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
Outline Problem: Combining Models from Different Contexts Theory: Solomonoff Operator Induction and Beta Distribution Practice: Inference Control Meta-Learning 3
Outline Problem: Combining Models from Different Contexts Theory: Solomonoff Operator Induction and Beta Distribution Practice: Inference Control Meta-Learning 4
Problem: Models from different contexts How to combine models obtained from different contexts? 5 Large Contexts → Underfit Small Contexts → Overfit
Problem: Preserve Uncertainty 6
Problem: Preserve Uncertainty Exploration vs Exploitation (Thompson Sampling) 7
Problem: ImplicationLink Second Order ImplicationLink <TV> ≡ P ( S | R ) R S Beta Distribution in disguise 8
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 9
Outline Problem: Combining Models from Different Contexts Theory: Solomonoff Operator Induction and Beta Distribution Practice: Inference Control Meta-Learning 10
Outline Problem: Combining Models from Different Contexts Theory: Solomonoff Operator Induction and Beta Distribution Practice: Inference Control Meta-Learning 11
Solomonoff Operator Induction Bayesian Model Averaging + Universal Distribution Probability Estimate: n + 1 ˆ a j � � O j ( A i | Q i ) P ( A n + 1 | Q n + 1 ) = 0 j i = 1 where: • Q i = i th question • A i = i th answer • O j = j th operator 0 = prior of j th operator • a j 12
Beta Distribution Operator Specialization of Solomonoff Operator Induction OpenCog implication link ImplicationLink <TV> R S ≡ Class of parameterized operators if A i = A n + 1 p , O j p ( A i | Q i ) = if R j ( Q i ) then 1 − p , otherwise 13
Beta Distribution Probability Density Function: pdf α,β ( x ) = x α − 1 ( 1 − x ) β − 1 B ( α, β ) Beta Function: � x p α − 1 ( 1 − p ) β − 1 dp B x ( α, β ) = 0 B ( α, β ) = B 1 ( α, β ) Conjugate Prior: pdf m + α, n − m + β ( x ) ∝ x m ( 1 − x ) n − m pdf α,β ( x ) 14
Artificial Completion p , if A i = A n + 1 O j if R j ( Q i ) then ( A i | Q i ) = p 1 − p , otherwise 15
Artificial Completion p , if A i = A n + 1 O j if R j ( Q i ) then p , C ( A i | Q i ) = 1 − p , otherwise else C ( A i | Q i ) 15
Second Order Solomonoff Operator Induction Probability Estimate: n + 1 ˆ a j � � O j ( A i | Q i ) P ( A n + 1 | Q n + 1 ) = 0 j i = 1 Probability Distribution Estimate: n ˆ a j � � O j ( A i | Q i ) cdf ( A n + 1 | Q n + 1 ) ( x ) = 0 O j ( A n + 1 | Q n + 1 ) ≤ x i = 1 16
Combing Solomonoff Operator Induction and Beta Distributions ˆ a j � 0 r j B x ( m j + α, n j − m j + β ) B ( m j + α, n j − m j + β ) cdf ( A n + 1 | Q n + 1 ) ( x ) ∝ j where • n j = number of observations explained by j th model • m j = number of true observations explained by j th model • r j = likelihood of the unexplained data r j =??? 17
Combing Solomonoff Operator Induction and Beta Distributions ˆ a j � 0 r j B x ( m j + α, n j − m j + β ) B ( m j + α, n j − m j + β ) cdf ( A n + 1 | Q n + 1 ) ( x ) ∝ j where • n j = number of observations explained by j th model • m j = number of true observations explained by j th model • r j = likelihood of the unexplained data r j =??? ≈ 2 − v ( 1 − c ) • v = n − n j = number of unexplained observations • c = compressability parameter • c = 1 → explains remaining data • c = 0 → can’t explain remaining data 17
Outline Problem: Combining Models from Different Contexts Theory: Solomonoff Operator Induction and Beta Distribution Practice: Inference Control Meta-Learning 18
Outline Problem: Combining Models from Different Contexts Theory: Solomonoff Operator Induction and Beta Distribution Practice: Inference Control Meta-Learning 19
Inference Control Meta-learning Learn how to reason efficiently 20
Inference Control Meta-learning Learn how to reason efficiently Methodology: 1. Solve sequence of problems (via reasoning) 20
Inference Control Meta-learning Learn how to reason efficiently Methodology: 1. Solve sequence of problems (via reasoning) 2. Store inference traces 20
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 20
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> 20
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 20
Combine Control Rules Implication <TV1> Implication <TV2> And And <inference-pattern-1> <inference-pattern-2> deduction-rule deduction-rule <good-inference> <good-inference> ⊕ c = 1 = 21
Combine Control Rules Implication <TV1> Implication <TV2> And And <inference-pattern-1> <inference-pattern-2> deduction-rule deduction-rule <good-inference> <good-inference> ⊕ c = 0 . 5 = 22
Combine Control Rules Implication <TV1> Implication <TV2> And And <inference-pattern-1> <inference-pattern-2> deduction-rule deduction-rule <good-inference> <good-inference> ⊕ c = 0 . 1 = 23
Conclusion Contribution: • Second Order Solomonoff Operator Induction • Specialized for Beta Distribution • Attempt to Deal with Partial Models 24
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) 24
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! 24
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