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Modern Exact and Approximate Combinatorial Optimization algorithms for Graphical Models
Rina DechterDonald Bren school of ICS, University of
California, Irvine
In the pursuit of a universal solver
A
D
B C
E
F
Main collaborator:Radu MarinescuLars OttenAlex IhlerKalev KaskRobert Mateescu CMU 2016
Combinatorial Optimization
Planning & Scheduling Computer Vision
Find an optimal schedule for the satellitethat maximizes the number of photographstaken, subject to on-board recording capacity
Image classification: label pixels inan image by their associated object class
[He et al. 2004; Winn et al. 2005]
grass
plane
sky
grass
cow
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Sample Applications for Graphical Models
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Sample Applications for Graphical Models
Learning: MAP
Reasoning: MAP
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How to Design a Good MAP SolverHeuristic SearchThe core of a good search algorithmA compact search spaceA good heuristic evaluation functionA good traversal strategy
Anytime search yields a good approximation.
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Outline Graphical models, Queries, Algorithms Inference Algorithms: bucket-elimination Bounded Inference: a) mini-bucket, b) cost-shifting AND/OR search spaces and AND/OR BnB Evaluation, Software Conclusion
CMU 2016
Graphical Models, Queries, Algorithms
Any collection of local functions over a subset of variableIs a graphical model
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Constraint Optimization Problems for Graphical Models
functionscost - },...,{ domains - },...,{ variables- },...,{
:where,, triplea is A
1
1
1
m
n
n
ffFDDDXXX
FDXRCOPfinite
===
=
A B D Cost1 2 3 31 3 2 22 1 3 02 3 1 03 1 2 53 2 1 0
G
A
B C
D F
( )∑ ==
m
i i XfXF1
)(
FunctionCost Global
Primal graph =Variables --> nodesFunctions, Constraints - arcs
f(A,B,D) has scope {A,B,D}
F(a,b,c,d,f,g)= f1(a,b,d)+f2(d,f,g)+f3(b,c,f)
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Bayesian Networks (Pearl, 1988)
P(S, C, B, X, D) = P(S) P(C|S) P(B|S) P(X|C,S) P(D|C,B)
lung Cancer
Smoking
X-ray
Bronchitis
DyspnoeaP(D|C,B)
P(B|S)
P(S)
P(X|C,S)
P(C|S)
Θ) (G,BN =
CPD:C B P(D| C,B)0 0 0.1 0.90 1 0.7 0.31 0 0.8 0.21 1 0.9 0.1
Belief Updating:P (lung cancer=yes | smoking=no, dyspnoea=yes ) = ?
MPE = find argmax P(S)· P(C|S)· P(B|S)· P(X|C,S)· P(D|C,B)
A graphical model (X,D,F): X = {X1,…Xn} variables D = {D1, … Dn} domains F = {f1,…,fr} functions
(constraints, CPTS, CNFs …)
Operators: combination : Sum, product, join Elimination: projection, sum, max/min
Tasks: Belief updating: ΣX\Y∏j Pi
MPE\MAP: maxX∏j Pj
Marginal MAP: maxY ∑𝑋𝑋\Y∏j Pj
CSP: ∏X ×j Cj
Max-CSP: minXΣj Fj
Graphical Models
)( : CAFfi +==
A
D
B C
E
F
All these tasks are NP-hard exploit problem structure identify special cases approximate
A C F P(F|A,C)0 0 0 0.140 0 1 0.960 1 0 0.400 1 1 0.601 0 0 0.351 0 1 0.651 1 0 0.721 1 1 0.68
Conditional ProbabilityTable (CPT)
Primal graph(interaction graph)
A C Fred green blueblue red redblue blue green
green red blue
Relation
(𝐴𝐴⋁𝐶𝐶⋁𝐹𝐹)
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Example Domains for Graphical Models
Natural Language Processing− Information extraction, semantic parsing, translation, summarization, ...
Computer Vision− Object recognition, scene analysis, segmentation, tracking, ...
Computational Biology− Pedigree analysis, protein folding / binding / design, sequence matching,
...
Networks− Webpage link analysis, social networks, communications, citations, ...
Robotics− Planning & decision making, ...
...CMU 2016
Max-product = min-sum max (A,B)
αα ϕϕ log' −=∑=α
αϕxxxAB
BA
x 'minarg,
*
∏=α
αϕxxxAB
BA
x,
* maxarg
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Tree-solving is easy
Belief updating (sum-prod)
MPE (max-prod)
CSP – consistency (projection-join)
#CSP (sum-prod)
P(X)
P(Y|X) P(Z|X)
P(T|Y) P(R|Y) P(L|Z) P(M|Z)
)(XmZX
)(XmXZ
)(ZmZM)(ZmZL
)(ZmMZ)(ZmLZ
)(XmYX
)(XmXY
)(YmTY
)(YmYT
)(YmRY
)(YmYR
Trees are processed in linear time and memoryCMU 2016
Inference vs conditioning-search Inferenceexp(w*) time/space
A
D
B C
E
F0 1 0 1 0 1 0 1
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0 1 0 1 0 1 0 10 1 01 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 01 0 1 0 1 0 1
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0 1 0 1
E
C
F
D
B
A 0 1 SearchExp(n) timeO(n) space
E K
F
L
H
C
BA
M
G
J
DABC
BDEF
DGF
EFH
FHK
HJ KLM
A=yellow A=green
B=blue B=red B=blueB=green
CK
G
LD
F H
M
J
EACB K
G
LD
F H
M
J
E
CK
G
LD
F H
M
J
EC
K
G
LD
F H
M
J
EC
K
G
LD
F H
M
J
ESearch+inference:Space: exp(w)Time: exp(w+c(w))
w: usercontrolled
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Inference vs conditioning-search Inferenceexp(w*) time/space
A
D
B C
E
F0 1 0 1 0 1 0 1
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0 1 0 1 0 1 0 10 1 01 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 01 0 1 0 1 0 1
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0 1 0 1
E
C
F
D
B
A 0 1 SearchExp(w*) timeO(w*) space
E K
F
L
H
C
BA
M
G
J
DABC
BDEF
DGF
EFH
FHK
HJ KLM
A=yellow A=green
B=blue B=red B=blueB=green
CK
G
LD
F H
M
J
EACB K
G
LD
F H
M
J
E
CK
G
LD
F H
M
J
EC
K
G
LD
F H
M
J
EC
K
G
LD
F H
M
J
ESearch+inference:Space: exp(q)Time: exp(q+c(q))
q: usercontrolled
Context minimal AND/OR search graph
18 AND nodes
AOR0ANDBOR
0ANDOR E
OR F FAND 0 1
AND 0 1
C
D D
0 1
0 1
1EC
D D0 1
1B
0E
F F0 1
C1
EC
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Outline Graphical models, Queries, Algorithms Inference Algorithms: bucket-elimination Bounded Inference: a) mini-bucket, b) cost-shifting AND/OR search spaces and AND/OR BnB Evaluation, Software Conclusions
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• Solving MAP by Inference:• Non-serial Dynamic programming• The induced-width/treewidth
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22
Computing the Optimal Cost SolutionThis image cannot currently be displayed.
0=emin
A
D E
CBB C
ED
Variable Elimination
bcde ,,,min0=
f(a,b)+f(a,c)+f(a,d)+f(b,c)+f(b,d)+f(b,e)+f(c,e)OPT =
f(a,c)+f(c,e) +c
min
),,,( ecdaCB→λ
f(a,b)+f(b,c)+f(b,d)+f(b,e)b
mind
min f(a,d) +
Combination
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Bucket Elimination
Elimination/Combination operators
bucket B:
bucket C:
bucket D:
bucket E:
bucket A:
Algorithm elim-opt [Dechter, 1996]Non-serial Dynamic Programming [Bertele & Briochi, 1973]
OPT =
F*=OPT
B
C
D
E
A
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Complexity of Bucket Elimination
Elimination/Combination operators
bucket B:
bucket C:
bucket D:
bucket E:
bucket A:
Algorithm elim-opt [Dechter, 1996]Non-serial Dynamic Programming [Bertele & Briochi, 1973]
OPT =
F*=OPT
B
C
D
E
A
exp(w*=4)“induced width” (max clique size)
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Generating the Optimal Assignment
C:
E:
B:
D:
A:
Return:
Time and space exponential in the induced-width / treewidth
O(n𝒌𝒌𝒘𝒘∗+𝟏𝟏)
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Complexity of Bucket Elimination
ddw ordering along graph of widthinduced the)(* −The effect of the ordering:
4)( 1* =dw 2)( 2
* =dw“Moral” graph
A
D E
CB
B
C
D
E
A
E
D
C
B
A
Bucket Elimination is time and space
r = number of functions
Finding the smallest induced width is hard!
Bucket Elimination
A
f(A,B)B
f(B,C)C f(B,F)F
f(A,G) f(F,G)
Gf(B,E) f(C,E)
Ef(A,D) f(B,D) f(C,D)
D
hG (A,F)
hF (A,B)
hB (A)
hE (B,C)hD (A,B,C)
hC (A,B)
A B
CD
E
F
G
A
B
C F
GD E
Ordering: (A, B, C, D, E, F, G)
=+++++
+++++
),(),(),(),(),(),(),(),(),(),(),(max ,,,,,,
gffgaffbfecfebfdcfdbfdafcbfdafbafgfedcba
MessagesFinding max Assignment
argmax
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Outline Graphical models, Queries, Algorithms Inference Algorithms: bucket-elimination Bounded Inference: a) mini-bucket, b) cost-shifting AND/OR search spaces and AND/OR BnB Evaluation, Software Conclusions
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Bounding approximations: Generating a heuristic evaluation function
• The mini-bucket scheme• The cost-shifting or re-parameterization scheme• Combining the two
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Mini-Bucket ApproximationSplit a bucket into mini-buckets => bound complexity
bucket (X) =
)()()O(e :decrease complexity lExponentia n rnr eOeO −+→
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Mini-Bucket Eliminationmini-buckets
bucket A:
bucket E:
bucket D:
bucket C:
bucket B:
L = lower bound
[Dechter and Rish, 1997; 2003]
.A
B C
D E
A
B D
CE
B’
Model relaxation: [Kask et al., 2001]
[Geffner et al., 2007][Choi et al., 2007]
[Johnson et al. 2007]CMU 2016
MBE-MPE(i) vs BE-MPE
Input: i – max number of variables allowed in a mini-bucket Output: [lower bound (P of a suboptimal solution), upper bound]
MBE-MPE(3) versus BE-MPE
A:
E:
D:
C:
B:
L = lower bound
A:
E:
D:
C:
B:
OPT
Max variables ina mini-bucket
3
3
3
2
1
[Dechter and Rish, 1997; 2003]CMU 2016
Mini-Bucket Decoding
C:
E:
B:
D:
A:
Return:
[Dechter and Rish, 1997; 2003]
Greedy configuration = upper bound L = lower bound
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Properties of MBE(i) Complexity: O(n exp(i)) time and O(exp(i)) space Yields a lower bound and an upper bound Accuracy estimatable by the upper/lower (U/L) bounds Possible use of mini-bucket approximations:
− As anytime algorithms− As heuristics in search
Other tasks (similar mini-bucket approximations):− Belief updating, Marginal MAP, MEU, WCSP, MaxCSP
[Dechter and Rish, 1997], [Liu and Ihler, 2011], [Liu and Ihler, 2013]
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Bounding from above and belowRelaxation upper bound by mini-bucket
Consistent solutions ( greedy search)
MAP
i=10 i=20
i=5
i=2
Relaxation provides upper boundAny configuration: lower bound
How to partition given an i-bound?• Scope-based greedy• Content-based greedy• (Rollon & Dechter 2010)
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Outline Graphical models, Queries, Algorithms Inference Algorithms: bucket-elimination Bounded Inference: a) mini-bucket, b) cost-shifting AND/OR search spaces and AND/OR BnB Evaluation, Software Conclusions
CMU 2016
Cost-Shifting
+
= 0 + 6
A B C f(A,B,C)
b b b 12
b b g 6
b g b 0
b g g 6
g b b 6
g b g 0
g g b 6
g g g 12
A B f(A,B)
b b 6 + 3
b g 0 - 1
g b 0 + 3
g g 6 - 1
B C f(B,C)
b b 6 - 3
b g 0 - 3
g b 0 + 1
g g 6 + 1
(Reparameterization)
B λ(B)
b 3
g -1
B B
Modify the individual functions
- but -
keep the sum of functions unchanged
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Dual Decomposition
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Dual Decomposition
Reparameterization:
(Convex dual: linear programming relaxation)
Bound solution using decomposed optimization Solve independently: optimistic bound
Tighten the bound by reparameterization− Enforce lost equality constraints via Lagrange multipliers
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Dual Decomposition
Many names for the same class of bounds:− Dual decomposition [Komodakis et al. 2007]
− TRW, MPLP [Wainwright et al. 2005, Globerson & Jaakkola 2007]
− Soft arc consistency [Cooper & Schiex 2004]
− Max-sum diffusion [Warner 2007]
Reparameterization:
(Convex dual: linear programming relaxation)CMU 2016
Various Update Schemes Can use any decomposition updates
(message passing, subgradient, augmented, etc.)
FGLP: Update the original factors
JGLP: Update clique function of the join graph
MBE-MM Update within each bucket only Apply cost-shifting within each bucket only
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Factor graph Linear Programming Update the original factors (FGLP)
Tighten all factors over over xi simultaneously Compute max-marginals & update:
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MBE-MM: MBE with moment matching
A
B C
D
E
P(A)
P(B|A) P(C|A)
P(E|B,C)
P(D|A,B)
Bucket B
Bucket C
Bucket D
Bucket E
Bucket A
P(B|A) P(D|A,B)P(E|B,C)
P(C|A)
E = 0
P(A)
maxB∏
hB (A,D)
MPE* is an upper bound on MPE --UGenerating a solution yields a lower bound--L
maxB∏
hD (A)
hC (A,E)
hB (C,E)
hE (A)W=2
m11
m12
m11,m12- moment-matchingmessages
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Anytime Approximation
Can tighten the bound in various ways
− Cost-shifting (improve consistency between cliques)
− Increase i-bound (higher order consistency)
Simple moment-matching step improves bound significantly
MBEMBE+MMMBE+JG
i = 1i = 3
i = 5i = 7
i = 9i = 11
i = 13i = 15
pedigree20
MBEMBE+MMMBE+JG
i = 1
i = 3
i = 5
i = 7i = 9 i = 11
i = 13
pedigree37
i = 1 + MM
i = 15 + MM
-124
-122
-120
-118
-116
-114
-112
-110
-108
-106
-340
-330
-320
-310
-300
-280
-270
-290
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Iterative tightening as bounding schemes
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Outline Graphical models, Queries, Algorithms Inference Algorithms: bucket-elimination Bounded Inference: a) mini-bucket, b) cost-shifting Generating heuristics using mini-bucket elimination AND/OR search spaces and AND/OR BnB Evaluation, Software Conclusions
CMU 2016
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Properties of the Heuristic MB heuristic is monotone, admissible Computed in linear time
IMPORTANT− Heuristic strength can vary by MB(i) & message passing− Higher i-bound → more pre-processing
→ more accurate heuristic → less search
Allows controlled trade-off between pre-processing and search
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Outline Graphical models, Queries , Algorithms Inference Algorithms Bounded Inference: mini-bucket, cost-shifting AND/OR search spaces and AND/OR BnB Evaluation, Software Conclusions
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Classic OR Search Space
( )∑=
Χ=9
1
* mini
iX fF
A
E
C
B
F
D
0 1 0 1 0 1 0 1
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0 1 0 1
C
D
F
E
B
A 0 1
Objective function:
A B f10 0 20 1 01 0 11 1 4
A C f20 0 30 1 01 0 01 1 1
A E f30 0 00 1 31 0 21 1 0
A F f40 0 20 1 01 0 01 1 2
B C f50 0 00 1 11 0 21 1 4
B D f60 0 40 1 21 0 11 1 0
B E f70 0 30 1 21 0 11 1 0
C D f80 0 10 1 41 0 01 1 0
E F f90 0 10 1 01 0 01 1 2
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The AND/OR Search TreeA
E
C
B
F
D
A
D
B
EC
F
Pseudo tree (Freuder & Quinn85)
OR
AND
OR
AND
OR
OR
AND
AND
A
0
B
0
E
F F
0 1 0 1
0 1
C
D D
0 1 0 1
0 1
1
E
F F
0 1 0 1
0 1
C
D D
0 1 0 1
0 1
1
B
0
E
F F
0 1 0 1
0 1
C
D D
0 1 0 1
0 1
1
E
F F
0 1 0 1
0 1
C
D D
0 1 0 1
0 1
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The AND/OR Search TreeA
E
C
B
F
D
A
D
B
EC
F
Pseudo tree
OR
AND
OR
AND
OR
OR
AND
AND
A
0
B
0
E
F F
0 1 0 1
0 1
C
D D
0 1 0 1
0 1
1
E
F F
0 1 0 1
0 1
C
D D
0 1 0 1
0 1
1
B
0
E
F F
0 1 0 1
0 1
C
D D
0 1 0 1
0 1
1
E
F F
0 1 0 1
0 1
C
D D
0 1 0 1
0 1
A solution subtree is (A=0, B=1, C=0, D=0, E=1, F=1)CMU 2016
AND/OR vs. OR Spaces
0 1 0 1 0 1 0 1
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0 1 0 1
C
D
F
E
B
A 0 1
OR
AND
OR
AND
OR
OR
AND
AND
A
0
B
0
E
F F
0 1 0 1
0 1
C
D D
0 1 0 1
0 1
1
E
F F
0 1 0 1
0 1
C
D D
0 1 0 1
0 1
1
B
0
E
F F
0 1 0 1
0 1
C
D D
0 1 0 1
0 1
1
E
F F
0 1 0 1
0 1
C
D D
0 1 0 1
0 1
A
E
C
B
F
D
A
D
B
EC
F54 nodes
126 nodes
Time O(𝒏𝒏𝒌𝒌𝒉𝒉)Space O(n)height is bounded by (log n) w*
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Weighted AND/OR Search Tree
( ) ( )∑=
Χ=Χ9
1min
iiX ff
A
E
C
B
F
D
Objective function:
A
D
B
EC
F
A
0
B
0
E
F F
0 1 0 1
OR
AND
OR
AND
OR
OR
AND
AND 0 1
C
D D
0 1 0 1
0 1
1
E
F F
0 1 0 1
0 1
C
D D
0 1 0 1
0 1
5 6 4 2 3 0 2 2
5 2 0 2
5 2 0 2
3 3
6
5
5
5
3 1 3 5
2
2 4 1 0 3 0 2 2
2 0 0 2
2 0 0 2
4 1
5
5 4 1 3
0
1
w(A,0) = 0 w(A,1) = 0
Node Value(bottom-up evaluation)
OR – minimizationAND – summation
A B f10 0 20 1 01 0 11 1 4
A C f20 0 30 1 01 0 01 1 1
A E f30 0 00 1 31 0 21 1 0
A F f40 0 20 1 01 0 01 1 2
B C f50 0 00 1 11 0 21 1 4
B D f60 0 40 1 21 0 11 1 0
B E f70 0 30 1 21 0 11 1 0
C D f80 0 10 1 41 0 01 1 0
E F f90 0 10 1 01 0 01 1 2
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Weighted AND/OR Search Tree
A
0
B
0
E
F F
0 1 0 1
OR
AND
OR
AND
OR
OR
AND
AND 0 1
C
D D
0 1 0 1
0 1
1
E
F F
0 1 0 1
0 1
C
D D
0 1 0 1
0 1
A
E
C
B
F
D
AB f10 0 201 010 111 4
A C f20 0 30 1 01 0 01 1 1
A E f30 0 00 1 31 0 21 1 0
A F f40 0 20 1 01 0 01 1 2
B C f50 0 00 1 11 0 21 1 4
B D f60 0 40 1 21 0 11 1 0
B E f70 0 30 1 21 0 11 1 0
C D f80 0 10 1 41 0 01 1 0
E F f90 0 10 1 01 0 01 1 2
5 6 4 2 3 0 2 2
5 2 0 2
5 2 0 2
3 3
6
5
5
5
3 1 3 5
2
2 4 1 0 3 0 2 2
2 0 0 2
2 0 0 2
4 1
5
5 4 1 3
0
0
1
B
0
E
F F
0 1 0 1
0 1
C
D D
0 1 0 1
0 1
1
E
F F
0 1 0 1
0 1
C
D D
0 1 0 1
0 1
5 6 4 2 1 2 0 4 2 4 1 0 1 2 0 4
6 4
7
7
5 2 1 0 2 0 1 0
5 2 1 0 2 0 1 0
4 2 4 0
0
0 2 5 2 2 5 3 0
1 4
A
D
B
EC
F
AND node = Combination operator (summation)
( )∑ =
9
1min :Goal
i iX Xf
OR node = Marginalization operator (minimization)CMU 2016
Pseudo-Trees(Freuder 85, Bayardo 95, Bodlaender and Gilbert, 91)
(a) Graph
4 61
3 2 7 5
(b) DFS treedepth=3
(c) pseudo- treedepth=2
(d) Chaindepth=6
4 6
1
3
2 7
5 2 7
1
4
3 5
6
4
6
1
3
2
7
5
m <= w* log n
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From Search Trees to Search Graphs
Any two nodes that root identical sub-trees or sub-graphs can be merged
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From Search Trees to Search Graphs
Any two nodes that root identical sub-trees or sub-graphs can be merged
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From AND/OR TreeA
D
B C
E
F
G H
J
K
A
D
B
CE
F
G
H
J
KAOR
0AND 1
BOR B
0AND 1 0 1
EOR C E C E C E C
OR D F D F D F D F D F D F D F D F
AND
AND 0 10 1 0 10 1 0 10 1 0 10 1
OR
OR
AND
AND
0
G
H H
0 1 0 1
0 1
1
G
H H
0 1 0 1
0 1
0
J
K K
0 1 0 1
0 1
1
J
K K
0 1 0 1
0 1
0
G
H H
0 1 0 1
0 1
1
G
H H
0 1 0 1
0 1
0
J
K K
0 1 0 1
0 1
1
J
K K
0 1 0 1
0 1
0
G
H H
0 1 0 1
0 1
1
G
H H
0 1 0 1
0 1
0
J
K K
0 1 0 1
0 1
1
J
K K
0 1 0 1
0 1
0
G
H H
0 1 0 1
0 1
1
G
H H
0 1 0 1
0 1
0
J
K K
0 1 0 1
0 1
1
J
K K
0 1 0 1
0 1
0
G
H H
0 1 0 1
0 1
1
G
H H
0 1 0 1
0 1
0
J
K K
0 1 0 1
0 1
1
J
K K
0 1 0 1
0 1
0
G
H H
0 1 0 1
0 1
1
G
H H
0 1 0 1
0 1
0
J
K K
0 1 0 1
0 1
1
J
K K
0 1 0 1
0 1
0
G
H H
0 1 0 1
0 1
1
G
H H
0 1 0 1
0 1
0
J
K K
0 1 0 1
0 1
1
J
K K
0 1 0 1
0 1
0
G
H H
0 1 0 1
0 1
1
G
H H
0 1 0 1
0 1
0
J
K K
0 1 0 1
0 1
1
J
K K
0 1 0 1
0 1
CMU 2016
An AND/OR GraphAOR
0AND 1
BOR B
0AND 1 0 1
EOR C E C E C E C
OR D F D F D F D F D F D F D F D F
AND
AND 0 10 1 0 10 1 0 10 1 0 10 1
OR
OR
AND
AND
0
G
H H
0 1 0 1
0 1
1
G
H H
0 1 0 1
0 1
0
J
K K
0 1 0 1
0 1
1
J
K K
0 1 0 1
0 1
A
D
B
CE
F
G
H
J
K
A
D
B C
E
F
G H
J
K
CMU 2016
Merging Based on Context One way of recognizing nodes that can be
merged (based on graph structure)context(X) = ancestors of X in the pseudo tree
that are connected to X, or to descendants of X
[ ]
[A]
[AB]
[AE][BC]
[AB]
A
D
B
EC
Fpseudo tree
A
E
C
B
F
D
A
E
C
B
F
D
CMU 2016
AND/OR Search Graph
( ) ( )∑=
Χ=Χ9
1min
iiX ff
A
E
C
B
F
D
Objective function:
A
D
B
EC
F
A B f10 0 20 1 01 0 11 1 4
A C f20 0 30 1 01 0 01 1 1
A E f30 0 00 1 31 0 21 1 0
A F f40 0 20 1 01 0 01 1 2
B C f50 0 00 1 11 0 21 1 4
B D f60 0 40 1 21 0 11 1 0
B E f70 0 30 1 21 0 11 1 0
C D f80 0 10 1 41 0 01 1 0
E F f90 0 10 1 01 0 01 1 2
context minimal graph
AOR
0AND
BOR
0AND
OR E
OR
AND
AND 0 1
C
D D
0 1 0 1
0 1
1
E
F F
0 1 0 1
0 1
C
D D
0 1 0 1
0 1
1
B
0
E
F F
0 1 0 1
0 1
C
0 1
1
E
0 1
C
0 1
B C Value0 00 11 01 1
Cache table for D
[]
[A]
[AB]
[AE]
[AB]
[BC]
CMU 2016
How Big Is The Context?Theorem: The maximum context size for a
pseudo tree is equal to the treewidth of the graph along the pseudo tree.
C
HK
D
M
F
G
A
B
E
J
O
L
N
P
[AB]
[AF][CHAE]
[CEJ]
[CD]
[CHAB]
[CHA]
[CH]
[C]
[ ]
[CKO]
[CKLN]
[CKL]
[CK]
[C]
(C K H A B E J L N O D P M F G)
B A
C
E
F G
H
J
D
K M
L
N
OP
max context size = treewidth
CMU 2016
69
All Four Search Spaces
Full OR search tree
126 nodes
0 1 0 1 0 1 0 1
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0 1 0 1 0 1 0 10 1 0 1 0 1 0 1 0 1 0 1 0 1 0 10 1 0 1 0 1 0 1 0 1 0 1 0 1 0 10 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0 1 0 1
CD
FE
BA 0 1
Full AND/OR search tree
54 AND nodes
AOR0ANDBOR
0AND
OR E
OR F F
AND 0 1 0 1
AND 0 1
C
D D
0 1 0 1
0 1
1
E
F F
0 1 0 1
0 1
C
D D
0 1 0 1
0 1
1
B
0
E
F F
0 1 0 1
0 1
C
D D
0 1 0 1
0 1
1
E
F F
0 1 0 1
0 1
C
D D
0 1 0 1
0 1
Context minimal OR search graph
28 nodes
0 1 0 1 0 1 0 1
0 1 0 1 0 1 0 1
0 1
0 1 0 1
0 1 0 1
CD
FE
BA 0 1
Context minimal AND/OR search graph
18 AND nodes
AOR0ANDBOR
0ANDOR E
OR F FAND 0 1
AND 0 1
C
D D
0 1
0 1
1
EC
D D
0 1
1
B
0
E
F F
0 1
C1
EC
A
E
C
B
F
D
Any query is best computedOver the c-minimal AO space
AND/OR graph OR graph
Space O(n kw*) O(n kpw*)
Time O(n kw*) O(n kpw*)
Computes any query:• Constraint satisfaction• Optimization• Weighted counting
A
D
B
EC
F
CMU 2016
The impact of the pseudo-tree C
0K
0H
0L
0 1N N
0 1 0 1
F F F
1 1
0 1 0 1F
G0 1
1
A0 1B B
0 1 0 1
E EE E0 1 0 1
J JJ J0 1 0 1
A0 1B B
0 1 0 1
E EE E0 1 0 1
J JJ J0 1 0 1
G0 1
G0 1
G01
M0 1
M0 1
M01
M01
P0 1
P0 1
O01
O01
O01
O01
L0 1N N
0 1 0 1
P0 1
P01
O01
O01
O01
O01
D0 1
D0 1
D0 1
D0 1
K0
H0
L0 1
N N0 1 0 1
1 1A
0 1B B
0 1 0 1
E EE E0 1 0 1
J JJ J0 1 0 1
A0 1B B
0 1 0 1
E EE E0 1 0 1
J JJ J0 1 0 1
P0 1
P0 1
O01
O01
O01
O01
L0 1
N N0 1 0 1
P0 1
P0 1
O01
O01
O01
O01
D01
D0 1
D0 1
D0 1
B A
C
E
F G
HJ
D
K ML
N
OP
C
HK
D
M
F
G
A
B
E
J
O
L
N
P[AB]
[AF][CHAE]
[CEJ]
[CD]
[CHAB]
[CHA]
[CH]
[C]
[ ]
[CKO]
[CKLN]
[CKL]
[CK]
[C]
(C K H A B E J L N O D P M F G)
(C D K B A O M L N P J H E F G)
F [AB]
G [AF]
J [ABCD]
D [C]
M [CD]
E [ABCDJ]
B [CD]
A [BCD]
H [ABCJ]
C [ ]
P [CKO]
O [CK]
N [KLO]
L [CKO]
K [C]
C
0 1D
0 1
N0 1
D0 1
K0 1
K0 1
B0 1
B0 1
B0 1
B0 1
M0 1
M0 1
M0 1
M0 1
A0 1
A0 1
A0 1
A0 1
A0 1
A0 1
A0 1
A0 1
F0 1
F0 1
F0 1
F0 1
G0 1
G0 1
G0 1
G0 1
J0 1
J0 1
J0 1
J0 1
J0 1
J0 1
J0 1
J0 1
J0 1
J0 1
J0 1
J0 1
J0 1
J0 1
J0 1
J0 1
E1
E0 1
E0 1
E0 1
E0 1
E0 1
E0 1
E0 1
E0 1
E0 1
E0 1
E0 1
E0 1
E0 1
E0 1
E0 1
E0 1
E0 1
E0 1
E0 1
E0 1
E0 1
E0 1
E0 1
E0 1
E0 1
E0 1
E0 1
E0 1
E0 1
E0 1
E0 1
H0 1
H0 1
H0 1
H0 1
H0 1
H0 1
H0 1
H0 1
H0 1
H0 1
H0 1
H0 1
H0 1
H0 1
H0 1
H0 1
O0 1
O0 1
O0 1
O0 1
L0 1
L0 1
L0 1
L0 1
L0 1
L0 1
L0 1
L0 1
P0 1
P0 1
P0 1
P0 1
P0 1
P0 1
P0 1
P0 1
N0 1
N0 1
N0 1
N0 1
N0 1
N0 1
N0 1
What is a good pseudo-tree?How to find a good one?
W=4,h=8
W=5,h=6
Min-Fill(Kjaerulff90)
HypergraphPartitioning
(h-Metis)
• Optimization• Choose pseudo-tree with a minimal search graph• But determinism is unpredictable• And pruning by BnB is even more unpredictable
CMU 2016
Outline Graphical models, Queries, Algorithms Inference Algorithms: bucket-elimination Bounded Inference: a) mini-bucket, b) cost-shifting AND/OR search spaces andAND/OR Branch & Bound Evaluation, Software Summary and future work
CMU 2016
Basic Heuristic Search SchemesHeuristic function f(xp) computes a lower bound on the
best extension of xp and can be used to guide a heuristic search algorithm. We focus on:
1. Branch-and-BoundUse heuristic function f(xp) to prune the depth-first search treeLinear space
2. Best-First SearchAlways expand the node with the highest heuristic value f(xp) needs lots of memory
f ≤ L
L
CMU 2016
Classic Branch-and-Bound
n
g(n)
h(n) - under-estimatesOptimal cost below n
f(n) = g(n) + h(n)f(n) = lower bound
Prune if f(n) ≥ UB
(UB) Upper Bound = best solution so far
Each node is a COP subproblem(defined by current conditioning)
CMU 2016
Mini-bucket Heuristics for BB search( Kask and dechterAIJ, 2001, Kask, Dechter and Marinescu 2004, 2005, 2009, Otten 2012)
…
B
A
E
D
C
L
B: P(E|B,C) P(D|A,B)P(B|A)
A:
E:
D:
C: P(C|A) hB(E,C)
hB(D,A)
hC(E,A)
P(A) hE(A) hD(A)
f(a,e,D) = P(a)·hB(D,a)· hC(e,a)
A
B C
DE
a
e
h(x) computed by MB(i)before or during search
CMU 2016
AND/OR Branch-and-Bound
Decompositionof independentsubproblems
Cache tablefor F (indepen-
dent of A)
Prune basedon current best
solution andheuristic estimate
(mini-bucket heuristic).
v=10+2=12
h=14v=10 2
B0011
E0101
cost106......
[Marinescu & Dechter, AIJ'09]CMU 2016
MAP: Anytime, Branch & Bound Best-First, Recursive Best-First Anytime:Breadth-Rotate AND/OR BnB (2011)Weighted heuristic AND/OR search (2014)
Finding m-best solutions Marginal map
CMU 2016
Empirical Evaluation(exact)
Grid and Pedigree benchmarks; Time limit 1 hour.
CMU 2016
Outline Graphical models, Queries, Algorithms Inference Algorithms: bucket-elimination Bounded Inference: a) mini-bucket, b) cost-shifting AND/OR search spaces and AND/OR BnB Evaluation, Software Conclusion
CMU 2016
DAOOPT: Improving AND/OR Branch-and-Bound for Graphical Models
(also won 2nd place uai 2014 as is)Lars Otten, Alexander Ihler,Kalev Kask, Rina Dechter
Dept. of Computer ScienceUniversity of California, Irvine
PASCAL 2012 Inference Challenge
CMU 2016
Our Solvers are being used:
Superlink online, software for linkage analysis (Geiger et. Al)
Figaro, probabilistic language (Avi Pfeffer)
CMU 2016
Software
aolib− http://graphmod.ics.uci.edu/group/Software(standalone AOBB, AOBF solvers)
daoopt− https://github.com/lotten/daoopt(distributed and standalone AOBB solver)
CMU 2016
UAI Probabilistic Inference Competitions
2006
2008
2011
2014
MPE/MAP MMAP
(aolib)
(aolib)
(daoopt)
(daoopt) (daoopt) (merlin)
CMU 2016
Conclusion Combining two “universal” lower-bounds scheme
(mpe/map) The mini-bucket scheme cost-shifting or re-parameterization, schemes
Exploiting bounds as heuristics in AND/OR search Yields BRAOBB that wins first place Pascal
competition 2011, over many benchmarks Future work: Dynamic heuristics and search
spaces
CMU 2016
Thank you
For publication see: http://www.ics.uci.edu/~dechter/publications.html
Kalev KaskIrina RishBozhena BidyukRobert MateescuRadu MarinescuVibhav GogateEmma RollonLars OttenNatalia FlerovaAndrew GelfandWilliam LamJunkyu LeeCMU 2016
![Outcome Optimization - Duke Universityconitzer/thesis_ch3.pdfinstance, in combinatorial auctions, the winner determination problem is NP-complete [Rothkopf et al., 1998] (even to approximate](https://img.dokumen.tips/doc/110x75/5f3b9fa8bdc8927a0b28797f/outcome-optimization-duke-university-conitzerthesisch3pdf-instance-in-combinatorial.jpg)
