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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TL;DR: It is shown that the recursive teaching dimension, recently introduced by Zilles et al. (2008), is strongly connected to known complexity notions in machine learning, e.g., the self-directed learning complexity and the VC-dimension.
Abstract: This paper is concerned with various combinatorial parameters of classes that can be learned from a small set of examples. We show that the recursive teaching dimension, recently introduced by Zilles et al. (2008), is strongly connected to known complexity notions in machine learning, e.g., the self-directed learning complexity and the VC-dimension. To the best of our knowledge these are the first results unveiling such relations between teaching and query learning as well as between teaching and the VC-dimension. It will turn out that for many natural classes the RTD is upper-bounded by the VCD, e.g., classes of VC-dimension 1, intersection-closed classes and finite maximum classes. However, we will also show that there are certain (but rare) classes for which the recursive teaching dimension exceeds the VC-dimension. Moreover, for maximum classes, the combinatorial structure induced by the RTD, called teaching plan, is highly similar to the structure of sample compression schemes. Indeed one can transform any repetition-free teaching plan for a maximum class C into an unlabeled sample compression scheme for C and vice versa, while the latter is produced by (i) the corner-peeling algorithm of Rubinstein and Rubinstein (2012) and (ii) the tail matching algorithm of Kuzmin and Warmuth (2007).
80 citations
Cites background from "Learnability and the Vapnik-Chervon..."
...The existence of the latter would mean that the combinatorial properties that determine the information complexity of PAC-learning (i.e., of learning from randomly drawn examples) are essentially the same as those that determine the information complexity of teaching (i.e., of learning from helpfully selected examples), at least when using the recursive teaching model....
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...For example, in PAC-learning (Valiant, 1984), the information complexity of a concept class C is the worst-case sample complexity a best possible PAC learner for C can achieve on all concepts in C....
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...This way sample bounds for PAC-learning of a class C can be obtained from the size of a smallest sample compression scheme for C (Littlestone and Warmuth, 1996; Floyd and Warmuth, 1995)....
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...Among many relevant properties, it provides bounds on the sample complexity of PAC-learning (Blumer et al., 1989)....
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01 Jun 1993
TL;DR: In this paper, it was shown that neural nets with piecewise-polynomial activation functions and arbitrary real weights can be simulated for Boolean inputs and outputs by neural nets of a somewhat larger size and depth with Heaviside gates and weights from {-1, 0, 1}.
Abstract: It is shown that high-order feedforward neural nets of constant depth with piecewise-polynomial activation functions and arbitrary real weights can be simulated for Boolean inputs and outputs by neural nets of a somewhat larger size and depth with Heaviside gates and weights from {-1, 0, 1}. This provides the first known upper bound for the computational power of the former type of neural nets. It is also shown that in the case of first-order nets with piecewise-linear activation functions one can replace arbitrary real weights by rational numbers with polynomially many bits without changing the Boolean function that is computed by the neural net. In order to prove these results, we introduce two new methods for reducing nonlinear problems about weights in multilayer neural nets to linear problems for a transformed set of parameters. These transformed parameters can be interpreted as weights in a somewhat larger neural net.
As another application of our new proof technique we show that neural nets with piecewise-polynomial activation functions and a constant number of analog inputs are probably approximately correct (PAC) learnable (in Valiant's model for PAC learning [Comm. Assoc. Comput. Mach., 27 (1984), pp. 1134--1142]).
80 citations
01 Jul 1992
TL;DR: The complexity of concept learning in various common models for on-line learning is considered, focusing on methods for proving lower bounds to the learning complexity of a concept class, among others, the model for learning with equivalence and membership queries.
Abstract: We consider the complexity of concept learning in various common models for on-line learning, focusing on methods for proving lower bounds to the learning complexity of a concept class. Among others, we consider the model for learning with equivalence and membership queries. For this model we give lower bounds on the number of queries that are needed to learn a concept class {\cal C} in terms of the Vapnik-Chervonenkis dimension of {\cal C}, and in terms of the complexity of learning {\cal C} with arbitrary equivalence queries. Furthermore, we survey other known lower bound methods and we exhibit all known relationships between learning complexities in the models considered and some relevant combinatorial parameters. As it turns out, the picture is almost complete. This paper has been written so that it can be read without previous knowledge of Computational Learning Theory.
80 citations
01 Aug 2008
TL;DR: This work shows how to build robust samples a priori, based on existing synopses for distinct value estimation, and shows the accuracy of the technique theoretically, and proves its performance experimentally.
Abstract: We study selectivity estimation techniques for set similarity queries. A wide variety of similarity measures for sets have been proposed in the past. In this work we concentrate on the class of weighted similarity measures (e.g., TF/IDF and BM25 cosine similarity and variants) and design selectivity estimators based on a priori constructed samples. First, we study the pitfalls associated with straightforward applications of random sampling, and argue that care needs to be taken in how the samples are constructed; uniform random sampling yields very low accuracy, while query sensitive realtime sampling is more expensive than exact solutions (both in CPU and I/O cost). We show how to build robust samples a priori, based on existing synopses for distinct value estimation. We prove the accuracy of our technique theoretically, and verify its performance experimentally. Our algorithm is orders of magnitude faster than exact solutions and has very small space overhead.
79 citations
TL;DR: This incomplete, personally biased survey, reflecting on some recent developments and prospects in the field of fixed parameter algorithms, concludes that fixed parameter tractability is under threat.
78 citations
Cites background from "Learnability and the Vapnik-Chervon..."
...Observe, however, that in some respects parameterized complexity appears to be, in a sense, ‘orthogonal’ to classical complexity: For instance, the so-called problem of computing the V-C dimension from learning theory [13,77], which is not known (and not believed) to be NP-hard, is W [1]-complete [30,31]....
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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
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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