RDM Chapter 3: Intro to Learning and Search

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RDM Chapter 3 Intro to Learning and Search

• prepared for COMP422/522-2008, Bernhard Pfahringer

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3.1 Representing Hypothesis and Instances

• language Le to represent examples
• language Lh to represent hypothesis
• h ? Lh h Le -gt Y, e.g. Y0,1
• cover relation over Lh x Le, c(h,e) true if and
  only if h(e) 1 (see Figures 3.1 and 3.2)

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3.2 Boolean data

• simplify
• item-sets, true/false, variable assignment
  Herbrand interpretation, sausage,beer,mustard,win
  e
• Le I I ? b,m,s,w
• Le Lh (single presentation trick)

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Machine Learning point of view

• given Le, Lh, unknown fLe -gt Y
• examples E-(e1,f(e1)), ..
• loss(h,E) measures quality of h wrt E
• find h argmin loss(h,E)
• zero-one loss (empirical risk)
• loss(h,E) 1/E ? f(e)-h(e)
• regression squared loss
• probabilistic settings log-likelihood

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Data Mining POV

• given Le, Lh, data D ? Le
• quality criterion Q(h,D), find set
• Th(Q,D,Lh) h? Lh Q(h,D) holds
• Q local,global, or heuristic
• e.g. freq(h,D) c(h,D) or
• rfreq(h,D) c(h,D) / D
• local rfreq(h,D) gt y
• acc(h,P,N) freq(h,P)/(freq(h,P)freq(h,N))
• global Q(h,P,N) argmax acc(h,P,N)

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Generate-and-test

• FORALL h ? Lh DO
• IF Q(h,D) true THEN output h
• Lh must be enumerable
• naïve, inefficient, but complete
• see Example 3.3

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3.6 Search space structure

• h1 is more general than h2, h1 ? h2, if c(h2) ?
  c(h1),
• proper generalization, if true subset
• reflexive, and transitive
• but syntactic variants -gt problematic
• (canonical forms, partial order, )
• see Example 3.5, Fig 3.5 Hasse diagram, top T and
  bottom ? element

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Monotonicity

• Q is monotonic (true for all specialisations)
  ?s,g ? Lh, ? D ? Le g?s ? Q(g,D) ? Q(s,D)
• Q is anti-monotonic (true for all
  generalisations) ?s,g ? Lh, ? D ? Le g?s ?
  Q(s,D) ? Q(g,D)

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Examples

• freq(h,D) x, minFreq, is anti-monotonic
• freq(h,D) x, maxFreq, is monotonic
• specific example e is covered (e ? c(h)), is
  anti-monotonic
• specific example e is not covered (e ?c(h)), is
  monotonic
• acc(h,P,N) x, is neither
• do Exercises 3.6, 3.7, 3.8.

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Pruning

• if monotonic Q is false for h, then Q is also
  false for all generalisations of h
• if anti-monotonic Q is false for h, then Q is
  false for all specialisations of h
• see examples 3.11/3.12, figures 3.6/3.7

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Min/max

• max(T) h ? T ?t ? T h lt t maximal
  elements are most specific
• min(T) h ? T ?t ? T t lt h minimal
  elements are most general
• if Lh is infinite, might not exist
• examples 3.13, 3.14

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Borders

• S border of maximally specific hypothesis for
  which Q holds
• S(Th(Q,D,Lh)) max(Th(Q,D,Lh))
• similarly G maximally general
• G(Th(Q,D,Lh)) max(Th(Q,D,Lh))
• example 3.15

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Border properties

• borders fully specify all solutions for
• anti-monotonic Q Th(Q,D,Lh) h ? Lh ?s ?
  S(Th(Q,D,Lh) h s
• monotonic Q Th(Q,D,Lh) h ? Lh ?g ?
  G(Th(Q,D,Lh) g h

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Version space

• Q is a conjunction of two criteria M? A, one
  monotonic (M), one anti-monotonic (A), then T is
  a version space if
• T h ? Lh ?s ?S(T), g ? G(T) g h s
• S and G are condensed representations, often
  much smaller
• example 3.18 / figure 3.8, example 3.20

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Negative borders

• the elements just outside the (positive) borders
• S-(Th) min(Lh-h ? Lh ?s ? S(Th) h s)
• G-(Th) max(Lh-h ? Lh ?g ? g(Th) g h)
• example 3.21
• border sets can be large item-sets -gt G
  exponentially large in N

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Refinement operators

• generalization op ?g Lh ? 2Lh with ?h ? Lh
  ?g(h) ? c?Lhch
• specialisation op ?s Lh ? 2Lh with ?h ? Lh
  ?s(h) ? c?Lhhc
• can be applied repeatedly

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Ideal refinement operator

• ideal specialisation ?h ? Lh ?s(h) min(
  h?Lhhh )
• returns exactly all children of a node in the
  Hasse diagram
• used in heuristic search (e.g. hill-climbing)

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Optimal operator

• no hypothesis is generated twice gt efficient
• used in complete search
• see example 3.22
• optimal operators define a canonical form and
  vice versa

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MGG minimally general gens

• mgg(h1,h2) min h?Lh hh1 ? h?h2
• if unique, than also called lgg (least general
  generalization) and lub (least upper bound)

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MGS maximally general specs

• mgs(h1,h2) max h?Lh h1h ? h2?h
• if unique, than also called glb (greatest lower
  bound)
• if lub and glb exist for h1,h2, than they form a
  lattice (e.g. item-sets do), example 3.23,
  exercises 3.24/3.25

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Generic learning algorithm

• Queue Init
• Th
• WHILE not Stop DO
• Delete h from Queue
• IF Q(h,D) true THEN
• add h to Th
• ELSE Queue Queue ??(h)
• Queue prune(Queue)
• return Th

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Generic algorithm continued

• lots of parameters
• Init defines start point
• Delete defines search strategy
• first-in-first-out (queue)gt breadth-first
• last-in-first-out (stack)gt depth-first
• best gt best-first search
• Stop Queue gt all
• Prune heuristic or sound

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Complete general-to-specific

• Queue T Q is anti-monotonic
• Th
• WHILE not Queue DO
• Delete h from Queue
• IF Q(h,D) true THEN
• add h to Th
• ELSE Queue Queue ??o(h)
• return Th
• see example 3.26, 3.27

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Heuristic general-to-specific

• Queue T
• Th
• WHILE Th DO
• Delete best h from Queue
• IF Q(h,D) true THEN
• add h to Th
• ELSE Queue Queue ??i(h)
• Queue prune(Queue)
• return Th
• useful when a single good solution suffices
  works for general Q if prune only keeps k best
  gt beam-search see also example 3.28

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Branch-and-bound

• assume bound b(h) exists
• ?h?Lh h?h ? b(h) ? f(h)
• then given current best v for bound we can prune
  all h with v ? b(h)
• can be viewed as a kind of combination of
  complete and heuristic
• see example 3.29

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(Cautious) Specific-to-general

• Queue ?
• Th
• WHILE Queue ? DO
• Delete some h from Queue
• IF Q(h,D) true THEN
• add h to Th
• ELSE select a d ? D such that ?(h?d)
• Queue Queue ? lgg(h,d)
• return Th
• see example 3.31 can be seen as computing S for
  (anti-monotonic) rfreq(h,D) ? 1

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Computing the G border general-to-specific

• Queue T
• Th
• WHILE Queue ? DO
• Delete h from Queue
• IF Q(h,D) true AND h?G THEN
• add h to Th
• ELSE IF Q(h,D) false THEN
• Queue Queue ??o(h)
• return Th
• similar for S possible when computing both S and
  G, more pruning is possible (see example 3.34)

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Computing S and G incrementally

• inrementally update a version space (SG), e.g.
  finding all correct h (rfreq(h,P) 1 ?
  rfreq(h,N)0)
• need msg(g,e) operation, which excludes e from
  g, i.e. minimally specialises g to not cover e
  (example 3.35)

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Mitchells candidate elimination

• S ? G T
• FOR ALL examples e DO
• IF e ? N THEN
• process negative example
• ELSE
• process positive example

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Process negative example

• S - s ? S e ? c(s)
• FOR ALL g ? G e ? c(g) DO
• ?g g ? ms(g,e) ?s?S g?s
• G G ? ?g
• G min(G)

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Process positive example

• G - g ? G e ? c(g)
• FOR ALL s ? S e ? c(s) DO
• ?s s ? lgg(s,e) ?g?G g?s
• S S ? ?s
• S max(S)
• see example 3.36, exercise 3.37

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Interesting properties

• S and G contain only one identical h gt converged
  on single solution
• S or G empty gt no solutions exists
• S and G can determine if any h is still possible
• S and G can predict some e, i.e. these e carry
  no additional information
• try exercise 3.39

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Intersection of version spaces

• two version spaces can be intersected by
  computing the new S as lgg(s1,s2) for all pairs
  of elements from S1 and S2, and by computing the
  new G as glb(g1,g2) for all pairs of elements
  from G1 and G2
• can use this to compute separate VS for every
  single positive example against all negative
  examples,
• then incrementally intersect these VSs