Stochastic%20Relaxation,%20Simulating%20Annealing,%20Global%20Minimizers

1
Stochastic Relaxation,Simulating
Annealing,Global Minimizers
2
Different types of relaxation

• Variable by variable relaxation strict
  minimization
• Changing a small subset of variables
  simultaneously Window strict minimization
  relaxation
• Stochastic relaxation may increase the energy
  should be followed by strict minimization

3
Complex landscape of E(X)
4
How to escape local minima?

• First go uphill, then may hit a lower basin
• In order to go uphill should allow increase in
  E(x)
• Add stochasticity allow E(x) to increase with
  probability which is governed by an external
  temperature-like parameter T
• The Metropolis Algorithm (Kirpartick et al.
  1983)
• Assume xold is the current state, define xnew
  to be a neighboring state and
  delEE(xnew)-E(xold) then
• If delElt0 replace xold by xnew
• else choose xnew with probability P(xnew)
• and xold with probability P(xold)1- P(xnew)

5
The probability to accept an increasing energy
move
6
The Metropolis Algorithm

• As T 0 and when delEgt0 P(xnew) 0
• At T0 strict minimization
• High T randomizes the configuration away from the
  minimum
• Low T cannot escape local minima
• Starting from a high T, the slower T is decreased
  the lower E(x) is achieved
• The slow reduction in T allows the material to
  obtain a more arranged configuration increase
  the size of its crystals and reduce their defects

7
Fast cooling amorphous solid
8
Slow cooling - crystalline solid
9
SA for the 2D Ising E-Sijsisj , i and j are
nearest neighbors




Eold-2
10
SA for the 2D Ising E-Sijsisj , i and j are
nearest neighbors







Eold-2
Enew2
11
SA for the 2D Ising E-Sijsisj , i and j are
nearest neighbors







Eold-2
Enew2
delEEnew- Eold4gt0 P(Enew)exp(-4/T)
12
SA for the 2D Ising E-Sijsisj , i and j are
nearest neighbors







Eold-2
Enew2
delEEnew- Eold4gt0 P(Enew)exp(-4/T) 0.3 gt
T-4/ln0.3 3.3
Reduce T by a factor a, 0ltalt1 Tn1aTn
13
Exc7 SA for the 2D Ising (see Exc1)

• Consider the following cases
• 1. For h1 h20 set a stripe of width 3,6 or 12
  with opposite sign
• 2. For h1-0.1, h20.4 set -1 at h1 and 1 at h2
• 3. Repeat 2. with 2 squares of 8x8 plus spins
  with h20.4 located apart from each other
• Calculate T0 to allow 10 flips of a spin
  surrounded by 4 neighbors of the same sign
• Use faster / slower cooling scheduling
• a. What was the starting T0 , E in each case
• b. How was T0 decreased, how many sweeps were
  employed
• c. What was the final configuration, was the
  global minimum achievable? If not try different
  T0
• d. Is it harder to flip a wider stripe?
• e. Is it harder to flip 2 squares than just one?

14
SA for the bisectioning problem
R
R
i
15
SA for the bisectioning problem individual
temperature
R
R
i
The probability of i to belong to R depends on
Si Sj in R aij / S aij
P(i in R)
1 delElt0 exp-delE/(TSi)
delEgt 0
16
SA for the bisectioning problem individual
temperature
R
R
i
The probability of i to belong to R should
increase if a bigger change along the cut line is
made
If delE is small enough it is expected that
further moves will indeed eventually produce a
lower E
17
SA for the bisectioning problem how to choose T

R
R
i

• Calculate delE/Si along the cut line and sort
  them
• Decide upon the of changes desired
• Find the appropriate T by demanding P()0.5

18
SA for the linear ordering problems multiple
choices for a variable

• Try to move node i up to k moves to the right and
  to the left
  choose between the 2k1 possibilities
• For j-k,..,-1,1,..,k , P(j)z min1 ,
  exp(-delE(j)/T(j))
• For k0 P(0)z minj1 - P(j)/z
• z is calculated from the normalization Sj P(j)1
• T(j) is calculating apriori for each j aiming at
  a certain acceptance rate (e.g. 60)

19
The Metropolis Algorithm (cont.)

• May result in a very slow processing
• Still, SA is considered to be a powerful global
  minimizer
• Instead of very slow cooling schedule, repeat
  heating-cooling several times

20
Heating-cooling scheduling
T
relaxation sweeps
21
The Metropolis Algorithm (cont.)

• May result in a very slow processing
• Still, SA is considered to be a powerful global
  minimizer
• Instead of very slow cooling schedule, repeat
  heating-cooling several times and keep track of
  the best-so-far configuration
• The best-so-far has a non-increasing E
• It is an outside observer
• The best-so-far is actually the calculated
  minimum

22
Heating-cooling scheduling
T
relaxation sweeps
Store the best-so-far
23
The Metropolis Algorithm (cont.)

• May result in a very slow processing
• Still, SA is considered to be a powerful global
  minimizer
• Instead of very slow cooling schedule, repeat
  heating-cooling several times and keep track of
  the best-so-far configuration
• The best-so-far has a non-increasing E
• It is an outside observer
• The best-so-far is actually the calculated
  minimum
• Problem heating may destroy already achieved
  minima in various subregions
• Add memory of the best-so-far for those
  subregions

24
Lowest Common Configuration
The global minimum
25
Lowest Common Configuration
C1
The global minimum
26
Lowest Common Configuration
C1
C2
The global minimum
27
Lowest Common Configuration
C1
C2
The global minimum
E(LCC(C1, C2))lt minE(C1), E(C2)
LCC(C1, C2)
28
Heating-cooling scheduling
T
relaxation sweeps
Apply LCC
29
Heating-cooling scheduling
T
relaxation sweeps
best-so-far ? LCC (best-so-far , the new T0)
30
Exc8 LCC for the bisectioning problem
R
R
i
Given 2 partitions, find a linear time algorithm
for the construction of their LCC
31
Exc8 LCC for linear ordering problems

• Find a (nearly) linear time algorithm
• (e.g. sorting is allowed) for the LCC of 2
• permutations, in which subpermutations are
• detected and chosen into the best-so-far

32
Multilevel Simulated Annealing

• Do not increase T by much
  avoid destroying the global solution inherited
  from the coarser levels
• Reduce T quickly typically 2-3 values of Tgt0
  (followed by strict minimization) are sufficient
• Repeat heating-cooling several times per level
• Accumulate the minimal solution into the
  best-so-far by applying the LCC at the end of
  T0
• Interpolate the best-so-far to the next level

33
Genetic algorithmA global minimizer
34
Genetic algorithm

• A global search technique inspired by
  evolutionary biology
• Start from a population of individuals (randomly
  generated) this is the 1st generation
• The next generation follows by
• 1. selection of individuals from the current
  generation to breed the next generation according
  to some fitness measure
• 2. crossover (recombination) of pair of (randomly
  chosen) parents to produce an offspring
• 3. mutations are applied randomly to enhance the
  diversity of the individuals in the generation

35
A genetic algorithm for the linear arrangement
problem P1

• Initial population 1. select a starting vertex
  2. built the permutation by the greedy frontal
  increase minimization algorithm

36
A genetic algorithm for the linear arrangement
problem P1

• Initial population 1. select a starting vertex
  2. built the permutation by the greedy frontal
  increase minimization algorithm

Fi
Choose a node
from Fi
The one which is mostly connected to the already
placed nodes
37
A genetic algorithm for the linear arrangement
problem P1

• Initial population 1. select a starting vertex
  2. built the permutation by the greedy frontal
  increase minimization algorithm
• Selection of survivals is based on the E(x)
• Recombinate two randomly chosen parents

Parent 1 5 7 2 3 8 6 9 1 4
Parent 2 6 1 2 4 3 5 9 8 7
38
A genetic algorithm for the linear arrangement
problem P1

• Initial population 1. select a starting vertex
  2. built the permutation by the greedy frontal
  increase minimization algorithm
• Selection of survivals is based on the E(x)
• Recombinate two randomly chosen parents

Parent 1 5 7 2 3 8 6 9 1 4
Parent 2 6 1 2 4 3 5 9 8 7
Offspring 2 9
39
A genetic algorithm for the linear arrangement
problem P1

• Initial population 1. select a starting vertex
  2. built the permutation by the greedy frontal
  increase minimization algorithm
• Selection of survivals is based on the E(x)
• Recombinate two randomly chosen parents

Parent 1 5 7 2 3 8 6 9 1 4
Parent 2 6 1 2 4 3 5 9 8 7
Offspring 2 9
_ 3 2 6 _ 7 9 5 _
40
A genetic algorithm for the linear arrangement
problem P1

• Initial population 1. select a starting vertex
  2. built the permutation by the greedy frontal
  increase minimization algorithm
• Selection of survivals is based on the E(x)
• Recombinate two randomly chosen parents

Parent 1 5 7 2 3 8 6 9 1 4
Parent 2 6 1 2 4 3 5 9 8 7
Offspring 2 9
_ 3 2 6 _ 7 9 5 _
_ 3 2 6 8 7 9 5 4
41
A genetic algorithm for the linear arrangement
problem P1

• Initial population 1. select a starting vertex
  2. built the permutation by the greedy frontal
  increase minimization algorithm
• Selection of survivals is based on the E(x)
• Recombinate two randomly chosen parents

Parent 1 5 7 2 3 8 6 9 1 4
Parent 2 6 1 2 4 3 5 9 8 7
Offspring 2 9
_ 3 2 6 _ 7 9 5 _
_ 3 2 6 8 7 9 5 4
1 3 2 6 8 7 9 5 4
42
A genetic algorithm for the linear arrangement
problem P1

• Initial population 1. select a starting vertex
  2. built the permutation by the greedy frontal
  increase minimization algorithm
• Selection of survivals is based on the E(x)
• The next generation is constructed by
• 1. Recombinations of 2 randomly chosen parents
• 2. Improving the E(x) of the offspring by local
  processing, e.g. by Simulated Annealing
• 3. Choose the best individuals from the pool of
  parents and children

43
Spectral Sequencing A global minimizer
44
Spectral Sequencing a global minimizer

• Given a weighted graph where wij is the edge
  weight between the nodes i and j
• Define the graph Laplacian A to be
• aij -wij
• aii Sjwij
• A is symmetric semipositive definite
• Consider the eigenvalue problem Axlx
• Arranging the nodes of the graph according to the
  eigenvector associated with the 2nd smallest
  eigenvalue has been shown by Hall (1970) to be
  the solution to the problem
  min Sjwij(xi - xj)2 for real variables x

45
Spectral Sequencing a global minimizer

• SS has been used extensively to solve a large
  variety of ordering problems
• Linear ordering problem P1,2,
• Partitioning problems
• Embedding to lower dimensions, etc.
• To calculate the eigenvectors use multilevel
• The direct use of multilevel to solve the
  original problem produces better results than
  using the ordering dictated by SS

46
P2 Multilevel approach vs. Spectral method
ratio
graphs
The results of the multilevel approach were
obtained without post-processing!
Ilya Safro, Dorit Ron, A. Brandt J. Graph Alg.
Appl. 10 (2006) 237-258