Introduction to Machine Learning and Knowledge Representation

1
Introduction to Machine Learning and Knowledge
Representation

• Florian Gyarfas
• COMP 790-072 (Robotics)

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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

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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

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Common Knowledge Representations

• Logic
• propositional
• predicate
• other
• Structured Knowledge Representations
• Frames
• Semantic nets

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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

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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)

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Logic Inference Rules

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

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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

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Semantic Nets

• Example (E. Rich, Artificial Intelligence)

Furniture
is-a
is-part
Chair
Person
Seat
is-a
is-a
owner
color
My-chair
Me
Tan
is-a
covering
Leather
Brown
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Types of Learning

• Deductive Inductive
• Supervised Unsupervised
• Symbolic Non-symbolic

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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

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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

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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

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Learning approaches

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

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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

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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)

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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)

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EBL example (3)

• Explanation

SafeToStack(Obj1,Obj2)
Lighter(Obj1,Obj2)
Weight(Obj1,Obj2)
Weight(Obj2,5)
Density(Obj1,0.3)
Type(Obj2,Table)
Volume(Obj1,2)
Equal(0.6,20.3)
LessThan(0.6,5)
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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)

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EBL - Remarks

• Knowledge Reformulation EBL just restates what
  the learner already knows
• You dont 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

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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

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Concept Learning Example

• Taken from Tom Mitchells 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 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
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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 lt?,Cold,High,?,?,?gt where ?
  means any value is acceptable for this attribute.
  We use ? to indicate that no value is
  acceptable for an attribute.

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Concept Learning

• Most general hypothesis
• lt?,?,?,?,?,?gt
• Most specific hypothesis
• lt?, ?, ?, ?, ?, ? gt
• 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 ltSunny,Warm,?,Strong,?,?gt

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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?

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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 ? HConsistent(h,D)

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Concept Learning (Version space)

• Version space for our example

ltSunny,Warm,?,Strong,?,?gt
ltSunny,?,?,Strong,?,?gt
ltSunny,Warm,?,?,?,?gt
lt?,Warm,?,Strong,?,?gt
ltSunny,?,?,?,?,?gt, lt?,Warm,?,?,?,?gt
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Concept LearningCandiate-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

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Concept LearningCandiate-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 cant 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.

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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?

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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

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Decision Tree Learning -Example

• Decision tree for the concept PlayTennis

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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?

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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

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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

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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

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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
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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
• GainSsunny,Humidity 0.97 (3/5)0 (2/5)0
  0.97
• GainSsunny,Temperature 0.97 (2/5)0
  (2/5)1 (1/5)0 0.57
• GainSsunny,Wind 0.97 (2/5)1 (3/5)0.918
  0.019

Outlook
Sunny
Rain
Overcast
?
?
Yes
What next?
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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. OSullivan, T.
  Mitchell and S. Thrun)

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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

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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.

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COBWEB algorithm

• Incremental clustering algorithm
• Data structure is a tree (Categorization tree)
• Root node represents entire dataset
• Leaves represent instances
• Inner nodes are clusters, subclusters etc.
• Add instances one by one (instances are feature
  vectors)

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COBWEB algorithm

• For every example e, the algorithm starts at root
  node. Then for every node, one of the following 4
  alternatives is chosen based on the category
  utility function
• Insert e into the best successor node
• Create a new leaf for e and make it a successor
  of the current node.
• Generate a new node n, which is predecessor of
  the two best successors of the actual node and
  insert n between the current node and the two
  succesors. Example e is inserted into n.
• Choose the best successor node, delete it and
  make its successors direct successors of the
  actual node. Afterwards insert example e in best
  successor node.
• Category utility function

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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?

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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.