Finding Ground States of Sherrington-Kirkpatrick Spin Glasses with Hierarchical BOA and Genetic...
-
Upload
martin-pelikan -
Category
Technology
-
view
1.744 -
download
0
description
Transcript of Finding Ground States of Sherrington-Kirkpatrick Spin Glasses with Hierarchical BOA and Genetic...
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
Institut fr Theoretische PhysikTechnische Universitat Dresden, Germany
Martin Pelikan, Helmut G. Katzgraber, Sigismund Kobe Finding Ground States of SK Spin Glasses with hBOA and GAs
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.
Martin Pelikan, Helmut G. Katzgraber, Sigismund Kobe Finding Ground States of SK Spin Glasses with hBOA and GAs
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.
Martin Pelikan, Helmut G. Katzgraber, Sigismund Kobe Finding Ground States of SK Spin Glasses with hBOA and GAs
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
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}.
Martin Pelikan, Helmut G. Katzgraber, Sigismund Kobe Finding Ground States of SK Spin Glasses with hBOA and GAs
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.
Martin Pelikan, Helmut G. Katzgraber, Sigismund Kobe Finding Ground States of SK Spin Glasses with hBOA and GAs
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).
Martin Pelikan, Helmut G. Katzgraber, Sigismund Kobe Finding Ground States of SK Spin Glasses with hBOA and GAs
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.
Martin Pelikan, Helmut G. Katzgraber, Sigismund Kobe Finding Ground States of SK Spin Glasses with hBOA and GAs
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.
Martin Pelikan, Helmut G. Katzgraber, Sigismund Kobe Finding Ground States of SK Spin Glasses with hBOA and GAs
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.
Martin Pelikan, Helmut G. Katzgraber, Sigismund Kobe Finding Ground States of SK Spin Glasses with hBOA and GAs
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).
Martin Pelikan, Helmut G. Katzgraber, Sigismund Kobe Finding Ground States of SK Spin Glasses with hBOA and GAs
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.
Martin Pelikan, Helmut G. Katzgraber, Sigismund Kobe Finding Ground States of SK Spin Glasses with hBOA and GAs
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.
Martin Pelikan, Helmut G. Katzgraber, Sigismund Kobe Finding Ground States of SK Spin Glasses with hBOA and GAs
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
Martin Pelikan, Helmut G. Katzgraber, Sigismund Kobe Finding Ground States of SK Spin Glasses with hBOA and GAs
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
Martin Pelikan, Helmut G. Katzgraber, Sigismund Kobe Finding Ground States of SK Spin Glasses with hBOA and GAs
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.
Martin Pelikan, Helmut G. Katzgraber, Sigismund Kobe Finding Ground States of SK Spin Glasses with hBOA and GAs
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.
Martin Pelikan, Helmut G. Katzgraber, Sigismund Kobe Finding Ground States of SK Spin Glasses with hBOA and GAs
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
Martin Pelikan, Helmut G. Katzgraber, Sigismund Kobe Finding Ground States of SK Spin Glasses with hBOA and GAs
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.
Martin Pelikan, Helmut G. Katzgraber, Sigismund Kobe Finding Ground States of SK Spin Glasses with hBOA and GAs
hBOA Results for n ≤ 300
101
102
10310
1
102
103
104
Problem size
Mea
n nu
mbe
r of
eva
luat
ions
Martin Pelikan, Helmut G. Katzgraber, Sigismund Kobe Finding Ground States of SK Spin Glasses with hBOA and GAs
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.)
Martin Pelikan, Helmut G. Katzgraber, Sigismund Kobe Finding Ground States of SK Spin Glasses with hBOA and GAs
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.
Martin Pelikan, Helmut G. Katzgraber, Sigismund Kobe Finding Ground States of SK Spin Glasses with hBOA and GAs
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
Martin Pelikan, Helmut G. Katzgraber, Sigismund Kobe Finding Ground States of SK Spin Glasses with hBOA and GAs
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
Martin Pelikan, Helmut G. Katzgraber, Sigismund Kobe Finding Ground States of SK Spin Glasses with hBOA and GAs
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.
Martin Pelikan, Helmut G. Katzgraber, Sigismund Kobe Finding Ground States of SK Spin Glasses with hBOA and GAs
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.
Martin Pelikan, Helmut G. Katzgraber, Sigismund Kobe Finding Ground States of SK Spin Glasses with hBOA and GAs