Probabilistic and Logistic Circuits: A New Synthesis of Logic and Machine Learning Guy Van den Broeck HRL/ACTIONS @ KR Oct 28, 2018
Foundation: Logical Circuit Languages
Negation Normal Form Circuits Δ = (sun ∧ rain ⇒ rainbow) [Darwiche 2002]
Decomposable Circuits Decomposable [Darwiche 2002]
Tractable for Logical Inference • Is there a solution? (SAT) ✓ – SAT( 𝛽 ∨ 𝛾 ) iff SAT( 𝛽 ) or SAT( 𝛾 ) ( always ) – SAT( 𝛽 ∧ 𝛾 ) iff SAT( 𝛽 ) and SAT( 𝛾 ) ( decomposable ) • How many solutions are there? (#SAT) • Complexity linear in circuit size
Deterministic Circuits Deterministic [Darwiche 2002]
How many solutions are there? (#SAT)
How many solutions are there? (#SAT) Arithmetic Circuit
Tractable for Logical Inference • Is there a solution? (SAT) ✓ ✓ • How many solutions are there? (#SAT) • Stricter languages (e.g., BDD, SDD): ✓ – Equivalence checking ✓ – Conjoin/disjoint/negate circuits • Complexity linear in circuit size • Compilation into circuit language by either – ↓ exhaustive SAT solver – ↑ conjoin/disjoin/negate
Learning with Logical Constraints
Motivation: Video [Lu, W. L., Ting, J. A., Little, J. J., & Murphy, K. P. (2013). Learning to track and identify players from broadcast sports videos.]
Motivation: Robotics [Wong, L. L., Kaelbling, L. P., & Lozano-Perez, T., Collision-free state estimation. ICRA 2012]
Motivation: Language • Non-local dependencies: At least one verb in each sentence • Sentence compression If a modifier is kept, its subject is also kept • Information extraction • Semantic role labeling … and many more! [Chang, M., Ratinov, L., & Roth, D. (2008). Constraints as prior knowledge],…, [ Chang, M. W., Ratinov, L., & Roth, D. (2012). Structured learning with constrained conditional models.], [https://en.wikipedia.org/wiki/Constrained_conditional_model]
Motivation: Deep Learning [Graves, A., Wayne, G., Reynolds, M., Harley, T., Danihelka, I., Grabska- Barwińska , A., et al.. (2016). Hybrid computing using a neural network with dynamic external memory. Nature , 538 (7626), 471-476.]
Running Example Courses: Data • Logic (L) • Knowledge Representation (K) • Probability (P) • Artificial Intelligence (A) Constraints • Must take at least one of Probability or Logic. • Probability is a prerequisite for AI. • The prerequisites for KR is either AI or Logic.
Structured Space unstructured structured L K P A L K P A 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 • Must take at least one of 0 0 1 0 0 0 1 0 Probability ( P ) or Logic ( L ). 0 0 1 1 0 0 1 1 • Probability is a prerequisite 0 1 0 0 0 1 0 0 for AI ( A ). 0 1 0 1 0 1 0 1 • 0 1 1 0 The prerequisites for KR ( K ) is 0 1 1 0 0 1 1 1 0 1 1 1 either AI or Logic. 1 0 0 0 1 0 0 0 1 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 7 out of 16 instantiations 1 0 1 1 1 0 1 1 are impossible 1 1 0 0 1 1 0 0 1 1 0 1 1 1 0 1 1 1 1 0 1 1 1 0 1 1 1 1 1 1 1 1
Boolean Constraints unstructured structured L K P A L K P A 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 1 0 0 0 1 1 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 1 0 1 0 1 0 1 1 0 0 1 1 0 0 1 1 1 0 1 1 1 1 0 0 0 1 0 0 0 1 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 7 out of 16 instantiations 1 0 1 1 1 0 1 1 are impossible 1 1 0 0 1 1 0 0 1 1 0 1 1 1 0 1 1 1 1 0 1 1 1 0 1 1 1 1 1 1 1 1
Learning in Structured Spaces + Data Constraints (Background Knowledge) (Physics) Learn ML Model Today‟s machine learning tools don‟t take knowledge as input!
Deep Learning with Logical Constraints
Deep Learning + Data Constraints with Deep Neural Learn Logical Knowledge Network Neural Network Logical Constraint Output Input Output is probability vector p , not Boolean logic!
Semantic Loss Q: How close is output p to satisfying constraint? Answer: Semantic loss function L( α , p ) • Axioms, for example: – If p is Boolean then L( p,p ) = 0 – If α implies β then L( α , p ) ≥ L(β , p ) ( α more strict ) • Properties: SEMANTIC – If α is equivalent to β then L( α , p ) = L( β , p ) Loss! – If p is Boolean and satisfies α then L( α , p ) = 0
Semantic Loss: Definition Theorem: Axioms imply unique semantic loss: Probability of getting x after flipping coins with prob. p Probability of satisfying α after flipping coins with prob. p
Example: Exactly-One • Data must have some label We agree this must be one of the 10 digits: • Exactly-one constraint 𝒚 𝟐 ∨ 𝒚 𝟑 ∨ 𝒚 𝟒 ¬𝒚 𝟐 ∨ ¬𝒚 𝟑 → For 3 classes: ¬𝒚 𝟑 ∨ ¬𝒚 𝟒 • Semantic loss: ¬𝒚 𝟐 ∨ ¬𝒚 𝟒 Only 𝒚 𝒋 = 𝟐 after flipping coins Exactly one true 𝒚 after flipping coins
Semi-Supervised Learning • Intuition: Unlabeled data must have some label • Minimize exactly-one semantic loss on unlabeled data Train with 𝑓𝑦𝑗𝑡𝑢𝑗𝑜 𝑚𝑝𝑡𝑡 + 𝑥 ∙ 𝑡𝑓𝑛𝑏𝑜𝑢𝑗𝑑 𝑚𝑝𝑡𝑡
MNIST Experiment Competitive with state of the art in semi-supervised deep learning
FASHION Experiment Same conclusion on CIFAR10 Outperforms Ladder Nets!
What about real constraints? Paths cf. Nature paper Good variable assignment Bad variable assignment (does not represent route) (represents route) 184 16,777,032 Unstructured probability space: 184+16,777,032 = 2 24 Space easily encoded in logical constraints [Nishino et al.]
How to Compute Semantic Loss? • In general: #P-hard • With a logical circuit for α : Linear! • Example: exactly-one constraint: L( α , p ) = L( , p ) = - log( ) • Why? Decomposability and determinism!
Predict Shortest Paths Add semantic loss for path constraint Is output Is prediction Are individual a path? the shortest path? edge predictions This is the real task! correct? (same conclusion for predicting sushi preferences, see paper)
Probabilistic Circuits
Logical Circuits L K L P A P L P A L K L P L P P K K A A A A Can we represent a distribution over the solutions to the constraint?
Recall: Decomposability L K L P A P L P A L K L P L P P K K A A A A AND gates have disjoint input circuits
Recall: Determinism L ⊥ ¬ P ⊥ ¬ L ⊥ L ⊥ ¬ P ⊥ ¬ L K P A L ¬ P ¬ A P ¬ L ¬ K P K ¬ K A ¬ A A ¬ A Input: L, K, P, A are true and ¬L, ¬K, ¬P, ¬A are false Property: OR gates have at most one true input wire
PSDD: Probabilistic SDD 0.1 0.6 0.3 1 0 1 0 1 0 0.6 0.4 1 0 1 0 L ⊥ ¬ P ⊥ L ⊥ ¬ P ⊥ ¬ L ⊥ ¬ L K P A L ¬ P ¬ A P ¬ L ¬ K P 0.8 0.2 0.25 0.75 0.9 0.1 K ¬ K A ¬ A A ¬ A Syntax: assign a normalized probability to each OR gate input
PSDD: Probabilistic SDD 0.1 0.6 0.3 0.6 1 0 1 0 1 0 0.4 1 0 1 0 L ⊥ ¬ P ⊥ ¬ L ⊥ L ⊥ ¬ P ⊥ ¬ L K P A L ¬ P ¬ A P ¬ L ¬ K P Input: L, K, P, A 0.8 0.2 0.75 0.25 0.9 0.1 are true A ¬ A A ¬ A K ¬ K Pr( L,K,P,A ) = 0.3 x 1 x 0.8 x 0.4 x 0.25 = 0.024
Each node represents a normalized distribution! 0.1 0.6 0.3 1 0 1 0 1 0 0.6 0.4 1 0 1 0 L K L P A P L K L P L P A L P P 0.8 0.2 0.25 0.75 0.9 0.1 A A A A A A Can read probabilistic independences off the circuit structure
Tractable for Probabilistic Inference • MAP inference : Find most-likely assignment to x given y (otherwise NP-hard) • Computing conditional probabilities Pr(x|y) (otherwise #P-hard) • Sample from Pr(x|y) • Algorithms linear in circuit size (pass up, pass down, similar to backprop)
Parameters are Interpretable 0.1 0.6 0.3 Probability of course P given L 1 0 1 0 1 0 0.6 0.4 1 0 1 0 L K L P L K L P A P L P A L P P 0.8 0.2 0.25 0.75 0.9 0.1 K K A A A A Student takes course P Student takes course L Explainable AI DARPA Program
Learning Probabilistic Circuit Parameters
Learning Algorithms • Closed form max likelihood from complete data • One pass over data to estimate Pr(x|y) Not a lot to say: very easy! • Where does the structure come from? For now: simply compiled from constraint…
Combinatorial Objects: Rankings rank sushi rank sushi 1 fatty tuna 1 shrimp 10 items : 2 sea urchin 2 sea urchin 3,628,800 3 salmon roe 3 salmon roe rankings 4 shrimp 4 fatty tuna 5 tuna 5 tuna 20 items : 6 squid 6 squid 2,432,902,008,176,640,000 7 tuna roll 7 tuna roll rankings 8 see eel 8 see eel 9 egg 9 egg cucumber cucumber 10 10 roll roll
Combinatorial Objects: Rankings • Predict Boolean Variables: rank sushi A ij - item i at position j 1 fatty tuna 2 sea urchin • Constraints: 3 salmon roe 4 shrimp each item i assigned to 5 tuna a unique position ( n constraints) 6 squid 7 tuna roll 8 see eel each position j assigned 9 egg a unique item ( n constraints) cucumber 10 roll
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