Reasoning with Deterministic and Probabilistic graphical models …frossi/rina2.pdf · 2012. 5....

Reasoning with Deterministic and Probabilistic graphical models

Class 2: Inference in Constraint Networks

Rina Dechter

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Road Map

n Graphical models

n Constraint networks Model n Inference

n Search

n Probabilistic Networks

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A B red green red yellow green red green yellow yellow green yellow red

Example: map coloring Variables - countries (A,B,C,etc.) Values - colors (red, green, blue) Constraints:

C

A

B

D E

F

G

A Constraint Networks

A

B

E

G

D F

C

Constraint graph

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etc. ,ED D, AB,A ≠≠≠

Example: map coloring Variables - countries (A,B,C,etc.) Values - colors (e.g., red, green, yellow) Constraints:

Constraint Satisfaction Tasks

Are the constraints consistent? Find a solution, find all solutions Count all solutions Find a good (optimal) solution

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etc. ,ED D, AB,A ≠≠≠

Constraint Network

n A constraint network is: R=(X,D,C)

n X variables

n D domain

n C constraints

n R expresses allowed tuples over scopes n A solution is an assignment to all variables that satisfies all constraints (join of all relations). n Tasks: consistency?, one or all solutions, counting, optimization

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},...,{ 1 nXXX =},...{},,...,{ 11 kin vvDDDD ==

),(},...{ 1

iii

t

RSCCCC

=

=

Crossword puzzle

n Variables: x1, …, x13 n Domains: letters n Constraints: words from

{HOSES, LASER, SHEET, SNAIL, STEER, ALSO, EARN, HIKE, IRON, SAME, EAT, LET, RUN, SUN, TEN, YES, BE, IT, NO, US}

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The Queen problem

The network has four variables, all with domains Di = {1, 2, 3, 4}. (a) The labeled chess board. (b) The constraints between variables.

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Constraint’s representations

n Relation: allowed tuples n Algebraic expression: n Propositional formula: n Semantics: by a relation

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YXYX ≠≤+ ,102

cba ¬→∨ )(

312231ZYX

Partial solutions

Not all consistent instantiations are part of a solution: (a) A consistent instantiation that is not part of a solution. (b) The placement of the queens corresponding to the solution (2, 4, 1, 3). (c) The placement of the queens corresponding to the solution (3, 1, 4, 2).

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Constraint Graphs (primal)

Queen problem

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Constraint Graphs: Primal, Dual and Hypergraphs

A (primal) constraint graph: a node per variable arcs connect constrained variables. A dual constraint graph: a node per constraint’s scope, an arc connect nodes sharing variables =hypergraph

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Graph Concepts Reviews: Primal, Hyper and Dual Graphs

n A hypergraph n Dual graphs n A primal graph

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Propositional Satisfiability

ϕ = {(¬C), (A v B v C), (¬A v B v E), (¬B v C v D)}.

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Constraint graphs of 3 instances of the Radio frequency assignment problem in CELAR’s benchmark

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Operations With Relations

n Intersection

n Union

n Difference

n Selection

n Projection

n Join

n Composition

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n Join : n Logical AND:

Local Functions

Combination

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=

gf

gf ∧

= ∧

Global View of the Problem

C1 C2 Global View

What about counting?

Number of true tuples Sum over all the tuples

true is 1 false is 0 logical AND?

TASK

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=

Road Map

n Graphical models

n Constraint networks Model n Inference

n Variable elimination: n Tree-clustering

n Constraint propagation

n Search

n Probabilistic Networks

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Bucket Elimination Adaptive Consistency (Dechter & Pearl, 1987)

Bucket E: E ≠ D, E ≠ C Bucket D: D ≠ A Bucket C: C ≠ B Bucket B: B ≠ A Bucket A:

A ≠ C

contradiction

=

D = C

B = A

= ≠

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widthinduced -*

*

w ))exp(w O(n :Complexity

The Idea of Elimination

3 value assignment

D

B

C RDBC

eliminating E

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project and join E variableEliminate

⇔

=∏ ECDBC EBEDDBC RRRR

A

E D C B

E

A

D C B

|| RDBE , || RE

|| RDB

|| RDCB || RACB || RAB

RA

RCBE

Bucket Elimination Adaptive Consistency (Dechter & Pearl, 1987)

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d ordering along widthinduced -(d) ,

*

*

w(d)))exp(w O(n :Complexity

≠

≠≠

≠

≠

≠

E

D

A

C

B

}2,1{

}2,1{}2,1{

}2,1{ }3,2,1{

:)(AB :)(BC :)(AD :)(

BE C,E D,E :)(

ABucketBBucketCBucketDBucketEBucket

≠≠≠

≠≠≠

:)(EB :)(

EC , BC :)(ED :)(

BA D,A :)(

EBucketBBucketCBucketDBucketABucket

≠≠≠

≠≠≠

Adaptive Consistency, Bucket-elimination

Initialize: partition constraints into For i=n down to 1 along d // process in reverse order

for all relations do // join all relations and “project-out”

If is not empty, add it to where k is the largest variable index in Else problem is unsatisfiable Return the set of all relations (old and new) in the buckets

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nbucketbucket ,...,1

im bucketRR ∈,...,1

) ()( jX jnew RRi

∏−←

iX

newR ,, ikbucketk <newR

Properties of bucket-elimination (adaptive consistency)

n Adaptive consistency generates a constraint network that is backtrack-free (can be solved without deadends). n The time and space complexity of adaptive consistency along ordering d is exponential in w* . n Therefore, problems having bounded induced width are tractable (solved in polynomial time).

n trees ( w*=1), n series-parallel networks ( w*=2 ), n and in general k-trees ( w*=k).

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Solving Trees (Mackworth and Freuder, 1985)

Adaptive consistency is linear for trees and equivalent to enforcing directional arc-consistency (recording only unary constraints)

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Tree Solving is Easy

1,2,3 1,2,3 1,2,3 1,2,3

1,2,3 1,2,3

1,2,3

<

<

Y

X

Z

T S R U

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Tree Solving is Easy

1,2,3 1,2,3 1,2,3 1,2,3

1,2,3 1,2,3

1,2,3

<

<

Y

X

Z

T S R U

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Tree Solving is Easy

1,2,3 1,2,3 1,2,3 1,2,3

1,2 1,2

1,2,3

<

<

Y

X

Z

T S R U

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Tree Solving is Easy

1,2,3 1,2,3 1,2,3 1,2,3

1,2 1,2

1,2,3

<

<

Y

X

Z

T S R U

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Tree Solving is Easy

1,2,3 1,2,3 1,2,3 1,2,3

1,2 1,2

1

<

<

Y

X

Z

T S R U

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Tree Solving is Easy

1,2,3 1,2,3 1,2,3 1,2,3

2 2

1

<

<

Y

X

Z

T S R U

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Tree Solving is Easy

3 3 3 3

2 2

1

<

<

Y

X

Z

T S R U

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Tree Solving is Easy

3 3 3 3

2 2

1

<

<

Y

X

Z

T S R U

Constraint propagation Solves trees in linear time

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Road Map

n Graphical models

n Constraint networks Model n Inference

n Variable elimination: n Tree-clustering

n Constraint propagation

n Search

n Probabilistic Networks

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Tree Decomposition

Each function in a cluster Satisfy running intersection property

G

E

F

C D

B

A A B C

R(a), R(b,a), R(c,a,b)

B C D F R(d,b), R(f,c,d)

B E F R(e,b,f)

E F G R(g,e,f)

2

4

1

3

EF

BC

BF

R(a), R(b,a), R(c,a,b) R(d,b), R(f,c,d) R(e,b,f), R(g,e,f)

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Cluster Tree Elimination

A B C R(a), R(b,a), R(c,a,b)

B C D F R(d,b), R(f,c,d),h(1,2)(b,c)

B E F R(e,b,f), h(2,3)(b,f)

E F G R(g,e,f)

2

4

1

3

EF

BC

BF sep(2,3)={B,F} elim(2,3)={C,D}

Computes the minimal domains

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),,(),()(),()2,1( bacRabRaRcbh a ⊗⊗=⇓

),(),,(),(),( )2,1(,)3,2( cbhdcfRbdRfbh dc ⊗⊗=⇓

CTE: Cluster Tree Elimination

ABC

2

4

1

3 BEF

EFG

EF

BF

BC

BCDF

Time: O ( exp(w*+1 )) Space: O ( exp(sep))

G

E

F

C D

B

A

Computes the minimal domains

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),,(),()(),()2,1( bacRabRaRcbh a ⊗⊗=⇓

),(),,(),(),( )2,3(,)1,2( fbhdcfRbdRcbh fd ⊗⊗=⇓

),(),,(),(),( )2,1(,)3,2( cbhdcfRbdRfbh dc ⊗⊗=⇓

),(),,(),( )3,4()2,3( fehfbeRfbh e ⊗=⇓

),(),,(),( )3,2()4,3( fbhfbeRfeh b ⊗=⇓

),,(),()3,4( fegRfeh g=⇓

Tree decompositions

A B C R(a), R(b,a), R(c,a,b)

B C D F R(d,b), R(f,c,d)

B E F R(e,b,f)

E F G R(g,e,f)

EF

BF

BC

G

E

F

C D

B A

Belief network Tree decomposition

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property) onintersecti (running subtree connected a forms set the variableeach For 2.

and thatsuch vertex oneexactly is there function each For 1.

:satisfying and x over vertelabelings are and and a tree is where,,, triple

a is for A

χ(v)}V|X{vXXχ(v))scope(Cψ(v)C

CCCψ(v)Xχ(v)Vv

ψχ(V,E)TTX,D,CRiondecomposit tree

ii

ii

i

∈∈∈

⊆∈

∈

⊆⊆∈

=><

>=<

ψχ

Induced-width and Tree-width

EDCB

DCBA

DBE

ADB

CBE

Tree-width =3

Tree-width =2

Induced-width Of ordering

Tree-width of a graph = smallest cluster in a cluster-tree Path-width of a graph = smallest cluster in a cluster-path

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E

D

A

C

B

Road Map

n Graphical models

n Constraint networks Model n Inference

n Variable elimination: n Tree-clustering

n Constraint propagation

n Search

n Probabilistic Networks

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Sudoku – Approximation: Constraint Propagation

Each row, column and major block must be alldifferent “Well posed” if it has unique solution: 27 constraints

2 3 4 6 2

〉 Variables: empty slots 〉 Domains = {1,2,3,4,5,6,7,8,9} 〉 Constraints:

〉 27 all-different

〉 Constraint 〉 Propagation 〉 Inference

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Approximating Inference: Local Constraint Propagation

n Problem: bucket-elimination/tree-clustering algorithms are intractable when induced width is large

n Approximation: bound the size of recorded dependencies, i.e. perform local constraint propagation (local inference)

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From Global to Local Consistency

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Arc-consistency

A binary constraint R(X,Y) is arc-consistent w.r.t. X is every value In x’s domain has a match in y’s domain.

Only domains are reduced:

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YXRR YX <== constraint },3,2,1{ },3,2,1{

∏←X YXYX DRR

3 2, 1,

3 2, 1, 3 2, 1,

1 ≤ X, Y, Z, T ≤ 3 X < Y Y = Z T < Z X ≤ T

X Y

T Z

3 2, 1, <

=

<

∧

Arc-consistency

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1 ≤ X, Y, Z, T ≤ 3 X < Y Y = Z T < Z X ≤ T

X Y

T Z

<

=

<

∧

1 3

2 3

Arc-consistency

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∏←X YXYX DRR

From Arc-consistency to relational arc-consistency

n Sound n Incomplete n Always converges

(polynomial)

A B

C D

3 2 1 A

3 2 1 B

3 2 1 D

3 2 1 C

<

<

< =

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Relational Distributed Arc-Consistency

Primal Dual

AB

AD

A

A B

C D

3 2 1 A

3 2 1 B

3 2 1 D

3 2 1 C

DC

BC

D

C

B

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AC-3

Complexity: Best case O(ek), since each arc may be processed in O(2k)

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)( 3ekO

Arc-consistency Algorithms

n AC-1: brute-force, distributed

n AC-3, queue-based

n AC-4, context-based, optimal n AC-5,6,7,…. Good in special cases

n Important: applied at every node of search n (n number of variables, e=#constraints, k=domain size) n Mackworth and Freuder (1977,1983), Mohr and Anderson, (1985)…

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)( 3ekO)( 3nekO

)( 2ekO

Path-consistency

n A pair (x, y) is path-consistent relative to Z, if every consistent assignment (x, y) has a consistent extension to z.

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Example: path-consistency

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Path-consistency Algorithms

n Apply Revise-3 (O(k^3)) until no change

n Path-consistency (3-consistency) adds binary constraints. n PC-1:

n PC-2:

n PC-4 optimal:

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)( kjkikijijij RDRRR ⊗⊗∩← π

)( 55knO)( 53knO

)( 33knO

Local i-consistency

i-consistency: Any consistent assignment to any i-1 variables is consistent with at least one value of any i-th variable

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Gausian and Boolean Propagation, Resolution

n Linear inequalities n Boolean constraint propagation, unit resolution

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2,2 ≤≤ yx

)( CA ¬∨

⇒≥≤++ 13,15 zzyx

⇒¬¬∨∨ )(),( BCBA

Directional Resolution ó Adaptive Consistency

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))exp(( :space and timeDR))(exp(||

*

*

wnOwObucketi =

Directional i-consistency

A

E

C D

B

D

C B

E

D

C B

E

D

C B

E

Adaptive d-arc d-path

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DCBR

≠

≠ ≠

≠ ≠≠≠

:AB A:BBC :C

AD C,D :DBE C,E D,E :E

≠

≠

≠≠

≠≠≠

DBDC RR ,CBR

DRCRDR

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Greedy Algorithms for Induced-Width

n Min-width ordering

n Min-induced-width ordering

n Max-cardinality ordering

n Min-fill ordering

n Chordal graphs

n Hypergraph partitionings (Project: present papers on induced-width, run algorithms for induced-width on new benchmarks…)

Min-width ordering

Proposition: algorithm min-width finds a min-width ordering of a graph Complexity:? O(e)

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Greedy orderings heuristics

min-induced-width (miw) input: a graph G = (V;E), V = {v1; :::; vn} output: A miw ordering of the nodes d = (v1; :::; vn). 1. for j = n to 1 by -1 do 2. r ß a node in V with smallest degree. 3. put r in position j. 4. connect r's neighbors: E ß E union {(vi; vj)| (vi; r) in E; (vj ; r) 2 in E}, 5. remove r from the resulting graph: V ßV - {r}. min-fill (min-fill) input: a graph G = (V;E), V = {v1; :::; vn} output: An ordering of the nodes d = (v1; :::; vn). 1. for j = n to 1 by -1 do 2. r ßa node in V with smallest fill edges for his parents. 3. put r in position j. 4. connect r's neighbors: E ßE union {(vi; vj)| (vi; r) 2 E; (vj ; r) in E}, 5. remove r from the resulting graph: V ßV –{r}.

Theorem: A graph is a tree iff it has both width and induced-width of 1.

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Example

We see again that G in the Figure (a) is not chordal since the parents of A are not connected in the max-cardinality ordering in Figure (d). If we connect B and C, the resulting induced graph is chordal.

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n A graph is cordal if every cycle of length at least 4 has a chord

n Finding w* over chordal graph is easy using the max-cardinality ordering

n If G* is an induced graph it is chordal n K-trees are special chordal graphs. n Finding the max-clique in chordal graphs is easy (just enumerate all cliques in a max-cardinality ordering

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Max-cardinality ordering

Graph Concepts: Hypergraphs and Dual Graphs

n A hypergraph is H = (V,S) , V= {v_1,..,v_n} and a set of subsets Hyperegdes: S={S_1, ..., S_l }. n Dual graphs of a hypergaph: The nodes are the hyperedges and a pair of nodes is connected if they share vertices in V. The arc is labeled by the shared vertices. n A primal (Markov, moral) graph of a hypergraph H = (V,S) has V as its nodes, and any two nodes are connected by an arc if they appear in the same hyperedge. n Factor Graphs. .

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Which greedy algorithm is best?

n MinFill, prefers a node who add the least number of fill-in arcs.

n Empirically, fill-in is the best among the greedy algorithms (MW,MIW,MF,MC)

n Complexity of greedy orderings?

n MW is O(?), MIW: O(?) MF (?) MC is O(mn)

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Road Map

n Graphical models

n Constraint networks Model n Inference

n Variable elimination: n Tree-clustering

n Constraint propagation

n Search

n Probabilistic Networks

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