A Practical Schema Theorem for
Genetic Algorithm Design and Tuning
David E. Goldberg
Kumara Sastry
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
Schema Theorem for GA Design & Tuning
Foreword
• Schema theorem has received wide range
of reaction
• ...from criticism to overemphasis
• But schema theorem is useful, and “easy”
to satisfy
Illinois Genetic Algorithms Laboratory
Department of General Engineering
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
D.E. Goldberg, K. Sastry
http://www-illigal.ge.uiuc.edu

2
Schema Theorem for GA Design & Tuning
Overview
• Background & Motivation
• Simplified practical schema theorem
• Designing for BB growth
• Selection schemes & BB growth
– Tournament selection
– Truncation selection
– Proportionate selection
• Summary
Illinois Genetic Algorithms Laboratory
Department of General Engineering
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
D.E. Goldberg, K. Sastry
http://www-illigal.ge.uiuc.edu

3
Schema Theorem for GA Design & Tuning
GA Design Decomposition
• Goldberg, Deb, & Clark, 1992
1. Know what GAs are processing—building blocks
(BBs)
2. Know thy BB challengers—BB-wise difficult
problems.
3. Ensure an adequate supply of raw BBs
4. Ensure increased market share for superior BBs
5. Know BB takeover and convergence time
6. Make decisions well among competing BBs
7. Mix BBs well
Illinois Genetic Algorithms Laboratory
Department of General Engineering
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
D.E. Goldberg, K. Sastry
http://www-illigal.ge.uiuc.edu

4
Schema Theorem for GA Design & Tuning
Motivation
• Ensuring superior BB growth
– Important, and necessary for GA success
• Usual approach: use schema theorem
• Which selection operator should I use ?
• What parameter values should I use?
• Decisions are usually made ad hoc
Illinois Genetic Algorithms Laboratory
Department of General Engineering
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
D.E. Goldberg, K. Sastry
http://www-illigal.ge.uiuc.edu

5
Schema Theorem for GA Design & Tuning
Objective
• Design for superior BB growth
• Utilize a simplified schema theorem
• Study its ramifications on
– choice of selection operator, and
– parameter tuning
• Verify with empirical results
Illinois Genetic Algorithms Laboratory
Department of General Engineering
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
D.E. Goldberg, K. Sastry
http://www-illigal.ge.uiuc.edu

6
Schema Theorem for GA Design & Tuning
Simplified Practical Schema Theorem
• Schema Theorem (Holland, 1975;
De Jong, 1975)
– Proportionate selection & one-point crossover
f (H, t) δ(H)
m(H, t + 1) ≥ m(H, t) 1 − pc
−1
f (t)
• Simplified practical schema theorem
– Goldberg & Deb, 1991
m(H, t + 1) ≥ m(H, t)sp (1 − pc )
Illinois Genetic Algorithms Laboratory
Department of General Engineering
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
D.E. Goldberg, K. Sastry
http://www-illigal.ge.uiuc.edu

7
Schema Theorem for GA Design & Tuning
Designing for Building Block Growth
1 − s−1
m(H, t + 1) p
≥ 1 ⇒ pc ≤
m(H, t)
1
ε = 0.25
ε = 0.5
0.9
ε = 0.75
ε = 1.0
0.8
0.7
Crossover probability, pc
0.6
0.5
0.4
0.3
0.2
0.1
0
0 1 2
10 10 10
Selection pressure, s p
Illinois Genetic Algorithms Laboratory
Department of General Engineering
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
D.E. Goldberg, K. Sastry
http://www-illigal.ge.uiuc.edu

8
Schema Theorem for GA Design & Tuning
BB Growth Design Model
1 − s−1
p
pc ≤
→∞
• Large selection pressure—s
– BB growth ensured even if crossover is fully
disruptive (pc = 1)
→0
• Small crossover probability—pc
– BB growth is ensured for any selection pressure
– Easiest way to obey schema theorem
– Does not guarantee mixing
Illinois Genetic Algorithms Laboratory
Department of General Engineering
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
D.E. Goldberg, K. Sastry
http://www-illigal.ge.uiuc.edu

9
Schema Theorem for GA Design & Tuning
Selection Schemes & Selection Pressure
• Initial growth of BBs is important
• Later in the GA run
– Aggressive growth wears down
– Loss of selection pressure
– Schema loss due to BB disruption
– Stall can occur
– Premature convergence
• Consider both early and late phases
Illinois Genetic Algorithms Laboratory
Department of General Engineering
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
D.E. Goldberg, K. Sastry
http://www-illigal.ge.uiuc.edu

10
Schema Theorem for GA Design & Tuning
Tournament Selection: Early Phase
• Goldberg & Deb, 1991
s
pt+1 = 1 − (1 − pt )
• Early in the GA run
– pt ≈ 0
– sp = s
– Good schemata grow exponentially
1 − s−1
pc ≤
Illinois Genetic Algorithms Laboratory
Department of General Engineering
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
D.E. Goldberg, K. Sastry
http://www-illigal.ge.uiuc.edu

11
Schema Theorem for GA Design & Tuning
Tournament Selection: Late Phase
pt > 1/s
•
• Account for self crosses (Shaffer, 1987)
• Effect loss due to crossover reduces
sp = p−1 [1 − pc (1 − pt )]
• t
sp ≥ 1 if
•
1 − s−1
pc ≤
Illinois Genetic Algorithms Laboratory
Department of General Engineering
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
D.E. Goldberg, K. Sastry
http://www-illigal.ge.uiuc.edu

12
Schema Theorem for GA Design & Tuning
Truncation Selection
• Growth is geometric
pt+1 = spt
• Early in the GA run
– sp = s, Same as tournament selection
• Late in the GA run
– sp = p−1 [1 − pc (1 − pt )]
t
– Same as tournament selection
1 − s−1
pc ≤
Illinois Genetic Algorithms Laboratory
Department of General Engineering
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
D.E. Goldberg, K. Sastry
http://www-illigal.ge.uiuc.edu

13
Schema Theorem for GA Design & Tuning
Results: Test Function
8-bit Trap
8
7
6
fitness Value
5
4
3
2
1
0 1 2 3456 7 8
No. of ones
• 8-bit deceptive trap function
• Single building block
• Global optimum: 00000000
• Local optimum: 11111111
Illinois Genetic Algorithms Laboratory
Department of General Engineering
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
D.E. Goldberg, K. Sastry
http://www-illigal.ge.uiuc.edu

14
Schema Theorem for GA Design & Tuning
Results: Run Conditions
• One-point crossover
• Schema disruption rate is known
• Crossover probability determined by
bisection method
10−5
• Tolerance:
• Population size: 5000
• Average of 25 runs
Illinois Genetic Algorithms Laboratory
Department of General Engineering
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
D.E. Goldberg, K. Sastry
http://www-illigal.ge.uiuc.edu

15
Schema Theorem for GA Design & Tuning
Results: Tournament & Truncation
1 1
Crossover probability, pc
c
Crossover probability, p
0.8 0.8
0.6 0.6
ε = 0.95
ε = 1.0
0.4 0.4
Truncation
Truncation
Tournament WOR
Tournament WOR
0.2 0.2
Tournament WR
Tournament WR
0 0
0 1 2 0 1
10 10 10 10 10
Selection pressure, s Selection pressure, s
p p
1 1
Crossover probability, pc
c
Crossover probability, p
0.8 0.8
0.6 0.6
ε = 0.9 ε = 0.85
0.4 0.4
Truncation Truncation
Tournament WOR Tournament WOR
0.2 0.2
Tournament WR Tournament WR
0 0
0 1 0 1
10 10 10 10
Selection pressure, sp Selection pressure, sp
Illinois Genetic Algorithms Laboratory
Department of General Engineering
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
D.E. Goldberg, K. Sastry
http://www-illigal.ge.uiuc.edu

16
Schema Theorem for GA Design & Tuning
Proportionate Selection: Early Phase
• Consider only two alternatives
– Average (f1 ) & Best (f2 ) individual
s
pt , s = f2 /f1
pt+1 =
(s − 1)pt + 1
• Early in the GA run
– sp = s, Similar to tournament & truncation
– s varies for different problems
– Difficult to determine s before the GA run
1 − s−1
pc ≤
Illinois Genetic Algorithms Laboratory
Department of General Engineering
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
D.E. Goldberg, K. Sastry
http://www-illigal.ge.uiuc.edu

17
Schema Theorem for GA Design & Tuning
Proportionate Selection: Late Phase
• Account for self crosses
s [1 − pc (1 − pt )]
sp =
(s − 1)pt + 1
1−s−1
sp ≥ 1 if pc ≤
•
s = f2 /f1 is not a constant
•
f1 increases during the run
•
s decreases during the run
•
• Best BB’s domination can stall
Illinois Genetic Algorithms Laboratory
Department of General Engineering
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
D.E. Goldberg, K. Sastry
http://www-illigal.ge.uiuc.edu

18
Schema Theorem for GA Design & Tuning
Results: Stall Demonstration
1 0.7
p = 0.225
(b)
(a) c
0.6
p = 0.2
0.8 c
Proportion of BBs
Proportion of BBs
0.5
0.6 0.4
0.3
0.4
0.2
all−zeros
0.2 all−zeros
0.1
all−ones all−ones
0 0
0 50 100 150 200 0 50 100 150 200
No. of generations No. of generations
0.7 1
(c) (d)
pc = 0.23 pc = 0.26
0.6
0.8
Proportion of BBs
Proportion of BBs
0.5
0.6
0.4
0.3 0.4
0.2
0.2 all−zeros
all−zeros
0.1 all−ones
all−ones
0 0
0 50 100 150 200 0 50 100 150 200
No. of generations No. of generations
Illinois Genetic Algorithms Laboratory
Department of General Engineering
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
D.E. Goldberg, K. Sastry
http://www-illigal.ge.uiuc.edu

19
Schema Theorem for GA Design & Tuning
Results: Proportionate Selection
1 1
(b)
(a)
Failure Failure
Proportion of optimal BB
Proportion of optimal BB
Success Success
0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2
pc = 0.215 pc = 0.22
0 0
0 50 100 150 0 50 100 150
No. of generations No. of generations
1 1
(c) (d)
Failure
Failure
Success
Proportion of optimal BB
Proportion of optimal BB
Success
0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2
pc = 0.225 p = 0.23
c
0 0
0 50 100 150 0 50 100 150
No. of generations No. of generations
Illinois Genetic Algorithms Laboratory
Department of General Engineering
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
D.E. Goldberg, K. Sastry
http://www-illigal.ge.uiuc.edu

20
Schema Theorem for GA Design & Tuning
Summary
• Schema theorem must be obeyed
– Does not guarantee BB mixing
• Schema theorem is “easy” to obey
• Selection operator & parameter tuning
• Use ordinal schemes
• Proportionate selection scheme
– BB growth can stall
– Use only in special cases
Illinois Genetic Algorithms Laboratory
Department of General Engineering
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
D.E. Goldberg, K. Sastry
http://www-illigal.ge.uiuc.edu

21
Schema Theorem for GA Design & Tuning
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.
D.E. Goldberg, K. Sastry
http://www-illigal.ge.uiuc.edu