Finding Ground States of Sherrington-Kirkpatrick Spin Glasses with Hierarchical BOA and Genetic Algorithms

Finding Ground States of Sherrington-KirkpatrickSpin 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/[email protected]

Theoretische PhysikETH Zurich, Switzerland

[email protected]

Institut fr Theoretische PhysikTechnische Universitat Dresden, Germany

[email protected]

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

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

[email protected]

[email protected]

[email protected]

Background

Background

I Spin glasses are prototypical models for disordered systems.

I Important topic in theoretical physics for several decades.I Popular also as test problem for evolutionary algorithms

I Can generate many random instances of varying difficulty.I Highly multimodal landscape.I Strong interactions between variables.I Similarities with other difficult NP-complete problems.

I Usually spins arranged on 2D or 3D lattices, but only fewstudies for the infinitely dimensional SK spin glass.

I Yet the infinitely dimensional systems are most difficult andinteresting.


Purpose

Purpose

I Develop and test a robust approach to reliably solving largeinstances of SK spin glass and other NP complete problems.

I Don’t compromise problem size or reliability.I Two target areas

I Computational physics.I Optimization.


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.


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 Instances of SK Spin Glass

Random spin glass instances

I Spin glass models usually studied over large sets of randominstances.

I Two most common distributions for couplingsI Gaussian: N(0, 1).I ±J : +1 or −1 with equal probability.

I Sometimes a distance metric is imposed and coupling strengthdecreases with distance.

Instances used in this work

I We use Gaussian couplings from N(0, 1).


Branch and Bound for SK Spin Glass

Basic idea

I Traverse the entire search space(try all spin configurations).

I Each level decides on one spin(+1 or -1).

I Each leaf encodes a unique spinconfiguration.

I Branches that lead to provablysuboptimal solutions are cut.

Why?

I BB is inefficient, but can verifythe global optimum.


Iterative Branch and Bound

Basic idea

I Hartwig, Daske, and Kobe (1984).I Reduce the system to consider only first i spins.I Solve for i = 2 to i = n with step 1.I Use previous results to provide better bounds.I Denote best energy for for first i spins by f∗i .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

I We must solve n− 1 problems instead of 1.I But the overall performance much better.


Current Situation and Goal

Current situation

I We have BB which is guaranteed solve small instances.I We have hBOA and other evolutionary algorithms which can

solve larger instances but we need to setI Population size.I Number of generations.

Goal

I Find reliable optima of relatively large instances.

I Don’t stick with small problems because of BB.

I Don’t compromise reliability by guessing EA parameters wildly.


Basic Approach

Step 1: Branch and bound

I Generate many instances for small problems solvable with BB.

I Solve each instance with iterative BB.

Step 2: hBOA with optimal settings

I Apply hBOA to each new instance.

I Find accurate statistical model for hBOA parameters.

I Use model to predict sufficient parameters for larger problems.

Step 3: Going to larger problems

I Apply hBOA with the conservative settings from step 2 tofind reliable global optima of larger instances.

I Go to step 2 (to get to larger and larger problems).


Step 1: Solve Small Problems with BB

Prepare instances

I Generate 10,000 random SK instances for n = 20 to 80.

I This gives a total of 310,000 unique problem instances.

I Solve each instance with BB to find global optimum.


Step 2: Run hBOA and Analyze Parameters

Basic setup

I hBOA with default parameters.

I Only population size and number of generations tuned.

I Deterministic 1-bit hill climber improves all solutions.

I Maximum number of generations is set to n.

I Population size set with bisection for each instance (10successes in 10 independent runs).

Analysis

I Total of 3,100,000 hBOA runs to analyze.I Analyze the distribution of the following

I Population size.I Number of generations.I Number of evaluations.I Number of flips of hill climber.


Step 2: Results

I Population size appears to follow log-normal distribution.

I Number of generations is very small in all cases.

n = 20 n = 80

0 20 40 60 800

500

1000

1500

2000

2500

Population size

Fre

quen

cy

0 100 200 300 400 5000

500

1000

1500

2000

Population size

Fre

quen

cy


Step 2: Results

I Estimate parameters of pop. size distribution for each n.

I Derive upper bound from 0.001% tail of the distribution,which sould solve 99.999% instances.

I Find a fit of this upper bound.

I 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 80100150200250300350400450500550600

Pop

ulat

ion

size

Problem size

Power−law fit95% prediction bounds99.999 percentile

20 40 60 80 100 120 140 160 180 2000

250

500

750

1000

1250

1500

1750

2000

Pop

ulat

ion

size

Problem size

Power−law fit95% prediction bounds


Step 3: Find Reliable Optima of Larger Instances

Starting point

I Predicted bound on pop. size to solve 99.999% instances.

Prepare larger instances

I Generate 1,000 instances for n = 100 to 200.I For each instance

I Use estimated upper bound of the population size.I Use maximum number of generations of n.I Make 10 hBOA runs on each instance to find global optimum.I Record the best solution found.I All runs should agree.


Step 2 Revisited: Run hBOA and Analyze Parameters

Run and analyze hBOA

I Run hBOA for n = 100 to 200 as for smaller instances.

I Repeat bisection 10 times for each instance.

Analysis

I Total of 2,100,000 successful hBOA runs.

I Do the analysis as for smaller problems.


Step 2 Revisited: Results

I Estimate parameters of pop. size distribution for each n.

I Derive upper bound from 0.001% tail of the distribution.

I Find a fit of this upper bound.

I 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 2000

250500750

1000125015001750200022502500

Pop

ulat

ion

size

Problem size

Power−law fit95% prediction bounds99.999 percentile

20 60 100 140 180 220 260 3000

500

1000

1500

2000

2500

3000

3500

4000

Pop

ulat

ion

size

Problem size

Power−law fit95% prediction bounds


So How Does It Work?

How does it work?

I Incrementally increase problem size.

I Set parameters using model based on smaller problems.

I If distributions are easy to model and the growth of differentparameters can be fit reliably, this allows us to reliably solvelarge instances even when no complete algorithm is tractable.

Ultimate goal

I Go to problems with 4,000 spins or so.

Important

I Don’t make too big steps to ensure tractability and reliability.


hBOA Results for n ≤ 300

101

102

10310

1

102

103

104

Problem size

Mea

n nu

mbe

r of

eva

luat

ions


Other Approaches: Fit Distribution Parameters

Basic idea

I Fit distribution of a quantity (e.g. pop. size).

I Fit a model to the parameters of the distribution.

I Estimate parameters for larger problems from the fit.

I Compute tails from estimated parameters.

20 40 60 80 100 120 140 160 180 2000

1

2

3

4

5

Problem size

Pop

ulat

ion

size

Log mean Power−law fit for meanLog standard deviation Power−law fit for std. dev.

20 40 60 80 100 120 140 160 180 200

00.20.40.60.8

11.21.41.61.8

Problem size

Num

ber

of it

erat

ions

Log mean Power−law fit (mean)Log standard deviation Power−law fit (std. dev.)


Other Approaches: Population Doubling

Basic idea

I Related to parameter-less genetic algorithms.

I Start with a reasonable population size.

I Make 10 runs (can change).

I Double the population and repeat.I Terminate doubling when

I All 10 runs result in the same solution.I Last couple of rounds resulted in the same solution.


Comparison: hBOA vs. GA (Uniform Crossover)

Number of evaluations Number of flips

20 40 60 80 100 120 140 160 180 2000.8

1

1.2

1.4

1.6

1.8

2

Number of spins

Num

. GA

(U

) ev

als.

/ nu

m. h

BO

A e

vals

.

20 40 60 80 100 120 140 160 180 2000.7

0.8

0.9

1

1.1

1.2

1.3

1.4

Number of spins

Num

. GA

(U

) fli

ps /

num

. hB

OA

flip

s


Comparison: Uniform vs. Two-Point Crossover

Number of evaluations Number of flips

20 40 60 80 100 120 140 160 180 2000.75

0.8

0.85

0.9

0.95

1

1.05

Number of spins

Num

. GA

(U

) ev

als.

/ nu

m. G

A (

2P)

eval

s.

20 40 60 80 100 120 140 160 180 2000.9

0.95

1

1.05

1.1

Number of spins

Num

. GA

(U

) fli

ps /

num

. GA

(2P

) fli

ps


Conclusions and Future Work

Conclusions

I The proposed approaches hold big promise for reliable solutionof extremely large problems.

I The proposed approaches can be used with other optimizationtechniques which require adequate parameter settings.

I SK spin glass closely related to other difficult problems, suchas protein folding.

Future workI Compare hBOA & GA to other techniques

I Extremal optimization (EO).I Hysteretic optimization (HO).

I Create efficient hybrids of hBOA, GA, EO, HO, and BB.

I Apply other efficiency enhancement techniques.

I Further increase problem size to 1,000–4,000 and more.


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.


Finding Ground States of Sherrington-Kirkpatrick Spin Glasses with Hierarchical BOA and Genetic Algorithms

Technology

kirkpatrick spin glasses

ising spin

spin glass instance

spin conguration c

ground states of sherrington

gas martin pelikan

values of spins

pairs of spins