1. Markov Logic Networks: A Step Towards a Unified Theory of Learning and Cognition
Pedro Domingos
Dept. of Computer Science & Eng.
University of Washington
Joint work with Jesse Davis, Stanley Kok, Daniel Lowd, Hoifung Poon, Matt Richardson, Parag Singla, Marc Sumner

2. One Algorithm
Observation: The cortex has the same architecture throughout
Hypothesis: A single learning/inference algorithm underlies all cognition
Let’s discover it!

3. The Neuroscience Approach
Map the brain and figure out how it works
Problem: We don’t know nearly enough

4. The Engineering Approach
Pick a task (e.g., object recognition)
Figure out how to solve it
Generalize to other tasks
Problem: Any one task is too impoverished

5. The Foundational Approach
Consider all tasks the brain does
Figure out what they have in common
Formalize, test and repeat
Advantage: Plenty of clues
Where to start?
The grand aim of science is to cover the greatest number of experimental facts by logical deduction from the smallest number of hypotheses or axioms.
Albert Einstein

6. Recurring Themes
Noise, uncertainty, incomplete information → Probability / Graphical models
Complexity, many objects & relations → First-order logic

7. Examples Field Logical approach Statistical approach
Knowledge representation
First-order logic
Graphical models
Automated reasoning
Satisfiability testing
Markov chain Monte Carlo
Machine learning
Inductive logic programming
Neural networks
Planning
Classical planning
Markov decision processes
Natural language
processing
Definite clause grammars
Prob. context-free grammars 7

8. Research Plan Unify graphical models and first-order logic
Develop learning and inference algorithms
Apply to wide variety of problems
This talk: Markov logic networks

9. Markov Logic Networks MLN = Set of 1st-order formulas with weights
Formula = Feature template (Vars→Objects)
E.g., Ising model:
Most graphical models are special cases
First-order logic is infinite-weight limit
Up(x) ^ Neighbor(x,y) => Up(y)
Weight of formula i
No. of true instances of formula i in x

10. MLN Algorithms: The First Three Generations
Problem
First generation
Second generation
Third generation
MAP inference
Weighted satisfiability
Lazy inference
Cutting planes
Marginal inference
Gibbs sampling
MC-SAT
Lifted inference
Weight learning
Pseudo-likelihood
Voted perceptron
Scaled conj. gradient
Structure learning
Inductive logic progr.
ILP + PL (etc.)
Clustering + pathfinding

11. Weighted Satisfiability
SAT: Find truth assignment that makes all formulas (clauses) true
Huge amount of research on this problem
State of the art: Millions of vars/clauses in minutes
MaxSAT: Make as many clauses true as possible
Weighted MaxSAT: Clauses have weights; maximize satisfied weight
MAP inference in MLNs is just weighted MaxSAT
Best current solver: MaxWalkSAT

12. MC-SAT Deterministic dependences break MCMC
In practice, even strong probabilistic ones do
Swendsen-Wang:
Introduce aux. vars. u to represent constraints among x
Alternately sample u | x and x | u.
But Swendsen-Wang only works for Ising models
MC-SAT: Generalize S-W to arbitrary clauses
Uses SAT solver to sample x | u.
Orders of magnitude faster than Gibbs sampling, etc.

13. Lifted Inference Consider belief propagation (BP)
Often in large problems, many nodes are interchangeable: They send and receive the same messages throughout BP
Basic idea: Group them into supernodes, forming lifted network
Smaller network → Faster inference
Akin to resolution in first-order logic

14. Belief Propagation
Features (f)
Nodes (x) 14

15. Lifted Belief Propagation
Features (f)
Nodes (x) 15

16. Lifted Belief Propagation
, :
Functions of edge counts  
Features (f)
Nodes (x) 16

17. Weight Learning
Pseudo-likelihood + L-BFGS is fast and robust but can give poor inference results
Voted perceptron: Gradient descent + MAP inference
Problem: Multiple modes
Not alleviated by contrastive divergence
Alleviated by MC-SAT
Start each MC-SAT run at previous end state

18. Weight Learning (contd.)
Problem: Extreme ill-conditioning
Solvable by quasi-Newton, conjugate gradient, etc.
But line searches require exact inference
Stochastic gradient not applicable because data not i.i.d.
Solution: Scaled conjugate gradient
Use Hessian to choose step size
Compute quadratic form inside MC-SAT
Use inverse diagonal Hessian as preconditioner

19. Structure Learning
Standard inductive logic programming optimizes the wrong thing
But can be used to overgenerate for L1 pruning
Our approach: ILP + Pseudo-likelihood + Structure priors
For each candidate structure change: Start from current weights & relax convergence
Use subsampling to compute sufficient statistics
Search methods: Beam, shortest-first, etc.

20. Applications to Date Natural language processing
Information extraction
Entity resolution
Link prediction
Collective classification
Social network analysis
Robot mapping
Activity recognition
Scene analysis
Computational biology
Probabilistic Cyc
Personal assistants
Etc.

21. Unsupervised Semantic Parsing
Goal
Microsoft buys Powerset.
BUYS(MICROSOFT,POWERSET)
Challenge
Microsoft buys Powerset
Microsoft acquires semantic search engine Powerset
Powerset is acquired by Microsoft Corporation
The Redmond software giant buys Powerset
Microsoft’s purchase of Powerset, …
USP
Recursively cluster expressions
composed of similar subexpressions
Evaluation
Extract knowledge from biomedical abstracts
and answer questions
Substantially outperforms state of the art
Three times as many correct answers; accuracy 88%

22. Research Directions Compact representations
Deep architectures
Boolean decision diagrams
Arithmetic circuits
Unified inference procedure
Learning MLNs with many latent variables
Tighter integration of learning and inference
End-to-end NLP system
Complete agent

23. Resources Open-source software/Web site: Alchemy
Learning and inference algorithms
Tutorials, manuals, etc.
MLNs, datasets, etc.
Publications
Book: Domingos & Lowd, Markov Logic, Morgan & Claypool, 2009.
alchemy.cs.washington.edu