![21
Inductive Bias
Consider
– concept learning algorithm L
– instances X, target concept c
– training examples Dc={<x,c(x)>}
– let L(xi,Dc) denote the classification assigned to the
instance xi by L after training on data Dc.
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
where A B means A logically entails B
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UNIT 1-INTRODUCTION-MACHINE LEARNING TECHNIQUES-AD
- 1. 1 UNIT I: Concept Learning • Learning from examples • General-to-specific ordering over hypotheses • Version Spaces and candidate elimination algorithm • Picking new examples • The need for inductive bias
- 2. 2 Features from Computer View Purple Green Green Yellow Yellow Yes Yes Yes Yes Yes Yes No No No Yes Round Round Round Square Round Square Round Square Square Round Eyes Fcolor Nose Hair? Head Smile? Triangle Square Triangle Square Triangle
- 3. 3 Representing Hypotheses Many possible representations for hypotheses h Idea: h as conjunctions of constraints on features Each constraint can be: – a specific value (e.g., Nose = Square) – don’t care (e.g., Eyes = ?) – no value allowed (e.g., Water=Ø) For example, Eyes Nose Head Fcolor Hair? <Round, ?, Round, ?, No> ?
- 4. 4 Prototypical Concept Learning Task Given: – Instances X: Faces, each described by the attributes Eyes, Nose, Head, Fcolor, and Hair? – Target function c: Smile? : X -> { no, yes } – Hypotheses H: Conjunctions of literals such as <?,Square,Square,Yellow,?> – Training examples D: Positive and negative examples of the target function Determine: a hypothesis h in H such that h(x)=c(x) for all x in D. ) ( , ,..., ) ( , , ) ( , 2 2 1 1 m m x c x x c x x c x
- 5. 5 Inductive Learning Hypothesis 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. • What are the implications? • Is this reasonable? • What (if any) are our alternatives? • What about concept drift (what if our views/tastes change over time)?
- 6. 6 Instances, Hypotheses, and More-General-Than Instances X Hypotheses H x1 =<Round,Square,Square,Purple,Yes> Specific General x2 =<Round,Square,Round,Green,Yes> h1 =<Round,?,Square,?,?> h3 =<Round,?,?,?,?> h2 =<Round,?,?,?,Yes> h3 h1 h2
- 7. 7 Find-S Algorithm 1. Initialize h to the most specific hypothesis in H 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 next more general constraint satisfied by x 3. Output hypothesis h
- 8. 8 Hypothesis Space Search by Find-S h3,4 Instances X Hypotheses H Specific General h1,2 h1=<Round,Triangle,Round,Purple,Yes> x1=<Round,Triangle,Round,Purple,Yes> + x2=<Square,Square,Square,Green,Yes> - x5=<Square,Square,Round,Yellow,Yes> + x4=<Round,Triangle,Round,Green,No> - x3=<Square,Triangle,Round,Yellow,Yes> + h2=<Round,Triangle,Round,Purple,Yes> h3=<?,Triangle,Round,?,Yes> h4=<?,Triangle,Round,?,Yes> h5=<?,?,Round,?,Yes> h0=< > h0 x 1 x 4 x 2 x 5 x 3 h5
- 9. 9 Complaints about Find-S • Cannot tell whether it has learned concept • Cannot tell when training data inconsistent • Picks a maximally specific h (why?) • Depending on H, there might be several! • How do we fix this?
- 10. 10 The List-Then-Eliminate Algorithm 1. Set VersionSpace equal to a list containing every hypothesis in H 2. For each training example, <x,c(x)> remove from VersionSpace any hypothesis h for which h(x) != c(x) 3. Output the list of hypotheses in VersionSpace • But is listing all hypotheses reasonable? • How many different hypotheses in our simple problem? – How many not involving “?” terms?
- 11. 11 Version Spaces 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 in D. The version space, VSH,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. ) ( ) ( ) ) ( , ( ) , ( x c x h D x c x D h Consistent )} , ( | { , D h Consistent H h VS D H
- 12. 12 Example Version Space S: { <?,Triangle,Round,?,Yes> } G: { <?,?,Round,?,?> <?,Triangle,?,?,?> } <?,?,Round,?,Yes> <?,Triangle,?,?,Yes> <?,Triangle,Round,?,?>
- 13. 13 Representing Version Spaces The General boundary, G, of version space VSH,D is the set of its maximally general members. The Specific boundary, S, of version space VSH,D is the set of its maximally specific members. Every member of the version space lies between these boundaries y x y x s h g G g S s H h VS D H to equal or general more is means where )} )( )( ( | { ,
- 14. 14 Candidate Elimination Algorithm G = maximally general hypotheses in H S = maximally specific hypotheses in H For each training example d, do If d is a positive example Remove from G any hypothesis that does not include d For each hypothesis s in S that does not include d Remove s from S Add to S all minimal generalizations h of s such that 1. h includes 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
- 15. 15 Candidate Elimination Algorithm (cont) For each training example d, do (cont) If d is a negative example Remove from S any hypothesis that does include d For each hypothesis g in G that does include d Remove g from G Add to G all minimal generalizations h of g such that 1. h does not include 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 If G or S ever becomes empty, data not consistent (with H)
- 16. 16 Example Trace S0: { <Ø,Ø,Ø,Ø,Ø> } G0: { <?,?,?,?,?> } X1=<R,T,R,P,Y> + S1: { <R,T,R,P,Y> } G1 X2=<S,S,S,G,Y> - G2: { <R,?,?,?,?>, <?,T,?,?,?>, <?,?,R,?,?>, <?,?,?,P,?> } S2 X3=<S,T,R,Y,Y> + S3: { <?,T,R,?,Y> } G3 X4=<R,T,R,G,N> - G4: { <?,T,?,?,Y>, <?,?,R,?,Y> } S4 X5=<S,S,R,Y,Y> + S5: { <?,?,R,?,Y> } G5
- 17. 17 What Training Example Next? S: { <?,Triangle,Round,?,Yes> } G: { <?,?,Round,?,?> <?,Triangle,?,?,?> } <?,?,Round,?,Yes> <?,Triangle,?,?,Yes> <?,Triangle,Round,?,?>
- 18. 18 How Should These Be Classified? S: { <?,Triangle,Round,?,Yes> } G: { <?,?,Round,?,?> <?,Triangle,?,?,?> } <?,?,Round,?,Yes> <?,Triangle,?,?,Yes> <?,Triangle,Round,?,?> ? ? ?
- 19. 19 What Justifies this Inductive Leap? + < Round, Triangle, Round, Purple, Yes > + < Square, Triangle, Round, Yellow, Yes > S: < ?, Triangle, Round, ?, Yes > Why believe we can classify the unseen? < Square, Triangle, Round, Purple, Yes > ?
- 20. 20 An UN-Biased Learner 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. For example: What are S, G, in this case? ? , ?, , , ?, , , ?, Purple Square Square Yes Round Triangle
- 21. 21 Inductive Bias Consider – concept learning algorithm L – instances X, target concept c – training examples Dc={<x,c(x)>} – let L(xi,Dc) denote the classification assigned to the instance xi by L after training on data Dc. 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 where A B means A logically entails B )] , ( ) )[( ( c i i c i D x L x D B X x
- 22. 22 Inductive Systems and Equivalent Deductive Systems Candidate Elimination Algorithm Using Hypothesis Space H Theorem Prover Training examples Training examples New instance New instance Assertion "H contains hypothesis" Classification of new instance, or "don't know" Classification of new instance, or "don't know" Inductive System Equivalent Deductive System