1. Analysis of Evolutionary Algorithms on the
One-Dimensional Spin Glass with Power-Law
Interactions
Martin Pelikan and Helmut G. Katzgraber
Missouri Estimation of Distribution Algorithms Laboratory (MEDAL)
University of Missouri, St. Louis, MO
http://medal.cs.umsl.edu/
pelikan@cs.umsl.edu
Download MEDAL Report No. 2009004
http://medal.cs.umsl.edu/files/2009004.pdf
Martin Pelikan and Helmut G. Katzgraber Analysis of EAs on 1D Spin Glass with Power-Law Interactions

2. Motivation
Testing evolutionary algorithms
Adversarial problems on the boundary of design envelope.
Random instances of important classes of problems.
Real-world problems.
This study
Use one-dimensional spin glass with power-law interactions.
This allows the user to tune the effective range of interactions.
Short-range to long-range interactions.
Generate large number of instances of proposed problem class.
Solve all instances with branch and bound and hybrids.
Test evolutionary algorithms on the generated instances.
Analyze the results.
Martin Pelikan and Helmut G. Katzgraber Analysis of EAs on 1D Spin Glass with Power-Law Interactions

3. Outline
1. Sherrington-Kirkpatrick (SK) spin glass.
2. Power-law interactions.
3. Problem instances.
4. Experiments.
5. Conclusions and future work.
Martin Pelikan and Helmut G. Katzgraber Analysis of EAs on 1D Spin Glass with Power-Law Interactions

4. SK Spin Glass
SK spin glass (Sherrington & Kirkpatrick, 1978)
Contains n spins s1 , s2 , . . . , sn .
Ising spin can be in two states: +1 or −1.
All pairs of spins interact.
Interaction of spins si and sj specified by
real-valued coupling Ji,j .
Spin glass instance is defined by set of couplings {Ji,j }.
Spin configuration is defined by the values of spins {si }.
Martin Pelikan and Helmut G. Katzgraber Analysis of EAs on 1D Spin Glass with Power-Law Interactions

5. Ground States of SK Spin Glasses
Energy
Energy of a spin configuration C is given by
H(C) = − Ji,j si sj
i<j
Ground states are spin configurations that minimize energy.
Finding ground states of SK instances is NP-complete.
Compare with other standard spin glass types
2D: Spin interacts with only 4 neighbors in 2D lattice.
3D: Spin interacts with only 6 neighbors in 3D lattice.
SK: Spin interacts with all other spins.
2D is polynomially solvable; 3D and SK are NP-complete.
Martin Pelikan and Helmut G. Katzgraber Analysis of EAs on 1D Spin Glass with Power-Law Interactions

6. Random Spin Glass Instances
Generating random spin glass instances
Generate couplings {Ji,j } using a specific distribution.
Study the properties of generated spin glasses.
Example study
Find ground states and analyze their properties.
Example coupling distributions
Each coupling is generated from N (0, 1).
Each coupling is +1 or -1 with equal probability.
Each coupling is generated from a power-law distribution.
Martin Pelikan and Helmut G. Katzgraber Analysis of EAs on 1D Spin Glass with Power-Law Interactions

7. Power-Law Interactions
Power-law interactions
Spins arranged on a circle.
Couplings generated according to
i,j
Ji,j = c(σ) σ ,
ri,j
i,j are generated according to N (0, 1),
c(σ) is a normalization constant,
σ > 0 is a parameter to control
effective range of interactions,
ri,j = n sin(π|i − j|/n)/π is geometric
Figure 1: One-dimensional spin glass of size n = 10 ar
distance between si and sj
Magnitude ofwhere ǫi,j are generated decreases with their distance. zero
spin-spin couplings according to normal distribution with
Effects of distance on magnitude of couplings increase withparameter t
is a normalization constant, σ > 0 is the user-specified σ.
interactions, and ri,j = n sin(π|i − j|/n)/π denotes the geometric d
figure 1). The magnitude of spin-spin couplings decreases with th
discussed shortly, the effects EAsdistance on the magnitude of coupli
Martin Pelikan and Helmut G. Katzgraber Analysis of of on 1D Spin Glass with Power-Law Interactions

8. Power-Law Interactions: Illustration
Example for n = 10 (normalized)
Distance on Coupling variance
circle σ = 0.0 σ = 0.5 σ = 2.0
1 1.00 1.00 1.00
2 1.00 0.73 0.28
3 1.00 0.62 0.15
4 1.00 0.57 0.11
5 1.00 0.56 0.10
Martin Pelikan and Helmut G. Katzgraber Analysis of EAs on 1D Spin Glass with Power-Law Interactions

9. Problem Instances
Parameters
n = 20 to 150.
σ ∈ {0.00, 0.55, 0.75, 0.83, 1.00, 1.50, 2.00}.
σ = 0 denotes standard SK spin glass with N(0,1) couplings.
σ = 2 enforces short-range interactions.
Variety of instances
For each n and σ, generate 10,000 random instances.
Overall 610,000 unique problem instances.
Finding optima
Small instances solved using branch and bound.
For large instances, use heuristic methods to find reliable (but
not guaranteed) optima.
Martin Pelikan and Helmut G. Katzgraber Analysis of EAs on 1D Spin Glass with Power-Law Interactions

10. Compared Algorithms
Basic algorithms
Hierarchical Bayesian optimization algorithm (hBOA).
Genetic algorithm with uniform crossover (GAU).
Genetic algorithm with twopoint crossover (G2P).
Local search
Single-bit-flip hill climbing (DHC) on each solution.
Improves performance of all methods.
Niching
Restricted tournament replacement (niching).
Martin Pelikan and Helmut G. Katzgraber Analysis of EAs on 1D Spin Glass with Power-Law Interactions

11. Experimental Setup
All algorithms
Bisection determines adequate population size for each
instance.
Ensure 10 successful runs out of 10 independent runs.
In RTR, use window size w = min{N/20, n}.
GA
Probability of crossover, pc = 0.6.
Probability of bit-flip in mutation, pm = 1/n.
Martin Pelikan and Helmut G. Katzgraber Analysis of EAs on 1D Spin Glass with Power-Law Interactions

12. Results: Evaluations until Optimum
Number of evaluations (GA, twopoint)
Number of evaluations (GA, twopoint)
Number of evaluations (GA, twopoint)
Number of evaluations (GA, twopoint)
Number of evaluations (GA, twopoint)
Number of evaluations (GA, twopoint)
5 5 5
10 10 10 σ=2.00 10 10 10 σ=2.00 10 10 10 σ=2.00
5 5 5
σ=2.00 σ=2.00 σ=2.00
Number of evaluations (hBOA)
5 5 5
σ=2.00 σ=2.00 σ=2.00
Number of evaluations (hBOA)
Number of evaluations (hBOA)
σ=1.50
σ=1.50σ=1.50 σ=1.50
σ=1.50
σ=1.50 σ=1.50
σ=1.50
σ=1.50
4 σ=1.00
4 4 σ=1.00 σ=1.00 4 σ=1.00
4 4 σ=1.00σ=1.00 4 4 σ=1.00
σ=1.00
4 σ=1.00
10 10 10 10 10 10 10 10 10
σ=0.83
σ=0.83σ=0.83 σ=0.83
σ=0.83
σ=0.83 σ=0.83
σ=0.83
σ=0.83
σ=0.75
σ=0.75σ=0.75 σ=0.75
σ=0.75
σ=0.75 σ=0.75
σ=0.75
σ=0.75
3 3 3 3 3 3 3 3 3
10 10 10 σ=0.55
σ=0.55σ=0.55 10 10 10 σ=0.55
σ=0.55
σ=0.55 10 10 10 σ=0.55
σ=0.55
σ=0.55
σ=0.00
σ=0.00σ=0.00 σ=0.00
σ=0.00
σ=0.00 σ=0.00
σ=0.00
σ=0.00
2 2 2 2 2 2 2 2 2
10 10 10 10 10 10 10 10 10
1 1 1 1 1 1 1 1 1
10 10 10 10 10 10 10 10 10
16 16 16 32 32 32 64 64 64 128 128
128 16 16 16 32 32 32 64 64 64 128 128
128 16 16 16 32 32 32 64 64 64 128128
128
Problem size
Problem size size
Problem Problem size
Problem size size
Problem Problem sizesize
Problem
Problem size
(a) hBOA
(a)(a) hBOA
hBOA (b) GA (twopoint)
(b) GA (twopoint)
(b) GA (twopoint) (c)(c) GA (uniform)
GA (uniform)
(c) GA (uniform)
Scalability of hBOA and GA with twopoint crossover better
forFigure 2:2: 2: Growththe the numberevaluations withwith problem size.
short-range interactions. of of evaluations problem size.
Figure Growth ofof of number of evaluations with problem size.
Figure Growth the number
6 6 σ=2.00 σ=2.00
10 10 10 σ=2.00
Linkage tightens 10 σ=2.00grows.
as σ
10 10 σ=2.00 σ=2.00 σ=2.00
σ=2.00
10 10 10 σ=2.00
Number of flips (GA, twopoint)
Number of flips (GA, twopoint)
Number of flips (GA, twopoint)
Number of flips (GA, twopoint)
Number of flips (GA, twopoint)
Number of flips (GA, twopoint)
6 6 6 6 6 6 6
σ=1.50σ=1.50
σ=1.50 σ=1.50σ=1.50
σ=1.50 σ=1.50
σ=1.50
σ=1.50
Number of flips (hBOA)
Number of flips (hBOA)
Tighter linkage makes problem easier (if good recombination).
Number of flips (hBOA)
5 5 σ=1.00 σ=1.00
5 σ=1.00 σ=1.00σ=1.00
σ=1.00 5 σ=1.00
σ=1.00
σ=1.00
5 5 5 5 5
10 10 10 10 10 10 10 10 10
σ=0.83σ=0.83 σ=0.83 σ=0.83
4 4 4
σ=0.83
σ=0.75σ=0.75
σ=0.75
Twopoint crossoverσ=0.75 respects tight linkage. σ=0.75σ=0.83
σ=0.75
σ=0.83
σ=0.83
σ=0.75 4
σ=0.83
4
σ=0.75
σ=0.75
4 4 4 4
σ=0.55σ=0.55
10 10 10 σ=0.55 σ=0.55
σ=0.55
10 10 10 σ=0.55 σ=0.55
σ=0.55
10 10 10 σ=0.55
3
σ=0.00
10 103 10
3
σ=0.00
σ=0.00 GA with uniform 10 10 10σ=0.00σ=0.00 with shorter-range σ=0.00
gets σ=0.00
worse 3
σ=0.00
10 10 10 3
interactions.
σ=0.00
3 3 3 3
2 2 2 2 2 2 2 2 2
10 10 10 10 10 10 10 10 10
16 16 16 32 32 32 64 64 64 128 128
128 16 16 16 32 32 32 64 64 64 128 128
128 16 16 16 32 32 32 64 64 64 128128
128
Problem size size
Problem
Problem size Problem size size
Problem
Problem size Problem sizesize
Problem size
Problem
(a) hBOA
(a)(a) hBOA
hBOA (b) GA (twopoint)
(b) GA (twopoint)
(b) GA (twopoint) (c)(c) GA (uniform)
GA (uniform)
(c) GA (uniform)
Martin Pelikan and Helmut G. Katzgraber Analysis of EAs on 1D Spin Glass with Power-Law Interactions

13. 10 2 2 2
1010 10 10 1010
2 2 2
Number
Numbe
10 10
Number of
Number of
Number
Number of
Number of
Number of
Number of
Results: LS Steps until Optimum (Flips)
10
1
10 10
16
1
16 16
1
32
32 32 64
64 64
Problem size
128
128 128
1010 10
1 1
1616 16
1
32 32 32 64 64 64 128 128
Problem size
128
10 1010
16 1616
1
1 1
32 3232 64 6464 128128
128
Problem size size
Problem Problem size size
Problem Problem sizesize
Problem size
Problem
(a) hBOA
(a) (a) hBOA
hBOA (b) GA (twopoint)
(b) GA GA (twopoint)
(b) (twopoint) (c)(c) GA (uniform)
GA GA (uniform)
(c) (uniform)
Figure 2: Growth ofofof the numberevaluations with problem size.
Figure 2: 2: Growththe number ofofof evaluations with problem size.
Figure Growth the number evaluations with problem size.
6 σ=2.00
σ=2.00 6 6 6 σ=2.00
σ=2.00 6 6 σ=2.00
σ=2.00
Number of flips (GA, twopoint)
Number of flips (GA, twopoint)
10 10 σ=2.00 1010 10σ=2.00 10 1010σ=2.00
6
Number of flips (GA, twopoint)
Number of flips (GA, twopoint)
Number of flips (GA, twopoint)
Number of flips (GA, twopoint)
6 6
10
σ=1.50
σ=1.50
σ=1.50 σ=1.50
σ=1.50
σ=1.50 σ=1.50
σ=1.50
σ=1.50
Number of flips (hBOA)
Number of flips (hBOA)
Number of flips (hBOA)
5 5 5 σ=1.00
σ=1.00
σ=1.00 5 σ=1.00
5 5 σ=1.00σ=1.00 5 5 σ=1.00
σ=1.00
5 σ=1.00
10
10 10 σ=0.83 1010 10 σ=0.83 10 1010 σ=0.83
σ=0.83
σ=0.83 σ=0.83
σ=0.83 σ=0.83
σ=0.83
σ=0.75
σ=0.75
σ=0.75 σ=0.75
σ=0.75
σ=0.75 σ=0.75
σ=0.75
σ=0.75
4 4 4 4 4 4 4 4 4
10 σ=0.55
σ=0.55
10 10 σ=0.55 σ=0.55
1010 10σ=0.55
σ=0.55 σ=0.55
10 1010σ=0.55
σ=0.55
σ=0.00
σ=0.00
σ=0.00 σ=0.00
σ=0.00
σ=0.00 σ=0.00
σ=0.00
σ=0.00
3 3 3 3 3 3 3 3 3
10
10 10 1010 10 10 1010
2 2 2 2 2 2 2 2 2
10
10 10 1010 10 10 1010
16
16 16 32
32 32 64
64 64 128
128 128 1616 16 32 32 32 64 64 64 128 128
128 16 1616 32 3232 64 6464 128128
128
Problem size
Problem size size
Problem Problem size
Problem size size
Problem Problem sizesize
Problem size
Problem
(a) (a) hBOA
hBOA
(a) hBOA (b) GA GA (twopoint)
(b) (twopoint)
(b) GA (twopoint) (c)(c) GA (uniform)
GA GA (uniform)
(c) (uniform)
Scalability 3:3:Growth ofofof the numberflipsflips with problem size. better
Figure
of hBOA and GA with twopoint crossover
Figure 3: Growththe number ofofof with problem size.
Figure Growth the number flips with problem size.
for short-range interactions.
and how thethe effects σuniform gets worse the algorithm under consideration; this is the topic
and how theeffects of ofchange depending on the algorithm under consideration; this is is the topic
effects of change depending on the algorithm under consideration; this the topic
and howGA with σ σ change depending on with shorter-range interactions.
discussed in thethe following few paragraphs.
discussed in in following few paragraphs.
discussed the following few paragraphs.
Based on thethe definitionthe the 1D spin glass with power-law interactions,the the value of σ grows,
Based onon definition of of 1D spin glass with power-law interactions, asas the value of grows,
Based the definition of the 1D spin glass with power-law interactions, as value of σ σ grows,
thethe rangethethe most significant interactions is reduced. With reduction of the range of interactions,
therange of of most significant interactions isis reduced. With reduction of the range of interactions,
range of the most significant interactions reduced. With reduction of the range of interactions,
thethe problem should become easier both for selectorecombinative GAs capablelinkage learning,
theproblem should become easier both for selectorecombinative GAs capable of ofof linkage learning,
problem should become easier both for selectorecombinative GAs capable linkage learning,
such as hBOA, as well as for for selectorecombinative GAs which rarely break interactionsbetween
such as as hBOA, as well as selectorecombinative GAs which rarely break interactions between
such hBOA, as well as for selectorecombinative GAs which rarely break interactions between
closely located bits, such as GAGA with twopoint crossover. This isis clearlydemonstrated by the
closely located bits, such asas with twopoint crossover. This is clearly demonstrated by the the
closely located bits, such GA with twopoint crossover. This clearly demonstrated by
results for for these two algorithms presentedfigures 2 2 2 andAlthough for many problem sizes, the the
results forthese two algorithms presented ininin figures and 3.3. Although for many problem sizes,
results these two algorithms presented figures and 3. Although for many problem sizes, the
Martinabsolute number evaluations and the the number flips are of EAs on smaller Glassfor larger valuesσ, σ,
absolute number of G. Katzgraber
Pelikan and Helmut of evaluations and number of of flips are factfact smaller larger values of of
Analysis in in 1D Spin for with Power-Law Interactions