L. G. Valiant. Learning Disjunctions of Conjunctions. In Proceedings of the 9 IJCAI, vol. 1, pp 560-566, Los Angeles, CA. August, 1985.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....although our theoretical understanding of computational learning has increased rapidly in the last decade, there is still no generic way to evaluate the difficulty of an arbitrary learning problem. Recent results in computational learning theory [6] extend earlier work by Valiant on PAC learning [7, 8] and rely in particular on the concept of VC dimension [9] Work in this area has tended to concentrate on distribution free (i.e. problem independent) analysis although some recent work has given attention to distribution specific analysis [10] However, even results in distribution specific ....

Valiant, L. (1985). Learning disjunctions of conjunctions. Proceedings of the Ninth International Joint Conference on Artificial Intelligence (pp. 560-566). Los Altos: Morgan Kaufmann.

....that can be viewed as hypothesis space search) is thus largely the same thing as measuring the inductive bias of the relevant learner. A general framework that enables us to quantify inductive bias has been provided by Haussler [4,5,6] This framework makes use of the PAC learning model of Valiant [7,8] and the concept of the VC (Vapnik Chervonenkis) dimension [9] A remarkable attribute of Haussler s framework is that it is essentially contextfree . It allows us to quantify the inductive bias of an algorithm in general, rather than with respect to some specific problem. Thus it allows us to ....

Valiant, L. (1985). Learning disjunctions of conjunctions. Proceedings of the Ninth International Joint Conference on Artificial Intelligence (pp. 560-566). Los Altos: Morgan Kaufmann.

....fail if even a small number of the labelled examples given to the learning algorithm were noisy. Two popular noise models for both theoretical and experimental research are the classification noise model introduced by Angluin and Laird [2, 21] and the malicious error model introduced by Valiant [35] and further studied by Kearns and Li [20] In the classification noise model, each example received by the learner is mislabelled randomly and independently with some fixed probability. In the malicious error model, an adversary is allowed, with some fixed probability, to substitute a labelled ....

....of the PAC model is that the data presented to the learner is required to be noise free. Two popular models of noise for both experimental and theoretical purposes are the classification noise model introduced by Angluin and Laird [2, 21] and the malicious error model introduced by Valiant [35]. The Classification Noise Model In the classification noise model, the example oracle EX(f, D) is replaced by a noisy example oracle EXcN(f, D) Each time this noisy example oracle is called, an instance x X is drawn according to D. The oracle then outputs Ix, f(x)l with probability 1 ] or ....

Leslie Valiant. Learning disjunctions of conjunctions. In Proceedings of the Ninth Interna- tional Joint Conference on Artificial Intelligence, 1985.

....fail if even a small number of the labelled examples given to the learning algorithm were noisy. Two popular noise models for both theoretical and experimental research are the classification noise model introduced by Angluin and Laird [2, 21] and the malicious error model introduced by Valiant [35] and further studied by Kearns and Li [20] In the classification noise model, each example received by the learner is mislabelled randomly and independently with some fixed probability. In the malicious error model, an adversary is allowed, with some fixed probability, to substitute a labelled ....

....of the PAC model is that the data presented to the learner is required to be noise free. Two popular models of noise for both experimental and theoretical purposes are the classification noise model introduced by Angluin and Laird [2, 21] and the malicious error model introduced by Valiant [35]. The Classification Noise Model In the classification noise model, the example oracle EX (f; D) is replaced by a noisy example CN (f; D) Each time this noisy example oracle is called, an instance x 2 X is drawn according to D. The oracle then outputs hx; f(x)i with probability 1 Gamma j or ....

Leslie G. Valiant. Learning disjunctions of conjunctions. In Proceedings IJCAI-85, pages 560--566. International Joint Committee for Artificial Intelligence, Morgan Kaufmann, August 1985.

....the base class. In both cases, the goal is to identify exactly the concept presented by the equivalence oracle. 1. 4 Related Work There is a considerable body of literature on errors in examples in the PAC model, starting with the first error tolerant algorithm in the PAC model, given by Valiant [24]. In this case the goal is PAC identification of the target concept, despite the corruption of the examples by one or another kind of error, for example, random or 5 malicious misclassification errors, random or malicious attribute errors, or malicious errors (in which both attributes and ....

L. G. Valiant. Learning disjunctions of conjunctions. In Proceedings of the 9th International Joint Conference on Artificial Intelligence, vol. 1, pages 560-- 566, Los Angeles, California, 1985. International Joint Committee for Artificial Intelligence.

....framework of computational learning theory. First, it falls within the framework of learning with persistent noise. Here one assumes that the function f is derived from some function in the class C by adding noise to it. Typical works in this direction either tolerate only small amounts of noise [2, 42, 21, 39] (i.e. that the function is modified only at a small fraction of all possible inputs) or assume that the noise is random [1, 26, 20, 25, 33, 13, 36] i.e. that the decision of whether or not to modify the function at any given input is made by a random process) In contrast, we take the setting ....

Leslie G. Valiant. Learning disjunctions of conjunctions. Proceedings of the 9th International Joint Conference on Artificial Intelligence, pp. 560--566, Morgan Kauffman Publishers, 1985.

.... boundary, and a negative result for the problem of tting appropriate probability distributions to the classes so that maximum likelihood gives rise to a decision boundary with the same performance guarantees We consider this question in the basic Probably Approximately Correct (PAC) setting of [20, 21], since it is well understood. In PAC learning, the usual algorithmic challenge is to separate the two classes of examples. It would be remarkable if it turned out that PAC learnability were equivalent to PAC learnability using unsupervised learners, in view of the way the PAC criterion seems to ....

L.G. Valiant (1985). Learning disjunctions of conjunctions. Procs. of 9th International Joint Conference on Articial Intelligence.

.... a biased coin; whenever the coin shows H , which happens with probability j, the classification of the example is flipped and so the algorithm is provided with the, wrongly classified, example (x; 1 Gamma c t (x) Another (stronger) model, called the Malicious Noise model, was introduced in [23], revisited in [17] and was further studied in [8, 10, 11, 20] In this model the adversary, whenever the j biased coin shows H , can replace the example (x; c t (x) by some arbitrary pair (x 0 ; b) where x 0 is any point in the input space and b is a boolean value. Note that this in ....

L. G. Valiant, "Learning Disjunctions of Conjunctions", IJCAI85, pp. 560--566, 1985.

....d dimensional patterns. 3 as negative examples. The idea is that we can learn a hypothesis from these examples that can accurately predict whether a new pattern came from near or not near the landmark. Chapters 3 and 4 work within the PAC (probably approximately correct) model of learning theory [79, 80, 54, 67, 3]. It is a batch model in that it takes a set of examples, processes them, and then outputs a hypothesis. A PAC algorithm is given inputs ffl and ffi and a set of examples of the target concept randomly drawn according to a fixed, arbitrary, and unknown probability distribution D. It outputs a ....

....viable solutions to the landmark matching problem. Section 9.1 discusses these issues. 2.2 Learning Theory Background In this part of the thesis we work within the PAC (probably approximately correct) model and the on line (or mistake bound) model. The PAC model was introduced by Valiant [79, 80], and details of it may be found in such textbooks as Kearns and 14 Vazirani [54] Natarajan [67] and Anthony and Biggs [3] Sections 2.2.1 2.2.5 review the basic definitions and results of the PAC model that are used here. Details on the on line learning model can be found in Angluin [2] and ....

L. G. Valiant. Learning disjunctions of conjunctions. In Proceedings of the 9th International Joint Conference on Artificial Intelligence, vol. 1, pages 560-- 566, Los Angeles, California, 1985. International Joint Committee for Artificial Intelligence.

....framework of computational learning theory. First, it falls within the framework of learning with persistent noise. Here one assumes that the function f is derived from some function in the class C by adding noise to it. Typical works in this direction either tolerate only small amounts of noise [2, 41, 21, 39] (i.e. that the function is modi ed only at a small fraction of all possible inputs) or assume that the noise is random [1, 26, 20, 25, 33, 13, 36] i.e. that the decision of whether or not to modify the function at any given input is made by a random process) In contrast, we take the setting ....

Leslie G. Valiant. Learning disjunctions of conjunctions. Proceedings of the 9th International Joint Conference on Articial Intelligence, pp. 560-566, Morgan Kauman Publishers, 1985.

.... and Raghavan, 1994 ] In 1984, Valiant studies the learnability of this class, and remarks that the attraction of this class is that humans appear to like it for representing knowledge as is evidenced, for example, by the success of the production system paradigm and of Horn clause logics [ Valiant, 1985 ] He proves that a subclass of DNF is learnable, and leaves as an open problem whether the whole class is learnable. Since then, many theoretical studies have investigated the learnability of DNF or subclasses [ Kearns et al. 1987; Pillaipakkamnatt and Raghavan, 1994; Aizenstein and Pitt, 1995; ....

L. G. Valiant. Learning disjunctions of conjunctions. In Proc. of the 9 th IJCAI, pages 560--566, 1985.

....in ( 1 ffl ; 1 ffi ; 1 1 Gamma2j ) is sufficient for pac learning. In particular, they show that k CNF formulas are pac learnable in polynomial time with a random noise rate of less than 1 2 . Other work has focused on the case of malicious misclassification errors in examples. Valiant (1985) poses the question of learning k CNF formulas despite an adversarial teacher that draws random positive or negative examples, but with error probability fi returns an arbitrary response instead of the correctly labeled example. Valiant shows that a small rate of error can be tolerated in this ....

Valiant, L. (1985). Learning disjunctions of conjunctions. In Proceedings of the 9th IJCAI, (pp. 560-566). Los Angeles, CA: Morgan Kaufmann.

....in ( 1 ffl ; 1 ffi ; 1 1 Gamma2j ) is sufficient for PAC learning. In particular, they show that k CNF formulas are PAClearnable in polynomial time with a random noise rate of less than 1 2 . Other work has focused on the case of malicious misclassification errors in examples. Valiant [109] poses the question of learning k CNF formulas despite an adversarial teacher that draws random positive or negative examples, but with error probability fi returns an arbitrary response instead of the correctly labeled example. Valiant shows that a small rate of error can be tolerated in this ....

Leslie G. Valiant. Learning disjunctions of conjunctions. In Proceedings IJCAI-85, pages 560--566. International Joint Committee for Artificial Intelligence, Morgan Kaufmann, August 1985.

....[5] and Kearns and Vazirani [47] provide more thorough introductions to computational learning theory. This first person to formally define PAC learning was Valiant, who was interested in learning some classes of propositional formulas [60] and some restricted classes of first order formulas [61]. A typical PAC learning problem (C; H) is described in terms of a concept class C and an hypothesis class H. The concept class is some collection of objects that we are interested in learning, and the hypothesis class contains the objects that may be output by a learning algorithm. Most work ....

L. G. Valiant. "Learning Disjunctions of Conjunctions". In Proceedings of the 9th International Joint Conference on Artificial Intelligence, pages 560--566, (1985).

....et al. 67] who obtain bounds on the learning curve in terms of , and a competition between the number of hypotheses at a given error (entropy term) vs. the error itself (energy term) where the error is indexed by . Malicious Noise The malicious error noise model was rst introduced by Valiant [113], revisited by Kearns and Li [74] and then further studied at [23, 27, 28] Essentially, the examples are generated by a noise oracle, such that: with probability 1 the noisy oracle returns hx; c t (x)i from EX (c t ; D) with probability the noisy oracle maliciously chooses x 2 X and l ....

L. G. Valiant. Learning disjunctions of conjunctions. In In Proceeding IJCAI-85, pages 560-566. International Joint Committee for Articial Intelligence, Morgan Kaufmann, August 1985.

....hidden units) In addition, if the weights are restricted to 1 and 0 (but the threshold is arbitrary) then linear threshold concepts on Boolean instances spaces are not properly PAC learnable [35] 3. The classes of k DNF, k CNF, and k decision lists are properly PAC learnable for each fixed k [41,37], but it is unknown whether the classes of all DNF functions, all CNF functions, or all decision trees are properly PAC learnable. Most of the difficulties in proper PAC learning are due to the computational difficulty of finding a hypothesis in the particular form specified by the target class. ....

L. G. Valiant. Learning disjunctions of conjunctions. In Proc. 9th IJCAI, pages 560--6, Los Angeles, August 1985.

....used as its weak hypothesis, and that such a term can be found by searching all possible conjunctions of at most k literals. From these observations, we obtain our PAC learning algorithm. In the following, we use d to denote the number of terms of a given target formula. Valiant s original papers [124, 125] have already analyzed PAC learnability of this class. He reduces the problem of learning k DNF to the problem of learning disjunctions of O(n k ) literals. Therefore its sample complexity is O(n k ) For simplicity, we omit the dependency on and ffi in the following discussion) Littlestone ....

L. G. Valiant. Learning disjunctions of conjunctions. In Proceedings of the 9th International Joint Conference on Artificial Intelligence, vol. 1, pages 560--566, 1985.

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L. G. Valiant. Learning Disjunctions of Conjunctions. In Proceedings of the 9 IJCAI, vol. 1, pp 560-566, Los Angeles, CA. August, 1985.

No context found.

Valiant, L.G., "Learning disjunctions of conjunctions" Proc. 9th IJCAI, Los Angeles, CA, 1985, pp. 560-566.

....strongly that for the basic inductive learning mechanism itself single layer networks with no hidden units sufficed. There exist systematic ways of constructing new compound features such as by forming conjunctions of features that have been observed to occur together frequently or even just once [40, 41]. The theoretical advantages that one layer nets offer, that we describe later, as well as some empirical evidence of their effectiveness, provide further support for this view. The notion of image units formalizes certain aspects of short term memory devices that have been widely discussed, ....

L.G. Valiant. Learning disjunctions of conjunctions. In International Joint Conference on Artificial Intelligence, pages 560--566, Los Angeles, CA, 1985. Morgan Kaufmann.

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L. G. Valiant. Learning disjunctions of conjunctions. In Proceedings of the 9th International Joint Conference on Artificial Intelligence, pages 560--566, August 1985.

No context found.

Valiant, L. (1985). Learning disjunctions of conjunctions. Proceedings of the Ninth International Joint Conference on Artificial Intelligence (pp. 560-566). Los Altos: Morgan Kaufmann.

No context found.

L. G. Valiant. Learning disjunctions of conjunctions. Proceedings of the Ninth International Joint Conference of Artificial Intelligence (pp. 560-566). Los Angeles, CA: Morgan Kaufmann.

No context found.

VALIANT, L.G. (1985), Learning Disjunctions of Conjunctions. in "Proceedings of the 9th International Joint Conference on Artificial Intelligence", Los Angeles, pp 560--566.

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L.G. Valiant (1985). Learning disjunctions of conjunctions. Procs. of 9th International Joint Conference on Articial Intelligence. 16

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