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which proves the theorem To summarize, the above theorem states that the number of mistakes made by the WEIGHTED-MAJORITY algorithm will never be greater than a constant factor times the number of mistakes made by the best member of the pool, plus a term that grows only logarithmically in the size of the pool This theorem is generalized by Littlestone and Warmuth (1991), who show 3 that for an arbitrary 0 5 j < 1 the above bound is
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76 SUMMARY AND FURTHER READING
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The main points of this chapter include:
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The probably approximately correct (PAC) model considers algorithms that learn target concepts from some concept class C, using training examples drawn at random according to an unknown, but fixed, probability distribution It requires that the learner probably (with probability at least [ l - 61) learn a hypothesis that is approximately (within error E) correct, given computational effort and training examples that grow only polynornially with I/ , 1/6, the size of the instances, and the size of the target concept Within the setting of the PAC learning model, any consistent learner using H will, with probability (1 - S), a finite hypothesis space H where C output a hypothesis within error E of the target concept, after observing m randomly drawn training examples, as long as
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This gives a bound on the number of training examples sufficient for successful learning under the PAC model One constraining assumption of the PAC learning model is that the learner knows in advance some restricted concept class C that contains the target concept to be learned In contrast, the agnostic learning model considers the more general setting in which the learner makes no assumption about the class from which the target concept is drawn Instead, the learner outputs the hypothesis from H that has the least error (possibly nonzero) over the training data Under this less restrictive agnostic learning model, the learner is assured with probability (1 -6) to output a hypothesis within error E of the
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best possible hypothesis in H, after observing rn randomly drawn training examples, provided
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The number of training examples required for successful learning is strongly influenced by the complexity of the hypothesis space considered by the learner One useful measure of the complexity of a hypothesis space H is its Vapnik-Chervonenkis dimension, VC(H) VC(H) is the size of the largest subset of instances that can be shattered (split in all possible ways) by H An alternative upper bound on the number of training examples sufficient for successful learning under the PAC model, stated in terms of VC(H) is
A lower bound is
An alternative learning model, called the mistake bound model, is used to analyze the number of training examples a learner will misclassify before it exactly learns the target concept For example, the HALVING algorithm will make at most Llog, 1 H 1J mistakes before exactly learning any target concept drawn from H For an arbitrary concept class C , the best worstcase algorithm will make Opt (C) mistakes, where log,(lCI) VC(C> 5 Opt(C) I The WEIGHTED-MAJORITY algorithm combines the weighted votes of multiple prediction algorithms to classify new instances It learns weights for each of these prediction algorithms based on errors made over a sequence of examples Interestingly, the number of mistakes made by WEIGHTED-MAJORITY can be bounded in terms of the number of mistakes made by the best prediction algorithm in the pool
Much early work on computational learning theory dealt with the question of whether the learner could identify the target concept in the limit, given an indefinitely long sequence of training examples The identification in the limit model was introduced by Gold (1967) A good overview of results in this area is (Angluin 1992) Vapnik (1982) examines in detail the problem of uniform convergence, and the closely related PAC-learning model was introduced by Valiant (1984) The discussion in this chapter of -exhausting the version space is based on Haussler's (1988) exposition A useful collection of results under the PAC model can be found in Blumer et al (1989) Kearns and Vazirani (1994) provide an excellent exposition of many results from computational learning theory Earlier texts in this area include Anthony and Biggs (1992) and Natarajan (1991)
Current research on computational learning theory covers a broad range of learning models and learning algorithms Much of this research can be found in the proceedings of the annual conference on Computational Learning Theory (COLT) Several special issues of the journal Machine Learning have also been devoted to this topic
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