Motivation Inductive Learning (1/2) � An AI agent operating in a complex Decision Tree Method world requires an awful lot of knowledge: state representations, state (If it’s not simple, axioms, constraints, action descriptions, it’s not worth learning it) heuristics, probabilities, ... R&N: Chap. 18, Sect. 18.1–3 � More and more, AI agents are designed to acquire knowledge through learning 1 2 What is Learning? Contents � Mostly generalization from experience: � Introduction to inductive learning � Logic-based inductive learning: “Our experience of the world is specific, yet we are able to formulate general • Decision-tree induction theories that account for the past and � Function-based inductive learning predict the future” M.R. Genesereth and N.J. Nilsson, • Neural nets in Logical Foundations of AI , 1987 � � Concepts, heuristics, policies � Supervised vs. un-supervised learning 3 4 Rewarded Card Example Logic-Based Inductive Learning � Background knowledge KB � Deck of cards, with each card designated by [r,s], its rank and suit, and some cards “rewarded” � Background knowledge KB: � Training set D (observed knowledge) ((r=1) v … v (r=10)) ⇔ NUM(r) that is not logically implied by KB ((r=J) v (r=Q) v (r=K)) ⇔ FACE(r) ((s=S) v (s=C)) ⇔ BLACK(s) ((s=D) v (s=H)) ⇔ RED(s) � Inductive inference : � Training set D: Find h such that KB and h imply D REWARD([4,C]) ∧ REWARD([7,C]) ∧ REWARD([2,S]) ∧ ¬ REWARD([5,H]) ∧ ¬ REWARD([J,S]) h = D is a trivial, but un-interesting solution (data caching) 5 6 1
Learning a Predicate Rewarded Card Example (Concept Classifier) � Deck of cards, with each card designated by [r,s], its � Set E of objects (e.g., cards) rank and suit, and some cards “rewarded” � Goal predicate CONCEPT(x), where x is an object in E, � Background knowledge KB: that takes the value True or False (e.g., REWARD) ((r=1) v … v (r=10)) ⇔ NUM(r) Example: ((r=J) v (r=Q) v (r=K)) ⇔ FACE(r) CONCEPT describes the precondition of an action, e.g., ((s=S) v (s=C)) ⇔ BLACK(s) There are several possible Unstack(C,A) ((s=D) v (s=H)) ⇔ RED(s) inductive hypotheses � Training set D: • E is the set of states REWARD([4,C]) ∧ REWARD([7,C]) ∧ REWARD([2,S]) ∧ • CONCEPT(x) ⇔ ¬ REWARD([5,H]) ∧ ¬ REWARD([J,S]) HANDEMPTY ∈ x, BLOCK(C) ∈ x, BLOCK(A) ∈ x, � Possible inductive hypothesis: CLEAR(C) ∈ x, ON(C,A) ∈ x h ≡ (NUM(r) ∧ BLACK(s) ⇔ REWARD([r,s])) Learning CONCEPT is a step toward learning an action description 7 8 Learning a Predicate Example of Training Set (Concept Classifier) � Set E of objects (e.g., cards) � Goal predicate CONCEPT(x), where x is an object in E, that takes the value True or False (e.g., REWARD) � Observable predicates A(x), B(X), … (e.g., NUM, RED) � Training set: values of CONCEPT for some combinations of values of the observable predicates 9 10 Learning a Predicate Example of Training Set (Concept Classifier) � Set E of objects (e.g., cards) � Goal predicate CONCEPT(x), where x is an object in E, that takes the value True or False (e.g., REWARD) Ternary attributes � Observable predicates A(x), B(X), … (e.g., NUM, RED) � Training set: values of CONCEPT for some combinations of values of the observable predicates Goal predicate is PLAY-TENNIS � Find a representation of CONCEPT in the form: CONCEPT(x) ⇔ S(A,B, …) where S(A,B,…) is a sentence built with the observable predicates, e.g.: CONCEPT(x) ⇔ A(x) ∧ ( ¬ B(x) v C(x)) Note that the training set does not say whether an observable predicate is pertinent or not 11 12 2
Learning an Arch Classifier Example set � These objects are arches: � An example consists of the values of CONCEPT (positive examples) and the observable predicates for some object x � These aren’t: � An example is positive if CONCEPT is True, (negative examples) else it is negative � The set X of all examples is the example set ARCH(x) ⇔ HAS-PART(x,b1) ∧ HAS-PART(x,b2) ∧ � The training set is a subset of X HAS-PART(x,b3) ∧ IS-A(b1,BRICK) ∧ IS-A(b2,BRICK) ∧ ¬ MEET(b1,b2) ∧ (IS-A(b3,BRICK) v IS-A(b3,WEDGE)) ∧ a small one! SUPPORTED(b3,b1) ∧ SUPPORTED(b3,b2) 13 14 Hypothesis Space Inductive Learning Scheme Inductive � An hypothesis is any sentence of the form: hypothesis h Training set D CONCEPT(x) ⇔ S(A,B, …) where S(A,B,…) is a sentence built using the - observable predicates - + - + - � The set of all hypotheses is called the - - - + + + + hypothesis space H - - + + � An hypothesis h agrees with an example if it + + - - - gives the correct value of CONCEPT + + Hypothesis space H Example set X {[CONCEPT(x) ⇔ S(A,B, …)]} {[A, B, …, CONCEPT]} 15 16 Size of Hypothesis Space Multiple Inductive Hypotheses � Deck of cards, with each card designated by [r,s], its rank and suit, and some cards “rewarded” � Background knowledge KB: � n observable predicates ((r=1) v … v (r=10)) ⇔ NUM(r) � 2 n entries in truth table defining ((r=J) v (r=Q) v (r=K)) ⇔ FACE(r) ((s=S) v (s=C)) ⇔ BLACK(s) ((s=D) v (s=H)) ⇔ RED(s) CONCEPT and each entry can be filled � Training set D: REWARD([4,C]) ∧ REWARD([7,C]) ∧ REWARD([2,S]) ∧ with True or False ¬ REWARD([5,H]) ∧ ¬ REWARD([J,S]) � In the absence of any restriction h 1 ≡ NUM(r) ∧ BLACK(s) ⇔ REWARD([r,s]) 2 n (bias), there are hypotheses to h 2 ≡ BLACK(s) ∧ ¬ (r=J) ⇔ REWARD([r,s]) 2 choose from h 3 ≡ ([r,s]=[4,C]) ∨ ([r,s]=[7,C]) ∨ [r,s]=[2,S]) ⇔ REWARD([r,s]) � n = 6 � 2x10 19 hypotheses! h 4 ≡ ¬ ([r,s]=[5,H]) ∨ ¬ ([r,s]=[J,S]) ⇔ REWARD([r,s]) agree with all the examples in the training set 17 18 3
Notion of Capacity Multiple Inductive Hypotheses � Deck of cards, with each card designated by [r,s], its � It refers to the ability of a machine to learn any rank and suit, and some cards “rewarded” training set without error � Background knowledge KB: Need for a system of preferences – called ((r=1) v … v (r=10)) ⇔ NUM(r) � A machine with too much capacity is like a botanist a bias – to compare possible hypotheses ((r=J) v (r=Q) v (r=K)) ⇔ FACE(r) with photographic memory who, when presented with ((s=S) v (s=C)) ⇔ BLACK(s) ((s=D) v (s=H)) ⇔ RED(s) a new tree, concludes that it is not a tree because it � Training set D: has a different number of leaves from anything he REWARD([4,C]) ∧ REWARD([7,C]) ∧ REWARD([2,S]) ∧ ¬ REWARD([5,H]) ∧ ¬ REWARD([J,S]) has seen before h 1 ≡ NUM(r) ∧ BLACK(s) ⇔ REWARD([r,s]) � A machine with too little capacity is like the h 2 ≡ BLACK(s) ∧ ¬ (r=J) ⇔ REWARD([r,s]) botanist’s lazy brother, who declares that if it’s green, it’s a tree h 3 ≡ ([r,s]=[4,C]) ∨ ([r,s]=[7,C]) ∨ [r,s]=[2,S]) ⇔ REWARD([r,s]) � Good generalization can only be achieved when the right balance is struck between the accuracy attained h 4 ≡ ¬ ([r,s]=[5,H]) ∨ ¬ ([r,s]=[J,S]) ⇔ REWARD([r,s]) on the training set and the capacity of the machine agree with all the examples in the training set 19 20 � Keep-It-Simple (KIS) Bias � Keep-It-Simple (KIS) Bias � Examples � Examples • Use much fewer observable predicates than the • Use much fewer observable predicates than the training set training set • Constrain the learnt predicate, e.g., to use only “high- • Constrain the learnt predicate, e.g., to use only “high- level” observable predicates such as NUM, FACE, level” observable predicates such as NUM, FACE, BLACK, and RED and/or to have simple syntax BLACK, and RED and/or to have simple syntax Einstein: “A theory must be as simple as possible, � Motivation � Motivation but not simpler than this” • If an hypothesis is too complex it is not worth • If an hypothesis is too complex it is not worth learning it (data caching does the job as well) learning it (data caching does the job as well) • There are much fewer simple hypotheses than • There are much fewer simple hypotheses than complex ones, hence the hypothesis space is smaller complex ones, hence the hypothesis space is smaller 21 22 � Keep-It-Simple (KIS) Bias Putting Things Together � Examples • Use much fewer observable predicates than the yes Test Object set If the bias allows only sentences S that are Evaluation no training set set conjunctions of k << n predicates picked from Example • Constrain the learnt predicate, e.g., to use only “high- set X Goal predicate the n observable predicates, then the size of level” observable predicates such as NUM, FACE, Training Induced BLACK, and RED and/or to have simple syntax H is O(n k ) set D hypothesis h Observable predicates � Motivation Learning Hypothesis procedure L • If an hypothesis is too complex it is not worth space H learning it (data caching does the job as well) Bias • There are much fewer simple hypotheses than complex ones, hence the hypothesis space is smaller 23 24 4
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