1. Chapter 2 - Concept learning
Learning from examples
Concept Learning
General-to specific ordering of hypotheses
Version spaces and candidate elimination algorithm
Inductive bias
Simple approach
Assumption: no noise
Illustrates key concepts

2. Concept learning task
Given examples, learn a general concept (category)
subset of some general domain
E.g., birds
boolean-valued function over the domain (characteristic function)
a function defined over all animals whose value is true for birds and false for every other animal
Definition
infer a boolean-valued function from samples of it (input, output)

3. Concept Learning
The problem is to learn a function mapping examples into two classes: positive and negative.
We are given a database of examples already classified as positive or negative.
Concept learning: the process of inducing a function mapping input examples into a Boolean output.
Examples:
Classifying objects in astronomical images as stars or galaxies
Classifying animals as vertebrates or invertebrates

4. Training Examples for Enjoy Sport
Concept: ”days on which my friend Aldo enjoys his favourite
water sports”
Task: predict the value of ”Enjoy Sport” for an arbitrary day
based on the values of the other attributes
attributes
Sky
Temp
Humid
Wind
Water
Fore-cast
Enjoy Sport
Sunny
Rainy
Warm
Cold
Normal
High
Strong
Cool
Same
Change
Yes No
instance

5. Representing Hypothesis
Hypothesis h is a conjunction of constraints on attributes
Each constraint can be:
A specific value : e.g. Water=Warm
A don’t care value : e.g. Water=?
No value allowed (null hypothesis): e.g. Water=Ø
Example: hypothesis h
Sky Temp Humid Wind Water Forecast
< Sunny ? ? Strong ? Same >
Instance x satisfies hypothesis h
all the constraints in h hold for x
define h(x) = 1

6. Prototypical Concept Learning Task
Given:
Instances X : Possible days decribed by the attributes Sky, Temp, Humidity, Wind, Water, Forecast
Target function c: EnjoySport X  {0,1}
Hypotheses H: conjunction of literals e.g.
< Sunny ? ? Strong ? Same >
Training examples D : positive and negative examples of the target function: <x1,c(x1)>,…, <xn,c(xn)>
Determine:
A hypothesis h in H such that h(x)=c(x) for all x in D.

7. Inductive Learning Hypothesis
All we know of c are the examples
Best we can do is to produce a h consistent with example data
Any hypothesis found to approximate the target function well over the training examples, will also approximate the target function well over the unobserved examples.
fundamental assumption in inductive learning

8. Concept learning as search
H = search space
find h best fitting to D
Selection of representation defines search space
size of space (syntactic/semantic)
Efficient search in very large (or infinite) spaces for best h

9. # of Instances, Concepts, Hypo.s
Sky: Sunny, Cloudy, Rainy
AirTemp: Warm, Cold
Humidity: Normal, High
Wind: Strong, Weak
Water: Warm, Cold
Forecast: Same, Change
#distinct instances : 3*2*2*2*2*2 = 96
#distinct concepts : 296
#syntactically distinct hypotheses : 5*4*4*4*4*4=5120
#semantically distinct hypotheses : 1+4*3*3*3*3*3=973

10. General-to-Specific order
Useful structure
organize the search process
exists for any concept learning task
search possible without enumerating all members of H (may be infinite)
A concept hj is more general than or equal to hk (written hj  hk) if and only if
x  X : [ (hk(x) = 1)  (hj(x) = 1)]

11. General to Specific Order
Consider two hypotheses:
h1=< Sunny,?,?,Strong,?,?>
h2=< Sunny,?,?,?,?,?>
Set of instances covered by h1 and h2:
h2 imposes fewer constraints than h1 and therefore classifies more instances x as positive.
h2 is more general than h1 or h2  h1
The relation  imposes a partial order over the hypothesis space H that is
independent of c!
utilized by many concept learning methods.

12. Insta., Hypo. and ”more general”
Instances x2 x1
Hypotheses h2 h3 h1
h2  h1
h2  h3
specific
general
x1=< Sunny,Warm,High,Strong,Cool,Same>
h1=< Sunny,?,?,Strong,?,?>
x2=< Sunny,Warm,High,Light,Warm,Same>
h2=< Sunny,?,?,?,?,?>
h3=< Sunny,?,?,?,Cool,?>

13. Find-S algorithm Example of using the ordering
begin: the most specific member of H
generalize it whenever h(x) = 0 and c(x) = 1 (examples)
Ignores negative examples
no problem assuming that
H contains a description of c and
D has no noise

14. Find-S Algorithm Initialize h to the most specific hypothesis in H
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
Output hypothesis h

15. Hypoth. Space Search by Find-S
Instances
Hypotheses h1
x1=<Sunny,Warm,Normal,Strong,Warm,Same>+ x1
h1=< Sunny,Warm,Normal,
Strong,Warm,Same>
specific
x3=<Rainy,Cold,High,Strong,Warm,Change> - x3 h0
h0=< Ø, Ø, Ø, Ø, Ø, Ø,>
h2,3
x2=<Sunny,Warm,High,Strong,Warm,Same>+ x2
h2,3=< Sunny,Warm,?,
Strong,Warm,Same> h4
x4=<Sunny,Warm,High,Strong,Cool,Change> + x4
h4=< Sunny,Warm,?,
Strong,?,?>
general

16. Find-S – Black board example
Size = large/small, Color = red/white/blue and Shape = cube/brick/round
Examples
Positive: obj(small, red, round)
Positive: obj(small, white, round)
Positive: obj(large, white, round)
What does Find-S find?
What if we add the following negative example ?
Negative: obj(large, blue, cube)
Negative: obj(large, blue, round)

17. Properties of Find-S
For Hypothesis space described by conjunctions of constraints, Find-S will output the most specific hypothesis within H that is consistent with the positve training examples
The output hypothesis will also be consistent with the negative examples, provided the target concept is contained in H.

18. Complaints about Find-S
Can’t tell if the learner has converged to the target concept, in the sense that it is unable to determine whether it has found the only hypothesis consistent with the training examples.
Can’t tell when training data is inconsistent, as it ignores negative training examples.
Why prefer the most specific hypothesis?
‘most specific’ is not always unique.
Find-S cannot backtrack!

19. Version Space and Cand. Elimi.
Another learning approach
outputs a description of all members of H that are consistent with D
possible without enumerating members of H (ordering!)
suffers from noise
useful framework for introducing many fundamental issues of ML

20. Version Spaces
A hypothesis h is consistent with a set of training examples D of target concept 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)
The version space, VSH,D , with respect to hypothesis space H, and training set D, is the subset of all hypotheses in H consistent with all training examples:
VSH,D = {h  H | Consistent(h,D) }

21. Elimination algorithms
List-then-Eliminate
list every hypothesis in H
at each example, remove inconsistent hypo
output the list of surviving hypotheses
something to start with
ineffective
Candidate elimination
same principle
more compact representation
maintain most specific and most general elements of VS(H,D)

22. Representing Version Spaces
The general boundary, G, of version space VSH,D is the set of maximally general members.
The specific boundary, S, of version space VSH,D is the set of maximally specific members.
Every member of the version space lies between these boundaries
VSH,D = {h  H| ( s  S) ( g  G) (g  h  s)
where x  y means x is more general or equal than y

23. Example Version Space S: G:
{<Sunny,Warm,?,Strong,?,?>}
<Sunny,?,?,Strong,?,?>
<Sunny,Warm,?,?,?,?>
<?,Warm,?,Strong,?,?> G:
{<Sunny,?,?,?,?,?>, <?,Warm,?,?,?>, }
x1 = <Sunny Warm Normal Strong Warm Same> +
x2 = <Sunny Warm High Strong Warm Same> +
x3 = <Rainy Cold High Strong Warm Change> -
x4 = <Sunny Warm High Strong Cool Change> +

24. Representation theorem
For any
set of instances X
set H of boolean-valued hypotheses over X
target concept c: X -> {0,1}
set of pairs <x,c(x)> D
VS(H,D) = {h | g >= h >= s}
where g & s range over G & S

25. Candidate Elimination Algorithm
G  maximally general hypotheses in H
S  maximally specific hypotheses in H
For each training example d=<x,c(x)>
If d is a positive example
Remove from G any hypothesis that is 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 consistent with d
Some member of G is more general than h
Remove from S any hypothesis that is more general than another hypothesis in S

26. Candidate Elimination Algorithm
If d is a negative example
Remove from S any hypothesis that is 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 consistent with d
Some member of S is more specific than h
Remove from G any hypothesis that is less general than another hypothesis in G

27. Candidate Elimination Algorithm

28. Example Candidate Elimination
x1 = <Sunny Warm Normal Strong Warm Same> +
{<, , , , ,  >} S:
{<?, ?, ?, ?, ?, ?>} G:
x2 = <Sunny Warm High Strong Warm Same> +
{< Sunny Warm Normal Strong Warm Same >} S:
{<?, ?, ?, ?, ?, ?>} G:
{< Sunny Warm ? Strong Warm Same >} S:
{<?, ?, ?, ?, ?, ?>} G:

29. Example Candidate Elimination..S:
{< Sunny Warm ? Strong Warm Same >}
x3 = <Rainy Cold High Strong Warm Change> - G:
{<?, ?, ?, ?, ?, ?>}
x4 = <Sunny Warm High Strong Cool Change> +
{< Sunny Warm ? Strong Warm Same >} S:
{<Sunny,?,?,?,?,?>, <?,Warm,?,?,?>, <?,?,?,?,?,Same>} G:
{< Sunny Warm ? Strong ? ? >} S:
{<Sunny,?,?,?,?,?>, <?,Warm,?,?,?> } G:

30. Example Candidate Elimination..S:
{<Sunny,Warm,?,Strong,?,?>}
<Sunny,?,?,Strong,?,?>
<Sunny,Warm,?,?,?,?>
<?,Warm,?,Strong,?,?> G:
{<Sunny,?,?,?,?,?>, <?,Warm,?,?,?>, }
x1 = <Sunny Warm Normal Strong Warm Same> +
x2 = <Sunny Warm High Strong Warm Same> +
x3 = <Rainy Cold High Strong Warm Change> -
x4 = <Sunny Warm High Strong Cool Change> +

31. Cand. Elim. – Black board example
Size = large/small, Color = red/white/blue and Shape = cube/brick/round
Examples
Positive (+): obj(small, red, round)
Negative (-): obj(small, blue, round)
Positive (+): obj(large, red, round)
Negative (-): obj(large, red, cube)
What does Candidate Elimination find?

32. Remarks on VS & CE Converges to correct h? Effect of errors (noise)
If no errors, H rich enough: yes
exact: G = S and both singletons
Effect of errors (noise)
example 2 negative -> CE removes the correct target from VS!
detection: VS gets empty
Similarly, VS gets empty when c can not be represented (e.g. disjunctions)

33. Classification of New Data
{<Sunny,Warm,?,Strong,?,?>}
<Sunny,?,?,Strong,?,?>
<Sunny,Warm,?,?,?,?>
<?,Warm,?,Strong,?,?> G:
{<Sunny,?,?,?,?,?>, <?,Warm,?,?,?>, }
x5 = <Sunny Warm Normal Strong Cool Change>
x6 = <Rainy Cold Normal Light Warm Same>
x7 = <Sunny Warm Normal Light Warm Same>
x8 = <Sunny Cold Normal Strong Warm Same>
+ 6/0
- 0/6
? 3/3
? 2/4

34. How to use partially learned concepts?
Classification with ambiguous VS
h(x) = 0/1 for every h in VS: ok
enough to check with G (0) & S (1)
3rd example: 50% support for both
4th: 66% support for 0
majority vote + confidence?
Ok if all h are equally likely

35. What example next? Three settings for learning:
Tasks are generated by a random process outside the learner
The learner can pose queries to be classified by a teacher
The learner explores its surroundings autonomously (conducts an experiment)
What to ask/check as next example/query?
Is there a good general strategy?
Should attempt to discriminate among the alternatives in VS

36. What to ask... Discriminating example Optimal choice halves VS
c(x) = 1: S will generalize
c(x) = 0: G will specialize
Optimal choice halves VS
x satisfies half of VS members
for example,
<Sunny, warm, Normal, Light, Warm, Same>
not possible to generate one in general
if we are able to do this, we can find an optimal hypothesis in log2 |VS| steps

37. Inductive Leap + <Sunny Warm Normal Strong Cool Change>
+ <Sunny Warm Normal Light Warm Same>
S : <Sunny Warm Normal ? ? ?>
How can we justify to classify the new example as
+ <Sunny Warm Normal Strong Warm Same>
Bias: We assume that the hypothesis space H contains
the target concept c. In other words that c can be
described by a conjunction of literals.

38. Biased hypothesis space
Assure H contains c
make H general enough
what if it’s not?
Example case
max specific h consistent with x1 & x2 (+) is too general for x3 (-) See next slide
x1 = <Sunny Warm Normal Strong Cool Change> +
x2 = <Cloudy Warm Normal Strong Cool Change> +
x3 = <Rainy Warm Normal Strong Cool Change> -
reason: we have biased the learner to consider only conjunctions

39. Biased Hypothesis Space
Our hypothesis space is unable to represent a simple disjunctive target concept : (Sky=Sunny) v (Sky=Cloudy)
x1 = <Sunny Warm Normal Strong Cool Change> +
x2 = <Cloudy Warm Normal Strong Cool Change> +
S : { <?, Warm, Normal, Strong, Cool, Change> }
x3 = <Rainy Warm Normal Strong, Cool, Change> -
S : {}

40. Unbiased Learner
Idea: Choose H that expresses every teachable concept, that means H is the set of all possible subsets of X called the power set P(X)
|X|=96, |P(X)|=296 ~ distinct concepts (too large!)
H = disjunctions, conjunctions, negations
e.g. <Sunny Warm Normal ? ? ?> v <? ? ? ? ? Change>
H surely contains the target concept.
Sky = Sunny or Sky = Cloudy
eliminates problems of expressibility
BUT, also eliminates the ability to generalize!

41. Unbiased Learner
Assume we have positive examples (x1, x2, x3) and negative examples (x4, x5)
S : { (x1 v x2 v x3) }
G : {  (x4 v x5) }
The only examples that are classified are the training examples themselves.
Each unobserved instance will be classified positive by precisely half the hypothesis in VS
For every concept classifying x positively, there is another concept that differs only on x, classifies x negatively.
That’s, in order to learn the target concept one would have to present every single instance in X as a training example.

42. Futility of Bias-Free LearningIf
learner makes no a priori assumptions on the target concept
Then
it has no rational basis for classifying any unseen instances
Inductive bias = prior assumptions
Inductive bias is the preference for a hypothesis space H and a search mechanism over H.

43. 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 instance xi by L after training on Dc.
Definition: The inductive bias of L is any minimal set of assertions B such that for any target concept c and corresponding training data Dc
(xi  X)[B  Dc  xi] |-- L(xi, Dc)
Where A |-- B means that A logically entails B.

44. Inductive and Deductive Systems
training
examples
classification of
new instance or
don’t know
candidate elimination
algorithm
using hypothesis space H
new instance
equivalent deductive system
training
examples
classification of
new instance or
don’t know
theorem prover
new instance
assertion ”H
contains target
concept”

45. Schematic summary Top: inductive system Bottom: deductive system
built-in (implicit) bias
Bottom: deductive system
explicit bias
non-procedural way for characterizing learning systems
allows comparisons of different learning algorithms

46. Three learners & their biases
Rote learner: no bias
Candidate Elimination
c is (representable) in H
Find-S
The hypothesis space contains the target concept and all instances are negative instances unless the opposite is entailed by its other knowledge.
c is the most specific consistent hypotheses of H

47. Summary
The input space is the space of all examples; the output space is the space of all classes.
A hypothesis maps examples into classes.
We want a hypothesis close to target concept c.
The General-to specific ordering establishes a natural partial ordering over the hypothesis space.
One can exploit this ordering to move along the corresponding lattice.

48. Summary
The candidate elimination algorithm exploits the general-specific ordering of hypotheses to find all hypotheses consistent with the training data.
The version space contains all consistent hypotheses and is simply represented by two lists: S and G.
Candidate elimination algorithm is not robust to noise and assumes the target concept is included in the hypothesis space.
Any inductive algorithm needs some assumptions about the hypothesis space, otherwise it would be impossible to perform predictions.