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  1. 1. Computational complexity and simulation of rare events of Ising spin glasses Pelikan, M., Ocenasek, J., Trebst, S., Troyer, M., Alet, F.
  2. 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. 3. Outline Hierarchical BOA (hBOA) Spin glasses Definition Difficulty Considered classes of spin glasses Experiments Summary and conclusions
  4. 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. 5. hBOA: Basic algorithm Bayesian New Current network Selection population population Restricted tournament replacement
  6. 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. 7. Finding ground states of SGs Energy E= ∑s J <i , j > i i, j sj 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. 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. 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. 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. 11. Coupling distribution 2-component normal mixture with overall σ2=1 Vary μ2-μ1 is from 0 to 2 N (μ1 , σ 12 ) + N (μ 2 , σ 2 ) 2 p(J ) = 2 Pure Gaussian (μ =0) μ = 0.60 μ = 0.80 -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. 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. 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 ξ<0: Frechet distribution (polynomial decay) Our case ξ=0: Gumbel distribution (exponential decay) ξ>0: Weibull distribution (faster than exponential decay) Frechet: mth moment exists iff |ξ|<m
  14. 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. 15. Number of evaluations
  16. 16. Location, μ
  17. 17. Shape, ξ
  18. 18. Transition: Location & Shape
  19. 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. 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. 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. 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/

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