Learning from and about complex energy landspaces Lenka Zdeborová (CNLS + T-4, LANL) in collaboration with: Florent Krzakala (ParisTech) Thierry Mora (Princeton Univ.)
Landscape
Landscape Santa Fe Institute
Visual Beauty
Art genre Even etymologically, from Dutch painter s term landschap The Hay Harvest by Bruegel the Elder (1565)
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
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
Glasses
Glass transition Almost any liquid when quenched fast enough undergoes a glass transition.
Angell s plot log(viscosity) η e T η e (T ) T T K inverse temperature
No apparent order Liquid Glass
David A. Weitz, a physics professor at Harvard, joked, There are more theories of the glass transition than there are theorists who propose them.
Hard Optimization Problems
Sometimes Easy
To Be Avoided
Graph Coloring NP-complete
Coloring Random Graphs probability of colorability T 1 0.8 0.6 0.4 p SAT N = 100 p SAT N = 71 p SAT N = 50 comp. time N = 100 comp. time N = 71 comp. time N = 50 4000 3500 3000 2500 2000 1500 time to decide 1000 0.2 500 0 0 2 4 6 8 10 0 average degree of the graph
Glasses and Coloring
Glasses and Coloring What do they have in common?
Glasses and Coloring What do they have in common? The Energy Landscape
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}
energy / cost space of q N configurations
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)
Random subcube model Consider strings of 0/1 of length N 00010010101110101 N = 17 For every position, mark it red with probability p 00010010101110101 Subcube: All strings where red positions are fixed and white positions can be both 0 or 1
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
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
s = arg max[σ(s) + s Σ(s) 0] s s tot = s + Σ(s ) complexity 0.25 0.2 0.15 0.1 0.05 p = 0.8 0.80 0.85 0.90 0.95 1.00 1.05 0-0.05-0.1 0 0.1 0.2 0.3 0.4 0.5 entropy
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)
Dynamical transition Ergodicity breaking (random walk in subcubes, dynamics in glasses, easy sampling in constraint satisfaction) Extreme slowing down of the dynamics. Diverging viscosity.
Condensation transition = Ideal Glass Transition (if such exists in real materials) Kauzmann (1948) transition in glasses
Colorability transition No configurations (string) at zero energy. Where the REALLY hard problems are? (Cheeseman, Kanefsky, Taylor, 1991) T 1 0.8 0.6 0.4 0.2 p SAT N = 100 p SAT N = 71 p SAT N = 50 comp. time N = 100 comp. time N = 71 comp. time N = 50 4000 3500 3000 2500 2000 1500 1000 500 But also glassiness (clustering) makes the problem hard (Mezard, Parisi, Zecchina, 2002) 0 0 0 2 4 6 8 10
Where the really hard problems REALLY are?
Where the really hard problems REALLY are? Canyon dominated vs. Valley dominated Positive energy states Positive energy states Zero energy states Zero energy states
0.03 Valleys 3-XOR-SAT with L=3 solvable only by Gauss E(S) 0.025 0.02 0.015 0.01 0.005 0-0.05 0 0.05 S Canyons 4-coloring of 9-regular random graphs solvable by reinforced belief propagation E(S) 0.02 0.018 0.016 0.014 0.012 0.01 0.008 0.006 0.004 0.002 0-0.1-0.05 0 0.05 0.1 S
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.