Adaptive annealing: a nearoptimal

1
Adaptive annealing a near-optimal connection
between sampling and counting
Daniel tefankovic (University of
Rochester) Santosh Vempala Eric Vigoda (Georgia
Tech)
2
Adaptive annealing a near-optimal connection
between sampling and counting
If you want to count using MCMC then
statistical physics is useful.
Daniel tefankovic (University of
Rochester) Santosh Vempala Eric Vigoda (Georgia
Tech)
3
Outline
1. Counting problems 2. Basic tools Chernoff,
Chebyshev 3. Dealing with large quantities
(the product method) 4. Statistical
physics 5. Cooling schedules (our work) 6.
More
4
Counting
independent sets spanning trees matchings per
fect matchings k-colorings
5
Counting
independent sets spanning trees matchings per
fect matchings k-colorings
6
Compute the number of
spanning trees
7
Compute the number of
spanning trees
det(D A)vv
Kirchhoffs Matrix Tree Theorem
-
det
D
A
8
Compute the number of
spanning trees
polynomial-time algorithm
G
number of spanning trees of G
9
?
Counting
independent sets spanning trees matchings per
fect matchings k-colorings
10
Compute the number of
independent sets
(hard-core gas model)
11
independent sets 7
independent set subset S of vertices
no two in S are neighbors
12
independent sets
G1
G2
G3
...
Gn-2
...
Gn-1
...
Gn
13
independent sets
2
G1
3
G2
5
G3
...
Gn-2
Fn-1
...
Gn-1
Fn
...
Gn
Fn1
14
independent sets 5598861
independent set subset S of vertices
no two in S are neighbors
15
Compute the number of
independent sets
?
polynomial-time algorithm
G
number of independent sets of G
16
Compute the number of
independent sets
(unlikely)
!
polynomial-time algorithm
G
number of independent sets of G
17
graph G ? independent sets in G
P
NP
FP
P
P-complete P-complete even for 3-regular
graphs
(Dyer, Greenhill, 1997)
18
graph G ? independent sets in G
?
approximation randomization
19
graph G ? independent sets in G
?
which is more important?
approximation randomization
20
graph G ? independent sets in G
My world-view (true) randomness is important
conceptually but NOT computationally (i.e., I
believe PBPP). approximation makes problems
easier (i.e., I believe PBPP)
?
which is more important?
approximation randomization
21
We would like to know Q
Goal random variable Y such that P( (1-?)Q
? Y ? (1?)Q ) ? 1-?
Y gives (1??)-estimate
22
We would like to know Q
Goal random variable Y such that P( (1-?)Q
? Y ? (1?)Q ) ? 1-?
(fully polynomial randomized approximation
scheme)
FPRAS
Y
polynomial-time algorithm
G,?,?
23
Outline
1. Counting problems 2. Basic tools Chernoff,
Chebyshev 3. Dealing with large quantities
(the product method) 4. Statistical
physics 5. Cooling schedules (our work) 6.
More...
24
We would like to know Q
1. Get an unbiased estimator X, i. e.,
EX Q
2. Boost the quality of X
25
The Bienaymé-Chebyshev inequality
P( Y gives (1??)-estimate )
1
VY
? 1 -
EY2
?2
26
The Bienaymé-Chebyshev inequality
P( Y gives (1??)-estimate )
1
VY
? 1 -
EY2
?2
squared coefficient of variation SCV
X1 X2 ... Xn
VY
VX
1
?
Y

n
EY2
EX2
n
27
The Bienaymé-Chebyshev inequality
Let X1,...,Xn,X be independent, identically
distributed random variables, QEX. Let
Then
P( Y gives (1??)-estimate of Q )
1
VX
? 1 -
?2
n EX2
28
Chernoffs bound
Let X1,...,Xn,X be independent, identically
distributed random variables, 0 ? X ? 1,
QEX. Let
Then
P( Y gives (1??)-estimate of Q )
- ?2 . n . EX / 3
? 1
e
29
(No Transcript)
30
1
1
VX
n
?2
EX2
?
Number of samples to achieve precision ? with
confidence ?.
3
1
ln (1/?)
n
?2
EX
0?X?1
31
BAD
1
1
VX
n
?2
EX2
?
Number of samples to achieve precision ? with
confidence ?.
3
1
ln (1/?)
n
?2
EX
GOOD
0?X?1
BAD
32
Median boosting trick
1
4
n
?2
EX
BY BIENAYME-CHEBYSHEV
?
P(
) ? 3/4
(1-?)Q
(1?)Q

Y
33
Median trick repeat 2T times
(1-?)Q
(1?)Q
BY BIENAYME-CHEBYSHEV
?
P(
) ? 3/4
?
BY CHERNOFF
-T/4
gt T out of 2T
P(
) ? 1 - e
?
-T/4
median is in
) ? 1 - e
P(
34
VX
32
n
ln (1/?)
?2
EX2
median trick
1
3
n
ln (1/?)
?2
EX
0?X?1
BAD
35
Creating approximator from X ?
precision ? confidence
36
Outline
1. Counting problems 2. Basic tools Chernoff,
Chebyshev 3. Dealing with large quantities
(the product method) 4. Statistical
physics 5. Cooling schedules (our work) 6.
More...
37
(approx) counting ? sampling
Valleau,Card72 (physical chemistry), Babai79
(for matchings and colorings),
Jerrum,Valiant,V.Vazirani86
the outcome of the JVV reduction
random variables X1 X2 ... Xt
such that
EX1 X2 ... Xt
1)
WANTED
2)
the Xi are easy to estimate
VXi
squared coefficient of variation (SCV)
O(1)
EXi2
38
(approx) counting ? sampling
EX1 X2 ... Xt
1)
WANTED
2)
the Xi are easy to estimate
VXi
O(1)
EXi2
Theorem (Dyer-Frieze91)
O(t2/?2) samples (O(t/?2) from each Xi) give
1?? estimator of WANTED with prob?3/4
39
JVV for independent sets
GOAL given a graph G, estimate the number of
independent sets of G
1
independent sets

P( )
40
P(A?B)P(A)P(BA)
JVV for independent sets
P( )
P( )
P( )
P( )
P( )
X1
X2
X3
X4
VXi
Xi ? 0,1 and EXi ?½ ?
O(1)
EXi2
41
P(A?B)P(A)P(BA)
JVV for independent sets
P( )
P( )
P( )
P( )
P( )
X1
X2
X3
X4
VXi
Xi ? 0,1 and EXi ?½ ?
O(1)
EXi2
42
Self-reducibility for independent sets
?
P( )
5

?
7
?
43
Self-reducibility for independent sets
?
P( )
5

?
7
?
7

5
44
Self-reducibility for independent sets
?
P( )
5

?
7
?
7
7


5
5
45
Self-reducibility for independent sets
P( )
3

?
5
?
5

3
46
Self-reducibility for independent sets
P( )
3

?
5
?
5
5


3
3
47
Self-reducibility for independent sets
5
7
7


3
5
5
5
7
3
7

3
5
2
48
JVV If we have a sampler oracle
random independent set of G
SAMPLER ORACLE
graph G
then FPRAS using O(n2) samples.
49
JVV If we have a sampler oracle
random independent set of G
SAMPLER ORACLE
graph G
then FPRAS using O(n2) samples.
VV If we have a sampler oracle
SAMPLER ORACLE
set from gas-model Gibbs at ?
?, graph G
then FPRAS using O(n) samples.
50
Application independent sets
O( V ) samples suffice for counting
Cost per sample (Vigoda01,Dyer-Greenhill01)
time O( V ) for graphs of degree ? 4.
Total running time O ( V2
).
51
Other applications
matchings O(n2m) (using
Jerrum, Sinclair89) spin systems Ising
model O(n2) for ?lt?C
(using Marinelli, Olivieri95)
k-colorings O(n2) for kgt2?
(using Jerrum95)
total running time
52
Outline
1. Counting problems 2. Basic tools Chernoff,
Chebyshev 3. Dealing with large quantities
(the product method) 4. Statistical
physics 5. Cooling schedules (our work) 6.
More
53
easy hot
hard cold
54
Hamiltonian
55
Big set ?
Hamiltonian H ? ? 0,...,n
Goal estimate H-1(0)
H-1(0) EX1 ... EXt
56
Distributions between hot and cold

• ? inverse temperature
• 0 ? hot ? uniform on ?
• ? ? cold ? uniform on H-1(0)

?? (x) ? exp(-H(x)?)
(Gibbs distributions)
57
Distributions between hot and cold
?? (x) ? exp(-H(x)?)
exp(-H(x)?)
?? (x)
Z(?)
Normalizing factor partition function
Z(?) ? exp(-H(x)?)
x??
58
Partition function
have Z(0) ? want Z(?) H-1(0)
59
Partition function - example
have Z(0) ? want Z(?) H-1(0)
4
Z(?) 1 e-4.? 4
e-2.? 4 e-1.?
7 e-0.?
2
1
0
Z(0) 16 Z(?)7
60
Assumption we have a sampler oracle for ??
exp(-H(x)?)
?? (x)
Z(?)
SAMPLER ORACLE
subset of V from ??
graph G ?
61
Assumption we have a sampler oracle for ??
exp(-H(x)?)
?? (x)
Z(?)
W ? ??
62
Assumption we have a sampler oracle for ??
exp(-H(x)?)
?? (x)
Z(?)
W ? ??
X exp(H(W)(? - ?))
63
Assumption we have a sampler oracle for ??
exp(-H(x)?)
?? (x)
Z(?)
W ? ??
X exp(H(W)(? - ?))
can obtain the following ratio
Z(?)
EX ? ??(s) X(s)

Z(?)
s??
64
Our goal restated
Partition function
Z(?) ? exp(-H(x)?)
x??
Goal estimate Z(?)H-1(0)
Z(?1) Z(?2) Z(?t)
Z(?)
Z(0)
...
Z(?0) Z(?1) Z(?t-1)
?0 0 lt ?1 lt ? 2 lt ... lt ?t ?
65
Our goal restated
Z(?1) Z(?2) Z(?t)
Z(?)
Z(0)
...
Z(?0) Z(?1) Z(?t-1)
Cooling schedule
?0 0 lt ?1 lt ? 2 lt ... lt ?t ?
How to choose the cooling schedule?
minimize length, while satisfying
Z(?i)
VXi
O(1)
EXi
Z(?i-1)
EXi2
66
Our goal restated
Z(?1) Z(?2) Z(?t)
Z(?)
Z(0)
...
Z(?0) Z(?1) Z(?t-1)
Cooling schedule
?0 0 lt ?1 lt ? 2 lt ... lt ?t ?
How to choose the cooling schedule?
minimize length, while satisfying
Z(?i)
VXi
O(1)
EXi
Z(?i-1)
EXi2
67
Outline
1. Counting problems 2. Basic tools Chernoff,
Chebyshev 3. Dealing with large quantities
(the product method) 4. Statistical
physics 5. Cooling schedules (our work) 6.
More...
68
Parameters A and n
Z(?) ? exp(-H(x)?)
x??
Z(0)
A
H? ? 0,...,n
ak H-1(k)
69
Parameters
Z(0)
A
H? ? 0,...,n
A
n
2V
E
independent sets matchings perfect matchings
k-colorings
V
? V!
V!
V
kV
E
70
Parameters
Z(0)
A
H? ? 0,...,n
A
n
2V
E
independent sets matchings perfect matchings
k-colorings
V
? V!
V!
V
matchings ways of marrying them so that no
unhappy couple
kV
E
71
Parameters
Z(0)
A
H? ? 0,...,n
A
n
2V
E
independent sets matchings perfect matchings
k-colorings
V
? V!
V!
V
matchings ways of marrying them so that no
unhappy couple
kV
E
72
Parameters
Z(0)
A
H? ? 0,...,n
A
n
2V
E
independent sets matchings perfect matchings
k-colorings
V
? V!
V!
V
matchings ways of marrying them so that no
unhappy couple
kV
E
73
Parameters
Z(0)
A
H? ? 0,...,n
A
n
2V
E
independent sets matchings perfect matchings
k-colorings
V
? V!
V!
V
marry ignoring compatibility
hamiltonian number of unhappy couples
kV
E
74
Parameters
Z(0)
A
H? ? 0,...,n
A
n
2V
E
independent sets matchings perfect matchings
k-colorings
V
? V!
V!
V
kV
E
75
Previous cooling schedules
Z(0)
A
H? ? 0,...,n
?0 0 lt ?1 lt ? 2 lt ... lt ?t ?
Safe steps

• ? ? 1/n
• ? ? (1 1/ln A)
• ln A ? ?

(Bezáková,tefankovic, Vigoda,V.Vazirani06)
Cooling schedules of length
O( n ln A)
(Bezáková,tefankovic, Vigoda,V.Vazirani06)
O( (ln n) (ln A) )
76
Previous cooling schedules
Z(0)
A
H? ? 0,...,n
?0 0 lt ?1 lt ? 2 lt ... lt ?t ?
Safe steps

• ? ? 1/n
• ? ? (1 1/ln A)
• ln A ? ?

(Bezáková,tefankovic, Vigoda,V.Vazirani06)
Cooling schedules of length
O( n ln A)
(Bezáková,tefankovic, Vigoda,V.Vazirani06)
O( (ln n) (ln A) )
77
Safe steps

• ? ? 1/n
• ? ? (1 1/ln A)
• ln A ? ?

(Bezáková,tefankovic, Vigoda,V.Vazirani06)
W ? ??
X exp(H(W)(? - ?))
1/e ? X ? 1
VX
1
?
? e
EX2
EX
78
Safe steps

• ? ? 1/n
• ? ? (1 1/ln A)
• ln A ? ?

(Bezáková,tefankovic, Vigoda,V.Vazirani06)
W ? ??
X exp(H(W)(? - ?))
Z(?) a0 ? 1 Z(ln A) ? a0 1
EX ? 1/2
79
Safe steps

• ? ? 1/n
• ? ? (1 1/ln A)
• ln A ? ?

(Bezáková,tefankovic, Vigoda,V.Vazirani06)
W ? ??
X exp(H(W)(? - ?))
EX ? 1/2e
80
Previous cooling schedules
1/n, 2/n, 3/n, .... , (ln A)/n, .... , ln A
Safe steps

• ? ? 1/n
• ? ? (1 1/ln A)
• ln A ? ?

(Bezáková,tefankovic, Vigoda,V.Vazirani06)
Cooling schedules of length
O( n ln A)
(Bezáková,tefankovic, Vigoda,V.Vazirani06)
O( (ln n) (ln A) )
81
No better fixed schedule possible
Z(0)
A
H? ? 0,...,n
THEOREM
A schedule that works for all
- ? n
Za(?) (1 a e )
(with a?0,A-1)
has LENGTH ? ?( (ln n)(ln A) )
82
Parameters
Z(0)
A
H? ? 0,...,n
Our main result
can get adaptive schedule of length O ( (ln
A)1/2 )
Previously
non-adaptive schedules of length ?( ln A )
83
Related work
can get adaptive schedule of length O ( (ln
A)1/2 )
Lovász-Vempala Volume of convex bodies in
O(n4) schedule of length O(n1/2)
(non-adaptive cooling schedule, using specific
properties of the volume partition functions)
84
Existential part
Lemma
for every partition function there exists a
cooling schedule of length O((ln A)1/2)
there exists
85
Cooling schedule (definition refresh)
Z(?1) Z(?2) Z(?t)
Z(?)
Z(0)
...
Z(?0) Z(?1) Z(?t-1)
Cooling schedule
?0 0 lt ?1 lt ? 2 lt ... lt ?t ?
How to choose the cooling schedule?
minimize length, while satisfying
Z(?i)
VXi
O(1)
EXi
Z(?i-1)
EXi2
86
Express SCV using partition function
(going from ? to ?)
W ? ??
X exp(H(W)(? - ?))
EX2
Z(2?-?) Z(?)
? C

EX2
Z(?)2

VX
1
EX2
87
?
?
2?-?
f(?)ln Z(?)
graph of f
(f(2?-?) f(?))/2 ? (ln C)/2 f(?)
Proof
? C(ln C)/2
88
Properties of partition functions
f is decreasing f is convex f(0)
? n f(0) ? ln A
f(?)ln Z(?)
89
Properties of partition functions
f is decreasing f is convex f(0)
? n f(0) ? ln A
f(?)ln Z(?)
n
?
f(?) ln ak e-? k
n
?
k0
ak k e-? k
-
k0
f(?)
n
?
f
ak e-? k
(ln f)
f
k0
90
f is decreasing f is convex f(0)
? n f(0) ? ln A
GOAL proving Lemma
for every partition function there exists a
cooling schedule of length O((ln A)1/2)
f(?)ln Z(?)
either f or f changes a lot
Proof
Let K?f
1
Then
1
?(ln f) ?
K
91
Let K?f
Then
Proof
1
1
?(ln f) ?
K
a
b
c
c (ab)/2, ? b-a have f(c)
(f(a)f(b))/2 1 (f(a) f(c)) /?
? f(a) (f(c) f(b)) /? ? f(b)
f is convex
92
Let K?f
f(b) f(a)
? 1-1/?f ? e-?f
Then
1
?(ln f) ?
K
c (ab)/2, ? b-a have f(c)
(f(a)f(b))/2 1 (f(a) f(c)) /?
? f(a) (f(c) f(b)) /? ? f(b)
f is convex
93
fa,b ? R, convex, decreasing can be
approximated using
f(a)
(f(a)-f(b))
f(b)
segments
94
Technicality getting to 2?-?
Proof
?
?
2?-?
95
Technicality getting to 2?-?
Proof
?i
?
?
2?-?
?i1
96
Technicality getting to 2?-?
Proof
?i
?
?
2?-?
?i2
?i1
97
Technicality getting to 2?-?
Proof
ln ln A extra steps
?i
?
?
2?-?
?i2
?i1
?i3
98
Existential ? Algorithmic
there exists
99
Algorithmic construction
Our main result
using a sampler oracle for ??
we can construct a cooling schedule of length
? 38 (ln A)1/2(ln ln A)(ln n)
Total number of oracle calls ? 107 (ln A) (ln
ln Aln n)7 ln (1/?)
100
Algorithmic construction
current inverse temperature ?
ideally move to ? such that
Z(?)
EX2
EX
? B2
B1 ?
Z(?)
EX2
101
Algorithmic construction
current inverse temperature ?
ideally move to ? such that
Z(?)
EX2
EX
? B2
B1 ?
Z(?)
EX2
X is easy to estimate
102
Algorithmic construction
current inverse temperature ?
ideally move to ? such that
Z(?)
EX2
EX
? B2
B1 ?
Z(?)
EX2
we make progress (where B1gt1)
103
Algorithmic construction
current inverse temperature ?
ideally move to ? such that
Z(?)
EX2
EX
? B2
B1 ?
Z(?)
EX2
need to construct a feeler for this
104
Algorithmic construction
current inverse temperature ?
ideally move to ? such that
Z(?)
EX2
EX
? B2
B1 ?
Z(?)
EX2

Z(?)
Z(2?-?)
Z(?)
Z(?)
need to construct a feeler for this
105
Algorithmic construction
current inverse temperature ?
bad feeler
ideally move to ? such that
Z(?)
EX2
EX
? B2
B1 ?
Z(?)
EX2

Z(?)
Z(2?-?)
Z(?)
Z(?)
need to construct a feeler for this
106
estimator for
ak e-? k
For W ? ?? we have P(H(W)k)
Z(?)
107
estimator for
If H(X)k likely at both ?, ? ? estimator
ak e-? k
For W ? ?? we have P(H(W)k)
Z(?)
ak e-? k
For U ? ?? we have P(H(U)k)
Z(?)
108
estimator for
If H(X)k likely at both ?, ? ? estimator
ak e-? k
For W ? ?? we have P(H(W)k)
Z(?)
ak e-? k
For U ? ?? we have P(H(U)k)
Z(?)
109
estimator for
ak e-? k
For W ? ?? we have P(H(W)k)
Z(?)
ak e-? k
For U ? ?? we have P(H(U)k)
Z(?)
P(H(U)k) P(H(W)k)

ek(?-?)
110
estimator for
ak e-? k
For W ? ?? we have P(H(W)k)
Z(?)
ak e-? k
For U ? ?? we have P(H(U)k)
Z(?)
P(H(U)k) P(H(W)k)

ek(?-?)
PROBLEM P(H(W)k) can be too small
111
Rough estimator for
interval instead of single value
d
?
ak e-? k
For W ? ?? we have
P(H(W)?c,d)
kc
Z(?)
d
?
ak e-? k
For U ? ?? we have
P(H(W)?c,d)
kc
Z(?)
112
Rough estimator for
If ?-?? d-c ? 1 then
1
P(H(U)?c,d) P(H(W)?c,d)
ec(?-?)
?
? e
e
We also need P(H(U) ? c,d)
P(H(W) ? c,d) to be large.
d
d
?
ak e-? k
?
ak e-? (k-c)
kc
kc
ec(?-?)

d
d
?
ak e-? k
?
ak e-? (k-c)
kc
kc
113
We will
Split 0,1,...,n into h ? 4(ln n) ln
A intervals 0,1,2,...,c,c(11/ ln
A),...
for any inverse temperature ? there exists a
interval with P(H(W)? I) ? 1/8h
We say that I is HEAVY for ?
114
We will
Split 0,1,...,n into h ? 4(ln n) ln
A intervals 0,1,2,...,c,c(11/ ln
A),...
for any inverse temperature ? there exists a
interval with P(H(W)? I) ? 1/8h
We say that I is HEAVY for ?
115
Algorithm
repeat
find an interval I which is heavy for
the current inverse temperature ? see how far I
is heavy (until some ?) use the interval I for
the feeler
Z(?)
Z(2?-?)
Z(?)
Z(?)
ANALYSIS
either make progress, or eliminate
the interval I or make a long move
116
I is heavy
?
distribution of h(X) where X???
...
I a heavy interval at ?
117
I is NOT heavy
I is heavy
?
?
distribution of h(X) where X???
!
no longer heavy at ?
...
I a heavy interval at ?
118
I is heavy
I is heavy
I is NOT heavy
?
?
?
distribution of h(X) where X???
heavy at ?
...
I a heavy interval at ?
119
I is heavy
I is heavy
I is NOT heavy
?
?
?
?1/(2n)
I is heavy
Ia,b
I is NOT heavy
?
use binary search to find ?
? min1/(b-a), ln A
120
I is heavy
I is heavy
I is NOT heavy
?
?
?
?1/(2n)
I is heavy
Ia,b
I is NOT heavy
?
use binary search to find ?
? min1/(b-a), ln A
How do you know that you can use binary search?
121
How do you know that you can use binary search?
I is NOT heavy
I is NOT heavy
I is heavy
I is heavy
Lemma the set of temperatures for which I
is h-heavy is an interval.
P(h(X)? I) ? 1/8h for X???
I is h-heavy at ?
n
?
?
1
ak e-? k
?
ak e-? k
8h
k?I
k0
122
How do you know that you can use binary search?
n
?
?
1
ak e-? k
?
ak e-? k
8h
k?I
k0
xe-?
Descartes rule of signs
c0 x0 c1 x1 c2 x2 .... cn xn

-


number of sign changes
number of positive roots
?
sign change
123
How do you know that you can use binary search?
n
-1xx2x3...xn 1xx20
?
?
1
ak e-? k
?
ak e-? k
h
k?I
k0
xe-?
Descartes rule of signs
c0 x0 c1 x1 c2 x2 .... cn xn

-


number of sign changes
number of positive roots
?
sign change
124
How do you know that you can use binary search?
n
?
?
1
ak e-? k
?
ak e-? k
8h
k?I
k0
xe-?
Descartes rule of signs
c0 x0 c1 x1 c2 x2 .... cn xn

-


number of sign changes
number of positive roots
?
sign change
125
Ia,b
I is heavy
?
I is heavy
?1/(2n)
?
I is NOT heavy
can roughly compute ratio of Z(?)/Z(?)
for ?, ?? ?,? if ?-?.b-a? 1
126
1. success
Ia,b
I is heavy
2. eliminate interval
?
I is heavy
3. long move
?1/(2n)
?
I is NOT heavy
find largest ? such that
can roughly compute ratio of Z(?)/Z(?)
for ?, ?? ?,? if ?-?.b-a? 1
Z(?)
Z(2?-?)
? C
Z(?)
Z(?)
127
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128
if we have sampler oracles for ?? then we can get
adaptive schedule of length tO ( (ln A)1/2 )
independent sets O(n2) (using
Vigoda01, Dyer-Greenhill01) matchings
O(n2m) (using Jerrum,
Sinclair89) spin systems Ising model
O(n2) for ?lt?C (using
Marinelli, Olivieri95) k-colorings
O(n2) for kgt2? (using Jerrum95)
129
Outline
1. Counting problems 2. Basic tools Chernoff,
Chebyshev 3. Dealing with large quantities
(the product method) 4. Statistical
physics 5. Cooling schedules (our work) 6.
More...
130
Outline
6. More a) proof of Dyer-Frieze b)
independent sets revisited c) warm starts
131
Appendix proof of
EX1 X2 ... Xt
1)
WANTED
2)
the Xi are easy to estimate
VXi
O(1)
EXi2
Theorem (Dyer-Frieze91)
O(t2/?2) samples (O(t/?2) from each Xi) give
1?? estimator of WANTED with prob?3/4
132
How precise do the Xi have to be?
First attempt term by term
Main idea
(1? )(1? )(1? )... (1? ) ? 1??
?(
)
VX
1
ln (1/?)
n
?2
EX2
each term ? (t2) samples ? ? (t3) total
133
How precise do the Xi have to be?
Analyzing SCV is better
(Dyer-Frieze1991)
XX1 X2 ... Xt
squared coefficient of variation (SCV)
GOAL SCV(X) ? ?2/4
P( X gives (1??)-estimate )
1
VX
? 1 -
EX2
?2
134
How precise do the Xi have to be?
Analyzing SCV is better
(Dyer-Frieze1991)
Main idea
?2/4
SCV(Xi) ?
?
SCV(X) lt ?2/4
t
?
proof
SCV(X) (1SCV(X1)) ... (1SCV(Xt)) - 1
135
How precise do the Xi have to be?
X1, X2 independent ? EX1 X2 EX1EX2
Analyzing SCV is better
X1, X2 independent ? X12,X22 independent
(Dyer-Frieze1991)
X1,X2 independent ?
SCV(X1X2)(1SCV(X1))(1SCV(X2))-1
Main idea
?2/4
SCV(Xi) ?
?
SCV(X) lt ?2/4
t
?
proof
SCV(X) (1SCV(X1)) ... (1SCV(Xt)) - 1
136
How precise do the Xi have to be?
Analyzing SCV is better
(Dyer-Frieze1991)
Main idea
?2/4
SCV(Xi) ?
?
SCV(X) lt ?2/4
?
t
each term O(t /?2) samples ? O(t2/?2) total
137
Outline
6. More a) proof of Dyer-Frieze b)
independent sets revisited c) warm starts
138
Hamiltonian
4
2
1
0
139
Hamiltonian many possibilities
2
1
0
(hardcore lattice gas model)
140
What would be a natural hamiltonian for planar
graphs?
141
What would be a natural hamiltonian for planar
graphs?
H(G) number of edges
natural MC
pick u,v uniformly at random
try G - u,v
1/(1?)
try G u,v
?/(1?)
142
1/(1?)
G
n(n-1)/2
G
v
v
u
u
?/(1?)
n(n-1)/2
natural MC
pick u,v uniformly at random
try G - u,v
1/(1?)
try G u,v
?/(1?)
143
1/(1?)
G
n(n-1)/2
G
v
v
u
u
?/(1?)
n(n-1)/2
?(G) ? ?number of edges
satisfies the detailed balance condition
?(G) P(G,G) ?(G) P(G,G)
(? exp(-?))
144
Outline
6. More a) proof of Dyer-Frieze b)
independent sets revisited c) warm starts
145
(n3)
Mixing time ?mix smallest t such that
?t - ? TV ?
1/e Relaxation time ?rel 1/(1-?2)
?(n ln n)
?(n)
?rel ? ?mix ? ?rel ln (1/?min)
(discrepancy may be substantially bigger for,
e.g., matchings)
146
Mixing time ?mix smallest t such that
?t - ? TV ?
1/e Relaxation time ?rel 1/(1-?2)
Estimating ?(S)
METHOD 1
X??

1 if X? S 0 otherwise
X1
Y
X2
EY?(S)
X3
...
Xs
147
Mixing time ?mix smallest t such that
?t - ? TV ?
1/e Relaxation time ?rel 1/(1-?2)
Estimating ?(S)
METHOD 1
X??

1 if X? S 0 otherwise
X1
Y
X2
EY?(S)
X3
...
METHOD 2 (Gillman98, Kahale96, ...)
Xs
...
Xs
X3
X2
X1
148
Mixing time ?mix smallest t such that
?t - ? TV ?
1/e Relaxation time ?rel 1/(1-?2)
Further speed-up
?t - ? TV ? exp(-t/?rel) Var?(?0/?)
(? ?(x)(?0(x)/?(x)-1)2)1/2
small ? called warm start
METHOD 2 (Gillman98, Kahale96, ...)
...
Xs
X3
X2
X1
149
Mixing time ?mix smallest t such that
?t - ? TV ?
1/e Relaxation time ?rel 1/(1-?2)
sample at ? can be used as a warm start for ?
? cooling schedule can step
from ? to ?
Further speed-up
?t - ? TV ? exp(-t/?rel) Var?(?0/?)
(? ?(x)(?0(x)/?(x)-1)2)1/2
small ? called warm start
METHOD 2 (Gillman98, Kahale96, ...)
...
Xs
X3
X2
X1
150
sample at ? can be used as a warm start for ?
? cooling schedule can step
from ? to ?
mO( (ln n)(ln A) )
?0
?3
?2
?1
?m
....
well mixed states
151
?0
?3
?2
?m
?1
....
well mixed states
run the our cooling-schedule algorithm with
METHOD 2 using well mixed states as starting
points
METHOD 2
Xs
...
Xs
X3
X2
X1
152
kO( (ln A)1/2 )
Output of our algorithm
?0
?1
?k
small augmentation (so that we can use sample
from current ? as a warm start at next)
still O( (ln A)1/2 )
?0
?3
?2
?1
?m
....
Use analogue of Frieze-Dyer for independent
samples from vector variables with slightly
dependent coordinates.
153
if we have sampler oracles for ?? then we can get
adaptive schedule of length tO ( (ln A)1/2 )
independent sets O(n2) (using
Vigoda01, Dyer-Greenhill01) matchings
O(n2m) (using Jerrum,
Sinclair89) spin systems Ising model
O(n2) for ?lt?C (using
Marinelli, Olivieri95) k-colorings
O(n2) for kgt2? (using Jerrum95)