MemoryEfficient Inference in Relational Domains

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Memory-Efficient Inference in Relational Domains

• Parag Singla Pedro Domingos
• Computer Science Engineering
• University of Washington

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Outline

• Motivation
• Satisfiability and Relational Inference
• Memory-Efficient Inference
• Experiments
• Discussion

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Motivation

• Most problems in AI are relational
• Multiple types of objects
• Relations between objects
• An efficient inference approach
  Propositionali-zation followed by satisfiability
  testing Kautz Selman 96
• MPE/MAP inference in a large class of SRL models
  can be done using weighted satisfiability
  Richardson Domingos 06

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Problem

• Exponential cost of propositionalization
• Memory cost is O(( objects)clause-arity)
• Applicability is severely limited
• Even moderately sized domains tend to blow out of
  memory

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Example Scientific Research Domain
1000 Papers , 100 Authors
Author(person,paper)
100,000 possible groundings .. But only a few
thousand true
Author(person1,paper) ? Author(person2,paper)
? Coauthor(person1,person2)
10 million possible groundings .. But only tens
of thousands unsatisfied
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Solution

• Most real-world domains characterized by extreme
  sparseness
• Majority of predicates are false
• Majority of clauses are satisfied
• Exploit sparseness
• Embodied in lazy variation of WalkSAT
• Create only potentially unsatisfied clause
  groundings
• LazySAT
• Memory-cost is O ( potentially unsatisfied
  clauses)

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Outline

• Motivation
• Satisfiability and Relational Inference
• Memory-Efficient Inference
• Experiments
• Discussion

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Satisfiability

• KB Set of formulas over Boolean variables
• Every KB can be converted to a CNF form
• (X1 ? X5 ? X7) ? .. (X12 ? X53 ? X5) ?
• SAT Problem of finding a satisfying assignment
• Protypical NP complete problem
• Weighted SAT
• Each clause given a weight
• Maximize sum of weights of satisfied clauses
• One of most efficient approaches is stochastic
  local search (e.g. WalkSAT Selman et al. 96)

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(Max)WalkSAT
for i ? 1 to max-tries do soln random truth
assignment to all atoms for j ? 1 to
max-flips do if ? weights(sat. clauses)
threshold then return soln c
? random unsatisfied clause with
probability p vf ? a randomly
chosen variable from c else
for each variable v in c do
compute DeltaGain(v)
vf ? v with highest DeltaGain(v) soln ?
soln with vf flipped return failure, best
solution found // DeltaGain(v) returns the
increase in ? weights(sat. clauses) // caused by
flipping v
initialization
random move
greedy move
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Relational Inference

• First-Order Logic (FOL) explicitly represents a
  domains relational structure
• Focus on function free finite FOL
• Propositionalize the first order KB
• Replace universal quantification by conjunction
• Replace existential quantification by disjunction
• Perform satisfiability testing over
  propositiona-lized KB

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Statistical Relational Learning

• Statistical relational models explicitly deal
  with
• Relational structure
• Probabilistic dependencies
• Markov Logic Richardson Domingos 06
• Weighted first-order formulas
• Subsumes many other SRL models
• MAP/MPE inference in Markov Logic is an instance
  of weighted satisfiability

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Outline

• Motivation
• Satisfiability and Relational Inference
• Memory-Efficient Inference
• Experiments
• Discussion

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Naïve Approach

• Create the groundings and keep in memory
• True atoms
• Unsatisfied clauses
• Memory-cost is O( unsatisfied clauses)
• Problem
• Need to go to the KB for each flip
• Too slow!
• Solution Idea Keep more things in memory
• A list of active atoms
• Potentially unsatisfied clauses (active clauses)

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LazySAT Definitions

• An atom is an Active Atom if
• It is in the initial set of active atoms
• It was flipped at some point during the search
• A clause is an Active Clause if
• It can be made unsatisfied by flipping zero or
  more active atoms in it

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LazySAT The Basics

• Activate all the atoms appearing in clauses
  unsatisfied by evidence DB
• Create the corresponding clauses
• Randomly assign truth value to all active atoms
• Activate an atom when it is flipped if not
  already so
• Potentially activate additional clauses
• No need to go to the KB for calculating the
  change in cost for flipping an active atom

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LazySAT
for i ? 1 to max-tries do active_atoms ?
atoms in clauses unsatisfied by DB
active_clauses ? clauses activated by
active_atoms soln random truth assignment to
active_atoms for j ? 1 to max-flips do
if ? weights(sat. clauses) threshold then
return soln c ? random unsatisfied
clause with probability p vf ? a
randomly chosen variable from c else
for each variable v in c do
compute DeltaGain(v), using weighted_KB if vf ?
active_atoms vf ? v with highest
DeltaGain(v) if vf ? active_atoms then
activate vf and add clauses activated by vf
soln ? soln with vf flipped return failure,
best soln found
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LazySAT
for i ? 1 to max-tries do active_atoms ?
atoms in clauses unsatisfied by DB
active_clauses ? clauses activated by
active_atoms soln random truth assignment to
active_atoms for j ? 1 to max-flips do
if ? weights(sat. clauses) threshold then
return soln c ? random unsatisfied
clause with probability p vf ? a
randomly chosen variable from c else
for each variable v in c do
compute DeltaGain(v), using weighted_KB if vf ?
active_atoms vf ? v with highest
DeltaGain(v) if vf ? active_atoms then
activate vf and add clauses activated by vf
soln ? soln with vf flipped return failure,
best soln found
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LazySAT
for i ? 1 to max-tries do active_atoms ?
atoms in clauses unsatisfied by DB
active_clauses ? clauses activated by
active_atoms soln random truth assignment to
active_atoms for j ? 1 to max-flips do
if ? weights(sat. clauses) threshold then
return soln c ? random unsatisfied
clause with probability p vf ? a
randomly chosen variable from c else
for each variable v in c do
compute DeltaGain(v), using weighted_KB if vf ?
active_atoms vf ? v with highest
DeltaGain(v) if vf ? active_atoms then
activate vf and add clauses activated by vf
soln ? soln with vf flipped return failure,
best soln found
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LazySAT
for i ? 1 to max-tries do active_atoms ?
atoms in clauses unsatisfied by DB
active_clauses ? clauses activated by
active_atoms soln random truth assignment to
active_atoms for j ? 1 to max-flips do
if ? weights(sat. clauses) threshold then
return soln c ? random unsatisfied
clause with probability p vf ? a
randomly chosen variable from c else
for each variable v in c do
compute DeltaGain(v), using weighted_KB if vf ?
active_atoms vf ? v with highest
DeltaGain(v) if vf ? active_atoms then
activate vf and add clauses activated by vf
soln ? soln with vf flipped return failure,
best soln found
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LazySAT Performance Analysis

• Solution Quality
• Performs the same sequence of flips
• Same result as WalkSAT
• Memory cost
• O( potentially unsatisfied clauses)
• Time cost
• Much lower initialization cost
• Cost of creating active clauses amortized over
  many flips

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Outline

• Motivation
• Satisfiability and Relational Inference
• Memory-Efficient Inference
• Experiments
• Discussion

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Experiments

• Two domains
• De-duplicating citation databases
• Cora
• BibServ
• Planning
• Blocks world
• Used the Alchemy system Kok et al. 2005
• Experiments run on 3 GHz machines with
  3.46 GB of RAM

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De-duplication

• Weighted KB consisting of 33 first-order rules
• High similarity score gt Same field
• Same record gt Same fields
• Cora
• Cleaned version of McCallums hand-labeled
  dataset
• 1295 citations to 132 different research papers
• BibServ
• Public repository of half-a-million citations
• Experimented on user donated subset (21,805
  citations)

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Methodology

• Compared using varying number of records
• Memory Number of clauses grounded
• Speed Average Flips/sec
• Results reported over an average of 5 randomly
  chosen subsets for a given record count

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Results Memory
Cora
BibServ
n2.97
n2.98
n1.75
n2.34
Memory Reduction on full DB 300X
Memory Reduction on full DB 400,000X
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Results Speed
Cora
BibServ
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Outline

• Motivation
• Satisfiability and Relational Inference
• Memory-Efficient Inference
• Experiments
• Discussion

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Conclusion

• Satisfiability testing is an effective approach
  to inference in relational domains
• Limitation Exponential cost of
  propositionali-zation
• LazySAT
• Exploits sparseness
• Reduces memory cost by many orders of magnitude
• Same solution with similar speed

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Future Work

• Extending LazySAT to other SAT solvers, MCMC
• Degrading gracefully when number of clauses
  exceeds available memory
• Combining LazySAT with KBMC
• LazySAT available in the Alchemy system
• http//www.cs.washington.edu/ai/alchemy