modern exact and approximate combinatorial optimization ...dechter/talks/cmu-v3.pdfmodern exact and...
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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
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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
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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)
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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
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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
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Inference vs conditioning-search Inferenceexp(w*) time/space
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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!
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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
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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
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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
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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
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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
CMU 2016
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Factor graph Linear Programming Update the original factors (FGLP)
Tighten all factors over over xi simultaneously Compute max-marginals & update:
CMU 2016
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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
CMU 2016
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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
CMU 2016
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Iterative tightening as bounding schemes
CMU 2016
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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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CMU 2016
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CMU 2016
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CMU 2016
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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
CMU 2016
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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
CMU 2016
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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
CMU 2016
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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
CMU 2016
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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
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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*
CMU 2016
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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
CMU 2016
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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
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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
CMU 2016
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From Search Trees to Search Graphs
Any two nodes that root identical sub-trees or sub-graphs can be merged
CMU 2016
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From Search Trees to Search Graphs
Any two nodes that root identical sub-trees or sub-graphs can be merged
CMU 2016
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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
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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
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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
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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
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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
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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
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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
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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 andAND/OR Branch & Bound Evaluation, Software Summary and future work
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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
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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)
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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
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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
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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
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Empirical Evaluation(exact)
Grid and Pedigree benchmarks; Time limit 1 hour.
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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
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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
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Our Solvers are being used:
Superlink online, software for linkage analysis (Geiger et. Al)
Figaro, probabilistic language (Avi Pfeffer)
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Software
aolib− http://graphmod.ics.uci.edu/group/Software(standalone AOBB, AOBF solvers)
daoopt− https://github.com/lotten/daoopt(distributed and standalone AOBB solver)
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UAI Probabilistic Inference Competitions
2006
2008
2011
2014
MPE/MAP MMAP
(aolib)
(aolib)
(daoopt)
(daoopt) (daoopt) (merlin)
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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
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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