Analysis of Evolutionary Algorithms on the One-Dimensional Spin Glass with Power-Law Interactions

Analysis of Evolutionary Algorithms on theOne-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/[email protected]

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

http://medal.cs.umsl.edu/

http://medal.cs.umsl.edu/files/2009004.pdf

Motivation

Testing evolutionary algorithms

I Adversarial problems on the boundary of design envelope.I Random instances of important classes of problems.I Real-world problems.

This study

I Use one-dimensional spin glass with power-law interactions.I This allows the user to tune the effective range of interactions.I Short-range to long-range interactions.

I Generate large number of instances of proposed problem class.I Solve all instances with branch and bound and hybrids.I Test evolutionary algorithms on the generated instances.I Analyze the results.


Outline

1. Sherrington-Kirkpatrick (SK) spin glass.

2. Power-law interactions.

3. Problem instances.

4. Experiments.

5. Conclusions and future work.


SK Spin Glass

SK spin glass (Sherrington & Kirkpatrick, 1978)

I Contains n spins s1, s2, . . . , sn.

I Ising spin can be in two states: +1 or −1.

I All pairs of spins interact.

I Interaction of spins si and sj specified byreal-valued coupling Ji,j .

I Spin glass instance is defined by set of couplings {Ji,j}.I Spin configuration is defined by the values of spins {si}.


Ground States of SK Spin Glasses

Energy

I Energy of a spin configuration C is given by

H(C) = −∑i<j

Ji,jsisj

I Ground states are spin configurations that minimize energy.I Finding ground states of SK instances is NP-complete.

Compare with other standard spin glass types

I 2D: Spin interacts with only 4 neighbors in 2D lattice.I 3D: Spin interacts with only 6 neighbors in 3D lattice.I SK: Spin interacts with all other spins.I 2D is polynomially solvable; 3D and SK are NP-complete.


Random Spin Glass Instances

Generating random spin glass instances

I Generate couplings {Ji,j} using a specific distribution.

I Study the properties of generated spin glasses.

Example study

I Find ground states and analyze their properties.

Example coupling distributions

I Each coupling is generated from N(0, 1).I Each coupling is +1 or -1 with equal probability.

I Each coupling is generated from a power-law distribution.


Power-Law Interactions

Power-law interactions

I Spins arranged on a circle.I Couplings generated according to

Ji,j = c(σ)εi,j

rσi,j

,

I εi,j are generated according to N(0, 1),I c(σ) is a normalization constant,I σ > 0 is a parameter to control

effective range of interactions,I ri,j = n sin(π|i − j|/n)/π is geometric

distance between si and sjFigure 1: One-dimensional spin glass of size n = 10 arranged on a ring.

where ǫi,j are generated according to normal distribution with zero mean and unit variance, c(σ)is a normalization constant, σ > 0 is the user-specified parameter to control the effective range ofinteractions, and ri,j = n sin(π|i − j|/n)/π denotes the geometric distance between si and sj (seefigure 1). The magnitude of spin-spin couplings decreases with their distance. Furthermore, asdiscussed shortly, the effects of distance on the magnitude of couplings increase with σ.

By varying σ one can tune the model from the infinite-range to the short-range universalityclass: For 0 < σ ≤ 1/2 the model is in the infinite-range universality in the sense that

∑j [J

2

ij ]avdiverges, and for σ = 0 it corresponds to the SK model. For 1/2 < σ ≤ 2/3 the model describesa mean-field long-range spin glass, corresponding—in the analogy with short-range systems—to ashort-range model in dimension above the upper critical dimension d ≥ du = 6. For 2/3 < σ ≤ 1the model has non-mean-field critical behavior with a finite transition temperature Tc, and forσ ≥ 1, the transition temperature is zero and the behavior of the model is short ranged.

To find guaranteed ground states, a branch-and-bound algorithm adopted from refs. [17, 24]was used. This allows us to find guaranteed ground states of instances of up to about n = 100 spinsfor larger values of σ and up to about n = 80 spins for smaller values of σ. For larger systems, weused the population-doubling approach proposed in ref. [42].

3 Compared Algorithms

The genetic algorithm (GA) [18, 11] evolves a population of candidate solutions typically repre-sented by binary strings of fixed length with the first population generated at random accordingto the uniform distribution over all binary strings. Each iteration starts by selecting promisingsolutions from the current population; we use binary tournament selection without replacement.New solutions are created by applying variation operators to the population of selected solutions.Specifically, crossover is used to exchange bits and pieces between pairs of candidate solutions andmutation is used to perturb the resulting solutions. Here we use uniform or twopoint crossover, andbit-flip mutation [11]. To maintain useful diversity in the population, the new candidate solutionsare incorporated into the original population using restricted tournament selection (RTS) [14]. Therun is terminated when termination criteria are met. In this paper, each run is terminated ei-ther when the global optimum has been found or when a maximum number of iterations has beenreached.

The hierarchical Bayesian optimization algorithm (hBOA) [37, 39, 36] is an estimation of dis-tribution algorithm (EDA) [1, 30, 26, 40, 27, 43]. EDAs—also called probabilistic model-buildinggenetic algorithms (PMBGAs) [40] and iterated density estimation algorithms (IDEAs) [6]—differ

3

I Magnitude of spin-spin couplings decreases with their distance.I Effects of distance on magnitude of couplings increase with σ.


Power-Law Interactions: Illustration

Example for n = 10 (normalized)

Distance on Coupling variancecircle σ = 0.0 σ = 0.5 σ = 2.0

1 1.00 1.00 1.002 1.00 0.73 0.283 1.00 0.62 0.154 1.00 0.57 0.115 1.00 0.56 0.10


Problem Instances

Parameters

I n = 20 to 150.I σ ∈ {0.00, 0.55, 0.75, 0.83, 1.00, 1.50, 2.00}.

I σ = 0 denotes standard SK spin glass with N(0,1) couplings.I σ = 2 enforces short-range interactions.

Variety of instances

I For each n and σ, generate 10,000 random instances.

I Overall 610,000 unique problem instances.

Finding optima

I Small instances solved using branch and bound.

I For large instances, use heuristic methods to find reliable (butnot guaranteed) optima.


Compared Algorithms

Basic algorithms

I Hierarchical Bayesian optimization algorithm (hBOA).

I Genetic algorithm with uniform crossover (GAU).

I Genetic algorithm with twopoint crossover (G2P).

Local search

I Single-bit-flip hill climbing (DHC) on each solution.

I Improves performance of all methods.

Niching

I Restricted tournament replacement (niching).


Experimental Setup

All algorithms

I Bisection determines adequate population size for eachinstance.

I Ensure 10 successful runs out of 10 independent runs.

I In RTR, use window size w = min{N/20, n}.

GA

I Probability of crossover, pc = 0.6.

I Probability of bit-flip in mutation, pm = 1/n.


Results: Evaluations until Optimum

16 32 64 12810

1

102

103

104

105

Problem size

Num

ber

of e

valu

atio

ns (

hBO

A)

σ=2.00σ=1.50σ=1.00σ=0.83σ=0.75σ=0.55σ=0.00

(a) hBOA

16 32 64 12810

1

102

103

104

105

Problem sizeN

umbe

r of

eva

luat

ions

(G

A, t

wop

oint

)

σ=2.00σ=1.50σ=1.00σ=0.83σ=0.75σ=0.55σ=0.00

(b) GA (twopoint)

16 32 64 12810

1

102

103

104

105

Problem size

Num

ber

of e

valu

atio

ns (

GA

, tw

opoi

nt)

σ=2.00σ=1.50σ=1.00σ=0.83σ=0.75σ=0.55σ=0.00

(c) GA (uniform)

Figure 2: Growth of the number of evaluations with problem size.

16 32 64 128

102

103

104

105

106

Problem size

Num

ber

of fl

ips

(hB

OA

)

σ=2.00σ=1.50σ=1.00σ=0.83σ=0.75σ=0.55σ=0.00

(a) hBOA

16 32 64 128

102

103

104

105

106

Problem size

Num

ber

of fl

ips

(GA

, tw

opoi

nt)

σ=2.00σ=1.50σ=1.00σ=0.83σ=0.75σ=0.55σ=0.00

(b) GA (twopoint)

16 32 64 128

102

103

104

105

106

Problem sizeN

umbe

r of

flip

s (G

A, t

wop

oint

)

σ=2.00σ=1.50σ=1.00σ=0.83σ=0.75σ=0.55σ=0.00

(c) GA (uniform)

Figure 3: Growth of the number of flips with problem size.

and how the effects of σ change depending on the algorithm under consideration; this is the topicdiscussed in the following few paragraphs.

Based on the definition of the 1D spin glass with power-law interactions, as the value of σ grows,the range of the most significant interactions is reduced. With reduction of the range of interactions,the problem should become easier both for selectorecombinative GAs capable of linkage learning,such as hBOA, as well as for selectorecombinative GAs which rarely break interactions betweenclosely located bits, such as GA with twopoint crossover. This is clearly demonstrated by theresults for these two algorithms presented in figures 2 and 3. Although for many problem sizes, theabsolute number of evaluations and the number of flips are in fact smaller for larger values of σ,the growth of the two statistics with problem size slows down substantially as σ is increased. Infact, for the smallest values of σ, the number of evaluations and the number of flips both appear togrow faster than polynomially, whereas for the largest values of σ, the growth appears to be upperbounded by a polynomial of low order. On the other hand, for the uniform crossover, which doesnot respect interactions between consequent bits, the growth appears to be faster than polynomialfor both the number of evaluations and the number of flips.

To better visualize the effects of σ on the performance of compared algorithms, figures 5 and 6show the ratio of the number of evaluations and the number of flips for σ < 2 and σ = 2. Thelarger the ratio, the better the results for σ = 2 compared to smaller values of σ. For both hBOAand GA with twopoint crossover, the ratios for σ ≤ 1 appear to grow relatively fast; in fact, thegrowth appears to be faster than polynomial; on the other hand, for σ = 1.5, the growth can stillbe observed but it is very slow in both cases. For GA with uniform crossover, the opposite behaviorcan be observed; this indicates that uniform crossover does not benefit from the decrease in the

5

16 32 64 12810

1

102

103

104

105

Problem size

Num

ber

of e

valu

atio

ns (

hBO

A)

σ=2.00σ=1.50σ=1.00σ=0.83σ=0.75σ=0.55σ=0.00

(a) hBOA

16 32 64 12810

1

102

103

104

105

Problem sizeN

umbe

r of

eva

luat

ions

(G

A, t

wop

oint

)

σ=2.00σ=1.50σ=1.00σ=0.83σ=0.75σ=0.55σ=0.00

(b) GA (twopoint)

16 32 64 12810

1

102

103

104

105

Problem size

Num

ber

of e

valu

atio

ns (

GA

, tw

opoi

nt)

σ=2.00σ=1.50σ=1.00σ=0.83σ=0.75σ=0.55σ=0.00

(c) GA (uniform)


16 32 64 128

102

103

104

105

106

Problem size

Num

ber

of fl

ips

(hB

OA

)

σ=2.00σ=1.50σ=1.00σ=0.83σ=0.75σ=0.55σ=0.00

(a) hBOA

16 32 64 128

102

103

104

105

106

Problem size

Num

ber

of fl

ips

(GA

, tw

opoi

nt)

σ=2.00σ=1.50σ=1.00σ=0.83σ=0.75σ=0.55σ=0.00

(b) GA (twopoint)

16 32 64 128

102

103

104

105

106

Problem size

Num

ber

of fl

ips

(GA

, tw

opoi

nt)

σ=2.00σ=1.50σ=1.00σ=0.83σ=0.75σ=0.55σ=0.00

(c) GA (uniform)





5

16 32 64 12810

1

102

103

104

105

Problem size

Num

ber

of e

valu

atio

ns (

hBO

A)

σ=2.00σ=1.50σ=1.00σ=0.83σ=0.75σ=0.55σ=0.00

(a) hBOA

16 32 64 12810

1

102

103

104

105

Problem sizeN

umbe

r of

eva

luat

ions

(G

A, t

wop

oint

)

σ=2.00σ=1.50σ=1.00σ=0.83σ=0.75σ=0.55σ=0.00

(b) GA (twopoint)

16 32 64 12810

1

102

103

104

105

Problem size

Num

ber

of e

valu

atio

ns (

GA

, tw

opoi

nt)

σ=2.00σ=1.50σ=1.00σ=0.83σ=0.75σ=0.55σ=0.00

(c) GA (uniform)


16 32 64 128

102

103

104

105

106

Problem size

Num

ber

of fl

ips

(hB

OA

)

σ=2.00σ=1.50σ=1.00σ=0.83σ=0.75σ=0.55σ=0.00

(a) hBOA

16 32 64 128

102

103

104

105

106

Problem size

Num

ber

of fl

ips

(GA

, tw

opoi

nt)

σ=2.00σ=1.50σ=1.00σ=0.83σ=0.75σ=0.55σ=0.00

(b) GA (twopoint)

16 32 64 128

102

103

104

105

106

Problem size

Num

ber

of fl

ips

(GA

, tw

opoi

nt)

σ=2.00σ=1.50σ=1.00σ=0.83σ=0.75σ=0.55σ=0.00

(c) GA (uniform)





5

I Scalability of hBOA and GA with twopoint crossover betterfor short-range interactions.

I Linkage tightens as σ grows.I Tighter linkage makes problem easier (if good recombination).I Twopoint crossover respects tight linkage.

I GA with uniform gets worse with shorter-range interactions.


Results: LS Steps until Optimum (Flips)16 32 64 128

101

102

103

104

105

Problem sizeN

umbe

r of

eva

luat

ions

(hB

OA

)

σ=2.00σ=1.50σ=1.00σ=0.83σ=0.75σ=0.55σ=0.00

(a) hBOA

16 32 64 12810

1

102

103

104

105

Problem size

Num

ber

of e

valu

atio

ns (

GA

, tw

opoi

nt)

σ=2.00σ=1.50σ=1.00σ=0.83σ=0.75σ=0.55σ=0.00

(b) GA (twopoint)

16 32 64 12810

1

102

103

104

105

Problem size

Num

ber

of e

valu

atio

ns (

GA

, tw

opoi

nt)

σ=2.00σ=1.50σ=1.00σ=0.83σ=0.75σ=0.55σ=0.00

(c) GA (uniform)


16 32 64 128

102

103

104

105

106

Problem size

Num

ber

of fl

ips

(hB

OA

)

σ=2.00σ=1.50σ=1.00σ=0.83σ=0.75σ=0.55σ=0.00

(a) hBOA

16 32 64 128

102

103

104

105

106

Problem sizeN

umbe

r of

flip

s (G

A, t

wop

oint

)

σ=2.00σ=1.50σ=1.00σ=0.83σ=0.75σ=0.55σ=0.00

(b) GA (twopoint)

16 32 64 128

102

103

104

105

106

Problem size

Num

ber

of fl

ips

(GA

, tw

opoi

nt)

σ=2.00σ=1.50σ=1.00σ=0.83σ=0.75σ=0.55σ=0.00

(c) GA (uniform)





5

16 32 64 12810

1

102

103

104

105

Problem sizeN

umbe

r of

eva

luat

ions

(hB

OA

)

σ=2.00σ=1.50σ=1.00σ=0.83σ=0.75σ=0.55σ=0.00

(a) hBOA

16 32 64 12810

1

102

103

104

105

Problem size

Num

ber

of e

valu

atio

ns (

GA

, tw

opoi

nt)

σ=2.00σ=1.50σ=1.00σ=0.83σ=0.75σ=0.55σ=0.00

(b) GA (twopoint)

16 32 64 12810

1

102

103

104

105

Problem size

Num

ber

of e

valu

atio

ns (

GA

, tw

opoi

nt)

σ=2.00σ=1.50σ=1.00σ=0.83σ=0.75σ=0.55σ=0.00

(c) GA (uniform)


16 32 64 128

102

103

104

105

106

Problem size

Num

ber

of fl

ips

(hB

OA

)

σ=2.00σ=1.50σ=1.00σ=0.83σ=0.75σ=0.55σ=0.00

(a) hBOA

16 32 64 128

102

103

104

105

106

Problem size

Num

ber

of fl

ips

(GA

, tw

opoi

nt)

σ=2.00σ=1.50σ=1.00σ=0.83σ=0.75σ=0.55σ=0.00

(b) GA (twopoint)

16 32 64 128

102

103

104

105

106

Problem size

Num

ber

of fl

ips

(GA

, tw

opoi

nt)

σ=2.00σ=1.50σ=1.00σ=0.83σ=0.75σ=0.55σ=0.00

(c) GA (uniform)





5

16 32 64 12810

1

102

103

104

105

Problem size

Num

ber

of e

valu

atio

ns (

hBO

A)

σ=2.00σ=1.50σ=1.00σ=0.83σ=0.75σ=0.55σ=0.00

(a) hBOA

16 32 64 12810

1

102

103

104

105

Problem size

Num

ber

of e

valu

atio

ns (

GA

, tw

opoi

nt)

σ=2.00σ=1.50σ=1.00σ=0.83σ=0.75σ=0.55σ=0.00

(b) GA (twopoint)

16 32 64 12810

1

102

103

104

105

Problem size

Num

ber

of e

valu

atio

ns (

GA

, tw

opoi

nt)

σ=2.00σ=1.50σ=1.00σ=0.83σ=0.75σ=0.55σ=0.00

(c) GA (uniform)


16 32 64 128

102

103

104

105

106

Problem size

Num

ber

of fl

ips

(hB

OA

)

σ=2.00σ=1.50σ=1.00σ=0.83σ=0.75σ=0.55σ=0.00

(a) hBOA

16 32 64 128

102

103

104

105

106

Problem size

Num

ber

of fl

ips

(GA

, tw

opoi

nt)

σ=2.00σ=1.50σ=1.00σ=0.83σ=0.75σ=0.55σ=0.00

(b) GA (twopoint)

16 32 64 128

102

103

104

105

106

Problem size

Num

ber

of fl

ips

(GA

, tw

opoi

nt)

σ=2.00σ=1.50σ=1.00σ=0.83σ=0.75σ=0.55σ=0.00

(c) GA (uniform)





5

I Scalability of hBOA and GA with twopoint crossover betterfor short-range interactions.

I GA with uniform gets worse with shorter-range interactions.


Comparison: Evaluations until Optimum16 32 64 128

0.5

1

2

4

Problem sizeSlo

wdo

wn

fact

or fo

r th

e nu

mbe

r of

flip

sfo

r hB

OA

(co

mpa

red

to σ

=2.

00)

σ=1.50σ=1.00σ=0.83σ=0.75σ=0.55σ=0.00

(a) hBOA

16 32 64 128

0.5

1

2

4

Problem sizeSlo

wdo

wn

fact

or fo

r th

e nu

mbe

r of

flip

sfo

r G

A w

ith tw

opoi

nt (

com

pare

d to

σ=

2.00

σ=1.50σ=1.00σ=0.83σ=0.75σ=0.55σ=0.00

(b) GA (twopoint)

16 32 64 128

0.0156

0.0312

0.0625

0.125

0.25

0.5

1

Problem sizeSlo

wdo

wn

fact

or fo

r th

e nu

mbe

r of

flip

sfo

r G

A w

ith u

nifo

rm (

com

pare

d to

σ=

2.00

σ=1.50σ=1.00σ=0.83σ=0.75σ=0.55σ=0.00

(c) GA (uniform)

Figure 6: Comparison of the number of flips for σ = 2.00 with that for σ < 2.00.

0 40 80 120 160

1

1.5

2

2.5

3

3.5

Problem size

Num

ber

of e

valu

atio

ns (

GA

, tw

opoi

nt)

/N

umbe

r of

eva

luat

ions

(hB

OA

)

σ=2.00σ=1.50σ=1.00σ=0.83σ=0.75σ=0.55σ=0.00

(a) GA (twopoint) and hBOA

0 40 80 120 1600

5

10

15

20

25

30

Problem sizeN

umbe

r of

eva

luat

ions

(G

A, u

nifo

rm)

/N

umbe

r of

eva

luat

ions

(hB

OA

)

σ=2.00σ=1.50σ=1.00σ=0.83σ=0.75σ=0.55σ=0.00

(b) GA (uniform) and hBOA

0 40 80 120 1600

2.5

5

7.5

10

12.5

15

17.5

Problem size

Num

ber

of e

valu

atio

ns (

GA

, uni

form

) /

Num

ber

of e

valu

atio

ns (

GA

, tw

opoi

nt)

σ=2.00σ=1.50σ=1.00σ=0.83σ=0.75σ=0.55σ=0.00

(c) GA (uniform) and GA (two-point)

Figure 7: Ratio for the number of evaluations for pairs of compared algorithms.

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[28] H. Muhlenbein and T. Mahnig. Convergence theory andapplications of the factorized distribution algorithm.Journal of Computing and Information Technology,7(1):19–32, 1998.

[29] H. Muhlenbein and T. Mahnig. FDA – A scalableevolutionary algorithm for the optimization of additivelydecomposed functions. Evolutionary Computation,7(4):353–376, 1999.

[30] H. Muhlenbein and G. Paaß. From recombination of genesto the estimation of distributions I. Binary parameters.Parallel Problem Solving from Nature, pages 178–187, 1996.

16 32 64 1280.5

1

2

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Problem sizeSlo

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fact

or fo

r th

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mbe

r of

flip

sfo

r hB

OA

(co

mpa

red

to σ

=2.

00)

σ=1.50σ=1.00σ=0.83σ=0.75σ=0.55σ=0.00

(a) hBOA

16 32 64 128

0.5

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sfo

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opoi

nt (

com

pare

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σ=

2.00

σ=1.50σ=1.00σ=0.83σ=0.75σ=0.55σ=0.00

(b) GA (twopoint)

16 32 64 128

0.0156

0.0312

0.0625

0.125

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Problem sizeSlo

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ith u

nifo

rm (

com

pare

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σ=

2.00

σ=1.50σ=1.00σ=0.83σ=0.75σ=0.55σ=0.00

(c) GA (uniform)


0 40 80 120 160

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σ=2.00σ=1.50σ=1.00σ=0.83σ=0.75σ=0.55σ=0.00


0 40 80 120 1600

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GA

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σ=2.00σ=1.50σ=1.00σ=0.83σ=0.75σ=0.55σ=0.00

























16 32 64 1280.5

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Problem sizeSlo

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sfo

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(co

mpa

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to σ

=2.

00)

σ=1.50σ=1.00σ=0.83σ=0.75σ=0.55σ=0.00

(a) hBOA

16 32 64 128

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Problem sizeSlo

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opoi

nt (

com

pare

d to

σ=

2.00

σ=1.50σ=1.00σ=0.83σ=0.75σ=0.55σ=0.00

(b) GA (twopoint)

16 32 64 128

0.0156

0.0312

0.0625

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nifo

rm (

com

pare

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σ=

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σ=1.50σ=1.00σ=0.83σ=0.75σ=0.55σ=0.00

(c) GA (uniform)


0 40 80 120 160

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0 40 80 120 1600

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σ=2.00σ=1.50σ=1.00σ=0.83σ=0.75σ=0.55σ=0.00


0 40 80 120 1600

2.5

5

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Num

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nt)

σ=2.00σ=1.50σ=1.00σ=0.83σ=0.75σ=0.55σ=0.00

























I hBOA outperforms both GA variants.

I Biggest differences for short-range interactions (expected).

I GA with uniform crossover performs worst.


Comparison: LS Steps until Optimum

0 40 80 120 160

0.8

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0 40 80 120 160

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0 40 80 120 160

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Figure 8: Ratio for the number of flips for pairs of compared algorithms.

0 40 80 120 1600.6

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0 40 80 120 1600

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0 40 80 120 1600

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Figure 9: Ratio for the required population sizes for pairs of compared algorithms.

[31] H. Muhlenbein and D. Schlierkamp-Voosen. Predictivemodels for the breeder genetic algorithm: I. Continuousparameter optimization. Evolutionary Computation,1(1):25–49, 1993.

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[33] B. Naudts, D. Suys, and A. Verschoren. Epistasis as a basicconcept in formal landscape analysis. Proc. of theInternational Conf. on Genetic Algorithms (ICGA-97),pages 65–72, 1997.

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[36] M. Pelikan. Hierarchical Bayesian optimization algorithm:Toward a new generation of evolutionary algorithms.Springer, 2005.

[37] M. Pelikan and D. E. Goldberg. Escaping hierarchical trapswith competent genetic algorithms. Proc. of the Geneticand Evolutionary Computation Conf. (GECCO-2001),pages 511–518, 2001.

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[39] M. Pelikan and D. E. Goldberg. A hierarchy machine:Learning to optimize from nature and humans. Complexity,8(5):36–45, 2003.

[40] M. Pelikan, D. E. Goldberg, and F. Lobo. A survey ofoptimization by building and using probabilistic models.Computational Optimization and Applications, 21(1):5–20,2002.

[41] M. Pelikan and A. K. Hartmann. Hierarchical BOA, clusterexact approximation, and Ising spin glasses. ParallelProblem Solving from Nature, pages 122–131, 2006.

[42] M. Pelikan, H. G. Katzgraber, and S. Kobe. Findingground states of sherrington-kirkpatrick spin glasses withhierarchical BOA and genetic algorithms. Proc. of theGenetic and Evolutionary Computation Conf.(GECCO-2008), pages 447–454, 2008.

[43] M. Pelikan, K. Sastry, and E. Cantu-Paz, editors. Scalableoptimization via probabilistic modeling: From algorithms toapplications. Springer-Verlag, 2006.

[44] M. Pelikan, K. Sastry, and D. E. Goldberg. Scalability ofthe Bayesian optimization algorithm. International Journalof Approximate Reasoning, 31(3):221–258, 2002.

[45] F. Rothlauf. Representations for genetic and evolutionaryalgorithms. Springer Verlag, Berlin, 2002.

[46] K. Sastry. Evaluation-relaxation schemes for genetic andevolutionary algorithms. Master’s thesis, University ofIllinois at Urbana-Champaign, Department of GeneralEngineering, Urbana, IL, 2001.

[47] D. Thierens, D. E. Goldberg, and A. G. Pereira. Dominoconvergence, drift, and the temporal-salience structure ofproblems. Proc. of the International Conf. on EvolutionaryComputation (ICEC-98), pages 535–540, 1998.

[48] C. Van Hoyweghen. Detecting spin-flip symmetry inoptimization problems. In L. Kallel et al., editors,Theoretical Aspects of Evolutionary Computing, pages423–437. Springer, Berlin, 2001.

[49] T.-L. Yu, K. Sastry, D. E. Goldberg, and M. Pelikan.Population sizing for entropy-based model building indiscrete estimation of distribution algorithms. Proc. of theGenetic and Evolutionary Computation Conf.(GECCO-2007), pages 601–608, 2007.

[50] Q. Zhang. On stability of fixed points of limit models of

0 40 80 120 160

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0 40 80 120 160

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0 40 80 120 160

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0 40 80 120 1600.6

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0 40 80 120 160

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σ=2.00σ=1.50σ=1.00σ=0.83σ=0.75σ=0.55σ=0.00























I hBOA outperforms both GA variants.

I Biggest differences for short-range of interactions (expected).

I GA with uniform crossover performs worst.


Comparison: Correlation w.r.t. LS Steps

Long range (σ = 0.55)

Short range (σ = 2.00)

I Long-range problems lead to stronger correlations.I Short-range problems deviate less than long-range ones.I Correlations between GAs stronger than for hBOA.


Comparison: Short-Range vs. Long-Range

hBOA GA (twopoint) GA (uniform)

16 32 64 1280.5

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(a) hBOA

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(b) GA (twopoint)

16 32 64 128

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(c) GA (uniform)


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ditionally, these results indicate that the correlations are almost consistently stronger for smallerproblem sizes.

5 Future Work

The numerous statistics provided by the experiments presented in this paper can be analyzedin more detail to provide a deeper insight into the strengths and weaknesses of the comparedevolutionary algorithms, and to provide inputs for further improvement of these algorithms on theone-dimensional spin glass with power-law interactions and on other difficult classes of additivelydecomposable problems with complex fitness landscape.

One of the important outputs of this paper was the large number of problem instances of theone-dimensional spin glass with power-law interactions. These instances can be used for additionalempirical studies in evolutionary computation and beyond. Most importantly, one should considertechniques that were shown to perform well on various classes of spin glass models and comparethe performance of these techniques to that of the evolutionary algorithms studied in this paper.Three algorithms are of special interest: extremal optimization [4], hysteretic optimization [34, 35],and GA with triadic crossover [34]. Hybrids of evolutionary algorithms, hysteretic optimization,and extremal optimization hold a great promise for solving large instances of both the general SKspin glass as well as the one-dimensional SK spin glass with power-law interactions. The key is toidentify the strengths and weaknesses of the different algorithms and to combine their strengths

7

I For hBOA and GA (twopoint), performance better withshort-range interactions.

I For GA (uniform), behavior is opposite.I Related to properties of recombination and problem structure.


Conclusions and Future Work

Summary and conclusions

I Considered class of SK spin glasses with power-lawinteractions as class of random test problems.

I Can tune effective size of overlap.I NP complete.

I Generated a broad range of problem instances.I Analyzed results using hybrids of GEAs.

Future work

I Use generated problems to test other algorithms.I Design new hybrid optimizers for broad range of spin glasses.I Relate results to existing theory (problem difficulty, landscape

analysis, population sizing, time to convergence).I Analyze obtained instances from computational physics

perspective.


Acknowledgments

Acknowledgments

I NSF; NSF CAREER grant ECS-0547013.

I U.S. Air Force, AFOSR; FA9550-06-1-0096.

I University of Missouri; High Performance ComputingCollaboratory sponsored by Information Technology Services;Research Award; Research Board.


Analysis of Evolutionary Algorithms on the One-Dimensional Spin Glass with Power-Law Interactions

Technology

d spin glass

magnitude of spin

powerlaw interactions

spin glass instance

ising spin

standard sk spin glass

theonedimensional spin

zerospinspin couplings