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As noted above, PAC-learnability is largely determined by the number of training examples required by the learner The growth in the number of required training examples with problem size, called the sample complexity of the learning problem, is the characteristic that is usually of greatest interest The reason is that in most practical settings the factor that most limits success of the learner is the limited availability of training data Here we present a general bound on the sample complexity for a very broad class of learners, called consistent learners A learner is consistent if it outputs hypotheses that perfectly fit the training data, whenever possible It is quite reasonable to ask that a learning algorithm be consistent, given that we typically prefer a hypothesis that fits the training data over one that does not Note that many of the learning algorithms discussed in earlier chapters, including all the learning algorithms described in 2, are consistent learners Can we derive a bound on the number of training examples required by any consistent learner, independent of the specific algorithm it uses to derive a consistent hypothesis The answer is yes To accomplish this, it is useful to recall the definition of version space from 2 There we defined the version space, V S H , D ,to be the set of all hypotheses h E H that correctly classify the training examples D
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The significance of the version space here is that every consistent learner outputs a hypothesis belonging to the version space, regardless of the instance space X, hypothesis space H, or training data D The reason is simply that by definition the version space V S H , Dcontains every consistent hypothesis in H Therefore, to bound the number of examples needed by any consistent learner, we need only bound the number of examples needed to assure that the version space contains no unacceptable hypotheses The following definition, after Haussler (1988), states this condition precisely
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Definition: Consider a hypothesis space H, target concept c, instance distribution V ,and set of training examples D of c The version space V S , , is said to be -exhaustedwith respect to c and V,if every hypothesis h in VSH,* has error less than 6 with respect to c and V
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This definition is illustrated in Figure 72 The version space is -exhausted just in the case that all the hypotheses consistent with the observed training examples (ie, those with zero training error) happen to have true error less than E Of course from the learner's viewpoint all that can be known is that these hypotheses fit the training data equally well-they all have zero training error Only an observer who knew the identity of the target concept could determine with certainty whether the version space is +exhausted Surprisingly, a probabilistic argument allows us to bound the probability that the version space will be -exhausted after a given number of training examples, even without knowing the identity of the target concept or the distribution from which training examples Hypothesis space H
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FIGURE 72 Exhausting the version space The version space VSH,Dis the subset of hypotheses h E H, which have zero training error (denoted by r = 0 in the figure) Of course the true errorv(h) (denoted by error in the figure) may be nonzero, even for hypotheses that commit zero errors over the training ~ data The version space is said to be -exhausted when all hypotheses h remaining in V S H ,have errorw(h) < E
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are drawn Haussler (1988) provides such a bound, in the form of the following theorem Theorem 71 -exhaustingthe version space If the hypothesis space H is finite, and D is a sequence of rn 1 independent randomly drawn examples of some target concept c, then for any 0 5 E 5 1, the probability that the version space V S H ,is ~ not -exhausted (with respect to c) is less than or equal to
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