We discuss the computational complexity of random 2D Ising spin glasses, which represent an interesting class of constraint satisfaction problems for black box optimization. Two extremal cases are considered: (1) the +/- J spin glass, and (2) the Gaussian spin glass. We also study a smooth transition between these two extremal cases. The computational complexity of all studied spin glass systems is found to be dominated by rare events of extremely hard spin glass samples. We show that complexity of all studied spin glass systems is closely related to Frechet extremal value distribution. In a hybrid algorithm that combines the hierarchical Bayesian optimization algorithm (hBOA) with a deterministic bit-flip hill climber, the number of steps performed by both the global searcher (hBOA) and the local searcher follow Frechet distributions. Nonetheless, unlike in methods based purely on local search, the parameters of these distributions confirm good scalability of hBOA with local search. We further argue that standard performance measures for optimization algorithms---such as the average number of evaluations until convergence---can be misleading. Finally, our results indicate that for highly multimodal constraint satisfaction problems, such as Ising spin glasses, recombination-based search can provide qualitatively better results than mutation-based search.

1. Computational complexity and
simulation of rare events of Ising spin glasses
Pelikan, M., Ocenasek, J., Trebst, S., Troyer, M., Alet, F.

2. Motivation
Spin glass
Origin in physics, but interesting for optimization as well
Huge number of local optima and plateaus
Local search fails miserably
Some classes can be scalably solved using analytical methods
Some classes provably NP-complete
This paper
Extends previous work to more classes of spin glasses
Provides a thorough statistical analysis of results

3. Outline
Hierarchical BOA (hBOA)
Spin glasses
Definition
Difficulty
Considered classes of spin glasses
Experiments
Summary and conclusions

4. Hierarchical BOA (hBOA)
Pelikan, Goldberg, and Cantu-Paz (2001, 2002)
Evolve population of candidate solutions
Operators
Selection
Variation
Build a Bayesian network with local structures for selected solutions
Sample the built network to generate new solutions
Replacement
Restricted tournament replacement for niching

5. hBOA: Basic algorithm
Bayesian New
Current network population
Selection
population
Restricted tournament replacement

6. Spin glass (SG)
Spins arranged on a lattice (1D, 2D, 3D)
Each spin si is +1 or -1
Neighbors connected
Periodic boundary conditions
Each connection (i,j) contains number Ji,j (coupling)
Couplings usually initialized randomly
+/- J couplings ~ uniform on {-1, +1}
Gaussian couplings ~ N(0,1)

7. Finding ground states of SGs
Energy
∑s J
E= sj
i i, j
<i , j >
Ground state
Configuration of spins that minimizes E for given couplings
Configurations can be represented with binary vectors
Finding ground states
Find ground states given couplings

8. 2-dimensional +/- J SG
As constraint satisfaction problem
≠ ≠
= Spins:
≠ =
≠
=
Constraints: ≠ =
≠
≠ ≠
= ≠
General case
Periodic boundary cond. (last and first connected)
Constraints can be weighted

9. SG Difficulty
1D
Trivial, deterministic O(n) algorithm
2D
Local search fails miserably (exponential scaling)
Good recombination-based EAs should scale-up
Analytical method exists, O(n3.5)
3D
NP-complete
But methods exist to solve SGs of 1000s spins

10. Test SG classes
Dimensions n=6x6 to n=20x20
1000 random instances for each n and distribution
2 basic coupling distributions
+/- J, where couplings are randomly +1 or -1
Gaussian, where couplings ~N(0,1)
Transition between the distributions for n=10x10
4 steps between the bounding cases

11. Coupling distribution
2-component normal mixture with overall σ2=1
N (μ1 , σ 12 ) + N (μ 2 , σ 2 )
Vary μ2-μ1 is from 0 to 2 2
p(J ) =
2
μ = 0.60 μ = 0.80
Pure Gaussian (μ=0)
-3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3
μ = 0.95 μ = 0.99 ±J
-3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3

12. Analysis of running times
Traditional approach
Run multiple times, estimate the mean
Often works well, but sometimes misleading
Performance on SGs
MCMC performance shown to follow Frechet distr.
All distribution moments ill-defined (incl. the mean)!
Here
Identify distribution of running times
Estimate parameters of the distribution

13. Frechet distribution
Central limit theorem for extremal values
⎞
⎛ 1
x−μ ⎞ ε⎟
⎜⎛
H ξ ;μ ;β = exp⎜ − ⎜1 + ξ ⎟⎟
⎜ β ⎟⎟
⎜⎝ ⎠
⎠
⎝
ξ = shape, μ = location, β = scale
ξ determines speed of tail decay
Our case
ξ<0: Frechet distribution (polynomial decay)
ξ=0: Gumbel distribution (exponential decay)
ξ>0: Weibull distribution (faster than exponential decay)
Frechet: mth moment exists iff |ξ|<m

14. Results
+/- J vs. Gaussian couplings
Distribution of the number of evaluations
Location scale-up
Shape
Transition
Location change
Shape change
10 independent runs for each instance
Minimum population size to converge in all runs

15. Number of evaluations

16. Location, μ

17. Shape, ξ

18. Transition: Location & Shape

19. Discussion
Performance on +/- J SGs
Number of evaluations grows approx. as O(n1.5)
Agrees with BOA theory for uniform scaling
Performance on Gaussian SGs
Number of evaluations grows approx. as O(n2)
Agrees with BOA theory for exponential scaling
Transition
Transition is smooth as expected

20. Important implications
Selection+Recombination scales up great
Exponential number of optima easily escaped
Global optimum found reliably
Overall time complexity similar to best analytical
method
Selection+Mutation fails to scale up
Easily trapped in local minima
Exponential scaling

21. Conclusions
Average running time anal. might be insufficient
In-depth statistical analysis confirms past results
hBOA scales up well on all tested classes of SGs
hBOA scalability agrees with theory
Promising direction for solving other
challenging constraint satisfaction problems

22. Contact
Martin Pelikan
Dept. of Math and Computer Science, 320 CCB
University of Missouri at St. Louis
8001 Natural Bridge Rd.
St. Louis, MO 63121
E-mail: pelikan@cs.umsl.edu
WWW: http://www.cs.umsl.edu/~pelikan/