Extremal Optimization for Low-Energy Excitations of very large Spin-Glasses

Transcription

1 Etremal Optimization for Low-Energy Ecitations of very large Spin-Glasses Stefan Boettcher Dept of Physics Collaborators: Senior: Allon Percus (Los Alamos) Paolo Sibani (Odense-DK) Michelangelo Grigni (Emory/CS) Students: Gabriel Charvat (Emory/Phy) Susannah Cooke (Emory/Phy) Martin Frank (Emory/Phy) Support: NSF-DMR (Pre-)Prints:

2 Summary: Applications of Etremal Optimization (EO) Here: EO for Mean-Field Glasses (SK) (n ) Spin Glass Ground States with ~6 Spins EO for Defect Energies of Glasses in d=3,..,7 Comparison with Mean-Field (d ) limit Result: Mean-Field Prediction fails Poster: EO vs Metropolis: Energy-Landscape Eplorations EO of MAX-3-Coloring at Phase Transition EO for the Thomson Problem Theory of EO

3 Hard Optimization Problems: Consider: ) System of n variables, i I. ) Configurations S=(,...,n) I n. 3) Cost Function C(S). 4) Local Search Neighborhood, S':=N(S). Problem:. Find Global Minimum of C(S) when )n is large, )C(S) has many local Etrema in N. C(S) =>Search-time t >> nk for any k. Need: Seach Heuristic to find approimate solution in t~nk. S

4 Spin Glass Ground States: Ferro-magnetic Bonds: = + Anti-ferro-mag Bonds: =- Spin Variables: i = - i =+ Hamiltonian (Cost): H = - <i,j> i,j i j i Fitness of i: λ = i i,j j i (or Graph) j Lattice FRUSTRATION H =- w/ fied, random Bonds i λi

5 Etremal Optimization (EO) Motivated by Self-Organized Criticality Emergent Structure without tuning any Control Parameters despite (or because of) Large Fluctuations How can we use it to optimize? Etremal Driving: Select and eliminate the bad, Replace it at random, Eventually, only the good is left! Evolutionary Search Heuristic

6 Fitness λ for various Problems: Spin Glasses (eg. MAX-CUT):. λi = i j i,j j i Partitioning (eg. MIN-CUT): λi = - (#-cut edges of i). i. Coloring (eg. Potts Anti-ferro): λi = - (#-monochrome edges of i). Cost H = - i λi i

7 >> Etremal Optimization (EO): << () Provide initial Configuration S=(,..., n), () Determine Fitness λi for each Variable i, (3) Rank all i= (k) according to λ λ... λ () () (n) (4) Select w w/ w= (), i.e. w has worst Fitness! (5) Update w unconditionally, (6) For tma times, Repeat at (), (7) Return: Best Cost(S) found along the way!

8 Typical Etremal Optimization Run: EO-run for Partitioning (n=5): random start t mop-up phase Co nve Range of CostFluctuations in t rge nc e

9 τ-eo - Searching at the Ergodic Edge : For Ranks λ ()... λ (n), update i= (k) with scale-free, power-law distribution <Cost> τ ~ τ random walk, too ergodic ln n ergodic edge P(k) k -τ τ greedy + frozen, non-ergodic τ

10 Results for τ-eo: Applications by Others: EO for Image Alignment (Batouche et al, [LNCS449(')33]) EO Aligned Images

11 Results for τ-eo: For Spin Glasses: EO for 3d-Lattice Spin Glasses [PRL86(')5] Genetic Algorithms by Pal [PhysicaA3('96)83] and by Hartmann [EPL4('97)49]

12 τ-eo for Sherrington-Kirkpatrick (SK): Mean-Field (d ) Spin Glasses: Kondor ('83) Parisi's RSB Scaling Correction ω=/3 Fluctuation Eponent θf=3/4

13 Lattice Spin Glasses (at T=): Defect-Energy: Reverse Bonds (Perturb) on scale L Measure Energy Fluctuations ΔE(L) Low-Energy Ecitations (like small Oscillations )

14 Lattice Spin Glasses (at T=): Defect-Energy: Reverse Bonds (Perturb) on scale Measure Stiffness : σ(lδe)~ y Energy Fluctuations ΔE(L) LMeasure Ground State Ground State E EE' Before: spins After: 5 spins Low-Energy Ecitations on bond-diluted Lattices (like small Oscillations )

15 Lattice Spin Glasses (at T=): Defect-Energy: Measure Stiffness : σ(δe)~ L Low-Energy Ecitations on bond-diluted Lattices y

16 Lattice Spin Glasses (at T=): Defect-Energy: Measure Stiffness : How bond-diluted σ(δe)~ L y Temp Lattices? Tg SG PM y> Why bond-diluted pc T= Bond Density p Lattices? Simpler Problem Larger Sizes L Better Scaling

17 / ) ( / ) ( = := E D H := H ) / H := H ( = E D v u )( ). v u w -- () w con: / ) ( := g := g w ** g ( g si ) = := E D g = : H H + ) -- H w ( - = E D H := H H w + -con: ( Reduction Method for sparse Graphs: Reduce low-connected Spins, optimize the Remainder (T= only!): -con: 3-con:

18 Lattice Spin Glasses (at T=): Applying the Reduction Rules: Before: spins After: 5 spins

19 Defect-Energy of diluted Lattices: Stiffness : σ(δe)~ L y

20 Stiffness Eponent y for Lattice Glasses: Reduction plus τ-eo for dilute-lattice ± Glasses: d=3 d=7 d=6 d= d=54(!)

21 Comparing with Mean-Field Theory: Stiffness : σ(δe)~ L y yrsb = d(-θf)

22 Cinderella Cluster: 8 Boes w/ AthlonXP (GHz), 5 Mb, no HD, Video, etc Server w/ P4, Gb, RAID5, Tape, etc Diskless NetBoot over fast Switch QUANTIAN_ (Debian) w/ TerminalServer OPENMOSIX loadbalancer MPI for parallel processing Cost: ~$8,

23 Conclusions: Etremal Optimization: Selection against etremely Bad Greedy! No Rejection Large Fluctuations No Trapping! Single, fied Parameter (τ) Simple! τ-eo: Optimizing at the Ergodic Edge. Problems: Definition of Fitness and Sorting Ranks. Results: Works well for Partitioning, Coloring, Spin Glasses, Satisfiability, Pattern Recognition (at least!). Works poorly for TSP, Polymer Folding, ie. highly connected problems! + Theory: amming Model, predicting τ. opt