http://www-illigal.ge.uiuc.edu
Genetic and Evolutionary Computation Conference
(GECCO-2001)
July 7-11, 2001
San Francisco, CA

1
Don’t Evaluate, Inherit
Foreword
• Fitness inheritance is simple
• Inheritance eliminates evaluation
• Can be easily modeled & predicted
Illinois Genetic Algorithms Laboratory
Department of General Engineering
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
K. Sastry, D.E. Goldberg, M. Pelikan
http://www-illigal.ge.uiuc.edu

2
Don’t Evaluate, Inherit
Overview
• Background & motivation
• Fitness inheritance
• Objective
• Facetwise models
– Constant solution quality
– Fixed population size
• Conclusions
Illinois Genetic Algorithms Laboratory
Department of General Engineering
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
K. Sastry, D.E. Goldberg, M. Pelikan
http://www-illigal.ge.uiuc.edu

3
Don’t Evaluate, Inherit
Background
• Design of competent GAs: A key challenge
– Solve hard problems quickly, reliably, and
accurately
– Significant progress made (Goldberg, 1999)
– Require subquadratic function evaluations
• High for large-scale problems
• Efficiency enhancement techniques (EETs)
– Parallelism, Hybridization, Time Continuation,
Evaluation Relaxation
Illinois Genetic Algorithms Laboratory
Department of General Engineering
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
K. Sastry, D.E. Goldberg, M. Pelikan
http://www-illigal.ge.uiuc.edu

4
Don’t Evaluate, Inherit
Motivation
• Fitness inheritance is an EET
– Offspring inherits fitness from parents
– Thus eliminates fitness evaluation
• Smith, Dike & Stegmann (1995)
– Offsprings inherit average parent fitness
– Reported very high speed-up
• Analytical investigation required
Illinois Genetic Algorithms Laboratory
Department of General Engineering
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
K. Sastry, D.E. Goldberg, M. Pelikan
http://www-illigal.ge.uiuc.edu

5
Don’t Evaluate, Inherit
Fitness Inheritance
• Offsprings assigned average fitness of
parents
• Computed during crossover
• Inheritance is a very simple operation
• Fitness evaluation eliminated
• Introduces noise
Parents Fitness Offsprings Inh Fit True Fit
110000 111111 2 3 6
001111 000000 4 3 0
Illinois Genetic Algorithms Laboratory
Department of General Engineering
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
K. Sastry, D.E. Goldberg, M. Pelikan
http://www-illigal.ge.uiuc.edu

6
Don’t Evaluate, Inherit
GA With Fitness Inheritance
Initialize Fitness Genetic
population evaluation Operators
Selection
Crossover
Mutation
No
Others
Are any
Yes stopping
criteria Inherit
met fitness
End
Evaluate
Replace
some
population
individuals
Illinois Genetic Algorithms Laboratory
Department of General Engineering
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
K. Sastry, D.E. Goldberg, M. Pelikan
http://www-illigal.ge.uiuc.edu

7
Don’t Evaluate, Inherit
Objective
• Model fitness inheritance in GAs
– Convergence time model
– Population-sizing model
• Optimize inheritance for greatest speedup
• Verify theory with empirical results
• Compare results with those of Smith et al
Illinois Genetic Algorithms Laboratory
Department of General Engineering
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
K. Sastry, D.E. Goldberg, M. Pelikan
http://www-illigal.ge.uiuc.edu

8
Don’t Evaluate, Inherit
Assumptions
• Fixed population size
• Non-overlapping population
• Generational GAs
• Binary encoding and fixed string length
• Stationary fitness functions
• models for OneMax domain
Illinois Genetic Algorithms Laboratory
Department of General Engineering
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
K. Sastry, D.E. Goldberg, M. Pelikan
http://www-illigal.ge.uiuc.edu

9
Don’t Evaluate, Inherit
Fitness Inheritance Model
• Actual fitness distribution: Gaussian
2
f ∼ N (µf,t , σf,t )
• Inherited fitness is average BB fitness
• Inherited fitness distribution: Gaussian
2
f ∼ N (µf,t , (1 − pi )σf,t )
pi is inheritance proportion: (ni /n)
•
– ni : No. of indivs. with inherited fitness
– n: Population size
Illinois Genetic Algorithms Laboratory
Department of General Engineering
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
K. Sastry, D.E. Goldberg, M. Pelikan
http://www-illigal.ge.uiuc.edu

10
Don’t Evaluate, Inherit
Convergence Time Model
• M¨hlenbein & Schlierkamp-Voosen (1993);
u
Thierens & Goldberg (1993); B¨ck (1994);
a
Miller & Goldberg (1996)
– Use Selection Intensity (Bulmer,1980)
• Prop. of correct BBs & convergence time
1 − pi
1
pt = 1 + sin It
2
π
tconv =
1 − pi
2I
Illinois Genetic Algorithms Laboratory
Department of General Engineering
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
K. Sastry, D.E. Goldberg, M. Pelikan
http://www-illigal.ge.uiuc.edu

11
Don’t Evaluate, Inherit
Proportion of Correct BBs
1
pi = 0.0
pi = 0.3
0.95
pi = 0.5
pi = 0.7
0.9
0.85
Proportion of correct allele, pt
0.8
0.75
0.7
0.65
0.6
0.55
0.5
0 5 10 15 20 25 30 35 40 45 50
No. of generations, t
Illinois Genetic Algorithms Laboratory
Department of General Engineering
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
K. Sastry, D.E. Goldberg, M. Pelikan
http://www-illigal.ge.uiuc.edu

12
Don’t Evaluate, Inherit
Convergence Time
90
80
70
Convergence time
60
50
40
30
20
−0.2 0 0.2 0.4 0.6 0.8 1 1.2
Proportion of inheritance, p
i
Illinois Genetic Algorithms Laboratory
Department of General Engineering
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
K. Sastry, D.E. Goldberg, M. Pelikan
http://www-illigal.ge.uiuc.edu

13
Don’t Evaluate, Inherit
Population-Sizing Model
• Gambler’s ruin population-sizing model
(Harik, Cant´-Paz, Goldberg & Miller,1997)
u
– Combines BB supply & decision making model
• Population sizing for noisy environments
(Miller,1997)
−2k−1 log α 2
πσf,t
n=
(1 − p3 )
i
Illinois Genetic Algorithms Laboratory
Department of General Engineering
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
K. Sastry, D.E. Goldberg, M. Pelikan
http://www-illigal.ge.uiuc.edu

14
Don’t Evaluate, Inherit
Population Size
170
160
150
140
Population size
130
120
110
100
90
80
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Proportion of inheritance, p i
Illinois Genetic Algorithms Laboratory
Department of General Engineering
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
K. Sastry, D.E. Goldberg, M. Pelikan
http://www-illigal.ge.uiuc.edu

15
Don’t Evaluate, Inherit
Optimal Inheritance Proportion
−1
, n ∝ (1 − p3 )−1
tconv ∝ (1 − pi )
• 2
i
pi does not yield high speed-up
• Low
pi
• High
– Requires large population size
– Longer convergence time
pi yields greatest speed-up
• Optimal
Illinois Genetic Algorithms Laboratory
Department of General Engineering
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
K. Sastry, D.E. Goldberg, M. Pelikan
http://www-illigal.ge.uiuc.edu

16
Don’t Evaluate, Inherit
Optimal Inheritance Proportion
• Neglect inheritance cost
• Minimize number of function evaluations
Nf e = n [tconv (1 − pi ) + pi ]
∂N
• Solve ∂pf e = 0
i
√
√
π
π
1 − pi +(1−p3 )
3p2 1 − pi −
(1 − pi ) + pi =0
i
i
2I 4I
• Solve numerically or Use approximations
Illinois Genetic Algorithms Laboratory
Department of General Engineering
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
K. Sastry, D.E. Goldberg, M. Pelikan
http://www-illigal.ge.uiuc.edu

17
Don’t Evaluate, Inherit
Optimal Inheritance Proportion
0.56
0.55
0.54
Optimal inheritance proportion, p *
i
0.53
0.52
0.51
0.5
0.49
0.48
0 100 200 300 400 500 600 700 800 900 1000
String length, l
0.54 ≤ p∗ ≤ 0.558
• Optimal Inheritance: i
Illinois Genetic Algorithms Laboratory
Department of General Engineering
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
K. Sastry, D.E. Goldberg, M. Pelikan
http://www-illigal.ge.uiuc.edu

18
Don’t Evaluate, Inherit
Results: 100-Bit OneMax
2250
2200
2150
2100
Function evaluation
2050
2000
1950
1900
1850
1800
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
proportion of inheritance, p
i
Illinois Genetic Algorithms Laboratory
Department of General Engineering
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
K. Sastry, D.E. Goldberg, M. Pelikan
http://www-illigal.ge.uiuc.edu

19
Don’t Evaluate, Inherit
Optimal Speed-Up
Nf e (p∗ ): No. of func. evals. with optimal
• i
inheritance
Nf e (pi = 0): No. of func. evals without
•
inheritance
ηs = Nf e (pi = 0)/Nf e (p∗ )
• Speed-Up, i
1.2422 ≤ ηs ≤ 1.2428
• Greatest speed-up:
Illinois Genetic Algorithms Laboratory
Department of General Engineering
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
K. Sastry, D.E. Goldberg, M. Pelikan
http://www-illigal.ge.uiuc.edu

20
Don’t Evaluate, Inherit
Result Comparison
• Smith, Dike & Stegmann (1995) reported
higher speed-up
ηs ≈ 1.24
• Our Study:
• Is something wrong?
• Fixed population size assumption
• Adjust population size for constant solution
quality
Illinois Genetic Algorithms Laboratory
Department of General Engineering
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
K. Sastry, D.E. Goldberg, M. Pelikan
http://www-illigal.ge.uiuc.edu

21
Don’t Evaluate, Inherit
Apparent Speed-Up
• Assume fixed population size
– Irrespective of pi value
• Speed-up obtained is apparent speed-up
• Results agree with Smith, Dike &
Stegmann (1995)
Illinois Genetic Algorithms Laboratory
Department of General Engineering
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
K. Sastry, D.E. Goldberg, M. Pelikan
http://www-illigal.ge.uiuc.edu

22
Don’t Evaluate, Inherit
Apparent Speed-Up
9000
Np = 100
N = 150
p
N = 200
8000 p
N = 250
p
Np = 300
7000
6000
Function evaluation
5000
4000
3000
2000
1000
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Proportion of inheritance, p i
Illinois Genetic Algorithms Laboratory
Department of General Engineering
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
K. Sastry, D.E. Goldberg, M. Pelikan
http://www-illigal.ge.uiuc.edu

23
Don’t Evaluate, Inherit
Optimal Apparent Speed-Up
1
κ 3
p∗ = 1−
i,app
n
κ a constant, depends on
•
– Building block size
– Failure probability
n < κ: Premature convergence
•
n > κ: Higher speed-up
•
Illinois Genetic Algorithms Laboratory
Department of General Engineering
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
K. Sastry, D.E. Goldberg, M. Pelikan
http://www-illigal.ge.uiuc.edu

24
Don’t Evaluate, Inherit
Optimal Apparent Speed-Up
0.95
pred
expt
0.9
0.85
i
Proportion of inheritance, p
0.8
0.75
0.7
0.65
0.6
0.55
100 120 140 160 180 200 220 240 260 280 300
Population size, N
p
Illinois Genetic Algorithms Laboratory
Department of General Engineering
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
K. Sastry, D.E. Goldberg, M. Pelikan
http://www-illigal.ge.uiuc.edu

25
Don’t Evaluate, Inherit
Summary
• Modeled fitness inheritance
– Convergence time model
– Semi-empirical population-sizing model
• Optimal inheritance for greatest speed-up
• Fitness inheritance: 20% reduction
• A loose upper bound
• Fixed population size—higher speed-up
Illinois Genetic Algorithms Laboratory
Department of General Engineering
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
K. Sastry, D.E. Goldberg, M. Pelikan
http://www-illigal.ge.uiuc.edu

26
Don’t Evaluate, Inherit
Future Work
• Extend analysis to other problem domain
• Analytical population-sizing model
• Model other inheritance techniques
• Apply to complex, real-world problems
Illinois Genetic Algorithms Laboratory
Department of General Engineering
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
K. Sastry, D.E. Goldberg, M. Pelikan
http://www-illigal.ge.uiuc.edu

27
Don’t Evaluate, Inherit
Acknowledgments
• Air Force Office of Scientific Research, Air Force
Materiel Command, USAF, under grant
F49620-00-1-0163.
• National Science Foundation under grant
DMI-9908252.
• U. S. Army Research Laboratory under the
Federated Laboratory Program, Cooperative
Agreement DAAL01-96-2-0003.
Illinois Genetic Algorithms Laboratory
Department of General Engineering
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
K. Sastry, D.E. Goldberg, M. Pelikan
http://www-illigal.ge.uiuc.edu

28
Don’t Evaluate, Inherit
Results: 40-Bit Trap, pc = 0.9
4
x 10
1.4
1.35
1.3
Function evaluation
1.25
1.2
1.15
1.1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
proportion of inheritance, p
i
Illinois Genetic Algorithms Laboratory
Department of General Engineering
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
K. Sastry, D.E. Goldberg, M. Pelikan
http://www-illigal.ge.uiuc.edu

29
Don’t Evaluate, Inherit
Results: 40-Bit Trap, s = 8
6600
6400
6200
6000
Function evaluation
5800
5600
5400
5200
5000
4800
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
proportion of inheritance, p
i
Illinois Genetic Algorithms Laboratory
Department of General Engineering
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
K. Sastry, D.E. Goldberg, M. Pelikan
http://www-illigal.ge.uiuc.edu