Finding Least Cost Proofs Using a Hierarchical PSO

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Finding Least Cost Proofs Using a Hierarchical PSO

• Shawn T. Chivers
• Gene A. Tagliarini
• Ashraf M. Abdelbar

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Cost-Based Abduction
Example modified figure fromEugene Santos, Jr.
A Linear Constraint Satisfaction Approach for
Abductive Reasoning. PhD thesis, Department of
Computer Science, Brown University, 1992.
http//citeseer.ist.psu.edu/santos92linear.html
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Abduction

• Process of proceeding from data describing a set
  of observations or events to hypotheses which
  accounts for data
• Useful for reasoning under uncertainty
• Finding Least Cost Proofs for CBA systems is
  known to be NP-Hard E. Charniak, and S.E. Shimony
  Cost-based abduction and MAP explanation,Artific
  ial Intelligence, Vol. 66, pp. 345-374, 1994.
• Objective is to find LCP for the given evidence

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Cost-Based Abduction 4-tuple model

• K(H,R,c,G)
• H is a set of hypotheses or propositions
• R is a rules set of the form
• (hi1 ? hi2 ? ? hin ) ? hiq , all members of H
• (antecedents ) ? consequence
• c is a function, c H ? ?, where h?H and c(h)
  is called the assumability cost
• G? H is the goal set or evidence

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Cost-Based Abduction

• An hypothesis h may be made true
• h may be assumed with cost c(H)
• Proven as the consequence of a rule, at no cost
• If h does not occur as the consequence of any
  rule it cannot be proven

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Partitioning Hypothesis Set

• HA assumable hypotheses do not appear as
  consequence of any rule
• HP infinite assumability cost hypotheses can
  only be proven

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RAA180

• RAA180 is a generated CBA problem
• Cost-Based Abduction Instance Library
  http//cbalib.org
• Dr. Ashraf Abdelbar
• 300 hypotheses
• 120 infinite cost
• 180 finite cost
• Hypothesis 300 is goal hypothesis (there is only
  one)
• 900 rules total
• Optimal solution is 10,821 obtained using Santos
  ILP method lp-solve

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Hierarchical PSO

• Introduced in 2003 S. Janson, and M. Middendorf,
  A hierarchical particle swarm optimizer,
  Proceedings IEEE Congress on Evolutionary
  Computation,2003.
• PSO arranged in tree topology
• Tree is process breadth-first starting with root
  node

• Better particles climb the tree (one level
  upward per iteration)
• However particles can fall many levels in one
  iteration

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Neighbors in Hierarchical PSO

• Neighbor is immediate parent in tree

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Hierarchical PSO

• Velocity vector is adjusted using

• For each dimension j1,,N we then apply

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Hierarchical PSO

• We then apply

• Where s is the sigmoid function

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Applying Hierarchical PSO to CBA

• Candidate solutions are represented as an n
  dimensional array, where n HA
• Each element in the array corresponds to a
  hypothesis
• Hypotheses included in the candidate solution are
  assigned a value of 1
• Hypotheses excluded from the candidate solution
  are assigned a value of 0

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Applying Hierarchical PSO to CBARepairing
Unfeasible Solutions

• We repair unfeasible solutions in the following
  way
• Choose a random element in the array with a value
  of 0
• Assign it a value of 1
• Check if the goal can be proven
• Repeat if goal is not proven
• After the goal is proven proceed with solution
  tuning

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Applying Hierarchical PSO to CBACandidate
Solution Tuning process

• Process candidate array elements in random
  orderwhile(elements remain)
• Select candidate array element that has a value
  of 1
• Assign it a value of 0
• If goal not still proven make element 1 again
• Repair and solution tuning are performed on
  initial population in addition to unfeasible
  solution

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Applying Hierarchical PSO to CBA Hierarchical
PSO parameters

• height h3
• degree d5
• number of particles m31
• F1F21.494
• Vmax6
• a starting at 0.729 and decreasing to 0.4 across
  500 iterations
• Based onI.C. Trelea, The particle swarm
  optimization algorithm convergence analysis and
  parameter selection, Information Processing
  Letters, Vol. 85, pp. 317-325, 2003. S.
  Janson, and M. Middendorf, A hierarchical
  particle swarm optimizer and its adaptive
  variant, IEEE Transactions on Systems, Man and
  Cybernetics, Part B Cybernetics, Vol. 35, No. 6,
  December 2005.

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Experimental Results

• Summary of 3,584 trials
• Median 12,119
• Std Dev 350.35
• Mean Time (sec) 125.88
• Std Dev Time 54.55

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Experimental Results
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Experimental ResultsCompared with Simulated
Annealing
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Future Work

• Improve performance using a adaptive HPSO S.
  Janson, and M. Middendorf, A hierarchical
  particle swarm optimizer and its adaptive
  variant, IEEE Transactions on Systems, Man and
  Cybernetics, Part B Cybernetics, Vol. 35, No. 6,
  December 2005.
• Hybrid HPSO and SA