Abstraction Sampling in Graphical Models Filjor Broka*, Rina - - PowerPoint PPT Presentation

Abstraction Sampling in Graphical Models

Filjor Broka*, Rina Dechter, Alexander Ihler, and Kalev Kask UCI

*In memory of Filjor (1985-2018)

Outline

❑ Background: Graphical models, search, sampling ❑ Motivation and the main idea ❑ Abstraction sampling algorithm – OR ❑ The AND/OR case, properness ❑ Properties ❑ Experiments ❑ Conclusion and Future Directions

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Graphical models

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Deep Boltzmann Machines

PCWP CO HRBP HREKG HRSAT ERRCAUTER HR HISTORY CATECHOL SAO2 EXPCO2 ARTCO2 VENTALV VENTLUNG VENITUBE DISCONNECT MINVOLSET VENTMACH KINKEDTUBE INTUBATION PULMEMBOLUS PAP SHUNT ANAPHYLAXIS MINOVL PVSAT FIO2 PRESS INSUFFANESTH TPR LVFAILURE ERRBLOWOUTPUT STROEVOLUME LVEDVOLUME HYPOVOLEMIA CVP BP

Bayesian Networks

Cancer(A) Smokes(A) Friends(A,A) Friends(B,A) Smokes(B) Friends(A,B) Cancer(B) Friends(B,B)

Markov Logic

Graphical models

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Example: The combination operator defines an overall function from the factors, e.g., “x” : A graphical model consists of:

• - variables
• - domains
• - functions or “factors”

and a combination operator

(w e’ll assume discrete)

Inference: compute quantities of interest about the distribution, e.g.,

• r

(partition function) (marginals)

A graphical model consists of:

• - variables
• - domains
• - functions or “factors”

and a combination operator

Primal graph

A B f(A,B) 2 1 4 1 3 1 1 1

Search trees & Enumeration

Full OR search tree 126 nodes

1 1 1 1 1 1 1 1 1 1 1 1 01 01 0 1 0 101 01 0 1 0 1 0 1 0 1 01 010 1 0 1 01 01 0 1 0 1 01 010 1 0 1 01 01 01 01 01 0101 01 01 01 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

C D F E B A

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Full AND/OR search tree 54 AND nodes

A

OR AND

B

OR AND OR

E

OR

F F

AND

0 1 0 1

AND

1 C D D 0 1 0 1 1 1 E F F 0 1 0 1 1 C D D 0 1 0 1 1 1 B E F F 0 1 0 1 1 C D D 0 1 0 1 1 1 E F F 0 1 0 1 1 C D D 0 1 0 1 1

Context minimal OR search graph 28 nodes

1 1 1 1 1 1 1 1 1 1 1 1 1

C D F E B A

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Context minimal AND/OR search graph 18 AND nodes

A

OR AND

B

OR AND OR

E

OR

F F

AND

0 1

AND

1 C D D 0 1 1 1 E C D D 1 1 B E F F 1 C 1 E C

A E C B F D

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Any query can be computed

• ver any of the search spaces

A D B E C F

pseudo tree

Search vs. Sampling

◼ Search

 Enumerate states; no stone unturned, none more than once.

◼ Sampling

 Exploit randomization “typicality”; concentration inequalities

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(Heuristic) Search

Structured enumeration over all possible states

1 1 1 1 1 1 1 1 1 1 1 1 01010101 010 1010101010101010101010101010101010101010101010 1010101 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 1 1

E C F D B A

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(Monte Carlo) Sampling

Use randomization to estimate averages over the state space

11 1 1 1 1 1 1 1 2 5 3 6 3 4 2 4 5 7 6 1 1 1

B A C

1 1 1 5 3 4 6 1 1 1 1 6 1 1 2 1 1 1 1 1 2 4 5 7 6 1 1 1 2 6 1 1 1 5 4 1 1 1 4 1

B A C

w1 w2 w3 S1 S2

1 1 1 1 1 5 3 4 5 7 6 1 1 1

Z estimate Z estimate Z estimate Importance sampling 2-config-subtree sampling 4-config-subtree sampling … …

Motivation 1: Sampling to Searching

More searching less sampling S2 S1

◼ Merge nodes that root identical subtrees

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Motivation 2: Searching to Sampling

Sampled subtree 1 Sampled subtree 2

similar

Stratified sampling

◼ Knuth 1975, Chen 1992 estimate search space size ◼ Partially enumerate, partially sample

Subdivide space into parts Enumerate over parts, sample within parts “Probe”: random draw corresponding to multiple states Theorem (Rizzo 2007): The variance reduction moving

from Importance Sampling (IS) to Stratified IS with k strata’s (under some conditions) is 𝑙 ∙ 𝑤𝑏𝑠(𝑎𝐾)

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Full OR Tree

A = 0 A = 1 A = 0 B = 0 C = 1 A = 0 B = 0 C = 0 A = 0 B = 1 C = 1 A = 1 B = 1 C = 1 A = 1 B = 0 C = 0 A = 0 B = 1 C = 0 A = 1 B = 0 C = 1 A = 1 B = 1 C = 0 A = 0 B = 0 A = 0 B = 1 A = 1 B = 0 A = 1 B = 1 0.6 0.4 0.3 0.7 0.1 0.9 0.2 0.8 0.2 0.8 0.4 0.6 0.4 0.6 Z(A=0,B=1,C=1) = 0.6*0.7*0.8

A = 0 A = 1 A = 0 B = 1 C = 1 A = 1 B = 0 C = 0 A = 0 B = 1 C = 0 A = 1 B = 0 C = 1 A = 0 B = 0 A = 0 B = 1 A = 1 B = 0 A = 1 B = 1 0.6 0.4 0.3 0.7 0.1 0.9 0.2 0.8 0.4 0.6 w = 1 w = 1 w = 1 w = 1 w = 2 w = 2 w = 2 w = 4 w = 2 w = 4

Method 1 – OR Tree

Zest = 4*(0.6*0.7*0.8) + 4*(0.4*0.1*0.6) =1.44 p = 1/2 p = 2/4 w = 1 w = 1 w = 2 w = 2

Abstraction Sampling - AND/OR

Improper Abstraction

B

OR AND

1 A

OR

C A C

OR AND

D 0 1

AND

1 1 1 1 D 1 D 0 1 D 0 1

Full AND/OR Search Tree

Sampled AND/OR Search Tree

B

OR AND

1 A

OR

C

OR AND AND

1 1 D D 1

Not a subset of solution trees Estimate ෡ 𝒂 is biased 16 Solution trees

25 D B C A

Not a proper abstraction

B

OR AND

1 A

OR

C A C

OR AND

D 1

AND

1 1 1 1 D D 0 1

a proper abstraction

The Proposal Distribution

❑ Our scheme is like any IS-based scheme where any

proposal can be used

❑ In our experiments we use a proposal

B A C

<𝒕, 𝒙 𝒕 > 1 1 5 3 4 15 20 1

g(s)

𝒒 ∝ 𝒙(𝒕) ∙ 𝒉(𝒕) ∙ 𝒊(𝒕)

𝒊 𝒕 ≥ 𝒂(𝒕)

Properties of AS

• Theorem. [unbiasedness] Estimate መ

𝑎 generated by AS is unbiased (𝐹 መ 𝑎 = 𝑎).

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• Theorem. [exact proposal] If ℎ 𝑜 = 𝑎(𝑜) then መ

𝑎 is exact for any choice of abstraction function 𝑏.

• Theorem. If 𝑢ℎ𝑓 𝑏𝑐𝑡𝑢𝑠𝑏𝑑𝑢𝑗𝑝𝑜 𝑏 is Z-isomorph, namely:

(𝑏 𝑜 = 𝑏(𝑜′)) ➔ (𝑎 𝑜 = 𝑎(𝑜′)) then መ 𝑎 is exact for any choice of proposal.

Experimental Setup

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◼ Use 4 classes of problems

Grids, DBN, Promedas, Pedigree

◼ Use weighted MB to generate the h ◼ Evaluate 2 context-based abstractions

Randomized, Relaxed

◼ Competing algorithms

AS-(OR,AO), WMB-IS, IJGP-SS

◼ Questions :

AS impact on variance, OR vs AO, vs competition

Abstractions Based on Context

◼ context(X) = ancestors of X in

pseudo tree, that disconnect its subtree from the rest of the problem

◼ Context-based (CB) Abstractions:

 assignments to context  Relaxed: most recent subset of context

variables

 Randomized : random subset of context

variables

A E C B F D A E C B F D

[ ]

[A]

[AB]

[AE]

[BC] [AB]

A D B E C F 35

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Future Directions

❑ Explore choice of abstraction in order to reduce

variance: relaxed-path based, relaxed-context based, heuristic based abstractions. Further explore tradeoffs between:

❑ Portion of search space sampled in a probe vs.

number of probes

❑ Accuracy of sampling probability (heuristic) vs.

time/memory needed to compute it

❑ Sampling in OR space vs. AND/OR space ❑ Sampling search trees vs. search graphs

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THANK YOU

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