Concept Learning and the General-to-Specific Ordering

1. Concept Learning and the General-to-Specific Ordering 이 종우 자연언어처리연구실

2. Concept Learning • Concepts or Categories • “birds” • “car” • “situations in which I should study more in order to pass the exam” • Concept • some subset of objects or events defined over a larger set, or a boolean valued function defined over this larger set.

3. Learning • inducing general functions from specific training examples • Concept Learning • acquiring the definition of a general category given a sample of positive and negative training examples of the category

4. A Concept Learning Task • Target Concept • “days on which Aldo enjoys water sport” • Hypothesis • vector of 6 constraints (Sky, AirTemp, Humidity, Wind, Water, Forecast, EnjoySport ) • Each attribute (“?”, single value or “0”) • e.g. <?, Cold, High, ?, ?, ?>

5. Instance Sky AirTemp Humidity Wind Water Forecast EnjoySport A Sunny Warm Normal Strong Warm Same No B Sumny Warm High Strong Warm Same Yes C Rainy Cold High Strong Warm Change No D Sunny Warm High Strong Cool Change Yes Training examples for the target concept EnjoySport

6. Given : • instances (X): set of iterms over which the concept is defined. • target concept (c) : c : X → {0, 1} • training examples (positive/negative) : <x,c(x)> • training set D: availabletraining examples • set of all possible hypotheses: H • Determine : • to find h(x) = c(x) (for all x in X)

7. Inductive Learning Hypothesis • Inductive Learning Hypothesis • Any good hypothesis over a sufficiently large set of training examples will also approximate the target function. well over unseen examples.

8. Concept Learning as Search • Issue of Search • to find training examples hypothesis that best fits training examples • Kinds of Space in EnjoySport • 3*2*2*2*2 = 96: instant space • 5*4*4*4*4 = 5120: syntactically distinct hypotheses within H • 1+4*3*3*3*3 = 973: semantically distinct hypotheses

9. Search Problem • efficient search in hypothesis space(finite/infinite)

10. General-to-Specific Ordering of Hypotheses • Hypotheses의 General-to-Specific Ordering • x satisfies h⇔h(x)=1 • more_general_than_or_equal_to relations • <Sunny,?,?,Strong,?,?> ≦ g <Sunny,?,?,?,?,?> • more_general_than_or_equal_to relations

11. Concept Learning as Search • partial order (reflexive,antisymmetric,transitive)

12. Find-S: Finding a Maximally Specific Hypothesis • algorithm • 1. Initialize h to the most specific hypothesis in H • 2. For each positive training example x • For each attribute constraint ai in h • If the constraint ai is satisfied by x • then do nothing • else replace aiin h by the next more general constraint that is satisfied by x • 3. Output hypothesis h • Property • guaranteed to output the most specific hypothesis • no way to determine unique hypothesis • not cope with inconsistent errors or noises

13. Find-S:Finding a Maximally Specific Hypothesis(2)

14. Version Spaces and the Candidate-Elimination Algorithm • output all hypotheses consistent with the training examples. • perform poorly with noisy training data. • Representation • Consistent(h,D) ⇔(∀<x,c(x)>  D) h(x) = c(x) • VSH,D ⇔ {h  H | Consistent(h,D)} • List-Then-Eliminate Algorithm • lists all hypotheses -> remove inconsistent ones. • Appliable to finite H

15. Version Spaces and the Candidate-Elimination Algorithm(2) • More Compact Representation for Version Spaces • general boundary G • specific boundary S • Version Space redefined with S and G

16. Version Spaces and the Candidate-Elimination Algorithm(3)

17. Version Spaces and the Candidate-Elimination Algorithm(4) • Condidate-Elimination Learning Algorithm • Initialize G to the set of maximally general hypotheses in H • Initialize S to the 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 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

18. Version Spaces and the Candidate-Elimination Algorithm(5) • If d is a negative example • 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 • 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

19. Version Spaces and the Candidate-Elimination Algorithm(6) • Illustrative Example

20. Version Spaces and the Candidate-Elimination Algorithm(7)

21. Version Spaces and the Candidate-Elimination Algorithm(8)

22. Version Spaces and the Candidate-Elimination Algorithm(9)

23. Remarks on Version Spaces and Candidate-Elimination • Will the Candidate-Elimination Algorithm Converge to the Correct Hypothesis? • Prerequisite • 1. No error in training examples • 2. Hypothesis exists which correctly describes c(x). • S and G boundary sets converge to an empty set => no hypothesis in H consistent with observed examples. • What Training Example Should the Learner Request Next? • Negative one specifies G , positive one generalizes S. • optimal query satisfy half the hypotheses.

24. Instance Sky AirTemp Humidity Wind Water Forecast EnjoySport A Sunny Warm Normal Strong Cool Change ? B Rainy Cold Normal Light Warm Same ? C Sunny Warm Normal Light Warm Same ? D Sunny Cold Normal Strong Warm Same ? Remarks on Version Spaces and Candidate-Elimination(2) • How Can Partially Learned Concepts Be Used? A : classified to positive B : classified to negative C : 3 positive , 3 negative D : 2 positive, 4 negative

25. Example Sky AirTemp Humidity Wind Water Forecast EnjoySport 1 Sunny Warm Normal Strong Cool Change Yes 2 Cloudy Warm Normal Strong Cool Change Yes 3 Rainy Warm Normal Strong Cool Change No Inductive Bias • A Biased Hypothesis Space - zero hypothesis in the version space - caused by only conjunctive hypothesis

26. Inductive Bias(2) • An Unbiased Learner • Power set of X : set of all subsets of a set X • number of size of power set : 2|X| • e.g. <Sunny,?,?,?,?,?>∨ <Cloudy,?,?,?,?,?> • new problem : unable to generalize beyond the observed examples. • Observed examples are only unambiguously classified. • Voting results in no majority or minority.

27. Inductive Bias(3) • The Futility of Bias-Free Learning • no inductive bias => cannot classify unseen data reasonably • inductive bias of L : any minimal set of assertions B such that • inductive bias of Candidate-Elimination algorithm • c∈ H • advantage of introducing inductive bias • generalizing beyond the observed data • allows comparison of different learners

28. Inductive Bias(4) • e.g • Rote-learner : no inductive bias • Candidate-Elimination algo : c ∈ H => more strong • Find-S : c ∈ H and that all are negative unless not proved positive

29. Inductive Bias(5)

30. Summary • Concept learning can be cast as a problem of searching through a large predefined space of potential hypotheses. • General-to-specific partial ordering of hypotheses provides a useful structure for search. • Find-S algorithm performs specific-to-general search to find the most specific hypothesis. • Candidate-Elimination algorithm computes version space by incrementally computing the sets of maximally specific (S) and maximally general (G) hypotheses. • S and G delimit the entire set of hypotheses consistent with the data.

31. Version spaces and Candidate-Elimination algorithm provide a useful conceptual framework for studying concept learning. • Candidate-Elimination algorithm not robust to noisy data or to situations where the unknown target concept is not expressible in the provided hypothesis space. • Inductive bias in Candidate-Elimination algorithm is that target concept exists in H • If hypothesis space be enriched so that there is a every possible hypothesis, that would remove the ability to classify any instance beyond the observed examples.