自旋玻璃与消息传递算法
Spin Glass and
Message-Passing Algorithms
周海军
http://www.itp.ac.cn/~zhouhj/
中国科学院理论物理研究所
提纲
1。自旋玻璃理论
2。消息传递算法
• 基本图像与平衡自
由能分布
• Vertex-Cover 问题, 3SAT 问题
• 空腔方法
• Survey Propagation算
法
2
部分参考文献
1.
Mezard, Parisi, Virasoro, “Spin Glass Theory and
Beyond” (World Scientific, 1987）
2.
Mezard, Parisi, “The Bethe lattice spin glass revisited”,
European Physics Journal B 20: 217-233 (2001)
3.
Mezard, Parisi, Zecchina, “Analytic and algorithmic
solution of random satisfiability problems”, Science
297: 812-815 (2002)
如果对报告中所涉及的具体模型的计算细节感兴趣，请参考
http://www.itp.ac.cn/~zhouhj/mainen.html
3
自旋玻璃理论：
自由能分
Statistical mechanics of a (simple) system in
equilibrium is well-established.
 Partition function, free energy, ….
Mean-field treatment.
 Phase transitions. Correlation length,
scaling exponents, … .
Renormalization flow.
5
…, but non-equilibrium dynamics of even a
simple system may be difficult to understand
 Formal framework.
 Connection with equilibrium.
 Glassy dynamics. Why relaxation becomes so
low and non-exponential?
 ……
6
Equilibrium (static) and dynamical properties
of complex systems are
both difficult and interesting
• Quenched randomness, frustration, nonself averaging, …, broken ergodicity.
• NP-complete combinatorial optimizations,
message-passing algorithms for
information science (CDMA, for example!),
econo-physics, …, biological systems.
7
自旋玻璃：无序与阻错系统的简单模型
• 3D regular lattice
(Edwards-Anderson, 1975)
• Complete graph
(Sherrington-Kirkpatrick 1975)
• Random Poisson graph
(Viana-Bray, 1985)
8
What we learned from an equilibrium
statistical mechanics course?
9
What we learned from an equilibrium
statistical mechanics course? (contl.)
10
What we learned from an equilibrium
statistical mechanics course? (contl.)
11
ergodic vs non-ergodic
?
12
repeated heating—annealing and
the equilibrium Gibbs measure
Complexity （复杂度）
13
distribution of equilibrium free-energies
1
14
distribution of equilibrium free-energies (contl.)
2
15
16
Which thermodynamic states contribute to
the equilibrium properties?
• If
Excited macrostates matter!
• If
Macrostates of minimal free
energy density matter!
17
3-spin-Interaction Ising model on a
complete graph
18
the mean free energy density
19
Overlap Distribution
20
自旋玻璃理论：
空腔方法（cavity method)
Let’s define an artificial system!
22
Some examples of the grand free energy：
2-body interactions
Beta=1.25
Beta=+infinity
The max-2-SAT problem
The +/- J spin-glass
model on a random
regular graph of
degree K=6
23
How to calculate the grand free energy?
The cavity approach
24
25
N
N+2
26
Population Dynamics Simulation
27
28
Message-Passing Algorithms
3-SAT 问题
30
顶点覆盖问题
31
This graph is covered, but not optimally covered.
32
Minimal Vertex Cover Problem
• A vertex cover of the global minimal size.
• Is a NP-hard optimization problem.
• Efficient algorithms for constructing nearoptimal solutions for a given graph.
33
There are many optimal
solutions for a given graph
34
Three types of vertices：
(1) vertices that are always covered
(frozen vertices,
)
(2) vertices that are always uncovered
(frozen vertices,
)
(3) vertices that are covered in some
solutions and uncovered in the
remaining solutions
(unfrozen vertices,
)
35
Mean-field analysis
of
the minimal vertex cover problem
on a random graph
36
The vertex cover problem
自洽的空穴场方法
覆盖还是不覆盖?
37
Weigt, Hartmann, PRL (2000), PRE (2001)
always uncovered
always covered
=
unfrozen
38
New vertex un-covered
New vertex partially covered
New vertex always covered
39
Mean-field theory result is lower than
experimental values for c > e=2.7183
2.7183
40
假定的相空间结构
41
引入参数 y
42
neighbors
vertex i
probability
VC size re-weighted
increase probability
all unfrozen
or always
covered
always
uncovered
0
at least one
always
uncovered
unfrozen or
always
covered
+1
43
44
45
同样的消息传递的算法可以用于解决
神经网络，信息系统，满足性问题，
…，
中的许多计算困难
46
Program and School in Beijing
2008
• ICTP-ITP Spring School on “Statistical
Physics and Interdisciplinary Applications”
March 03-14, 2008
• KITPC Program “Collective Dynamics in
Information Systems” March 01-April 15,
2008
47