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Approximation Techniques Iterative bounded inference COMPSCI 276, Fall 2014 Set 11: Rina Dechter (Reading: class notes chapter 8,9, Darwiche chapters 14)
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Approximation Techniques Iterative bounded inference

COMPSCI 276, Fall 2014

Set 11: Rina Dechter

(Reading: class notes chapter 8,9, Darwiche chapters 14)

Agenda

Mini-bucket elimination

Mini-clustering

Iterative Belief propagation

Iterative-join-graph propagation

IJGP complexity

Convergence and pair-wise consistency

Accuracy when converged

Belief Propagation and constraint propagation

Using Mini-bucket as heuristics for optimization

2

Iterative Belief Proapagation Belief propagation is exact for poly-trees

IBP - applying BP iteratively to cyclic networks

No guarantees for convergence

Works well for many coding networks

3

)( 11uX

1U 2U 3U

2X1X

)( 12xU

)( 12uX

)( 13xU)bel(U-CTE

:step One

1

IJGP - Example

4

A

D

I

B

E

J

F

G

C

H

Belief network

A

ABDE

FGI

ABC

BCE

GHIJ

CDEF

FGH

C

H

A C

A AB BC

BE

C

C DE CE

F H

F FG GH H

GI

Loopy BP graph

5

CPCS422 - Absolute error

evidence=0 evidence=10

CPCS 422, evid=0, w*=23, 1 instance

i-bound

2 4 6 8 10 12 14 16 18

Absolu

te e

rror

0.00

0.01

0.02

0.03

0.04

0.05

MC

IBP

CPCS 422, evid=10, w*=23, 1 instance

i-bound

2 4 6 8 10 12 14 16 18

Absolu

te e

rror

0.00

0.01

0.02

0.03

0.04

0.05

MC

IBP

MBE-mpe vs. IBP

6

c o d e s *w-lo w o nb e t t e r is m p e-m b e

Bit error rate (BER) as a function of noise (sigma):

Iterative Join Graph Propagation

Loopy Belief Propagation Cyclic graphs Iterative Converges fast in practice (no guarantees though) Very good approximations (e.g., turbo decoding, LDPC codes, SAT

– survey propagation)

Mini-Clustering(i) Tree decompositions Only two sets of messages (inward, outward) Anytime behavior – can improve with more time by increasing the

i-bound

We want to combine:

Iterative virtues of Loopy BP Anytime behavior of Mini-Clustering(i)

7

IJGP - The basic idea

Apply Cluster Tree Elimination to any join-graph

We commit to graphs that are I-maps

Avoid cycles as long as I-mapness is not violated

Result: use minimal arc-labeled join-graphs

8

Minimal arc-labeled join-graph

9

IJGP - Example

10

A

D

I

B

E

J

F

G

C

H

Belief network

A

ABDE

FGI

ABC

BCE

GHIJ

CDEF

FGH

C

H

A C

A AB BC

BE

C

C DE CE

F H

F FG GH H

GI

Loopy BP graph

Arc-Minimal Join-Graph

11

A

ABDE

FGI

ABC

BCE

GHIJ

CDEF

FGH

C

H

A C

A AB BC

BE

C

C DE CE

F H

F FG GH H

GI

A

ABDE

FGI

ABC

BCE

GHIJ

CDEF

FGH

C

H

A C

A AB BC

BE

C

C DE CE

F H

F FG GH H

GI

A

ABDE

FGI

ABC

BCE

GHIJ

CDEF

FGH

C

H

A C

AB BC

BE

C

C DE CE

F H

F FG GH H

GI

A

ABDE

FGI

ABC

BCE

GHIJ

CDEF

FGH

C

H

A C

AB BC

BE

C

C DE CE

F H

F FG GH H

GI

A

ABDE

FGI

ABC

BCE

GHIJ

CDEF

FGH

C

H

A

AB BC

BE

C DE CE

F H

F FG GH H

GI

A

ABDE

FGI

ABC

BCE

GHIJ

CDEF

FGH

C

H

A

AB BC

BE

C DE CE

F H

F FG GH H

GI

A

ABDE

FGI

ABC

BCE

GHIJ

CDEF

FGH

C

H

A

AB BC

BE

C DE CE

F H

F FG GH H

GI

A

ABDE

FGI

ABC

BCE

GHIJ

CDEF

FGH

C

H

A

AB BC

C DE CE

F H

F FG GH H

GI

A

ABDE

FGI

ABC

BCE

GHIJ

CDEF

FGH

C

H

A

AB BC

BE

C DE CE

H

F FG GH H

GI

A

ABDE

FGI

ABC

BCE

GHIJ

CDEF

FGH

C

H

A

AB BC

BE

C DE CE

H

F FG GH H

GI

A

ABDE

FGI

ABC

BCE

GHIJ

CDEF

FGH

C

H

A

AB BC

BE

C DE CE

H

F FG GH

GI

A

ABDE

FGI

ABC

BCE

GHIJ

CDEF

FGH

C

H

A

AB BC

BE

C DE CE

H

F FG GH

GI

A

ABDE

FGI

ABC

BCE

GHIJ

CDEF

FGH

C

H

A

AB BC

BE

C DE CE

H

F FG GH

GI

Arcs labeled with any single variable should form a TREE

A

ABDE

FGI

ABC

BCE

GHIJ

CDEF

FGH

C

H

A

AB BC

BE

C DE CE

H

F FG GH

GI

Collapsing Clusters

12

ABCDE

FGI

BCE

GHIJ

CDEF

FGH

BC

CDE CE

F FG GH

GI

ABCDE

FGI

BCE

GHIJ

CDEF

FGH

BC

CDE CE

F FG GH

GI

ABCDE

FGI

BCE

GHIJ

CDEF

FGH

BC

CDE CE

F FG GH

GI

ABCDE

FGHI GHIJ

CDEF

CDE

F

GHI

Join-Graphs

13

A

ABDE

FGI

ABC

BCE

GHIJ

CDEF

FGH

C

H

A C

A AB BC

BE

C

C DE CE

F H

F FG GH H

GI

A

ABDE

FGI

ABC

BCE

GHIJ

CDEF

FGH

C

H

A

AB BC

C DE CE

H

F F GH

GI

ABCDE

FGI

BCE

GHIJ

CDEF

FGH

BC

DE CE

F F GH

GI

ABCDE

FGHI GHIJ

CDEF

CDE

F

GHI

more accuracy

less complexity

Message propagation

14

ABCDE

FGI

BCE

GHIJ

CDEF

FGH

BC

CDE

CE

F F GH

GI

ABCDE

p(a), p(c), p(b|ac),

p(d|abe),p(e|b,c)

h(3,1)(bc)

BCD

CDEF

BC

CDE CE

1 3

2

h(3,1)(bc)

h(1,2)

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}

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

15

Constructing Join-Graphs

16

a) schematic mini-bucket(i), i=3 b) arc-labeled join-graph decomposition

CDB

CAB

BA

A

CB

P(D|B)

P(C|A,B)

P(A)

BA

P(B|A)

FCD P(F|C,D)

GFE

EBF

BF

EF P(E|B,F)

P(G|F,E)

B

CD

BF

A

F

G: (GFE)

E: (EBF) (EF)

F: (FCD) (BF)

D: (DB) (CD)

C: (CAB) (CB)

B: (BA) (AB) (B)

A: (A)

G

E

F

C D

B

A

Empirical evaluation

17

Algorithms: Exact

IBP

MC

IJGP

Measures: Absolute error

Relative error

Kulbach-Leibler (KL) distance

Bit Error Rate

Time

Networks (all variables are binary):

Random networks

Grid networks (MxM)

CPCS 54, 360, 422

Coding networks

Coding networks - BER

18

Coding, N=400, 500 instances, 30 it, w*=43, sigma=.32

i-bound

0 2 4 6 8 10 12

BE

R

0.00237

0.00238

0.00239

0.00240

0.00241

0.00242

0.00243

IBP

IJGP

Coding, N=400, 500 instances, 30 it, w*=43, sigma=.51

i-bound

0 2 4 6 8 10 12

BE

R

0.0745

0.0750

0.0755

0.0760

0.0765

0.0770

0.0775

0.0780

0.0785

IBP

IJGP

Coding, N=400, 500 instances, 30 it, w*=43, sigma=.65

i-bound

0 2 4 6 8 10 12

BE

R

0.1900

0.1902

0.1904

0.1906

0.1908

0.1910

0.1912

0.1914

IBP

IJGP

sigma=.22 sigma=.32

sigma=.51 sigma=.65

Coding, N=400, 1000 instances, 30 it, w*=43, sigma=.22

i-bound

0 2 4 6 8 10 12

BE

R

1e-5

1e-4

1e-3

1e-2

1e-1

IJGP

MC

IBP

CPCS 422, evid=0, w*=23, 1instance

i-bound

2 4 6 8 10 12 14 16 18

KL

dis

tan

ce

0.0001

0.001

0.01

0.1

IJGP 30 it (at convergence)

MC

IBP 10 it (at convergence)

CPCS 422 – KL Distance

19

CPCS 422, evid=30, w*=23, 1instance

i-bound

3 4 5 6 7 8 9 10 11 12 13 14 15 16

KL

dis

tan

ce

0.0001

0.001

0.01

0.1

IJGP at convergence

MC

IBP at convergence

evidence=0 evidence=30

CPCS 422 – KL vs. Iterations

20

CPCS 422, evid=0, w*=23, 1instance

number of iterations

0 5 10 15 20 25 30 35

KL

dis

tan

ce

0.0001

0.001

0.01

0.1

IJGP (3)

IJGP(10)

IBP

CPCS 422, evid=30, w*=23, 1instance

number of iterations

0 5 10 15 20 25 30 35

KL d

ista

nce

0.0001

0.001

0.01

0.1

1

IJGP(3)

IJGP(10)

IBP

evidence=0 evidence=30

Coding networks - Time

21

Coding, N=400, 500 instances, 30 iterations, w*=43

i-bound

0 2 4 6 8 10 12

Tim

e (

se

co

nd

s)

0

2

4

6

8

10

IJGP 30 iterations

MC

IBP 30 iterations

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•(n+N) •d i+1)

space: O(N•d)

22

Join-graph decomposition

23

Agenda

Mini-bucket elimination

Mini-clustering

Iterative Belief propagation

Iterative-join-graph propagation

IJGP complexity

Convergence and pair-wise consistency

Accuracy when converged

BP and constraint propagation

24

More On the Power of Belief Propagation

BP as local minima of KL distance

BP’s power from constraint propagation perspective.

25

Summary of IJGP so far

Agenda

Mini-bucket elimination

Mini-clustering

Iterative Belief propagation

Iterative-join-graph propagation

IJGP complexity

Convergence and pair-wise consistency

Accuracy when converged

Belief Propagation and constraint propagation

Weighted Mini-bucket

Using Mini-bucket as heuristics for optimization

38

Weighted Mini-Bucket [Liu & Ihler]

Builds upon Mini-Bucket Elimination (MBE)

Bounds log Zψ using Hölder’s Inequality

Parameterized by set of weights

Weights optimized to ‘tighten’ bound

Standard MBE is specific setting of weights

Complexity controlled by iBound parameter

(Slides by Gellfand and from Qiang thesis)

Review of Mini-Bucket Elimination Markov Network

Cost is O(k4)

In Bucket 1:

Cost is O(k3)

Copies of Variable x1

Bucket 1

Hölder’s Inequality

The weighted summation operator is:

where , are positive functions and

are weights

Hölder’s Inequality

Let and all weights be positive, then

Weighted Power Sum

42

Holder Inequality

43

Weighted Mini-Bucket Elimination

Markov Network

In Bucket 1: Let

Gives the mini-bucket bound:

Weighted Mini-Bucket Elimination Markov Network

In Bucket 1:

Let

Gives the mini-bucket bound:

What happens when?

Weighted Mini-Bucket Elimination

Markov Network

In Bucket 1:

Let

Gives the mini-bucket bound:

What happens when?

“Standard “ mini-Bucket

Example: MBE and WMBE

47

48

Process each mini-bucket

Tightening the bound

Note that bound written as

Let

Optimization problem is:

Weights are optimized by iterative algorithm

Messages passed up/down the bucket tree

Weights updated on each pass

Experiments

Run on 15-by-15 grid with binary variables

for

WMB -

iBounds

WMB -

iBounds

WMB -

iBounds

Summary

Variational methods formulate inference as an optimization problem

e.g. given p, find distribution in Q closest to p

Provides new perspective for analysis

e.g. equivalence between fixed points of sum-product message passing and stationary points

Lead to development of many new algorithms

e.g. Liu & Ihler’s weighted mini-bucket


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