Iterative Join Graph Propagation

1
Iterative Join Graph Propagation

• Rina Dechter
• Kalev Kask
• Robert Mateescu

2
What is IJGP?

• IJGP is an approximate algorithm for belief
  updating in Bayesian networks
• IJGP is a version of join-tree clustering which
  is both anytime and iterative
• IJGP applies message passing along a join-graph,
  rather than a join-tree
• Empirical evaluation shows that IJGP is almost
  always superior to other approximate schemes
  (IBP, MC)

3
Outline

• IBP - Iterative Belief Propagation
• MC - Mini Clustering
• IJGP - Iterative Join Graph Propagation
• Empirical evaluation
• Conclusion

4
Iterative Belief Propagation - IBP

• Belief propagation is exact for poly-trees
• IBP - applying BP iteratively to cyclic networks
• No guarantees for convergence
• Works well for many coding networks

U1
U2
U3
One step update BEL(U1)
X2
X1
5
Mini-Clustering - MC
Cluster Tree Elimination
Mini-Clustering, i3
A B C p(a), p(ba), p(ca,b)
1
BC
B C D F p(db), p(fc,d)
B C D F p(db), h(1,2)(b,c), p(fc,d)
2
2
BF
sep(2,3) B,F elim(2,3) C,D
A
B
B E F p(eb,f)
3
E
C
D
EF
F
E F G p(ge,f)
4
G
6
CTE vs. MC
ABC
ABC
1
1
BC
BC
BCDF
BCDF
2
2
BF
BF
BEF
BEF
3
3
EF
EF
EFG
EFG
4
4
Cluster Tree Elimination
Mini-Clustering, i3
7
IJGP - Motivation

• IBP is applied to a loopy network iteratively
• not an anytime algorithm
• when it converges, it converges very fast
• MC applies bounded inference along a tree
  decomposition
• MC is an anytime algorithm controlled by i-bound
• IJGP combines
• the iterative feature of IBP
• the anytime feature of MC

8
IJGP - The basic idea

• Apply Cluster Tree Elimination to any join-graph
• We commit to graphs that are minimal I-maps
• Avoid cycles as long as I-mapness is not violated
• Result use minimal arc-labeled join-graphs

9
IJGP - Example
A
B
C
A
C
A
ABC
C
A
AB
BC
C
D
E
ABDE
BCE
BE
C
DE
CE
F
CDEF
G
H
H
FGH
H
F
F
FG
GH
H
GI
I
J
FGI
GHIJ
a) Belief network
a) The graph IBP works on
10
Arc-minimal join-graph
A
C
A
A
ABC
C
A
ABC
C
A
AB
BC
C
AB
BC
ABDE
BCE
ABDE
BCE
BE
C
C
DE
CE
DE
CE
CDEF
CDEF
H
H
FGH
H
FGH
H
F
F
F
FG
GH
H
FG
GH
GI
GI
FGI
GHIJ
FGI
GHIJ
11
Minimal arc-labeled join-graph
A
A
A
ABC
C
A
ABC
C
AB
BC
AB
BC
ABDE
BCE
ABDE
BCE
C
C
DE
CE
DE
CE
CDEF
CDEF
H
H
FGH
H
FGH
H
F
F
FG
GH
F
GH
GI
GI
FGI
GHIJ
FGI
GHIJ
12
Join-graph decompositions
A
A
ABC
C
AB
BC
BC
BC
ABDE
BCE
ABCDE
BCE
ABCDE
BCE
C
DE
CE
CDE
CE
DE
CE
CDEF
CDEF
CDEF
H
FGH
H
FGH
FGH
F
F
F
F
GH
F
GH
F
GH
GI
GI
GI
FGI
GHIJ
FGI
GHIJ
FGI
GHIJ
a) Minimal arc-labeled join graph
b) Join-graph obtained by collapsing nodes of
graph a)
c) Minimal arc-labeled join graph
13
Tree decomposition
BC
ABCDE
BCE
ABCDE
DE
CE
CDE
CDEF
CDEF
FGH
F
F
F
GH
GHI
GI
FGI
GHIJ
FGHI
GHIJ
a) Minimal arc-labeled join graph
a) Tree decomposition
14
Join-graphs
more accuracy
less complexity
15
Minimal arc-labeled decomposition
BC
BC
ABCDE
BCE
ABCDE
BCE
CDE
CE
DE
CE
CDEF
CDEF
a) Fragment of an arc-labeled join-graph
a) Shrinking labels to make it a minimal
arc-labeled join-graph

• Use a DFS algorithm to eliminate cycles relative
  to each variable

16
Message propagation
BC
ABCDE
BCE
ABCDE p(a), p(c), p(bac), p(dabe),p(eb,c)
h(3,1)(bc)
h(3,1)(bc)
CDE
BCD
1
3
CE
BC
CDEF
FGH
h(1,2)
CDE
CE
F
F
GH
CDEF
2
GI
FGI
GHIJ
Minimal arc-labeled sep(1,2)D,E
elim(1,2)A,B,C
Non-minimal arc-labeled sep(1,2)C,D,E
elim(1,2)A,B
17
Bounded decompositions

• We want arc-labeled decompositions such that
• the cluster size (internal width) is bounded by i
  (the accuracy parameter)
• the width of the decomposition as a graph
  (external width) is as small as possible
• Possible approaches to build decompositions
• partition-based algorithms - inspired by the
  mini-bucket decomposition
• grouping-based algorithms

18
Partition-based algorithms
GFE
P(GF,E)
EF
EBF
P(EB,F)
P(FC,D)
BF
F
FCD
BF
CD
CDB
P(DB)
CB
B
CAB
P(CA,B)
BA
BA
P(BA)
A
A
P(A)
a) schematic mini-bucket(i), i3 b) arc-labeled
join-graph decomposition
19
IJGP properties

• Theorem
• If IJGP is applied to a join-tree decomposition
  it is guaranteed to compute the exact beliefs in
  one iteration (two phases, up and down)
• The time complexity of one iteration of IJGP is
  O(deg(nN) d i1) and the space complexity is
  O(Nd?), where deg max degree of a node in
  the decomposition n number of variables N
  number of nodes in the decomposition d max
  domain size of a variable i max cluster
  size ? max label size

20
IJGP properties

• IJGP(i) applies BP to min arc-labeled
  join-graph, whose cluster size is bounded by i
• On join-trees IJGP finds exact beliefs
• IJGP is a Generalized Belief Propagation
  algorithm (Yedidia, Freeman, Weiss 2001)
• Complexity of one iteration
• time O(deg(nN) d i1)
• space O(Nd?)

21
Empirical evaluation

• Algorithms
• Exact
• IBP
• MC
• IJGP

• Measures
• Absolute error
• Relative error
• Kullback-Leibler (KL) distance
• Bit Error Rate
• Time

• Networks (all variables are binary)
• Random networks
• Grid networks (MxM)
• CPCS 54, 360, 422
• Coding networks

22
Random networks - KL at convergence
evidence0
evidence5
23
Random networks - KL vs. iterations
evidence0
evidence5
24
Random networks - Time
25
Grid 81 KL distance at convergence
26
Grid 81 KL distance vs. iterations
27
Grid 81 Time vs. i-bound
28
Coding networks
N400, P4, 500 instances, 30 iterations, w43
29
Coding networks - BER
sigma.22
sigma.32
sigma.51
sigma.65
30
Coding networks - Time
31
CPCS 422 KL distance
evidence0
evidence30
32
CPCS 422 KL vs. iterations
evidence0
evidence30
33
CPCS 422 Relative error
evidence30
34
CPCS 422 Time vs. i-bound
evidence0
35
Conclusion

• IJGP borrows the iterative feature from IBP and
  the anytime virtues of bounded inference from MC
• Empirical evaluation showed the potential of
  IJGP, which improves with iteration and most of
  the time with i-bound, and scales up to large
  networks
• IJGP is almost always superior, often by a high
  margin, to IBP and MC
• Based on all our experiments, we think that IJGP
  provides a practical breakthrough to the task of
  belief updating