This study focuses on the problem of finding ground states of random instances of the Sherrington-Kirkpatrick (SK) spin-glass model with Gaussian couplings. While the ground states of SK spin-glass instances can be obtained with branch and bound, the computational complexity of branch and bound yields instances of not more than about 90 spins. We describe several approaches based on the hierarchical Bayesian optimization algorithm (hBOA) to reliably identifying ground states of SK instances intractable with branch and bound, and present a broad range of empirical results on such problem instances. We argue that the proposed methodology holds a big promise for reliably solving large SK spin-glass instances to optimality with practical time complexity. The proposed approaches to identifying global optima reliably can also be applied to other problems and they can be used with many other evolutionary algorithms. Performance of hBOA is compared to that of the genetic algorithm with two common crossover operators.

1. Finding Ground States of Sherrington-Kirkpatrick
Spin Glasses with hBOA and GAs
Martin Pelikan, Helmut G. Katzgraber, & Sigismund Kobe
Missouri Estimation of Distribution Algorithms Laboratory (MEDAL)
University of Missouri, St. Louis, MO
http://medal.cs.umsl.edu/
pelikan@cs.umsl.edu
Theoretische Physik
ETH Z¨urich, Switzerland
katzgraber@phys.ethz.ch
Institut fr Theoretische Physik
Technische Universit¨at Dresden, Germany
kobe@physik.tu-dresden.de
Martin Pelikan, Helmut G. Katzgraber, Sigismund Kobe Finding Ground States of SK Spin Glasses with hBOA and GAs

2. Background
Background
Spin glasses are prototypical models for disordered systems.
Important topic in theoretical physics for several decades.
Popular also as test problem for evolutionary algorithms
Can generate many random instances of varying difficulty.
Highly multimodal landscape.
Strong interactions between variables.
Similarities with other difficult NP-complete problems.
Usually spins arranged on 2D or 3D lattices, but only few
studies for the infinitely dimensional SK spin glass.
Yet the infinitely dimensional systems are most difficult and
interesting.
Martin Pelikan, Helmut G. Katzgraber, Sigismund Kobe Finding Ground States of SK Spin Glasses with hBOA and GAs

3. Purpose
Purpose
Develop and test a robust approach to reliably solving large
instances of SK spin glass and other NP complete problems.
Don’t compromise problem size or reliability.
Two target areas
Computational physics.
Optimization.
Martin Pelikan, Helmut G. Katzgraber, Sigismund Kobe Finding Ground States of SK Spin Glasses with hBOA and GAs

4. Outline
1. Sherrington-Kirkpatrick (SK) spin glass.
2. Branch and bound for SK spin glass.
3. Approaches to reliable solution of large SK instances.
4. Future work.
5. Summary and conclusions.
Martin Pelikan, Helmut G. Katzgraber, Sigismund Kobe Finding Ground States of SK Spin Glasses with hBOA and GAs

5. 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, Helmut G. Katzgraber, Sigismund Kobe Finding Ground States of SK Spin Glasses with hBOA and GAs

6. Ground States of SK Spin Glasses
Energy
Energy of a spin configuration C is given by
H(C) = −
i<j
Ji,jsisj
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, Helmut G. Katzgraber, Sigismund Kobe Finding Ground States of SK Spin Glasses with hBOA and GAs

7. Random Instances of SK Spin Glass
Random spin glass instances
Spin glass models usually studied over large sets of random
instances.
Two most common distributions for couplings
Gaussian: N(0, 1).
±J: +1 or −1 with equal probability.
Sometimes a distance metric is imposed and coupling strength
decreases with distance.
Instances used in this work
We use Gaussian couplings from N(0, 1).
Martin Pelikan, Helmut G. Katzgraber, Sigismund Kobe Finding Ground States of SK Spin Glasses with hBOA and GAs

8. Branch and Bound for SK Spin Glass
Basic idea
Traverse the entire search space
(try all spin configurations).
Each level decides on one spin
(+1 or -1).
Each leaf encodes a unique spin
configuration.
Branches that lead to provably
suboptimal solutions are cut.
Why?
BB is inefficient, but can verify
the global optimum.
Martin Pelikan, Helmut G. Katzgraber, Sigismund Kobe Finding Ground States of SK Spin Glasses with hBOA and GAs

9. Iterative Branch and Bound
Basic idea
Hartwig, Daske, and Kobe (1984).
Reduce the system to consider only first i spins.
Solve for i = 2 to i = n with step 1.
Use previous results to provide better bounds.
Denote best energy for for first i spins by f∗
i .
Lower bound on best energy for first j spins given by
f∗
j ≥ f∗
j−1 −
j−1
i=1
|Ji,j|.
Effects of iterative approach
We must solve n − 1 problems instead of 1.
But the overall performance much better.
Martin Pelikan, Helmut G. Katzgraber, Sigismund Kobe Finding Ground States of SK Spin Glasses with hBOA and GAs

10. Current Situation and Goal
Current situation
We have BB which is guaranteed solve small instances.
We have hBOA and other evolutionary algorithms which can
solve larger instances but we need to set
Population size.
Number of generations.
Goal
Find reliable optima of relatively large instances.
Don’t stick with small problems because of BB.
Don’t compromise reliability by guessing EA parameters wildly.
Martin Pelikan, Helmut G. Katzgraber, Sigismund Kobe Finding Ground States of SK Spin Glasses with hBOA and GAs

11. Basic Approach
Step 1: Branch and bound
Generate many instances for small problems solvable with BB.
Solve each instance with iterative BB.
Step 2: hBOA with optimal settings
Apply hBOA to each new instance.
Find accurate statistical model for hBOA parameters.
Use model to predict sufficient parameters for larger problems.
Step 3: Going to larger problems
Apply hBOA with the conservative settings from step 2 to
find reliable global optima of larger instances.
Go to step 2 (to get to larger and larger problems).
Martin Pelikan, Helmut G. Katzgraber, Sigismund Kobe Finding Ground States of SK Spin Glasses with hBOA and GAs

12. Step 1: Solve Small Problems with BB
Prepare instances
Generate 10,000 random SK instances for n = 20 to 80.
This gives a total of 310,000 unique problem instances.
Solve each instance with BB to find global optimum.
Martin Pelikan, Helmut G. Katzgraber, Sigismund Kobe Finding Ground States of SK Spin Glasses with hBOA and GAs

13. Step 2: Run hBOA and Analyze Parameters
Basic setup
hBOA with default parameters.
Only population size and number of generations tuned.
Deterministic 1-bit hill climber improves all solutions.
Maximum number of generations is set to n.
Population size set with bisection for each instance (10
successes in 10 independent runs).
Analysis
Total of 3,100,000 hBOA runs to analyze.
Analyze the distribution of the following
Population size.
Number of generations.
Number of evaluations.
Number of flips of hill climber.
Martin Pelikan, Helmut G. Katzgraber, Sigismund Kobe Finding Ground States of SK Spin Glasses with hBOA and GAs

14. Step 2: Results
Population size appears to follow log-normal distribution.
Number of generations is very small in all cases.
n = 20 n = 80
0 20 40 60 80
0
500
1000
1500
2000
2500
Population size
Frequency
0 100 200 300 400 500
0
500
1000
1500
2000
Population size
Frequency
Martin Pelikan, Helmut G. Katzgraber, Sigismund Kobe Finding Ground States of SK Spin Glasses with hBOA and GAs

15. Step 2: Results
Estimate parameters of pop. size distribution for each n.
Derive upper bound from 0.001% tail of the distribution,
which sould solve 99.999% instances.
Find a fit of this upper bound.
Predict pop. size for larger problems (up to n = 200).
Fit of 99.999% percentile Prediction for larger instances
20 30 40 50 60 70 80
100
150
200
250
300
350
400
450
500
550
600
Populationsize
Problem size
Power−law fit
95% prediction bounds
99.999 percentile
20 40 60 80 100 120 140 160 180 200
0
250
500
750
1000
1250
1500
1750
2000
Populationsize
Problem size
Power−law fit
95% prediction bounds
Martin Pelikan, Helmut G. Katzgraber, Sigismund Kobe Finding Ground States of SK Spin Glasses with hBOA and GAs

16. Step 3: Find Reliable Optima of Larger Instances
Starting point
Predicted bound on pop. size to solve 99.999% instances.
Prepare larger instances
Generate 1,000 instances for n = 100 to 200.
For each instance
Use estimated upper bound of the population size.
Use maximum number of generations of n.
Make 10 hBOA runs on each instance to find global optimum.
Record the best solution found.
All runs should agree.
Martin Pelikan, Helmut G. Katzgraber, Sigismund Kobe Finding Ground States of SK Spin Glasses with hBOA and GAs

17. Step 2 Revisited: Run hBOA and Analyze Parameters
Run and analyze hBOA
Run hBOA for n = 100 to 200 as for smaller instances.
Repeat bisection 10 times for each instance.
Analysis
Total of 2,100,000 successful hBOA runs.
Do the analysis as for smaller problems.
Martin Pelikan, Helmut G. Katzgraber, Sigismund Kobe Finding Ground States of SK Spin Glasses with hBOA and GAs

18. Step 2 Revisited: Results
Estimate parameters of pop. size distribution for each n.
Derive upper bound from 0.001% tail of the distribution.
Find a fit of this upper bound.
Predict pop. size for larger problems (up to n = 300).
Fit of 99.999% percentile Prediction for larger instances
20 40 60 80 100 120 140 160 180 200
0
250
500
750
1000
1250
1500
1750
2000
2250
2500
Populationsize
Problem size
Power−law fit
95% prediction bounds
99.999 percentile
20 60 100 140 180 220 260 300
0
500
1000
1500
2000
2500
3000
3500
4000
Populationsize
Problem size
Power−law fit
95% prediction bounds
Martin Pelikan, Helmut G. Katzgraber, Sigismund Kobe Finding Ground States of SK Spin Glasses with hBOA and GAs

19. So How Does It Work?
How does it work?
Incrementally increase problem size.
Set parameters using model based on smaller problems.
If distributions are easy to model and the growth of different
parameters can be fit reliably, this allows us to reliably solve
large instances even when no complete algorithm is tractable.
Ultimate goal
Go to problems with 4,000 spins or so.
Important
Don’t make too big steps to ensure tractability and reliability.
Martin Pelikan, Helmut G. Katzgraber, Sigismund Kobe Finding Ground States of SK Spin Glasses with hBOA and GAs

20. hBOA Results for n ≤ 300
10
1
10
2
10
3
10
1
10
2
10
3
10
4
Problem size
Meannumberofevaluations
Martin Pelikan, Helmut G. Katzgraber, Sigismund Kobe Finding Ground States of SK Spin Glasses with hBOA and GAs

21. Other Approaches: Fit Distribution Parameters
Basic idea
Fit distribution of a quantity (e.g. pop. size).
Fit a model to the parameters of the distribution.
Estimate parameters for larger problems from the fit.
Compute tails from estimated parameters.
20 40 60 80 100 120 140 160 180 200
0
1
2
3
4
5
Problem size
Populationsize
Log mean
Power−law fit for mean
Log standard deviation
Power−law fit for std. dev.
20 40 60 80 100 120 140 160 180 200
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Problem size
Numberofiterations
Log mean
Power−law fit (mean)
Log standard deviation
Power−law fit (std. dev.)
Martin Pelikan, Helmut G. Katzgraber, Sigismund Kobe Finding Ground States of SK Spin Glasses with hBOA and GAs

22. Other Approaches: Population Doubling
Basic idea
Related to parameter-less genetic algorithms.
Start with a reasonable population size.
Make 10 runs (can change).
Double the population and repeat.
Terminate doubling when
All 10 runs result in the same solution.
Last couple of rounds resulted in the same solution.
Martin Pelikan, Helmut G. Katzgraber, Sigismund Kobe Finding Ground States of SK Spin Glasses with hBOA and GAs

23. Comparison: hBOA vs. GA (Uniform Crossover)
Number of evaluations Number of flips
20 40 60 80 100 120 140 160 180 200
0.8
1
1.2
1.4
1.6
1.8
2
Number of spins
Num.GA(U)evals./num.hBOAevals.
20 40 60 80 100 120 140 160 180 200
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
Number of spins
Num.GA(U)flips/num.hBOAflips
Martin Pelikan, Helmut G. Katzgraber, Sigismund Kobe Finding Ground States of SK Spin Glasses with hBOA and GAs

24. Comparison: Uniform vs. Two-Point Crossover
Number of evaluations Number of flips
20 40 60 80 100 120 140 160 180 200
0.75
0.8
0.85
0.9
0.95
1
1.05
Number of spins
Num.GA(U)evals./num.GA(2P)evals.
20 40 60 80 100 120 140 160 180 200
0.9
0.95
1
1.05
1.1
Number of spins
Num.GA(U)flips/num.GA(2P)flips
Martin Pelikan, Helmut G. Katzgraber, Sigismund Kobe Finding Ground States of SK Spin Glasses with hBOA and GAs

25. Conclusions and Future Work
Conclusions
The proposed approaches hold big promise for reliable solution
of extremely large problems.
The proposed approaches can be used with other optimization
techniques which require adequate parameter settings.
SK spin glass closely related to other difficult problems, such
as protein folding.
Future work
Compare hBOA & GA to other techniques
Extremal optimization (EO).
Hysteretic optimization (HO).
Create efficient hybrids of hBOA, GA, EO, HO, and BB.
Apply other efficiency enhancement techniques.
Further increase problem size to 1,000–4,000 and more.
Martin Pelikan, Helmut G. Katzgraber, Sigismund Kobe Finding Ground States of SK Spin Glasses with hBOA and GAs

26. Acknowledgments
Acknowledgments
NSF; NSF CAREER grant ECS-0547013.
U.S. Air Force, AFOSR; FA9550-06-1-0096.
University of Missouri; High Performance Computing
Collaboratory sponsored by Information Technology Services;
Research Award; Research Board.
Martin Pelikan, Helmut G. Katzgraber, Sigismund Kobe Finding Ground States of SK Spin Glasses with hBOA and GAs