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Learning and Lower Bounds for AC 0 with Threshold Gates

  • Conference paper
Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (RANDOM 2010, APPROX 2010)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6302))

Abstract

In 2002 Jackson [JKS02] asked whether AC 0 circuits augmented with a threshold gate at the output can be efficiently learned from uniform random examples. We answer this question affirmatively by showing that such circuits have fairly strong Fourier concentration; hence the low-degree algorithm of Linial, Mansour and Nisan [LMN93] learns such circuits in sub-exponential time. Under a conjecture of Gotsman and Linial [GL94] which upper bounds the total influence of low-degree polynomial threshold functions, the running time is quasi-polynomial. Our results extend to AC 0 circuits augmented with a small super-constant number of threshold gates at arbitrary locations in the circuit. We also establish some new structural properties of AC 0 circuits augmented with threshold gates, which allow us to prove a range of separation results and lower bounds.

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Author information

Authors and Affiliations

  1. Microsoft Research Silicon Valley, Mountain View, CA, 94203, U.S.A.

    Parikshit Gopalan

  2. Columbia University, New York, NY, 10027, U.S.A.

    Rocco A. Servedio

Authors
  1. Parikshit Gopalan
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  2. Rocco A. Servedio
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Editor information

Editors and Affiliations

  1. Universitat Politecnica de Catalunya, Departament de Llenguatges i Sistemes Informatics, Campus Nord - Ed Omega, Desp. 235, Jordi Girona, 1-3, 08034, Barcelona, Spain

    Maria Serna

  2. Department of Computer Science, University of Haifa, Mount Carmel, 31905, Haifa, Israel

    Ronen Shaltiel

  3. Department of Computer Science, University of Kiel, Olshausenstrasse 40, 24098, Kiel, Germany

    Klaus Jansen

  4. University of Geneva, Centre Universitaire d’Informatique, Battelle Bat. A, 7 rte de Drize, 1227, Carouge, Switzerland

    José Rolim

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© 2010 Springer-Verlag Berlin Heidelberg

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Gopalan, P., Servedio, R.A. (2010). Learning and Lower Bounds for AC 0 with Threshold Gates. In: Serna, M., Shaltiel, R., Jansen, K., Rolim, J. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. RANDOM APPROX 2010 2010. Lecture Notes in Computer Science, vol 6302. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15369-3_44

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  • DOI: https://doi.org/10.1007/978-3-642-15369-3_44

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-15368-6

  • Online ISBN: 978-3-642-15369-3

  • eBook Packages: Computer ScienceComputer Science (R0)

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