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Vapnik-Chervonenkis Dimension of Parallel Arithmetic Computations

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Algorithmic Learning Theory (ALT 2007)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 4754))

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Abstract

We provide upper bounds for the Vapnik-Chervonenkis dimension of concept classes parameterized by real numbers whose membership tests are programs described by bounded-depth arithmetic networks. Our upper bounds are of the kind O(k 2 d 2), where d is the depth of the network (representing the parallel running time) and k is the number of parameters needed to codify the concept. This bound becomes O(k 2 d) when membership tests are described by Boolean-arithmetic circuits. As a consequence we conclude that families of concepts classes having parallel polynomial time algorithms expressing their membership tests have polynomial VC dimension.

Partially supported by spanish grant TIN2007-67466-C02-02.

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Author information

Authors and Affiliations

  1. Centro de Inteligencia Artificial, Universidad de Oviedo, Campus de Viesques, 33271 Gijón, Spain

    César L. Alonso

  2. Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria, 39005 Santander, Spain

    José Luis Montaña

Authors
  1. César L. Alonso
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  2. José Luis Montaña
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Editor information

Editors and Affiliations

  1. RSISE @ ANU and SML @ NICTA, Canberra,, ACT, 0200, Australia

    Marcus Hutter

  2. Columbia University, NY, P.O. Box, New York, USA

    Rocco A. Servedio

  3. Graduate School of Information Sciences, Tohoku University,, Sendai 980-8579, Japan

    Eiji Takimoto

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© 2007 Springer-Verlag Berlin Heidelberg

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Alonso, C.L., Montaña, J.L. (2007). Vapnik-Chervonenkis Dimension of Parallel Arithmetic Computations. In: Hutter, M., Servedio, R.A., Takimoto, E. (eds) Algorithmic Learning Theory. ALT 2007. Lecture Notes in Computer Science(), vol 4754. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-75225-7_12

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  • DOI: https://doi.org/10.1007/978-3-540-75225-7_12

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-75224-0

  • Online ISBN: 978-3-540-75225-7

  • eBook Packages: Computer ScienceComputer Science (R0)

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