Adaptive annealing: a near-optimal

1
Adaptive annealing a near-optimal connection
between sampling and counting
Daniel Štefankovic (University of
Rochester) Santosh Vempala Eric Vigoda (Georgia
Tech)
2
Counting
independent sets spanning trees matchings per
fect matchings k-colorings
3
Compute the number of
independent sets
(hard-core gas model)
4
independent sets 7
independent set subset S of vertices
no two in S are neighbors
5
independent sets 5598861
independent set subset S of vertices
no two in S are neighbors
6
graph G ? independent sets in G
P-complete P-complete even for 3-regular
graphs
(Dyer, Greenhill, 1997)
7
graph G ? independent sets in G
?
approximation randomization
8
We would like to know Q
Goal random variable Y such that P( (1-?)Q
? Y ? (1?)Q ) ? 1-?
Y gives (1??)-estimate
9
(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
10
(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
11
JVV for independent sets
GOAL given a graph G, estimate the number of
independent sets of G
1
independent sets

P( )
12
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
13
Self-reducibility for independent sets
?
P( )
5

?
7
?
14
Self-reducibility for independent sets
?
P( )
5

?
7
?
7

5
15
Self-reducibility for independent sets
?
P( )
5

?
7
?
7
7


5
5
16
Self-reducibility for independent sets
P( )
3

?
5
?
5

3
17
Self-reducibility for independent sets
P( )
3

?
5
?
5
5


3
3
18
Self-reducibility for independent sets
5
7
7


3
5
5
5
7
3
7

3
5
2
19
JVV If we have a sampler oracle
random independent set of G
SAMPLER ORACLE
graph G
then FPRAS using O(n2) samples.
20
JVV If we have a sampler oracle
random independent set of G
SAMPLER ORACLE
graph G
then FPRAS using O(n2) samples.
ŠVV If we have a sampler oracle
SAMPLER ORACLE
set from gas-model Gibbs at ?
?, graph G
then FPRAS using O(n) samples.
21
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
).
22
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
23
easy hot
hard cold
24
Hamiltonian
4
2
1
0
25
Big set ?
Hamiltonian H ? ? 0,...,n
Goal estimate H-1(0)
H-1(0) EX1 ... EXt
26
Distributions between hot and cold

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

?? (x) ? exp(-H(x)?)
(Gibbs distributions)
27
Distributions between hot and cold
?? (x) ? exp(-H(x)?)
exp(-H(x)?)
?? (x)
Z(?)
Normalizing factor partition function
Z(?) ? exp(-H(x)?)
x??
28
Partition function
have Z(0) ? want Z(?) H-1(0)
29
Assumption we have a sampler oracle for ??
exp(-H(x)?)
?? (x)
Z(?)
SAMPLER ORACLE
subset of V from ??
graph G ?
30
Assumption we have a sampler oracle for ??
exp(-H(x)?)
?? (x)
Z(?)
W ? ??
31
Assumption we have a sampler oracle for ??
exp(-H(x)?)
?? (x)
Z(?)
W ? ??
X exp(H(W)(? - ?))
32
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??
33
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 ?
34
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
35
Parameters A and n
Z(?) ? exp(-H(x)?)
x??
Z(0)
A
H? ? 0,...,n
ak H-1(k)
36
Parameters
Z(0)
A
H? ? 0,...,n
A
n
2V
E
independent sets matchings perfect matchings
k-colorings
V
? V!
V!
V
kV
E
37
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,Štefankovic, Vigoda,V.Vazirani06)
Cooling schedules of length
O( n ln A)
(Bezáková,Štefankovic, Vigoda,V.Vazirani06)
O( (ln n) (ln A) )
38
No better fixed schedule possible
Z(0)
A
H? ? 0,...,n
A schedule that works for all
- ? n
Za(?) (1 a e )
(with a?0,A-1)
has LENGTH ? ?( (ln n)(ln A) )
39
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 )
40
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)
41
Existential part
Lemma
for every partition function there exists a
cooling schedule of length O((ln A)1/2)
there exists
42
Express SCV using partition function
(going from ? to ?)
W ? ??
X exp(H(W)(? - ?))
EX2
Z(2?-?) Z(?)
? C

EX2
Z(?)2
43
?
?
2?-?
f(?)ln Z(?)
Proof
? C(ln C)/2
44
f is decreasing f is convex f(0)
? n f(0) ? ln A
f(?)ln Z(?)
either f or f changes a lot
Proof
Let K?f
1
1
?(ln f) ?
K
45
fa,b ? R, convex, decreasing can be
approximated using
f(a)
(f(a)-f(b))
f(b)
segments
46
Technicality getting to 2?-?
Proof
?
?
2?-?
47
Technicality getting to 2?-?
Proof
?i
?
?
2?-?
?i1
48
Technicality getting to 2?-?
Proof
?i
?
?
2?-?
?i2
?i1
49
Technicality getting to 2?-?
Proof
ln ln A extra steps
?i
?
?
2?-?
?i2
?i1
?i3
50
Existential ? Algorithmic
there exists
51
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/?)
52
Algorithmic construction
current inverse temperature ?
ideally move to ? such that
Z(?)
EX2
EX
? B2
B1 ?
Z(?)
EX2
53
Algorithmic construction
current inverse temperature ?
ideally move to ? such that
Z(?)
EX2
EX
? B2
B1 ?
Z(?)
EX2
X is easy to estimate
54
Algorithmic construction
current inverse temperature ?
ideally move to ? such that
Z(?)
EX2
EX
? B2
B1 ?
Z(?)
EX2
we make progress (assuming B1gt1)
55
Algorithmic construction
current inverse temperature ?
ideally move to ? such that
Z(?)
EX2
EX
? B2
B1 ?
Z(?)
EX2
need to construct a feeler for this
56
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
57
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
58
Rough estimator for
ak e-? k
For W ? ?? we have P(H(W)k)
Z(?)
59
Rough estimator for
If H(X)k likely at both ?, ? ? rough

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(?)
60
Rough 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(?-?)
61
Rough estimator for
d
?
ak e-? k
For W ? ?? we have
P(H(W)?c,d)
kc
Z(?)
62
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.
63
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 ?
64
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(?)
either make progress, or eliminate
the interval I
65
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(?)
either make progress, or eliminate
the interval I or make a long move
66
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)
67
(No Transcript)
68
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
69
The Bienaymé-Chebyshev inequality
P( Y gives (1??)-estimate )
1
VY
? 1 -
?2
EY2
70
The Bienaymé-Chebyshev inequality
P( Y gives (1??)-estimate )
1
VY
? 1 -
?2
EY2
squared coefficient of variation SCV
VY
VX
1

EY2
EX2
n
71
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
72
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
73
1
1
VX
n
?2
EX2
?
3
1
ln (1/?)
n
?2
EX
0?X?1
74
0?X?1
1
1
1
n
?2
EX
?
3
1
ln (1/?)
n
?2
EX
0?X?1
75
Median boosting trick
1
4
n
?2
EX
?
P(
) ? 3/4
(1-?)Q
(1?)Q

Y
76
Median trick repeat 2T times
(1-?)Q
(1?)Q
?
P(
) ? 3/4
?
-T/4
gt T out of 2T
P(
) ? 1 - e
?
-T/4
median is in
) ? 1 - e
P(
77
0?X?1
1
32
n
ln (1/?)
?2
EX
median trick
1
3
n
ln (1/?)
?2
EX
0?X?1
78
VX
32
n
ln (1/?)
?2
EX2
median trick
1
3
n
ln (1/?)
?2
EX
0?X?1
79
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
80
How precise do the Xi have to be?
First attempt Chernoffs bound
81
How precise do the Xi have to be?
First attempt Chernoffs bound
Main idea
(1? )(1? )(1? )... (1? ) ? 1??
82
How precise do the Xi have to be?
First attempt Chernoffs bound
Main idea
(1? )(1? )(1? )... (1? ) ? 1??
each term ? (t2) samples ? ? (t3) total
83
How precise do the Xi have to be?
Bienaymé-Chebyshev 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
84
How precise do the Xi have to be?
Bienaymé-Chebyshev is better
(Dyer-Frieze1991)
Main idea
?2/4
SCV(Xi) ?
?
SCV(X) lt ?2/4
?
t
SCV(X) (1SCV(X1)) ... (1SCV(Xt)) - 1
85
How precise do the Xi have to be?
Bienaymé-Chebyshev 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
86
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)