Learnability and the Vapnik-Chervonenkis dimension
TL;DR: This paper shows that the essential condition for distribution-free learnability is finiteness of the Vapnik-Chervonenkis dimension, a simple combinatorial parameter of the class of concepts to be learned.
Abstract: Valiant's learnability model is extended to learning classes of concepts defined by regions in Euclidean space En. The methods in this paper lead to a unified treatment of some of Valiant's results, along with previous results on distribution-free convergence of certain pattern recognition algorithms. It is shown that the essential condition for distribution-free learnability is finiteness of the Vapnik-Chervonenkis dimension, a simple combinatorial parameter of the class of concepts to be learned. Using this parameter, the complexity and closure properties of learnable classes are analyzed, and the necessary and sufficient conditions are provided for feasible learnability.
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Proceedings Article•
13 Jul 2008TL;DR: A theory that ties the expressiveness of mechanisms to their efficiency in a domain-independent manner and derives an upper bound on the expected efficiency of any mechanism under its most efficient Nash equilibrium, which depends only on the mechanism's expressiveness.
Abstract: A key trend in (electronic) commerce is a demand for higher levels of expressiveness in the mechanisms that mediate interactions. We develop a theory that ties the expressiveness of mechanisms to their efficiency in a domain-independent manner. We introduce two new expressiveness measures, 1) maximum impact dimension, which captures the number of ways that an agent can impact the outcome, and 2) shatterable outcome dimension, which is based on the concept of shattering from computational learning theory. We derive an upper bound on the expected efficiency of any mechanism under its most efficient Nash equilibrium. Remarkably, it depends only on the mechanism's expressiveness. We prove that the bound increases strictly as we allow more expressiveness. We also show that in some cases a small increase in expressiveness yields an arbitrarily large increase in the bound. Finally, we study channel-based mechanisms, which subsume most combinatorial auctions, multi-attribute mechanisms, and the Vickrey-Clarke-Groves scheme. We show that our domain-independent measures of expressiveness appropriately relate to the natural measure of expressiveness of channel-based mechanisms: the number of channels allowed. Using this bridge, our general results yield interesting implications. For example, any (channel-based) multi-item auction that does not allow rich combinatorial bids can be arbitrarily inefficient--unless agents have no private information.
38 citations
TL;DR: It is concluded that guidance and recognition in the context of search are not separate processes mediated by different features, and that what the literature knows as guidance is really recognition performed on blurred objects viewed in the visual periphery.
Abstract: Search is commonly described as a repeating cycle of guidance to target-like objects, followed by the recognition of these objects as targets or distractors. Are these indeed separate processes using different visual features? We addressed this question by comparing observer behavior to that of support vector machine (SVM) models trained on guidance and recognition tasks. Observers searched for a categorically defined teddy bear target in four-object arrays. Target-absent trials consisted of random category distractors rated in their visual similarity to teddy bears. Guidance, quantified as first-fixated objects during search, was strongest for targets, followed by target-similar, medium-similarity, and target-dissimilar distractors. False positive errors to first-fixated distractors also decreased with increasing dissimilarity to the target category. To model guidance, nine teddy bear detectors, using features ranging in biological plausibility, were trained on unblurred bears then tested on blurred versions of the same objects appearing in each search display. Guidance estimates were based on target probabilities obtained from these detectors. To model recognition, nine bear/nonbear classifiers, trained and tested on unblurred objects, were used to classify the object that would be fixated first (based on the detector estimates) as a teddy bear or a distractor. Patterns of categorical guidance and recognition accuracy were modeled almost perfectly by an HMAX model in combination with a color histogram feature. We conclude that guidance and recognition in the context of search are not separate processes mediated by different features, and that what the literature knows as guidance is really recognition performed on blurred objects viewed in the visual periphery.
37 citations
01 Jan 2007
TL;DR: It is shown that boosting can be used to solve large scale constrained optimization problems, whose solutions are well characterizable and derive convergence guarantees for a quite general family of boosting algorithms.
Abstract: In this work we consider statistical learning problems. A learning machine aims to extract information from a set of training examples such that it is able to predict the associated label on unseen examples. We consider the case where the resulting classification or regression rule is a combination of simple rules – also called base hypotheses. The so-called boosting algorithms iteratively find a weighted linear combination of base hypotheses that predict well on unseen data. We address the following issues: The statistical learning theory framework for analyzing boosting methods. We study learning theoretic guarantees on the prediction performance on unseen examples. Recently, large margin classification techniques emerged as a practical result of the theory of generalization, in particular Boosting and Support Vector Machines. A large margin implies a good generalization performance. Hence, we analyze how large the margins in boosting are and find an improved algorithm that is able to generate the maximum margin solution. How can boosting methods be related to mathematical optimization techniques? To analyze the properties of the resulting classification or regression rule, it is of high importance to understand whether and under which conditions boosting converges. We show that boosting can be used to solve large scale constrained optimization problems, whose solutions are well characterizable. To show this, we relate boosting methods to methods known from mathematical optimization, and derive convergence guarantees for a quite general family of boosting algorithms. How to make Boosting noise robust? One of the problems of current boosting techniques is that they are sensitive to noise in the training sample. In order to make boosting robust, we transfer the soft margin idea from support vector learning to boosting. We develop theoretically motivated regularized algorithms that exhibit a high noise robustness. How to adapt boosting to regression problems? Boosting methods are originally designed for classification problems. To extend the boosting idea to regression problems, we use the previous convergence results and relations to semi-infinite programming to design boosting-like algorithms for regression problems. We show that these leveraging algorithms have desirable theoretical and practical properties. Can boosting techniques be useful in practice? The presented theoretical results are guided by simulation results either to illustrate properties of the proposed algorithms or to show that they work well in practice. We report on successful applications in a non-intrusive power monitoring system, chaotic time series analysis and a drug discovery process. Preface I started my research …
37 citations
Cites methods from "Learnability and the Vapnik-Chervon..."
...In a refined version one uses polynomial growth in the parameters of the concept class Blumer et al. [1989]....
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...Then the VC dimension of the function class of linear combinations of T functions from H isO(T# log(T )) [Baum and Haussler, 1989, Blumer et al., 1989]....
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21 Aug 2009
TL;DR: The key to the proof is a "Brunn-Minkowski" inequality for log-concave densities that may be of independent interest.
Abstract: In 1990, E. Baum gave an elegant polynomial-time algorithm for learning the intersection of two origin-centered halfspaces with respect to any symmetric distribution (i.e., any ${\cal D}$ such that ${\cal D}(E) = {\cal D}(-E)$) [3]. Here we prove that his algorithm also succeeds with respect to any mean zero distribution with a log-concave density (a broad class of distributions that need not be symmetric). As far as we are aware, prior to this work, it was not known how to efficiently learn any class of intersections of halfspaces with respect to log-concave distributions.
The key to our proof is a "Brunn-Minkowski" inequality for log-concave densities that may be of independent interest.
37 citations
Cites methods from "Learnability and the Vapnik-Chervon..."
...We assume the reader is familiar with basic notions in computational learning theory such as Valiant’s PAC model of learning and VC-dimension (see Kearns & Vazirani for an in-depth treatment [11]). Theorem 1 ([15, 6 ])....
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IBM1
TL;DR: It is found that, with zero knowledge built in, the network is able to learn from scratch to play the entire game at a fairly strong intermediate level of performance, which is clearly better than conventional commercial programs, and which in fact surpasses comparable networks trained on a massive human expert data set.
Abstract: This paper examines whether temporal difference methods for training connectionist networks, such as Suttons's TD(λ) algorithm, can be successfully applied to complex real-world problems. A number of important practical issues are identified and discussed from a general theoretical perspective. These practical issues are then examined in the context of a case study in which TD(λ) is applied to learning the game of backgammon from the outcome of self-play. This is apparently the first application of this algorithm to a complex nontrivial task. It is found that, with zero knowledge built in, the network is able to learn from scratch to play the entire game at a fairly strong intermediate level of performance, which is clearly better than conventional commercial programs, and which in fact surpasses comparable networks trained on a massive human expert data set. The hidden units in these network have apparently discovered useful features, a longstanding goal of computer games research. Furthermore, when a set of hand-crafted features is added to the input representation, the resulting networks reach a near-expert level of performance, and have achieved good results against world-class human play.
37 citations
References
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01 Jan 1979
42,654 citations
Book•
01 Jan 1979
TL;DR: The second edition of a quarterly column as discussed by the authors provides a continuing update to the list of problems (NP-complete and harder) presented by M. R. Garey and myself in our book "Computers and Intractability: A Guide to the Theory of NP-Completeness,” W. H. Freeman & Co., San Francisco, 1979.
Abstract: This is the second edition of a quarterly column the purpose of which is to provide a continuing update to the list of problems (NP-complete and harder) presented by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NP-Completeness,’’ W. H. Freeman & Co., San Francisco, 1979 (hereinafter referred to as ‘‘[G&J]’’; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed. Readers having results they would like mentioned (NP-hardness, PSPACE-hardness, polynomial-time-solvability, etc.), or open problems they would like publicized, should send them to David S. Johnson, Room 2C355, Bell Laboratories, Murray Hill, NJ 07974, including details, or at least sketches, of any new proofs (full papers are preferred). In the case of unpublished results, please state explicitly that you would like the results mentioned in the column. Comments and corrections are also welcome. For more details on the nature of the column and the form of desired submissions, see the December 1981 issue of this journal.
40,020 citations
Book•
01 Jan 1968
TL;DR: The arrangement of this invention provides a strong vibration free hold-down mechanism while avoiding a large pressure drop to the flow of coolant fluid.
Abstract: A fuel pin hold-down and spacing apparatus for use in nuclear reactors is disclosed. Fuel pins forming a hexagonal array are spaced apart from each other and held-down at their lower end, securely attached at two places along their length to one of a plurality of vertically disposed parallel plates arranged in horizontally spaced rows. These plates are in turn spaced apart from each other and held together by a combination of spacing and fastening means. The arrangement of this invention provides a strong vibration free hold-down mechanism while avoiding a large pressure drop to the flow of coolant fluid. This apparatus is particularly useful in connection with liquid cooled reactors such as liquid metal cooled fast breeder reactors.
17,939 citations
14,948 citations
Book•
01 Jan 1973
TL;DR: In this article, a unified, comprehensive and up-to-date treatment of both statistical and descriptive methods for pattern recognition is provided, including Bayesian decision theory, supervised and unsupervised learning, nonparametric techniques, discriminant analysis, clustering, preprosessing of pictorial data, spatial filtering, shape description techniques, perspective transformations, projective invariants, linguistic procedures, and artificial intelligence techniques for scene analysis.
Abstract: Provides a unified, comprehensive and up-to-date treatment of both statistical and descriptive methods for pattern recognition. The topics treated include Bayesian decision theory, supervised and unsupervised learning, nonparametric techniques, discriminant analysis, clustering, preprosessing of pictorial data, spatial filtering, shape description techniques, perspective transformations, projective invariants, linguistic procedures, and artificial intelligence techniques for scene analysis.
13,647 citations