Machine Learning - DISCo

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50 MACHINE LEARNING 2.8. 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 in D, that if H is the power set of X, then exactly half the hypotheses in VSH,D will classify x as positive and half will classify it as negative. 2.9. Consider a learning problem where each instance is described by a conjunction of n boolean attributes a1 . . .a,. Thus, a typical instance would be (al = T) A (az = F) A . . . A (a, = T) 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 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. 2.10. Implement the FIND-S algorithm. First verify that it successfully produces the trace in Section 2.4 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: (Sunny, Warm, ?, ?, ?, ?) Consider training your FIND-S program 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? REFERENCES 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.
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), 546. 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: Mc- Graw 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'h.D. 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, (Ph.D. dissertation). [MIT Technical Report AI-TR-2311.
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Machine Learning Tom M. Mitchell Pr
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xvi PREFACE A third principle that
Page 13 and 14: CHAPTER INTRODUCTION Ever since com
Page 15 and 16: CHAPTER 1 INTRODUCITON 3 0 Learning
Page 17 and 18: CHAFTlB 1 INTRODUCTION 5 that allow
Page 19 and 20: 1.2.2 Choosing the Target Function
Page 21 and 22: CHAPTER I INTRODUCTION 9 x5: the nu
Page 23 and 24: Thus, we seek the weights, or equiv
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Page 29 and 30: 0 Chapter 10 covers algorithms for
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Page 33 and 34: CHAFER 2 CONCEm LEARNING AND THE GE
Page 35 and 36: When learning the target concept, t
Page 37 and 38: CHAPTER 2 CONCEPT LEARNING AND THE
Page 39 and 40: is needed. In the general case, as
Page 41 and 42: 2.5 VERSION SPACES AND THE CANDIDAT
Page 43 and 44: { 1 FIGURE 2.3 A version space w
Page 45 and 46: CHAPTER 2 CONCEET LEARNJNG AND THE
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Page 79 and 80: easonable strategy, in fact it can
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Page 83 and 84: multiple ways, then averaging the r
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CHAPTER 4 ARTIFICIAL NEURAL NETWORK
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and combining this with Equations (
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Inputs Outputs Input 10000000 0 100
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FIGURE 4.8 Learning the 8 x 3 x 8 N
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30 x 32 resolution input images lef
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left, (0,1,0,O) to encode a face lo
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close to zero, as the bright face w
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4.8.2 Alternative Error Minimizatio
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BACKPROPAGATION searches the space
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4.6. Explain informally why the del
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Mitchell, T. M., & Thrun, S. B. (19
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the impact of possible pruning step
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Here the notation Pr denotes that t
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a A random variable can be viewed a
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5.3.2 The Binomial Distribution A g
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In case the random variable Y is go
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which is wide enough to contain 95%
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intervals for discrete-valued hypot
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Then as n + co, the distribution go
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we might observe this difference in
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1. Partition the available data Do
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perfectly fits the form of the abov
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CHAFER 5 EVALUATING HYPOTHESES 151
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Geman, S., Bienenstock, E., & Dours
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CHAFER 6 BAYESIAN LEARNING 155 U Fo
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CHAPTER 6 BAYESIAN LEARNING 157 As
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. - Product rule: probability P(A A
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CHAPTER 6 BAYESIAN LEARNING 161 Rec
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CHAPTER 6 BAYESIAN LEARNING 163 of
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CHAPTER 6 BAYESIAN LEARNING 165 a l
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CHAPTER 6 BAYESIAN LEARNING 167 the
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CHAPTER 6 BAYESIAN LEARNING 169 doe
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CIUPlER 6 BAYESIAN LEARNING 171 sub
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CHAPTER 6 BAYESIAN LEARNING 173 0 T
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CHAFER 6 BAYESIAN LEARNING 175 the
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6.9 NAIVE BAYES CLASSIFIER CHAPTJZR
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CHAETER 6 BAYESIAN LEARNING 179 Sim
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-a- CHAPTER 6 BAYESIAN LEARNING 181
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CHAPTER 6 BAYESIAN LEARNING 183 Exa
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CHAPTER 6 BAYESIAN LEARNING 185 In
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C H m R 6 BAYESIAN LEARNING 187 Fir
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CHAPTER 6 BAYESIAN LEARNING 189 For
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CHAPTER 6 BAYESIAN LEARNING 191 ord
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CHAPTER 6 BAYESIAN LEARNING 193 max
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CHAPTER 6 BAYESIAN LEARNING 195 Ste
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6.13 SUMMARY AND FURTHER READING Th
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CHAPTER 6 BAYESIAN LEARNING 199 pro
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CHAPTER COMPUTATIONAL LEARNING THEO
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CHAPTER 7 COMPUTATIONAL LEARNING TH
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C Instance space X Where c and h di
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usually more concerned with the num
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are drawn. Haussler (1988) provides
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CHAPTER 7 COMPUTATIONAL LEARNING TH
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C that contains every teachable con
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Instance space X FIGURE 73 A set of
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can show that it is at least 3, as
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The following theorem bounds the VC
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FIND-S: C CHAPTER 7 COMPUTATIONAL L
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We define the optimal mistake bound
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log' Rearranging terms yields which
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Current research on computational l
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(b) You now draw a new set of 100 i
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CHAPTER 8 INSTANCE-BASED LEARNING 2
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(i.e., on all axes in the Euclidean
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8.3.1 Locally Weighted Linear Regre
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C H m R 8 INSTANCE-BASED LEARNING 2
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CHAPTER 8 INSTANCEBASED LEARNING 24
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CHAPTER 8 INSTANCE-BASED LEARMNG 24
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A thorough discussion of radial bas
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CHAPTER GENETIC ALGORITHMS Genetic
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Fitness: A function that assigns an
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can then be represented by the foll
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the crossover point n is chosen at
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value of the target attribute c. Th
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In the above experiment, the two ne
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The second step above follows from
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FIGURE 9.2 Crossover operation appl
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0 (DU x y) (do until), which execut
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9.6.2 Baldwin Effect Although Lamar
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iteration, the most fit members of
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Belew, R. K., & Mitchell, M. (Eds.)
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Tumey, P. D., Whitley, D., & Anders
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As an example of first-order rule s
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10.2.1 General to Specific Beam Sea
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A few remarks on the LEARN-ONE-RULE
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decisions regarding preconditions o
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10.4 LEARNING FIRST-ORDER RULES In
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Every well-formed expression is com
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eralize the current disjunctive hyp
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Here let us also make the closed wo
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In the case of noise-free training
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(e.g., neural network) fits the dat
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C : KnowMaterial v -Study C: KnowMa
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CHAPTER 10 LEARNING SETS OF RULES 2
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In contrast, the generate-and-test
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which enable the user to specify th
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Wrobel (1992) use rule schemata in
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De Raedt, L., & Bruynooghe, M. (199
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CHAPTER ANALYTICAL LEARNING Inducti
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What is interesting about this ches
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Given: rn Instance space X: Each in
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theories to automatically improve p
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Explanation: Training Example: FIGU
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FIGURE 11.3 Computing the weakest p
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example that is not yet covered by
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contain no description of such a pr
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additional detail that must be cons
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CHAPTER 11 ANALYTICAL LEARNING 325
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explaining to itself the reason for
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that fits the learner's prior knowl
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Height (Joe, Short) Height(Sue, Sho
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CHAF'TER 11 ANALYTICAL LEARNING 333
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CHAPTER 12 COMBINING INDUCTIVE AND
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CHAF'TER 12 COMBINING INDUCTIVE AND
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12.2.2 Hypothesis Space Search CAAP
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- KBANN(Domain-Theory, Training_Exa
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CHAF'TER 12 COMBINING INDUCTIVE AND
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Hypothesis Space Hypotheses that ma
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CHAFER 12 COMBINING INDUCTIVE AND A
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CHAPTER 12 COMBmG INDUCTIVE AND ANA
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Hypothesis Space 1 Hypotheses thatf
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y first order Horn clauses to learn
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Craven, M. W., & Shavlik, J. W. (19
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CHAPTER REINFORCEMENT LEARNING Rein
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has or does not have prior knowledg
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CHAPTER 13 REINFORCEMENT LEARNING 3
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CHAPTER 13 REINFORCEMENT LEARNJNG 3
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While Q learning and related reinfo
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a neural network reinforcement lear
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CHAFER 13 REINFORCEMENT LEARNING 38
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CHaPrER 13 REINFORCEMENT LEARNING 3
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APPENDIX NOTATION Below is a summar
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INDEXES
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in KBANN algorithm, 343-344 optimiz
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leave-one-out, 235 in neural networ
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extensions to, 258-259 ID5R algorit
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of LMS algorithm, 64 of ROTE-LEARNE
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prediction of probabilities with, 1
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Probability distribution, 133. See
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Split infomation, 73-74 Squashing f
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