Qi Lou Rina Dechter Alexander Ihler Feb 8 2017 1 Guideline Anytime bounds for the partition function of a graphical model Estimate bound the partition function as a heuristic search problem on ANDOR search trees ID: 675127
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Slide1
Anytime Anyspace AND/OR Search for Bounding the Partition Function
Qi Lou, Rina Dechter, Alexander IhlerFeb. 8, 2017
1Slide2
Guideline
Anytime bounds for the partition function of a graphical model.Estimate (bound) the partition function as a heuristic search problem on AND/OR search treesBest-first search using pre-compiled variational heuristics and carefully designed priorities.
Operate with limited-memory settings
2Slide3
Background
3Slide4
Graphical Models
A graphical model (X, D, F):
Bayesian
networks, Markov
random fields (MRF),
factor
graphs, etc.
Example model (MRF):
A
B
C
Primal Graph
4
A
B
f(A,B)
0
0
0.24
010.56101.2111.2
BCf(B,C)000.12010.36100.3111.7
…Slide5
Graphical Models
Typical QueriesMaximum A Posteriori (MAP) / Most Probable Explanation (MPE)The Partition Function
5Slide6
Bounding the Partition Function
Z 6
Deterministic methods
Elimination based (e.g., mini-bucket elimination
[
Dechter
and
Rish
2001
]
)
Variational approaches (e.g., tree-reweighted belief propagation [Wainwright et al. 2003]
)Search based (e.g., [Viricel et al. 2016])Monte
Carlo methodsImportance sampling based (e.g., [Liu et al. 2015,
Bidyuk and Dechter 2007]
)Approximate hash-based counting (e.g., [Chakraborty et al. 2016]
)Slide7
Algorithm
7Slide8
OR Search Tree
8
B
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1G01F01
G01G01E01F01G01G01
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A
Solution path
: corresponds to a complete configuration of all variables
A
B
C
D
E
F
G
p
rimal graphSlide9
AND/OR Search Tree
[Nillson 1980
,
Dechter
and
Mateescu
2007]
OR
AND
OR
AND
OR
OR
AND
AND
9
A
B
B010101F0
1G01G01F01G01G01C01E
0
1
D
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A
B
C
D
E
F
G
Solution tree
: corresponds to a complete configuration of all variables
p
rimal graph
A
B
C
F
G
D
E
p
seudo tree
[
Freuder
and Quinn 1985
]Slide10
Cost
10
A
B
B
0
1
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F
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G
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F01G01G01C01E01D01E
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Cost
g
(
n
)
:
Product
of
functions
fully instantiated
by the path from the root to node
n
.Slide11
Value and Heuristic
11
Value
v
(
n
)
:
M
ass of the subtree rooted at node
n
Heuristic
h
(
n): Estimate of
v(
n)
A
BB010101F
01G01G01F01G01G01C01E
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1Slide12
Estimate
Z via Search on AND/OR Trees
15
A
B
C
F
G
D
E
p
seudo tree
A
0
1Slide13
Estimate
Z
via Search on AND/OR Trees
15
A
B
C
F
G
D
E
p
seudo tree
B
0
1
A
0
1Slide14
Key Issues
How to design high-quality heuristics?Quality of the estimate largely depends on the quality of heuristics14Slide15
λ
G (A,F)
15
Weighted Mini-Bucket (WMB) Heuristics
A
f(A,B)
B
f(B,C)
C
f(B,F)
F
f(A,G) f(F,G)
G
f(B,E) f(C,E)
E
λ
F
(A,B)
λB (A)λE (B,C)λD (B,C)λC (B)f(A)DDλD (A)f(B,D) f(C,D)f(A,D)…WMB heuristics are formed by messages from descendants.WMB heuristics are upper (or lower) bounds of the node value.WMB heuristics are monotonic.Resolving mini-buckets makes heuristics more (no less) accurate.[Liu and Ihler 2011]Slide16
Key Issues (revisited)
How to design high-quality heuristics?Quality of the estimate largely depends on the quality of heuristicsHow to design an effective search strategy?Our goal: quickly close the bound gap U - L
U
:= upper
bound of
Z
L
:= lower
bound of
Z
Equivalent: how to set priority for frontier nodes.
16Slide17
F
0
1
C
0
1
0
B
1
B
1
0
Intuition: expand the frontier node that potentially reduces the bound gap
U
- L
most
bounds the global
bound gap reduction
on
Z if the subtree beneath n is fully expanded.Priority17A01Strategy: Expand the frontier node with the largest priority value!
gap priorityupper prioritySlide18
Overcome the Memory Limit
Main strategy (SMA*-like [Russell 1992])Keep track of the lowest-priority node as wellWhen reach the memory limit, delete the lowest-priority nodes, and keep
expanding the
top-priority
ones
18Slide19
Experiment
19Slide20
AND/OR Best-first Search (AOBFS)
Variants of our algorithmA-G: AND/OR search tree with gap priorityA-U: AND/OR search tree with upper priorityO-G: OR search tree with gap priority [Henrion 1991]
20Slide21
Baselines
[Viricel et al. 2016] A recent depth-first branch-and-bound search algorithm that
provides deterministic upper
and
lower bounds on
Z.
For a given
ɛ
,
it
returns (non-anytime)
bounds on
lnZ whose gap is at most
ln(1+ɛ).
VEC [Dechter
1999, 2013] Variable Elimination with Conditioning, also known as custet-conditioning)
Apply variable elimination to each assignment of the cutset.MMAP-DFS [
Marinescu et al. 2014] (abbreviated M-D)A
state-of-the-art method for marginal MAP using AND/OR searchSolve the internal summation problem exactly using depth-first search aided by
WMB heuristics.21Slide22
Benchmarks
PIC’11: 23 instances selected by [Viricel et al. 2016] from the 2012 UAI competitionBN
: Bayesian networks from
the 2006
competition (50 randomly selected
instances)
Protein
:
made from the “small” protein side-chains of
[
Yanover
and Weiss 2002] (50 randomly selected
instances)CPD: computational protein design problems from
[Viricel et al. 2016] (100 randomly selected instances)
22Slide23
Parameters
Implementationall methods are implemented in C++ by the original authorsRuntime: 1 hourMemory:1GB, 4GB and
16GB
i
-bound determined by memory
23Slide24
Results
24
(a) PIC’11/queen5_5_4 (b)
Protein/1g6x
Anytime
behavior of AOBFSSlide25
Results
Number of instances solved to “tight” tolerance interval. The best (most solved) for each setting is
bolded
.
25Slide26
Results
Number of instances solved to “loose” tolerance interval. The best (most solved) for each setting is
bolded
.
26Slide27
Results
A-G sometimes effectively solve (<1e-3) instances with very high width given relatively few high-weight configurations of the model. “1who” is solved in 12 seconds and 1GB memory, while the corresponding junction tree requires about 150GB memory;
“2fcr”
is solved in 21 minutes and 16GB memory, while junction tree would require approximately 3.5PB.
27Slide28
Summary
Best-first search algorithm for bounding the partition functionBounds in an anytime fashion within limited memory resources.Search runs on AND/OR trees that enable exploiting conditional independencePriority-driven
best-first search scheme b
ased
on precompiled variational heuristics
Outperform state-of-the-art baselines on
multiple
benchmark &
memory
setups
28Slide29
Thank You!
Q&A29