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CANDIDATE-ELIMINATION The positive training example generalizes the S boundary, from Trace 3 S to S4 One member of Gg must also be deleted, because it is no longer more general than the S 3 4 boundary
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bounded by S4 and G4, is shown in Figure 27 This learned version space is independent of the sequence in which the training examples are presented (because in the end it contains all hypotheses consistent with the set of examples) As further training data is encountered, the S and G boundaries will move monotonically closer to each other, delimiting a smaller and smaller version space of candidate hypotheses
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FIGURE 27 The final version space for the EnjoySport concept learning problem and training examples described earlier
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26 REMARKS ON VERSION SPACES AND CANDIDATE-ELIMINATI 261 Will the CANDIDATE-ELIMINATION Converge to the Algorithm Correct Hypothesis
The version space learned by the CANDIDATE-ELIMINATION will conalgorithm verge toward the hypothesis that correctly describes the target concept, provided (1) there are no errors in the training examples, and (2) there is some hypothesis in H that correctly describes the target concept In fact, as new training examples are observed, the version space can be monitored to determine the remaining ambiguity regarding the true target concept and to determine when sufficient training examples have been observed to unambiguously identify the target concept The target concept is exactly learned when the S and G boundary sets converge to a single, identical, hypothesis What will happen if the training data contains errors Suppose, for example, that the second training example above is incorrectly presented as a negative example instead of a positive example Unfortunately, in this case the algorithm is certain to remove the correct target concept from the version space! Because, it will remove every hypothesis that is inconsistent with each training example, it will eliminate the true target concept from the version space as soon as this false negative example is encountered Of course, given sufficient additional training data the learner will eventually detect an inconsistency by noticing that the S and G boundary sets eventually converge to an empty version space Such an empty version space indicates that there is no hypothesis in H consistent with all observed training examples A similar symptom will appear when the training examples are correct, but the target concept cannot be described in the hypothesis representation (eg, if the target concept is a disjunction of feature attributes and the hypothesis space supports only conjunctive descriptions) We will consider such eventualities in greater detail later For now, we consider only the case in which the training examples are correct and the true target concept is present in the hypothesis space
262 What Training Example Should the Learner Request Next
Up to this point we have assumed that training examples are provided to the learner by some external teacher Suppose instead that the learner is allowed to conduct experiments in which it chooses the next instance, then obtains the correct classification for this instance from an external oracle (eg, nature or a teacher) This scenario covers situations in which the learner may conduct experiments in nature (eg, build new bridges and allow nature to classify them as stable or unstable), or in which a teacher is available to provide the correct classification (eg, propose a new bridge and allow the teacher to suggest whether or not it will be stable) We use the term query to refer to such instances constructed by the learner, which are then classified by an external oracle Consider again the version space learned from the four training examples of the Enjoysport concept and illustrated in Figure 23 What would be a good query for the learner to pose at this point What is a good query strategy in
general Clearly, the learner should attempt to discriminate among the alternative competing hypotheses in its current version space Therefore, it should choose an instance that would be classified positive by some of these hypotheses, but negative by others One such instance is
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Note that this instance satisfies three of the six hypotheses in the current version space (Figure 23) If the trainer classifies this instance as a positive example, the S boundary of the version space can then be generalized Alternatively, if the trainer indicates that this is a negative example, the G boundary can then be specialized Either way, the learner will succeed in learning more about the true identity of the target concept, shrinking the version space from six hypotheses to half this number In general, the optimal query strategy for a concept learner is to generate instances that satisfy exactly half the hypotheses in the current version space When this is possible, the size of the version space is reduced by half with each new example, and the correct target concept can therefore be found with only rlog2JVS11experiments The situation is analogous to playing the game twenty questions, in which the goal is to ask yes-no questions to determine the correct hypothesis The optimal strategy for playing twenty questions is to ask questions that evenly split the candidate hypotheses into sets that predict yes and no While we have seen that it is possible to generate an instance that satisfies precisely half the hypotheses in the version space of Figure 23, in general it may not be possible to construct an instance that matches precisely half the hypotheses In such cases, a larger number of queries may be required than rlog21VS(1
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