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FOCL uses the domain theory to increase the number of candidate specializations considered at each step of the search for a single Horn clause Figure 129 compares the hypothesis space search performed by FOCL to that performed by the purely inductive FOIL algorithm on which it is based FOCL's theory-suggested specializations correspond to "macro" steps in FOIL'S search, in which several literals are added in a single step This process can be viewed as promoting a hypothesis that might be considered later in the search to one that will be considered immediately If the domain theory is correct, the training data will bear out the superiority of this candidate over the others and it will be selected If the domain theory is incorrect, the empirical evaluation of all the candidates should direct the search down an alternative path To summarize, FOCL uses both a syntactic generation of candidate specializations and a domain theory driven generation of candidate specializations at each step in the search The algorithm chooses among these candidates based solely on their empirical support over the training data Thus, the domain theory is used in a fashion that biases the learner, but leaves final search choices to be made based on performance over the training data The bias introduced by the domain theory is a preference in favor of Horn clauses most similar to operational, logically sufficient conditions entailed by the domain theory This bias is combined with
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Hypotheses thatfit training data equally well
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FIGURE 1 29 Hypothesis space search in FOCL FOCL augments the set of search operators used by FOIL Whereas FOIL considers adding a single new literal at each step, FOIL also considers adding multiple literals derived from the domain theory
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the bias of the purely inductive FOIL program, which is a preference for shorter hypotheses FOCL has been shown to generalize more accurately than the purely inductive FOIL algorithm in a number of application domains in which an imperfect domain theory is available For example, Pazzani and Kibler (1992) explore learning the concept "legal chessboard positions" Given 60 training examples describing 30 legal and 30 illegal endgame board positions, FOIL achieved an accuracy of 86% over an independent set of test examples FOCL was given the same 60 training examples, along with an approximate domain theory with an accuracy of 76% FOCL produced a hypothesis with generalization accuracy of 94%-less than half the error rate of FOIL Similar results have been obtained in other domains For example, given 500 training examples of telephone network problems and their diagnoses from the telephone company NYNEX, FOIL achieved an accuracy of 90%, whereas FOCL reached an accuracy of 98% when given the same training data along with a 95% accurate domain theory
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126 STATE OF THE ART
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The methods presented in this chapter are only a sample of the possible approaches to combining analytical and inductive learning While each of these methods has been demonstrated to outperform purely inductive learning methods in selected domains, none of these has been thoroughly tested or proven across a large variety of problem domains The topic of combining inductive and analytical learning remains a very active research area
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127 SUMMARY AND FURTHER READING
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The main points of this chapter include:
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Approximate prior knowledge, or domain theories, are available in many practical learning problems Purely inductive methods such as decision tree induction and neural network BACKPROPAGATION utilize such domain fail to theories, and therefore perform poorly when data is scarce Purely analytical learning methods such as PROLOG-EBG utilize such domain theories, but produce incorrect hypotheses when given imperfect prior knowledge Methods that blend inductive and analytical learning can gain the benefits of both approaches: reduced sample complexity and the ability to overrule incorrect prior knowledge One way to view algorithms for combining inductive and analytical learning is to consider how the domain theory affects the hypothesis space search In this chapter we examined methods that use imperfect domain theories to (1) create the initial hypothesis in the search, (2) expand the set of search operators that generate revisions to the current hypothesis, and (3) alter the objective of the search A system that uses the domain theory to initialize the hypothesis is KBANN This algorithm uses a domain theory encoded as propositional rules to analytically construct an artificial neural network that is equivalent to the domain theory This network is then inductively refined using the BACKPROPAGATION algorithm, to improve its performance over the training data The result is a network biased by the original domain theory, whose weights are refined inductively based on the training data TANGENTPROP prior knowledge represented by desired derivatives of uses the target function In some domains, such as image processing, this is a natural way to express prior knowledge TANGENTPROP incorporates this knowledge by altering the objective function minimized by gradient descent search through the space of possible hypotheses EBNN uses the domain theory to alter the objective in searching the hypothesis space of possible weights for an artificial neural network It uses a domain theory consisting of previously learned neural networks to perform a neural network analog to symbolic explanation-based learning As in symbolic explanation-based learning, the domain theory is used to explain individual examples, yielding information about the relevance of different example features With this neural network representation, however, information about relevance is expressed in the form of derivatives of the target function value with respect to instance features The network hypothesis is trained using a variant of the TANGENTPROP algorithm, in which the error to be minimized includes both the error in network output values and the error in network derivatives obtained from explanations FOCL uses the domain theory to expand the set of candidates considered at each step in the search It uses an approximate domain theory represented
by first order Horn clauses to learn a set of Horn clauses that approximate the target function FOCL employs a sequential covering algorithm, learning each Horn clause by a general-to-specific search The domain theory is used to augment the set of next more specific candidate hypotheses considered at each step of this search Candidate hypotheses are then evaluated based on their performance over the training data In this way, FOCL combines the greedy, general-to-specific inductive search strategy of FOIL with the rule-chaining, analytical reasoning of analytical methods The question of how to best blend prior knowledge with new observations remains one of the key open questions in machine learning
There are many more examples of algorithms that attempt to combine inductive and analytical learning For example, methods for learning Bayesian belief networks discussed in 6 provide one alternative to the approaches discussed here The references at the end of this chapter provide additional examples and sources for further reading
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