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28 In this chapter, we commented that given an unbiased hypothesis space (the power set of the instances), the learner would find that each unobserved instance would match exactly half the current members of the version space, regardless of which training examples had been observed Prove this In particular, prove that for any instance space X, any set of training examples D, and any instance x E X not present will in D, that if H is the power set of X, then exactly half the hypotheses in V S H , D classify x as positive and half will classify it as negative 29 Consider a learning problem where each instance is described by a conjunction of n boolean attributes a1 a, Thus, a typical instance would be
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Now consider a hypothesis space H in which each hypothesis is a disjunction of constraints over these attributes For example, a typical hypothesis would be
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Propose an algorithm that accepts a sequence of training examples and outputs a consistent hypothesis if one exists Your algorithm should run in time that is polynomial in n and in the number of training examples 210 Implement the FIND-Salgorithm First verify that it successfully produces the trace in Section 24 for the Enjoysport example Now use this program to study the number of random training examples required to exactly learn the target concept Implement a training example generator that generates random instances, then classifies them according to the target concept:
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Consider training your FIND-Sprogram on randomly generated examples and measuring the number of examples required before the program's hypothesis is identical to the target concept Can you predict the average number of examples required Run the experiment at least 20 times and report the mean number of examples required How do you expect this number to vary with the number of " " in the target concept How would it vary with the number of attributes used to describe instances and hypotheses
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Bruner, J S, Goodnow, J J, & Austin, G A (1957) A study of thinking New York: John Wiey & Sons Buchanan, B G (1974) Scientific theory formation by computer In J C Simon (Ed), Computer Oriented Learning Processes Leyden: Noordhoff Gunter, C A, Ngair, T, Panangaden, P, & Subramanian, D (1991) The common order-theoretic structure of version spaces and ATMS's Proceedings of the National Conference on Artijicial Intelligence (pp 500-505) Anaheim Haussler, D (1988) Quantifying inductive bias: A1 learning algorithms and Valiant's learning framework Artijicial Intelligence, 36, 177-221 Hayes-Roth, F (1974) Schematic classification problems and their solution Pattern Recognition, 6, 105-113 Hirsh, H (1990) Incremental version space merging: A general framework for concept learning Boston: Kluwer
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Hirsh, H (1991) Theoretical underpinnings of version spaces Proceedings of the 12th IJCAI (pp 665-670) Sydney Hirsh, H (1994) Generalizing version spaces Machine Learning, 17(1), 5 4 6 Hunt, E G, & Hovland, D I (1963) Programming a model of human concept formation In E Feigenbaum & J Feldman (Eds), Computers and thought (pp 310-325) New York: McGraw Hill Michalski, R S (1973) AQVALI1: Computer implementation of a variable valued logic system VL1 and examples of its application to pattern recognition Proceedings of the 1st International Joint Conference on Pattern Recognition (pp 3-17) Mitchell, T M (1977) Version spaces: A candidate elimination approach to rule learning Fijlh International Joint Conference on AI @p 305-310) Cambridge, MA: MIT Press Mitchell, T M (1979) Version spaces: An approach to concept learning, (F'hD dissertation) Electrical Engineering Dept, Stanford University, Stanford, CA Mitchell, T M (1982) Generalization as search ArtQcial Intelligence, 18(2), 203-226 Mitchell, T M, Utgoff, P E, & Baneji, R (1983) Learning by experimentation: Acquiring and modifying problem-solving heuristics In Michalski, Carbonell, & Mitchell (Eds), Machine Learning (Vol 1, pp 163-190) Tioga Press Plotkin, G D (1970) A note on inductive generalization In Meltzer & Michie (Eds), Machine Intelligence 5 (pp 153-163) Edinburgh University Press Plotkin, G D (1971) A further note on inductive generalization In Meltzer & Michie (Eds), Machine Intelligence 6 (pp 104-124) Edinburgh University Press Popplestone, R J (1969) An experiment in automatic induction In Meltzer & Michie (Eds), Machine Intelligence 5 (pp 204-215) Edinburgh University Press Sebag, M (1994) Using constraints to build version spaces Proceedings of the 1994 European Conference on Machine Learning Springer-Verlag Sebag, M (1996) Delaying the choice of bias: A disjunctive version space approach Proceedings of the 13th International Conference on Machine Learning (pp 444-452) San Francisco: Morgan Kaufmann Simon, H A,, & Lea, G (1973) Problem solving and rule induction: A unified view In Gregg (Ed), Knowledge and Cognition (pp 105-127) New Jersey: Lawrence Erlbaum Associates Smith, B D, & Rosenbloom, P (1990) Incremental non-backtracking focusing: A polynomially bounded generalization algorithm for version spaces Proceedings of the 1990 National Conference on ArtQcial Intelligence (pp 848-853) Boston Subramanian, D, & Feigenbaum, J (1986) Factorization in experiment generation Proceedings of the I986 National Conference on ArtQcial Intelligence (pp 518-522) Morgan Kaufmann Vere, S A (1975) Induction of concepts in the predicate calculus Fourth International Joint Conference on AI (pp 281-287) Tbilisi, USSR Winston, P H (1970) Learning structural descriptions from examples, (PhD dissertation) [MIT Technical Report AI-TR-2311