Spin Glas Dynamics and Stochastic Optimization Schemes. Karl Heinz Hoffmann TU Chemnitz

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1 Spin Glas Dynamics and Stochastic Optimization Schemes Karl Heinz Hoffmann TU Chemnitz 1

2 Spin Glasses spin glass e.g. AuFe AuMn CuMn nobel metal (no spin) transition metal (spin) at% ferromagnetic coupling? antiferromagnetic coupling Frustration disorder + frustration complex system 2

3 Spin Glas Models Mimic spatial disorder and interaction variation by interaction disorder Edwards-Anderson model H = J i, j s i s j + h j s j <i, j > j Interaction is chosen randomly: Usually symmetric: Gauss, uniform, ±J 3

4 Complex Systems Complex state space = many, many local minima in the energy function E rough energy landscape state: variable to describe degrees of freedom, e.g. spin configuration: important: neighborhood relation states 4

5 ZFC - Experiment H The Zero-Field-Cooled- Experiment T 0 0 t Spinglasübergang T m t w 0 t 5

6 ZFC - Experiment: Aging Phenomena (Fe 0.15 Ni 0.85 ) 75 P 16 B 6 Al 3 measurements depend on waiting time t w S(t = t w ) = MAX 6

7 Thermally activated Relaxation Dynamics E hopping on the energy landscape heat bath = random number generator P P t 1 t 2 > t 1 description by a probability density P states 7

8 Thermally Activated Relaxation Dynamics simulation of the thermally activated hopping by the Metropolis-Algorithm: choose a neighbor of state j at random, for instance i accept i with probabilty P metropolis = { 1 E i E j e (E i E j ) /T E i > E j random walk on the energy landscape 8

9 Master Equation modelling the time evolution leads to a stochastic (Marcov) process: P i (t + Δt ) = P i (t )+ W (i j)p j (t ) j W ( j i )P i (t ) j transition probabilities W (i j) W ij = (P ac N ) ij matrix notation P i (t + Δt) =W ij P j (t) P(t + Δt) =W P(t) 9

10 Coarse Graining the State Space E Problem: too many states Solution: coarse graining the state space coarse graining states tree structures 10

11 Complex Relaxation Dynamics Problem: How does the relaxation proceed? E Result: the relaxation is a sequence of partial (or quasi) equilibria in larger and larger regions of the state space due to energy barriers on all scales: power laws 11

12 Spin Glass Modelling: Simple Tree Models E Δ W d W u Detailed Ballance: W u W d = κ e βδ Relaxation can be treated analytically Power Laws: P(i,t j) t γ Result: Relaxation proceeds through a sequence of Quasiequilibria 12

13 Spin Glass Modelling: Simple Tree Models dynamics M 1 M k + magnetic properties: magnetic disorder correlation function <M k M l > + linear response theory: out of equilibrium = model results 13

14 ZFC - Experiment: Model Results experiment model 14

15 Complex Systems: the Ground State Problem E How does one find the ground state in a complex system? states 15

16 Stochastic Optimization: Simulated Annealing E simulated annealing: Metropolis algorithm with T (t ) 0 P * min states T(t 1 ) P * (i ) e E i /T P * T(t 2 ) < T(t 1 ) δ i,min 16

17 Universal Optimization Scheme Ω = { i } state spin configuration E : i E i objective function energy { N i } neighborhood relation, move class spin configuration differing by one spin flip T (t ) external parameter temperature simulated annealing: E E MIN 17

18 Examples of Complex Optimization Problems chip design graph partitioning neural networks seismic deconvolution travelling salesman problem patern recognition... NP-hard optimization problems 18

19 Chip Design Chip Design Find best layout for chip typical constraint: minimize number of layers needed for connections 19

20 Seismic Deconvolution typical example for a parameter fit t t S in i= M S out th S out = a i S in ( t - t i ) i Ω = { ( a t ) } M th E: ( a t ) E = (S out - S out ) 2 N( ( a t ) ) = { (a' t' ) a' = a ± a, t' = t ± t } 20

21 The Travelling Salesman Problem The TSP is a typical NP-hard optimization problem Ω = { tour = permutation of cities } Kiel Rostock E: tour lenght of tour Bremen Hamburg Berlin N(tour) = { tour' two cities in tour exchanged } or: Hannover start, finish Kassel Halle distance matrix given shortest tour? 21

22 The Grötschel Problem: a TSP The Grötschel Problem an instance of a travelling salesman problem (TSP) 443 cities to visit 22

23 Simulated Annealing: Optimal Annealing Schedule E E MIN t guarantee: for if there is only a finite time available: τ P(i, τ) δ i,min then determine such that P(i, τ) E (τ) = P(i, τ)e i MIN i dynamics T(t) aim: determine such that T(t) E (τ) MIN optimal annealing schedule 23

24 Simulated Annealing: Optimal Annealing Schedule example: a single barrier E τ linear: exponentiell: optimal T = T 0 εt T = T 0 α t 24

25 Optimality Criteria Criteria based on final distribution at time S : Mean final energy (MIN) E (S ) = i Final probability in ground state (MAX) P (i,s )E i MIN Other features of the final distribution P (i,s ) Criteria based on the Best So Far Energy: E (S ) = MIN 0 t S [ E (S )] Features of the BSFE distribution B S (E ) 25

26 B(est) S(o) F(ar) E(nergy) Distribution Modify transition probabilities: make states below absorbing Master equation Probability to have seen energies below at time 26

27 B(est) S(o) F(ar) E(nergy) Distribution Finite state space: finite number of energy values Sort energies Probability that the lowest energy visited is Mean BSF energy 27

28 B(est) S(o) F(ar) E(nergy) Distribution Create enlarged state space Use new probabilties 28

29 General Optimality Criterion In general Control: acceptance probabilities at each point in time 29

30 Simulated Annealing: the Ensemble Approach problem: the barrier structure is unknown solution: adaptive schedules method: the ensemble approach to simulated annealing take N copies of the system, anneal with the same schedule use ensemble properties (mean, variance) to control schedule adaptively 30

31 The Ensemble Approach to SA many at the same time with communication come here! parallel implementation: task states compute farm result 31

32 Adaptive Schedules for SA idea: anneal as fast as possible but stay close to equilibrium distribution γ σ the algorithm: determine ensemble mean and variance anneal if ensemble mean starts to fluctuate: lower temperature etc 32

33 The Ensemble Approach to SA 80 Typical distribution of an ensemble Note progress 40 in time towards lower energies Energy 33

34 Threshold Accepting Simulated Annealing P metropolis = { 1 E i E j 0 e (E i E j ) /T E i E j > 0 Threshold Accepting P threshold = { 1 E i E j T 0 E i E j > T computationally more effective: no exponentials required! 34

35 Rényi Entropy Another information measure: R q = 1 1 q ln i P q i for : P Shannon i q 1 R q 1 = P i ln i interesting property: non-extensive minimum "Rényi information": No Boltzmann distribution!! 35

36 Tsallis Entropy A new, non-extensive information measure: S q = 1 q q (1 P 1 P i = P i ) i 1 q 1 q i i for q 1 : S q 1 = P Shannon i lnp i i interesting properties: non-extensive minimum "Tsallis information": No Boltzmann distribution!! 36

37 Tsallis Annealing Tsallis Annealing acceptance probability P tsallis = 1 { (1 1 ΔE 0 ΔE (1 q) T ) 1 1 q ΔE > 0 and (1 q) ΔE T 1 0 ΔE > 0 and (1 q) ΔE T > q Quench

38 Modified Tsallis Annealing Modified Tsallis Annealing (Astrid Franz & KHH) P MT = { 1 ΔE 0 (1 1 q ΔE 2 q T ) 1 1 q ΔE > 0 and 1 q ΔE 2 q T 1 0 ΔE > 0 and 1 q ΔE 2 q T > 1 important property: 1 q 1 q P MT = P metropolis P MT = P threshold

39 Optimal MT Annealing threshold accepting is best!! q test many different systems threshold accepting is always best!! 39

40 Optimal Strategies for Finding Ground States What is the best acceptance probability?? Theorem: on finite state spaces within the class of acceptance probabilities with these features: P acceptance = P acceptance (ΔE) P acceptance (ΔE) ΔE is monotone decreasing as a function of Threshold Accepting is the best possible strategy (for linear criteria) Note that Metropolis, TA, Tsalis, MT have these features 40

41 Optimizing Acceptance Probabilities At each step, an acceptance probability controls the further time evolution of the distribution in the state space tr Linear criterion: (P ac N ) ij F i P i (t final ) = F i W ij W jk W kl KW mn P n (t initial ) 41

42 Optimal Strategies for Finding Ground States finite state space finite number of ΔE α 0 < P ac (ΔE α ) < 1 P acceptance (ΔE ) ΔE is montone decreasing as a function of P ac (ΔE 1 ) P ac (ΔE 2 ) P ac (ΔE 3 )... for ΔE 1 < ΔE 2 < ΔE 3 <... P ac (ΔE 3 ) optimization over a simplex P ac (ΔE 2 ) (P ac (ΔE 1 ),P ac (ΔE 2 ),P ac (ΔE 3 ),...) P ac (ΔE 1 ) 42

43 Optimizing Acceptance Probabilities a typical vertex of the simplex (1,1,1,...,1, 0, 0,K, 0) linear criterion: optimum lies on vertex of simplex, i.e. threshold accepting iterate along path: tr 43

44 Extremal Optimization invented by Böttcher and Percus a state has internal structure i = { i 1,i 2,i 3,L,i j,l,i n 1,i n } each DoF may have several values i j i ( j ) = i ( j ) 1,i ( j ) 2,i ( j ) 3,L,i ( j ) m 1,i ( j ) m example: spin configuration each spin is a DoF Ising spins: only two values s j s ( j ) = { 1,1} 44

45 Extremal Optimization: the Algorithm In a state i = { i 1,i 2,i 3,L,i n 1,i n,} determine a fitness for each DoF: λ j (i j ) example: for spins take the local field H = J i, j s i s j + h j s j = λ j s j <i, j > j j rank the DoFs with respect to their fitness change the worst one to one of its other values randomly 45

46 Extremal Optimization: the Algorithm EO creates Marcov-process EO walks fast through state space: there are no rejected moves tau-eo: choose DoF according to a rank selection probabilty 0,6 0,5 d(rank) 0,4 0,3 0,2 d t (k) k τ 0, rank k 46

47 Optimal Rank Selection Probability for EO What is the best rank selection probability for Extremal Optimization?? Is a power law distribution (with no scale) on the ranks the best?? Constraints: all time steps independent normalization 47

48 Optimal Rank Selection Probability for EO Problem: optimize linear criterion over simplex 48

49 Optimal Rank Selection Probability for EO result: for linear criteria a step disribution is the best!!... Fitness Threshold Accepting 49

50 Outlook and Open Problems Optimal thresholds Threshold Accepting Fitness Threshold Accepting Controlled dynamics on energy landscapes Protein folding Functional relevant minima 50

51 Summary Spin glas dynamics aging experiments Complex state space dynamics Optimal schedules for SA Best strategy Threshold Accepting Fitness Threshold Accepting 51