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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

--> output a hypothesis (correct, cover observed

positive training examples)

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PROLOG-EBG AlgorithmPROLOG-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 system’s 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) (Humidity=x)) THEN ((PlayTennis = YES) (Humidity <= 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