1.3.Concept Learning

1.3.Concept Learning - Aims 11s1: COMP9417 Machine Learning...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Aims 11s1: COMP9417 Machine Learning and Data Mining Fundamentals of Concept Learning This lecture aims to develop your understanding of representing and searching hypothesis spaces for concept learning. Following it you should be able to: • define a representation for concepts March 1, 2011 • define a hypothesis space in terms of generality ordering on concepts • describe an algorithm to search a hypothesis space • express the framework of version spaces Acknowledgement: Material derived from slides for the book Machine Learning, Tom Mitchell, McGraw-Hill, 1997 http://www-2.cs.cmu.edu/~tom/mlbook.html • describe an algorithm to search a hypothesis space using the framework of version spaces • explain the role of inductive bias in concept learning COMP9417: March 1, 2011 Overview Fundamentals of Concept Learning: Slide 1 Training Examples for EnjoySport Concept Learning inferring a Boolean-valued function from training examples of its input and output. • Learning from examples Sky Sunny Sunny Rainy Sunny Temp Warm Warm Cold Warm Humid Normal High High High Wind Strong Strong Strong Strong Water Warm Warm Warm Cool Forecst Same Same Change Change EnjoySpt Yes Yes No Yes • General-to-specific ordering over hypotheses • Version spaces and candidate elimination algorithm What is the general concept? • Picking new examples • The need for inductive bias Note: simple approach assuming no noise, illustrates key concepts COMP9417: March 1, 2011 Fundamentals of Concept Learning: Slide 2 COMP9417: March 1, 2011 Fundamentals of Concept Learning: Slide 3 Representing Hypotheses The Prototypical Concept Learning Task Many possible representations . . . • Given: Here, h is a conjunction of constraints on attributes. – Instances X : Possible days, each described by the attributes Each constraint can be: Attribute Sky AirTemp Humid Wind Water Forecast • a specific value (e.g., W ater = W arm) • don’t care (e.g., “W ater =?”) • no value allowed (e.g.,“Water=∅”) Values Sunny, Cloudy, Rainy Warm, Cold Normal, High Strong, Weak Warm, Cool Same, Change For example, Sky ￿Sunny AirTemp ? COMP9417: March 1, 2011 Humid ? Wind Strong Water ? Forecst Same￿ Fundamentals of Concept Learning: Slide 4 COMP9417: March 1, 2011 Fundamentals of Concept Learning: Slide 5 The inductive learning hypothesis The Prototypical Concept Learning Task – Target function c: EnjoySport : X → {0, 1} – Hypotheses H : Conjunctions of literals. E.g. ￿?, Cold, High, ?, ?, ?￿. – Training examples D: Positive and negative examples of the target function ￿ x 1 , c( x 1 ) ￿ , . . . ￿ x m , c( x m ) ￿ Any hypothesis found to approximate the target function well over a sufficiently large set of training examples will also approximate the target function well over other unobserved examples. • Determine: A hypothesis h in H such that h(x) = c(x) for all x in D (usually called the target hypothesis). COMP9417: March 1, 2011 Fundamentals of Concept Learning: Slide 6 COMP9417: March 1, 2011 Fundamentals of Concept Learning: Slide 7 Concept Learning as Search Concept Learning as Search Hypothesis space Question: What can be learned ? Answer: (only) what is in the hypothesis space Sky × AirTemp × . . . × Forecast = 5 × 4 × 4 × 4 × 4 × 4 How big is the hypothesis space for EnjoySport ? = 5120 Instance space (semantically distinct∗ only) = 1 + (4 × 3 × 3 × 3 × 3 × 3) = 973 Sky × AirTemp × . . . × Forecast = 3 × 2 × 2 × 2 × 2 × 2 = 96 ∗ any hypothesis with an ∅ constraint covers no instances, hence all are semantically equivalent. The learning problem ≡ searching a hypothesis space. How ? COMP9417: March 1, 2011 Fundamentals of Concept Learning: Slide 8 Instances, Hypotheses, and More-General-Than Instances X COMP9417: March 1, 2011 Fundamentals of Concept Learning: Slide 9 A generality order on hypotheses Hypotheses H Specific h 1 x1 x h h 2 Definition: Let hj and hk be Boolean-valued functions defined over instances X . Then hj is more general than or equal to hk (written hj ≥g hk ) if and only if 3 (∀x ∈ X )[(hk (x) = 1) → (hj (x) = 1)] 2 General Intuitively, hj is more general than or equal to hk if any instance satisfying hk also satisfies hj . x1= <Sunny, Warm, High, Strong, Cool, Same> h 1= <Sunny, ?, ?, Strong, ?, ?> x = <Sunny, Warm, High, Light, Warm, Same> 2 h = <Sunny, ?, ?, ?, ?, ?> 2 h = <Sunny, ?, ?, ?, Cool, ?> 3 hj is (strictly) more general than hk (written hj >g hk ) if and only if (hj ≥g hk ) ∧ (hk ￿≥g hj ). hj is more specific than hk when hk is more general than hj . COMP9417: March 1, 2011 Fundamentals of Concept Learning: Slide 10 COMP9417: March 1, 2011 Fundamentals of Concept Learning: Slide 11 The Find-S Algorithm Hypothesis Space Search by Find-S Instances X Hypotheses H 1. Initialize h to the most specific hypothesis in H h0 - x3 2. For each positive training instance x • For each attribute constraint ai in h If the constraint ai in h is satisfied by x Then do nothing Else replace ai in h by the next more general constraint that is satisfied by x + x1 2 + x4 x 2 = <Sunny Warm High Strong Warm Same>, + x 3 = <Rainy Cold High Strong Warm Change>, - x = <Sunny Warm High Strong Cool Change>, + 4 Fundamentals of Concept Learning: Slide 12 h 2,3 x+ x 1 = <Sunny Warm Normal Strong Warm Same>, + COMP9417: March 1, 2011 Specific h1 h General h = <∅, ∅, ∅, ∅, ∅, ∅> 0 h1 = <Sunny Warm Normal Strong Warm Same> h2 = <Sunny Warm ? Strong Warm Same> h = <Sunny Warm ? Strong Warm Same> 3 h = <Sunny Warm ? Strong ? ? > 4 COMP9417: March 1, 2011 Find-S - does it work ? 4 Fundamentals of Concept Learning: Slide 13 Complaints about Find-S Assume: a hypothesis hc ∈ H describes target function c, and training data is error-free. • Can’t tell whether it has learned concept learned hypothesis may not be the only consistent hypothesis By definition, hc is consistent with all positive training examples and can never cover a negative example. • Can’t tell when training data inconsistent For each h generated by Find-S, hc is more general than or equal to h. • Picks a maximally specific h (why?) cannot handle noisy data might require maximally general h So h can never cover a negative example. COMP9417: March 1, 2011 Fundamentals of Concept Learning: Slide 14 COMP9417: March 1, 2011 Fundamentals of Concept Learning: Slide 15 Version Spaces The List-Then-Eliminate Algorithm A hypothesis h is consistent with a set of training examples D of target concept c if and only if h(x) = c(x) for each training example ￿x, c(x)￿ in D. Consistent(h, D) ≡ (∀￿x, c(x)￿ ∈ D) h(x) = c(x) 1. V ersionSpace ← a list containing every hypothesis in H 2. For each training example, ￿x, c(x)￿ remove from V ersionSpace any hypothesis h for which h(x) ￿= c(x) 3. Output the list of hypotheses in V ersionSpace The version space, V SH,D , with respect to hypothesis space H and training examples D, is the subset of hypotheses from H consistent with all training examples in D. V SH,D ≡ {h ∈ H |Consistent(h, D)} COMP9417: March 1, 2011 Fundamentals of Concept Learning: Slide 16 COMP9417: March 1, 2011 Example Version Space Fundamentals of Concept Learning: Slide 17 Representing Version Spaces The General boundary, G, of version space V SH,D is the set of its maximally general members S: { <Sunny, Warm, ?, Strong, ?, ?> } The Specific boundary, S, of version space V SH,D is the set of its maximally specific members <Sunny, ?, ?, Strong, ?, ?> <Sunny, Warm, ?, ?, ?, ?> <?, Warm, ?, Strong, ?, ?> Every member of the version space lies between these boundaries G: V SH,D = {h ∈ H |(∃s ∈ S )(∃g ∈ G)(g ≥ h ≥ s)} { <Sunny, ?, ?, ?, ?, ?>, <?, Warm, ?, ?, ?, ?> } COMP9417: March 1, 2011 where x ≥ y means x is more general or equal to y Fundamentals of Concept Learning: Slide 18 COMP9417: March 1, 2011 Fundamentals of Concept Learning: Slide 19 The Candidate Elimination Algorithm The Candidate Elimination Algorithm G ← maximally general hypotheses in H • If d is a negative example S ← 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 s in S that is not consistent with d ∗ Remove s from S ∗ Add to S all minimal generalizations h of s such that 1. h is consistent with d, and 2. some member of G is more general than h ∗ Remove from S any hypothesis that is more general than another hypothesis in S COMP9417: March 1, 2011 Fundamentals of Concept Learning: Slide 20 – Remove from S any hypothesis inconsistent with d – For each hypothesis g in G that is not consistent with d ∗ Remove g from G ∗ Add to G all minimal specializations h of g such that 1. h is consistent with d, and 2. some member of S is more specific than h ∗ Remove from G any hypothesis that is less general than another hypothesis in G COMP9417: March 1, 2011 Fundamentals of Concept Learning: Example Trace S: 0 Slide 21 Example Trace S : {< ∅, ∅, ∅, ∅, ∅, ∅ >} 0 {<Ø, Ø, Ø, Ø, Ø, Ø>} S 1 : {<Sunny, Warm, Normal, Strong, Warm, Same> } S 2 : {<Sunny, Warm, ?, Strong, Warm, Same>} G ,G ,G : 012 {<?, ?, ?, ?, ?, ?> } Training examples: G 0: COMP9417: March 1, 2011 1 . <Sunny, Warm, Normal, Strong, Warm, Same>, Enjoy Sport = Yes {<?, ?, ?, ?, ?, ?>} 2 . <Sunny, Warm, High, Strong, Warm, Same>, Enjoy Sport = Yes Fundamentals of Concept Learning: Slide 22 COMP9417: March 1, 2011 Fundamentals of Concept Learning: Slide 23 Example Trace Example Trace S 2 , S 3 : { <Sunny, Warm, ?, Strong, Warm, Same> } S 3 : {<Sunny, Warm, ?, Strong, Warm, Same>} S 4: G 3: { <Sunny, Warm, ?, Strong, ?, ?> } {<Sunny, ?, ?, ?, ?, ?> <?, Warm, ?, ?, ?, ?> <?, ?, ?, ?, ?, Same> } G 4: {<Sunny, ?, ?, ?, ?, ?> <?, Warm, ?, ?, ?, ?>} G 2: {<?, ?, ?, ?, ?, ?> } G 3: {<Sunny, ?, ?, ?, ?, ?> <?, Warm, ?, ?, ?, ?> <?, ?, ?, ?, ?, Same>} Training Example: Training Example: 3. <Rainy, Cold, High, Strong, Warm, Change>, EnjoySport=No 4. <Sunny, Warm, High, Strong, Cool, Change>, EnjoySport = Yes COMP9417: March 1, 2011 Fundamentals of Concept Learning: Slide 24 COMP9417: March 1, 2011 Fundamentals of Concept Learning: Slide 25 Which Training Example Is Best To Choose Next ? Example Trace S 4 : {<Sunny, Warm, ?, Strong, ?, ?>} S: { <Sunny, Warm, ?, Strong, ?, ?> } <Sunny, ?, ?, Strong, ?, ?> <Sunny, Warm, ?, ?, ?, ?> <?, Warm, ?, Strong, ?, ?> <Sunny, ?, ?, Strong, ?, ?> <Sunny, Warm, ?, ?, ?, ?> <?, Warm, ?, Strong, ?, ?> G : {<Sunny, ?, ?, ?, ?, ?>, <?, Warm, ?, ?, ?, ?>} 4 G: COMP9417: March 1, 2011 Fundamentals of Concept Learning: Slide 26 { <Sunny, ?, ?, ?, ?, ?>, <?, Warm, ?, ?, ?, ?> } COMP9417: March 1, 2011 Fundamentals of Concept Learning: Slide 27 How Should New Instances Be Classified ? Which Training Example To Choose Next ? S: { <Sunny, Warm, ?, Strong, ?, ?> } S: { <Sunny, Warm, ?, Strong, ?, ?> } <Sunny, ?, ?, Strong, ?, ?> <Sunny, ?, ?, Strong, ?, ?> G: <Sunny, Warm, ?, ?, ?, ?> G: { <Sunny, ?, ?, ?, ?, ?>, <?, Warm, ?, ?, ?, ?> } <?, Warm, ?, Strong, ?, ?> Fundamentals of Concept Learning: { <Sunny, ?, ?, ?, ?, ?>, <?, Warm, ?, ?, ?, ?> } ￿ Sunny Warm Normal Strong Cool Change ￿ ￿ Rainy Cold Normal Light Warm Same ￿ ￿ Sunny Warm Normal Light Warm Same ￿ ￿Sunny W arm N ormal Light W arm Same￿ COMP9417: March 1, 2011 <Sunny, Warm, ?, ?, ?, ?> <?, Warm, ?, Strong, ?, ?> Slide 28 COMP9417: March 1, 2011 Fundamentals of Concept Learning: Slide 29 What Justifies this Inductive Leap ? How Should New Instances Be Classified ? S: { <Sunny, Warm, ?, Strong, ?, ?> } + ￿Sunny W arm N ormal Strong Cool Change￿ + ￿Sunny W arm N ormal Light W arm Same￿ <Sunny, ?, ?, Strong, ?, ?> <Sunny, Warm, ?, ?, ?, ?> <?, Warm, ?, Strong, ?, ?> S : ￿Sunny W arm N ormal ? ? ?￿ G: { <Sunny, ?, ?, ?, ?, ?>, <?, Warm, ?, ?, ?, ?> } Why believe we can classify this unseen instance ? ￿Sunny W arm N ormal Strong W arm Same￿ ￿ Sunny Warm Normal Strong Cool Change ￿ (6 + /0−) ￿ Rainy Cold Normal Light Warm Same ￿ (0 + /6−) ￿ Sunny Warm Normal Light Warm Same ￿ (3 + /3−) COMP9417: March 1, 2011 Fundamentals of Concept Learning: Slide 30 COMP9417: March 1, 2011 Fundamentals of Concept Learning: Slide 31 An UNBiased Learner Inductive Bias Consider Idea: Choose H that expresses every teachable concept (i.e. H is the power set of X ) ￿ Consider H = disjunctions, conjunctions, negations over previous H . E.g. • concept learning algorithm L • instances X , target concept c • training examples Dc = {￿x, c(x)￿} ￿Sunny W arm N ormal ? ? ?￿ ∨ ¬￿? ? ? ? ? Change￿ • let L(xi, Dc) denote the classification assigned to the instance xi by L after training on data Dc. What are S , G in this case? Definition: The inductive bias of L is any minimal set of assertions B such that for any target concept c and corresponding training examples Dc (∀xi ∈ X )[(B ∧ Dc ∧ xi) ￿ L(xi, Dc)] where A ￿ B means A logically entails B S← G← COMP9417: March 1, 2011 Fundamentals of Concept Learning: Slide 32 COMP9417: March 1, 2011 Inductive Systems and Equivalent Deductive Systems Fundamentals of Concept Learning: Slide 33 Three Learners with Different Biases Inductive system Training examples Candidate Elimination Algorithm New instance Classification of new instance, or "don’t know" Using Hypothesis Space H 1. Rote learner: Store examples, Classify x iff it matches previously observed example. 2. Version space candidate elimination algorithm 3. Find-S Equivalent deductive system Classification of new instance, or "don’t know" Training examples New instance Theorem Prover Assertion " H contains the target concept" Inductive bias made explicit COMP9417: March 1, 2011 Fundamentals of Concept Learning: Slide 34 COMP9417: March 1, 2011 Fundamentals of Concept Learning: Slide 35 Summary Points 1. Concept learning as search through H 2. General-to-specific ordering over H 3. Version space candidate elimination algorithm 4. S and G boundaries characterize learner’s uncertainty 5. Learner can generate useful queries 6. Inductive leaps possible only if learner is biased 7. Inductive learners can be modelled by equivalent deductive systems [Suggested reading: Mitchell, Chapter 2] COMP9417: March 1, 2011 Fundamentals of Concept Learning: Slide 36 ...
View Full Document

Ask a homework question - tutors are online