Partial Operator Induction with Beta Distribution Nil Geisweiller - - PowerPoint PPT Presentation

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Partial Operator Induction with Beta Distribution Nil Geisweiller - - PowerPoint PPT Presentation

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Partial Operator Induction with Beta Distribution Nil Geisweiller AGI-18 Prague 1 Problem: Combining Models from Different Contexts Theory: Solomonoff Operator Induction and Beta Distribution Practice: Inference Control Meta-Learning 2

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Partial Operator Induction with Beta Distribution

Nil Geisweiller AGI-18 Prague

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Problem: Combining Models from Different Contexts Theory: Solomonoff Operator Induction and Beta Distribution Practice: Inference Control Meta-Learning

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Outline

Problem: Combining Models from Different Contexts Theory: Solomonoff Operator Induction and Beta Distribution Practice: Inference Control Meta-Learning

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Outline

Problem: Combining Models from Different Contexts Theory: Solomonoff Operator Induction and Beta Distribution Practice: Inference Control Meta-Learning

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Problem: Models from different contexts

How to combine models obtained from different contexts? Large Contexts → Underfit Small Contexts → Overfit

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Problem: Preserve Uncertainty

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Problem: Preserve Uncertainty

Exploration vs Exploitation (Thompson Sampling)

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Problem: ImplicationLink

ImplicationLink <TV> R S

≡ Second Order P(S|R) Beta Distribution in disguise

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Solution

Bayesian Model Averaging / Solomonoff Operator Induction, modified to:

  • 1. Support partial models
  • 2. Produce a probability distribution estimate, rather than

probability estimate.

  • 3. Specialize for Beta distributions

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Outline

Problem: Combining Models from Different Contexts Theory: Solomonoff Operator Induction and Beta Distribution Practice: Inference Control Meta-Learning

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Outline

Problem: Combining Models from Different Contexts Theory: Solomonoff Operator Induction and Beta Distribution Practice: Inference Control Meta-Learning

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Solomonoff Operator Induction

Bayesian Model Averaging + Universal Distribution Probability Estimate: ˆ P(An+1|Qn+1) =

  • j

aj

n+1

  • i=1

Oj(Ai|Qi) where:

  • Qi = ith question
  • Ai = ith answer
  • Oj = jth operator
  • aj

0 = prior of jth operator 12

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Beta Distribution Operator

Specialization of Solomonoff Operator Induction OpenCog implication link

ImplicationLink <TV> R S

≡ Class of parameterized operators Oj

p(Ai|Qi) = if Rj(Qi) then

   p, if Ai = An+1 1 − p,

  • therwise

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Beta Distribution

Probability Density Function: pdfα,β(x) = xα−1(1 − x)β−1 B(α, β) Beta Function: Bx(α, β) = x pα−1(1−p)β−1dp B(α, β) = B1(α, β) Conjugate Prior: pdfm+α,n−m+β(x) ∝ xm(1−x)n−mpdfα,β(x)

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Artificial Completion

Oj

p

(Ai|Qi) = if Rj(Qi) then    p, if Ai = An+1 1 − p,

  • therwise

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Artificial Completion

Oj

p,C(Ai|Qi) =

if Rj(Qi) then    p, if Ai = An+1 1 − p,

  • therwise

else C(Ai|Qi)

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Second Order Solomonoff Operator Induction

Probability Estimate: ˆ P(An+1|Qn+1) =

  • j

aj

n+1

  • i=1

Oj(Ai|Qi) Probability Distribution Estimate: ˆ cdf (An+1|Qn+1)(x) =

  • Oj(An+1|Qn+1)≤x

aj

n

  • i=1

Oj(Ai|Qi)

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Combing Solomonoff Operator Induction and Beta Distributions

ˆ cdf (An+1|Qn+1)(x) ∝

  • j

aj

0r jBx(mj+α, nj−mj+β)B(mj+α, nj−mj+β)

where

  • nj = number of observations explained by jth model
  • mj = number of true observations explained by jth model
  • r j = likelihood of the unexplained data

r j =???

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Combing Solomonoff Operator Induction and Beta Distributions

ˆ cdf (An+1|Qn+1)(x) ∝

  • j

aj

0r jBx(mj+α, nj−mj+β)B(mj+α, nj−mj+β)

where

  • nj = number of observations explained by jth model
  • mj = number of true observations explained by jth model
  • r j = likelihood of the unexplained data

r j =??? ≈ 2−v(1−c)

  • v = n − nj = number of unexplained observations
  • c = compressability parameter
  • c = 1 → explains remaining data
  • c = 0 → can’t explain remaining data

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Outline

Problem: Combining Models from Different Contexts Theory: Solomonoff Operator Induction and Beta Distribution Practice: Inference Control Meta-Learning

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Outline

Problem: Combining Models from Different Contexts Theory: Solomonoff Operator Induction and Beta Distribution Practice: Inference Control Meta-Learning

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Inference Control Meta-learning

Learn how to reason efficiently

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Inference Control Meta-learning

Learn how to reason efficiently Methodology:

  • 1. Solve sequence of problems (via reasoning)

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Inference Control Meta-learning

Learn how to reason efficiently Methodology:

  • 1. Solve sequence of problems (via reasoning)
  • 2. Store inference traces

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Inference Control Meta-learning

Learn how to reason efficiently Methodology:

  • 1. Solve sequence of problems (via reasoning)
  • 2. Store inference traces
  • 3. Mine traces to discover patterns

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Inference Control Meta-learning

Learn how to reason efficiently Methodology:

  • 1. Solve sequence of problems (via reasoning)
  • 2. Store inference traces
  • 3. Mine traces to discover patterns
  • 4. Build control rules

Implication <TV> And <inference-pattern> <rule> <good-inference>

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Inference Control Meta-learning

Learn how to reason efficiently Methodology:

  • 1. Solve sequence of problems (via reasoning)
  • 2. Store inference traces
  • 3. Mine traces to discover patterns
  • 4. Build control rules

Implication <TV> And <inference-pattern> <rule> <good-inference>

  • 5. Combine control rules to guide future reasoning

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Combine Control Rules

Implication <TV1> And <inference-pattern-1> deduction-rule <good-inference>

⊕

c = 1 Implication <TV2> And <inference-pattern-2> deduction-rule <good-inference>

=

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Combine Control Rules

Implication <TV1> And <inference-pattern-1> deduction-rule <good-inference>

⊕

c = 0.5 Implication <TV2> And <inference-pattern-2> deduction-rule <good-inference>

=

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Combine Control Rules

Implication <TV1> And <inference-pattern-1> deduction-rule <good-inference>

⊕

c = 0.1 Implication <TV2> And <inference-pattern-2> deduction-rule <good-inference>

=

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Conclusion

Contribution:

  • Second Order Solomonoff Operator Induction
  • Specialized for Beta Distribution
  • Attempt to Deal with Partial Models

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Conclusion

Contribution:

  • Second Order Solomonoff Operator Induction
  • Specialized for Beta Distribution
  • Attempt to Deal with Partial Models

Future Work:

  • Improve Likelihood of Unexplained Data
  • More Experiments (Inference Control Meta-learning)

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Conclusion

Contribution:

  • Second Order Solomonoff Operator Induction
  • Specialized for Beta Distribution
  • Attempt to Deal with Partial Models

Future Work:

  • Improve Likelihood of Unexplained Data
  • More Experiments (Inference Control Meta-learning)

Thank you!

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