1. Introduction to Machine Learning and Knowledge Representation
Florian Gyarfas
COMP (Robotics)

2. Outline Introduction, definitions Common knowledge representations
Types of learning
Deductive Learning (rules of inference)
Explanation-based learning
Inductive Learning – some approaches
Concept Learning
Decision-Tree Learning
Clustering
Summary, references

3. Introduction What is Knowledge Representation?
Formalisms that represent knowledge (facts about the worlds) and mechanisms to manipulate such knowledge (for example, derive new facts from existing knowledge)
What is Machine Learning?
Mitchell: “A computer program is said to learn from experience E with respect to some class of tasks T and performance measure P, if its performance at tasks in T, as measured by P, improves with experience E.”
Example: T: playing checkers P: percent of games won against opponents E: playing practice games against itself

4. Common Knowledge Representations
Logic
propositional
predicate
other
Structured Knowledge Representations:
Frames
Semantic nets …

5. Propositional Logic Consists of: Constants: true/false
Set of elements called symbols, variables or atomic formulas (typically letters: a,b,c,…)
Operators (, , , , )
Axioms
Examples:
a  b  a
a  a  true
Inference rule(s)
Modus ponens

6. Predicate logic In many cases, propositional logic is too weak.
For example, how can we express something like this in propositional logic:
Every person is mortal. Tom is a person. Tom is mortal.
How can we represent these sentences in such a way that we can infer the third sentence from the first two?
Need quantifiers and predicates…
x: Person(x)  Mortal(x) Person(Tom)
We can infer: Mortal(Tom)
In addition to propositional logic, predicate logic has:
Functions
Predicates
Quantifiers (,)
More axioms
One more inference rule (Generalization)

7. Logic: Inference Rules
Used for deductive learning
Propositional Logic: Modus Ponens is all you need
If P, then Q. P Therefore, Q.
Meta-rule, not the same as axioms
Predicate Logic: Modus Ponens and Generalization Rule:

8. Structured Knowledge Representations
Semantic nets
Really just graphs that represent knowledge
Nodes represent concepts
Arcs represent binary relationships between concepts
Frames
Extension of semantic nets
Entities have attributes
Class/subclass hierarchy that supports inheritance; classes inherit attributes from superclasses

9. Semantic Nets Example (E. Rich, Artificial Intelligence) Furniture
is-a
is-part
Person
Chair
Seat
is-a
is-a
owner
color Me
My-chair
Tan
covering
is-a
Leather
Brown

10. Types of Learning Deductive – Inductive Supervised – Unsupervised
Symbolic – Non-symbolic

11. Deductive vs. inductive
Deductive learning
Knowledge is deduced from existing knowledge by means of truth-preserving transformations (this is nothing more than reformulation of existing knowledge).
If the premises are true, the conclusion must be true!
Example: propositional/predicate logic with rules of inference

12. Deductive vs. inductive
Inductive learning
Generalization from examples
Example: “All observed crows are black”  “All crows are black”
process of reasoning in which the premises of an argument are believed to support the conclusion but do not ensure it

13. Unsupervised vs. Supervised
Supervised Learning:
There exists a “teacher” that for each training example tells the learner how it is classified (training data consists of pairs of input vectors and desired outputs)
Reinforcement Learning:
No input/output pairs; “reward function” tells agent how good its action was
Unsupervised Learning:
No a priori output; also no reward; training data just feature vectors; the system needs to form concepts (classes) by itself

14. Learning approaches Deductive/Analytical Inductive
Explanation-based learning
Inductive
Supervised:
Concept Learning
Decision-Tree Learning
Neural networks
Naive Bayes classifier
Support Vector Machines …
Unsupervised:
Clustering
Expectation-Maximization

15. Explanation-based learning (EBL)
Deductive
assumes prior knowledge (“domain theory”) in addition to training examples
assumes domain theory is given as a set of horn clauses
Horn clause: Disjunction of literals with at most one positive literal
Example: NOT a OR NOT b OR NOT c OR d which is the same as (a AND b AND c)  d
Tries to explain training examples using the domain theory

16. EBL example Consider multiple physical objects
Which are the pairs of objects such that one can be stacked safely on the other?
Target concept:
premise(x,y)  SafeToStack(x,y)
where premise is a conjunctive expression containing the variables x and y.
Domain Theory:
SafeToStack(x,y)  NOT Fragile(y)
SafeToStack(x,y)  Lighter(x,y)
Lighter(x,y)  Weight(x,wx) AND Weight(y,wy) AND LessThan(wx,wy)
Weight(x,w)  Volume(x,v) AND Density(x,d) AND Equal(w,times(v,d))
Weight(x,5)  Type(x,Table)
Fragile(x)  Material(x,Glass)

17. EBL example (2) Training example: On(Obj1,Obj2) Type(Obj1,Box)
Type(Obj2,Table)
Color(Obj1,Red)
Color(Obj2,Blue)
Volume(Obj1,2)
Density(Obj1,0.3)
Material(Obj1,Cardboard)
Material(Obj2,Wood)
SafeToStack(Obj1,Obj2)

18. EBL example (3) Explanation: SafeToStack(Obj1,Obj2) Lighter(Obj1,Obj2)
Weight(Obj1,Obj2)
Weight(Obj2,5)
Volume(Obj1,2)
Density(Obj1,0.3)
Equal(0.6,2*0.3)
LessThan(0.6,5)
Type(Obj2,Table)

19. EBL algorithm Explain training example Analyze/Generalize Explanation
Add Explanation to Learned Rules (Domain Theory)
Use for example REGRESS algorithm (Mitchell, p. 318) for step (2)
In our example most general rule that can be justified by the explanation is:
SafeToStack(x,y)  Volume(x,vx) AND Density(x,dx) AND Equal(wx,times(vx,dx)) AND LessThan(wx,5) AND Type(y,Table)

20. EBL - Remarks
Knowledge Reformulation: EBL just restates what the learner already knows
You don’t really gain new knowledge!
Why do we need it then? In principle, we can compute everything we need using just the domain theory
In practice, however, this might not work. Consider chess: Does knowing all the rules make you a perfect player?
So EBL reformulates existing knowledge into a more operational form which might be much more effective especially under certain constraints

21. Concept Learning Inductive
learn general concept definition from specific training examples
search through predefined space of potential hypotheses for target concept
pick the one that best fits training examples

22. Concept Learning Example
Taken from Tom Mitchell’s book “Machine Learning”
Concept to learn: “Days on which Tom enjoys his favorite water sport”
Training examples D (every row is an instance, every column an attribute): #
Sky
AirTemp
Humidity
Wind
Water
Forecast
EnjoySport (Classification) 1
Sunny
Warm
Normal
Strong
Same
Yes 2
High 3
Rainy
Cold
Change No 4
Cool

23. Concept Learning
X = set of all instances (instance = combination of attributes), D = set of all training examples (D  X)
The “target concept” is a function (target function) c(x) that for a given instance x is either 0 or 1 (in our example 0 if EnjoySport = No, 1 if EnjoySport = Yes)
We do not know the target concept, but we can come up with hypotheses. We would like to find a hypothesis h(x) such that h(x) = c(x) at least for all x in D (D = training examples)
Hypotheses representation:
Let us assume that the target concept is expressed as a conjunction of constraints on the instance attributes. Then we can write a hypothesis like this: AirTemp = Cold  Humidity = High. Or, in short: <?,Cold,High,?,?,?> where “?” means any value is acceptable for this attribute. We use “” to indicate that no value is acceptable for an attribute.

24. Concept Learning Most general hypothesis: <?,?,?,?,?,?>
Most specific hypothesis:
<, , , , ,  >
General-to-specific ordering of hypotheses: ≥g (more general than or equal to) defines partial order over hypothesis space H
FIND-S algorithm
finds the most specific hypothesis
For our example : <Sunny,Warm,?,Strong,?,?>

25. Concept Learning (FIND-S)
FIND-S algorithm
Initialize h to the most specific hypothesis in H
For each positive training instance x
For each atttribute constraint ai in h If the constraint ai is satisfied by x Then do nothing Else replace ai in h by the next more general constraint that is satisfied by x
Output hypothesis h
Why can we ignore negative examples?

26. Concept Learning FIND-S only computes the most specific hypothesis
Another approach to concept learning: CANDIDATE-ELIMINATION
CANDIDATE-ELIMINATION finds all hypotheses in the the version space
The version space, denoted VSH,D, with respect to hypothesis space H and training examples D, is a subset of hypotheses from H consistent with the training examples in D.
VSH,D = {h  H|Consistent(h,D)}

27. Concept Learning (Version space)
Version space for our example
{<Sunny,Warm,?,Strong,?,?>}
{<Sunny,?,?,Strong,?,?>}
{<Sunny,Warm,?,?,?,?>}
{<?,Warm,?,Strong,?,?>}
{<Sunny,?,?,?,?,?>}, {<?,Warm,?,?,?,?>}

28. Concept Learning: Candiate-Elimination algorithm
Initialize G to the set of maximally general hypotheses in H
Initialize S to set of maximally specific hypotheses in H
For each training example d, do
If d is a positive example
Remove from G any hypothesis inconsistent with d
For each hypothesis in s in S that is not consistent with d
Remove s from S
Add to S all minimal generalizations h of s such that
h is consistent with d, and some member of G is more general than h
Remove from S any hypothesis that is more general than another hypothesis in S

29. Concept Learning: Candiate-Elimination algorithm
If d is a negative example
Remove from S any hypothesis inconsistent with d
For each hypothesis in g in G that is not consistent with d
Remove g from G
Add to G all minimal specializations h of g such that
h is consistent with d, and some member of S is more specific than h
Remove from G any hypothesis that is less general than another hypothesis in G
How to use version space for classification of new instances?
Both algorithms can’t handle noisy training data, i.e. they assume none of the training examples is incorrect
For more complex Concept Learning algorithms see Mitchell – Machine Learning, Chapter 10.

30. Inductive bias
For both algorithms, we assumed that the target concept was contained in the hypothesis space
Our hypothesis space was the set of all hypotheses than can be expressed as a conjunction of attributes
Such an assumption is called an inductive bias
What if target concept not a conjunction of constraints? Why not consider all possible hypotheses?

31. Decision Tree Learning
Another inductive, supervised learning method for approximating discrete-valued target functions
Learned function represented by a tree
Leaf nodes provide classification
Each node specifies a test of some attribute of the instance

32. Decision Tree Learning -Example
Decision tree for the concept “PlayTennis”

33. Decision Tree Learning
Classification starts at the root node
Example tree corresponds to the expression:
(Outlook = Sunny AND Humidity = Normal) OR (Outlook = Overcast) OR (Outlook = Rain AND Wind = Weak)
Using the tree to classify new instances is easy, but how do we construct a decision tree from training examples?

34. Decision Tree Learning: ID3 Algorithm
Constructs tree top-down
Order of attributes?
Evaluate each attribute using a statistical test (see next slide) to determine how well it alone classifies the training examples
Use the attribute that best classifies the training example attribute at the root node. Create descendants for each possible value of that attribute.
Repeat process for each descendant…

35. Decision Tree Learning: Entropy/Gain
Given a collection S, containing positive and negative examples of some target concept, the entropy of S is:
Entropy(S) = -p+ log2p+ – p- log2p-
Information gain is then defined as:
ID3 Algorithm picks attribute with highest gain

36. Decision Trees - Remarks
Tree represents hypothesis; thus, ID3 determines only a single hypothesis, unlike Candidate-Elimination
Unlike both Concept Learning approaches, ID3 makes no assumptions regarding the hypothesis space; every possible hypothesis can be represented by some tree
Inductive Bias: Shorter trees are preferred over larger trees

37. Decision Trees – Example (Mitchell, p. 59)
Day
Outlook
Temp.
Humid.
Wind
PlayTennis D1
Sunny
Hot
High
Weak No D2
Strong D3
Overcast
Yes D4
Rain
Mild D5
Cool
Normal D6 D7 D8 D9
D10
D11
D12
D13
D14

38. Decision Tree - Example
Gain(S, Outlook) = 0.246
Gain(S, Humidity) = 0.151
Gain(S, Wind) = 0.048
Gain(S, Temperature) = 0.029
So “Outlook” is the first attribute of the tree:
Ssunny = {D1,D2,D8,D9,D11}
Gain{Ssunny,Humidity} = 0.97 – (3/5)*0 – (2/5)*0 = 0.97
Gain{Ssunny,Temperature} = 0.97 – (2/5)*0 – (2/5)*1 – (1/5)*0 = 0.57
Gain{Ssunny,Wind} = 0.97 – (2/5)*1 – (3/5)*0.918 = 0.019
 Humidity should be tested next
Outlook
Sunny
Overcast
Rain ? ?
Yes
What next?

39. Learning algorithms in robotics
EBL has been used for Planning Algorithms (example: PRODIGY system (Carbonell et al., 1990))
Could use EBL, concept learning, decision tree learning for things like object recognition (CL and DTL can easily be extended to more than 2 categories)
However, while symbolic learning algorithms are simple and easy to understand, they are not very flexible and powerful
In many applications, robots need to learn to classify something they perceive (using cameras, sensors etc.)
Sensor inputs etc. are normally numeric
 Non-symbolic approaches such as NN, Reinforcement Learning or HMM are better suited and thus more commonly used in practice
Combined approaches exist, for example EBNN (Explanation-based Neural Networks) (See for example paper “Explanation Based Learning for Mobile Robot Perception” by J. O’Sullivan, T. Mitchell and S. Thrun)

40. Unsupervised Learning
Training data only consists of feature vectors, does not include classifications
Unsupervised Learning algorithms try to find patterns in the data
Classic examples:
Clustering
Fitting Gaussian Density functions to data
Dimensionality reduction

41. Clustering: k-means algorithm
Cluster objects based on attributes into k partitions
Tries to minimize intra-cluster variance:
Algorithm:
Algorithm starts by partitioning input points into k initial sets, either at random or using some heuristic data.
Then it calculates the mean point, or centroid, of each set.
Constructs new partition by associating each point with the closest centroid.
Recalculates centroids for new clusters
Repeats this until convergence, which is when points no longer switch clusters.

42. Summary
This presentation mainly covered symbol-based learning algorithms
Deductive
EBL
Inductive
Concept Learning
Decision-Tree Learning
Unsupervised
Clustering (not necessarily symbol-based), COBWEB
Non symbol-based learning algorithms such as neural networks part of the next lecture?

43. References (Books) Tom Mitchell: Machine Learning. McGraw-Hill, 1997.
Stuart Russell, Peter Norvig: Artificial Intelligence – a modern approach. Prentice Hall, 2003.
George Luger: Artificial Intelligence. Addison-Wesley, 2002.
Elaine Rich: Artificial Intelligence. McGraw-Hill, 1983.