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Introduction to Machine Learning and
Knowledge Representation
Florian Gyarfas
COMP 790-072 (Robotics)
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
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
Common Knowledge Representations
Logic
propositional
predicate
other
Structured Knowledge Representations:
Frames
Semantic nets
…
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
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)
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:
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
Semantic Nets
Example (E. Rich, Artificial Intelligence)
Furniture
is-a
Person
Chair
is-a
Me
is-part
Seat
is-a
owner
My-chair
covering
Leather
color
Tan
is-a
Brown
Types of Learning
Deductive
–
Inductive
Supervised
–
Unsupervised
Symbolic
–
Non-symbolic
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
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
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
Learning approaches
Deductive/Analytical
Explanation-based learning
Inductive
Supervised:
•
•
•
•
•
•
Concept Learning
Decision-Tree Learning
•
•
•
•
Clustering
Neural networks
Naive Bayes classifier
Support Vector Machines
…
Unsupervised:
Neural networks
Expectation-Maximization
…
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
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)
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)
EBL example (3)
Explanation:
SafeToStack(Obj1,Obj2)
Lighter(Obj1,Obj2)
Weight(Obj1,Obj2)
Volume(Obj1,2) Density(Obj1,0.3) Equal(0.6,2*0.3)
Weight(Obj2,5)
LessThan(0.6,5)
Type(Obj2,Table)
EBL algorithm
1) Explain training example
2) Analyze/Generalize Explanation
3) 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)
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
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
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
1
Sunny
Warm
Normal
Strong
Warm
Same
Yes
2
Sunny
Warm
High
Strong
Warm
Same
Yes
3
Rainy
Cold
High
Strong
Warm
Change
No
4
Sunny
Warm
High
Strong
Cool
Change
Yes
(Classification)
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.
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,?,?>
Concept Learning (FIND-S)
FIND-S algorithm
1. Initialize h to the most specific hypothesis in H
2. 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
3. Output hypothesis h
Why can we ignore negative examples?
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)}
Concept Learning
(Version space)
Version space for our example
{<Sunny,Warm,?,Strong,?,?>}
{<Sunny,?,?,Strong,?,?>}
{<Sunny,Warm,?,?,?,?>}
{<Sunny,?,?,?,?,?>}, {<?,Warm,?,?,?,?>}
{<?,Warm,?,Strong,?,?>}
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
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.
•
•
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?
Decision Tree Learning
Another inductive, supervised learning
method for approximating discretevalued target functions
Learned function represented by a tree
Leaf nodes provide classification
Each node specifies a test of some
attribute of the instance
Decision Tree Learning Example
Decision tree for the concept
“PlayTennis”
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?
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…
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- log2pInformation gain is then defined as:
Gain( S , A)  Entropy( S ) 

vValues( A )
Sv
S
Entropy( S v )
ID3 Algorithm picks attribute with highest gain
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
Decision Trees – Example
(Mitchell, p. 59)
Day
Outlook
Temp.
Humid.
Wind
PlayTennis
D1
Sunny
Hot
High
Weak
No
D2
Sunny
Hot
High
Strong
No
D3
Overcast
Hot
High
Weak
Yes
D4
Rain
Mild
High
Weak
Yes
D5
Rain
Cool
Normal
Weak
Yes
D6
Rain
Cool
Normal
Strong
No
D7
Overcast
Cool
Normal
Strong
Yes
D8
Sunny
Mild
High
Weak
No
D9
Sunny
Cool
Normal
Weak
Yes
D10
Rain
Mild
Normal
Weak
Yes
D11
Sunny
Mild
Normal
Strong
Yes
D12
Overcast
Mild
High
Strong
Yes
D13
Overcast
Hot
Normal
Weak
Yes
D14
Rain
Mild
High
Strong
No
Decision Tree - Example
Gain(S,
Gain(S,
Gain(S,
Gain(S,
Outlook)
Humidity)
Wind)
Temperature)
=
=
=
=
0.246
0.151
0.048
0.029
So “Outlook” is the first attribute of the tree:
Outlook
Sunny
?
Overcast
Yes
Rain
?
What next?
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
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 (Explanationbased Neural Networks) (See for example paper “Explanation
Based Learning for Mobile Robot Perception” by J. O’Sullivan, T.
Mitchell and S. Thrun)
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
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.
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?
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.