Learning from and about complex energy landspaces

Transcription

1 Learning from and about complex energy landspaces Lenka Zdeborová (CNLS + T-4, LANL) in collaboration with: Florent Krzakala (ParisTech) Thierry Mora (Princeton Univ.)

2 Landscape

3 Landscape Santa Fe Institute

4 Visual Beauty

5 Art genre Even etymologically, from Dutch painter s term landschap The Hay Harvest by Bruegel the Elder (1565)

6 Tool of abstraction in sciences Fitness landscape - visualization of the relation between genotypes and reproductive success (S. Wright, 1932) (Potential, Free) Energy landscape - crucial in protein folding, glass theory. Cost function landscape - optimization

7 This talk about Energy Landscape of Models of glasses & Hard optimization problems Definition of gra State: each node has a color Rule (energy cost): neighbors have d

8 Glasses

9 Glass transition Almost any liquid when quenched fast enough undergoes a glass transition.

10 Angell s plot log(viscosity) η e T η e (T ) T T K inverse temperature

11 No apparent order Liquid Glass

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15 David A. Weitz, a physics professor at Harvard, joked, There are more theories of the glass transition than there are theorists who propose them.

16 Hard Optimization Problems

17 Sometimes Easy

18 To Be Avoided

19 Graph Coloring NP-complete

20 Coloring Random Graphs probability of colorability T p SAT N = 100 p SAT N = 71 p SAT N = 50 comp. time N = 100 comp. time N = 71 comp. time N = time to decide average degree of the graph

21 Glasses and Coloring

22 Glasses and Coloring What do they have in common?

23 Glasses and Coloring What do they have in common? The Energy Landscape

24 Glasses and Coloring What do they have in common? The Energy Landscape Cost function of Potts glass and graph coloring: H = (ij) δ Si,S j S i {1,..., q}

25 energy / cost space of q N configurations

26 Energy Landscape Visualization 2D or 3D, but complex energy landscapes often many dimensional Energy landscape of problems on random graphs: description via the cavity or replica method (Parisi, 1980; Mezard, Parisi, 1999). Many features reproduced in the simple random subcube model (Mora, Zdeborova, 2007)

27 Random subcube model Consider strings of 0/1 of length N N = 17 For every position, mark it red with probability p Subcube: All strings where red positions are fixed and white positions can be both 0 or 1

28 Random subcube model Take 2 (1 α)n random subcubes Define energy 0 for every string that belongs to at least one subcube, and 1 to all other strings. Study this energy landscape when N

29 Probability that a subcube has size ( N ) 2 Ns sn (1 p) sn p (1 s)n Number of subcubes of a given size ( 2 NΣ(s) = 2 (1 α)n N (1 p) sn p (1 s)n sn ) In large N limit: Σ(s) = (1 α) + s log 2 s 1 p + (1 s) log 2 1 s p

30 s = arg max[σ(s) + s Σ(s) 0] s s tot = s + Σ(s ) complexity p = entropy

31 Liquid phase Clustered phase Condensed phase Uncolorable phase s tot = 1 s tot < 1 s tot < 1 Σ(s ) > 0 Σ(s ) = 0 Σ(s) < 0 s α d α c α s = 1 α d = log 2 (2 p) α c = p/(2 p) + log 2 (2 p)

32 Dynamical transition Ergodicity breaking (random walk in subcubes, dynamics in glasses, easy sampling in constraint satisfaction) Extreme slowing down of the dynamics. Diverging viscosity.

33 Condensation transition = Ideal Glass Transition (if such exists in real materials) Kauzmann (1948) transition in glasses

34 Colorability transition No configurations (string) at zero energy. Where the REALLY hard problems are? (Cheeseman, Kanefsky, Taylor, 1991) T p SAT N = 100 p SAT N = 71 p SAT N = 50 comp. time N = 100 comp. time N = 71 comp. time N = But also glassiness (clustering) makes the problem hard (Mezard, Parisi, Zecchina, 2002)

35 Where the really hard problems REALLY are?

36 Where the really hard problems REALLY are? Canyon dominated vs. Valley dominated Positive energy states Positive energy states Zero energy states Zero energy states

37 0.03 Valleys 3-XOR-SAT with L=3 solvable only by Gauss E(S) S Canyons 4-coloring of 9-regular random graphs solvable by reinforced belief propagation E(S) S

38 Conclusions Many dimensional configurational space makes problem complex and interesting. In particular glasses and hard optimization problems - exponentially many zones of attraction / valleys / states / clusters. Methods for understanding models on random graphs - a lot of surprising features.