Chapter 11: Analytical Learning

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Chapter 11 Analytical Learning

• Inductive learning
• training examples
• Analytical learning
• prior knowledge deductive reasoning
• Explanation based learning
• - prior knowledge analyze, explain how each
  training
• example
  satisfies the target concept
• - distinguish relevant features
• generalization based on logical reasoning
• - applied to learning search control rules

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Introduction

• Inductive learning
• poor performance when insufficient data
• Explanation based learning
• (1) accept explicit prior knowledge
• (2) generalize more accurately than inductive
  system
• (3) prior knowledge reduce complexity of
  hypotheses space
• reduce sample complexity
• improve generalization accuracy

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• Task of learning play chess
• - target concept
• chessboard positions in which black will
  lose its queen
• within two moves
• - human
• explain/analyze training examples by prior
  knowledge
• - knowledge legal rules of chess

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• Chapter summary
• - Learning algorithm that automatically
  construct/learn
• from explanation
• - Analytical learning problem definition
• - PROLOG-EBG algorithm
• - General properties, relationship to
  inductive learning algorithm
• - Application to improving performance at
  large state-space
• search problems

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Inductive Generalization Problem

• Given
• Instances
• Hypotheses
• Target Concept
• Training examples of target concept
• Determine
• Hypotheses consistent with the training
  examples

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Analytical Generalization Problem

• Given
• Instances
• Hypotheses
• Target Concept
• Training Examples
• Domain Theory
• Determine
• Hypotheses consistent with training
  examples and
• domain theory

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Example of an analytical learning problem

• Instance space describe a pair of objects
• Color, Volume, Owner, Material, Density, On
• Hypothesis space H
• SafeToStack(x,y) Volume(x,vx)
  Volume(y,vy) LessThan(vx,vy)
• Target concept SafeToStack(x,y)
• pairs of physical objects, such that one
  can be stacked safely on the other

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• Training Examples
• On(Obj1, Obj2) Owner(Obj1, Fred)
• Type(Obj1, Box) Owner(Obj2,Louise)
• Type(Obj2, Endtable) Density(Obj1,0.3)
• Color(Obj1,Red) Material(Obj1,Cardboard)
• Color(Obj2,Blue) Material(Obj2,Wood)
• Volume(Obj1,2)
• Domain Theory B
• SafeToStack(x,y) Fragile(y)
• SafeToStack(x,y) Lighter(x,y)
• Lighter(x,y) Weight(x,wx)
  Weight(y,wy) LessThan(wx,wy)
• Weight(x,w) Volume(x,v)
  Density(x,d) Equal(w,times(v,d))
• Weight(x,5) Type(x,Endtable)
• Fragile(x) Material(x,Glass)
• .......

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• Domain Theory B
• - explain why certain pairs of objects can be
  safely stacked
• on one another
• - described by a collection of Horn clause
• enable system to incorporate any learned
  hypotheses into
• subsequent domain theories

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Learning with Perfect Domain Theories PROLOG-EBG

• Correct assertions are truthful statements
• Complete covers every positive examples
• Perfect domain theory is available?
• Yes
• Why does it need to learn when perfect domain
  theory is given?

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PROLOG-EBG

• Operation
• (1) Leaning a single Horn clause rule
• (2) Removing positive examples covered
• (3) Iterating this process
• Given a complete/correct domain theory
• --gt output a hypothesis (correct, cover
  observed
• positive training examples)

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PROLOG-EBG Algorithm

• PROLOG-EBG(Target Concept,Training
  Examples,Domain Theory)
• Learned Rules
• Pos the positive examples from Training
  Examples
• for each Positive Examples in Pos that is not
  covered by Learned Rules, do
• 1. Explain
• - Explanation explanation in
  terms of Domain Theory that Positive
• Examples satisfies the Target
  Concept
• 2. Analyze
• - Sufficient Conditions the most
  general set of features of Positive
• Examples sufficient to satisfy the
  Target Concept according to the
• Explanation
• 3. Refine
• - Learned Rules Learned Rules
  NewHornClause
• Target Concept
  Sufficient Conditions
• Return Learned Rules

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• Weakest Preimage
• The weakest preimage of a conclusion C with
  respect to a
• proof P is the most general set of initial
  assertions A,
• such that A entails C according to P
• the most general rules
• SafeToStack(x,y) Volume(x,vx)
  Density(x,dx)
• Equal(wx,times(vx,dx))
  LessThan(wx,5)
• Type(y,Endtable)

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Remarks on Explanation-Based Learning

• Properties
• (1) justified general hypotheses by using
  prior knowledge
• (2) explanation determines relevant
  attributes
• (3) regressing the target concept allows
  deriving more general
• constraints
• (4) learned Horn clause sufficient
  condition to satisfy target
• concept
• (5) implicitly assume the complete/correct
  domain theory
• (6) generality of the Horn clause depends on
  the formulation of
• the domain theory

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• Perspectives on example based learning
• (1) EBL as theory-guided generalization
• (2) EBL as example-guided reformation of
  theories
• (3) EBL as just restating what the learner
  already knows
• Knowledge compilation
• - reformulate the domain theory to produce
  general rules that
• classify examples in a single inference step
• - transformation efficiency improving task
  without altering
• correctness of systems knowledge

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Characteristics

• Discovering New Features
• - learned feature feature by hidden units
  of neural networks
• Deductive Learning
• - background knowledge of ILP enlarge the
  set of hypotheses
• - domain theory reduce the set of
  acceptable hypotheses
• Inductive Bias
• - inductive bias of PROLOG-EBG domain
  theory B
• - Approximate inductive bias of PROLOG-EBG
• domain theory B
• preference for small sets of maximally
  general Horn clauses

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• LEMMA-ENUMERATOR algorithm
• - enumerate all proof trees
• - for each proof tree, calculate the weakest
  preimage and
• construct a Horn clause
• - ignore the training data
• - output a superset of Horn clauses output by
  PROLOG-EBG
• Role of training data
• focus algorithm on generating rules that
  cover the distribution
• of instances that occur in practice
• Observed positive example allow generalizing
  deductively
• to other unseen instances
• IF ((PlayTennis YES)
  (Humidityx))
• THEN ((PlayTennis YES)
  (Humidity lt x)

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• Knowledge-level learning
• - the learned hypothesis entails predictions
  that go beyond
• those entailed by the domain theory
• - deductive closure set of all predictions
  entailed by a set of
• assertions
• Determinations
• - some attribute of the instance is fully
  determined by certain
• other attributes, without specifying the
  exact nature of the
• dependency
• - example
• target concept people who speak
  Portuguese
• domain theory the language spoken by a
  person is determined
• by their nationality
• training example Joe, 23-year-old
  Brazilian, speaks Portuguese
• conclusion all Brazilians speak
  Portuguese

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Explanation-based Learning of Search Control
Knowledge

• Speed up complex search programs
• Complete/Correct domain theory for learning
  search control knowledge
• definition of legal search operator
• definition of the search objective
• Problem
• find a sequence of operators that will
  transform an arbitrary initial
• state S to some final state F that satisfies
  the goal predicate G

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PRODIGY

• Domain-independent planning system
• find a sequence of operators that leads from S to
  O
• means-ends planner
• decompose problems into subgoals
• solve them
• combine their solution into a solution for
  the full problem

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SORA System

• Support a broad variety of problem-solving
  strategies
• Learned by explaining situations in which its
  current strategy leads to inefficiencies

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Practical Problems applying EBL to learning
search control

• The number of control rules that must be learned
  is very large
• (1) efficient algorithms for matching rules
• (2) utility analysis estimating the
  computational cost and
• benefit of each rule
• (3) identify types of rules that will be
  costly to match
• re-expressing such rules in
  more efficient forms
• optimizing rule-matching algorithm

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• Constructing the explanations for the desired
  target
• concept is intractable
• (1) example
• - states for which operator A leads toward
  the optimal solution
• (2) lazy or incremental explanation
• - heuristics are used to produce
  partial/approximate and
• tractable explanation
• - learned rules may be imperfect
• - monitoring performance of the rule on
  subsequent cases
• - when error, original explanation is
  elaborated to cover new case,
• - more refined rules is extracted