On PAC learning of functions with smoothness properties using feedforward sigmoidal networks

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We consider Probably and Approximately Corrct (PAC) learning of an unknown function f: [0,1]{sup d} {r_arrow} [0,1], based on finite samples using feedforward sigmoidal networks. The unknown function f is chosen from the family F{intersection}C([0,1]{sup d}) or F{intersection}L{sup {infinity}}([0,1]{sup d}), where F has either bounded modulus of smoothness or bounded capacity or both. The learning sample is given by (X{sub 1},f(X{sub 1})),(X{sub 2},f(X{sub 2})),{hor_ellipsis},(X{sub n},f(X{sub n})), where X{sub 1},X{sub 2},{hor_ellipsis},X{sub n} are independently and identically distributed according to an unknown distribution. We consider the feedforward networks with a a single hidden layer of 1/(1 + e{sup {minus}{gamma}z})-units and bounded parameters, ... continued below

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12 p.

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Rao, N.S.V. & Protopopescu, V.A. April 1, 1996.

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Description

We consider Probably and Approximately Corrct (PAC) learning of an unknown function f: [0,1]{sup d} {r_arrow} [0,1], based on finite samples using feedforward sigmoidal networks. The unknown function f is chosen from the family F{intersection}C([0,1]{sup d}) or F{intersection}L{sup {infinity}}([0,1]{sup d}), where F has either bounded modulus of smoothness or bounded capacity or both. The learning sample is given by (X{sub 1},f(X{sub 1})),(X{sub 2},f(X{sub 2})),{hor_ellipsis},(X{sub n},f(X{sub n})), where X{sub 1},X{sub 2},{hor_ellipsis},X{sub n} are independently and identically distributed according to an unknown distribution. We consider the feedforward networks with a a single hidden layer of 1/(1 + e{sup {minus}{gamma}z})-units and bounded parameters, but the results can be extended to other neural networks where the hidden units satisfy suitable smoothness conditions. We analyze three function estimators based on nearest neighbor rule, local averaging, and Nadaraya-Watson estimator, all computed using the Haar system. It is shown that given a sufficiently large sample, each of these estimators approximates the best neural network to any given error with arbitrarily high probability. This result is crucical for establishing the essentially equivalent capabilities of neural networks and the above estimators for PAC learning from finite samples. Practical importance of this ``equivalence`` stems from the fact that computing a neural network which approximates the best possible one is computationally difficult, whereas the three estimators are linear-time computable in terms of sample size.

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12 p.

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OSTI as DE96009545

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  • Other Information: PBD: [1996]

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  • Other: DE96009545
  • Report No.: ORNL--96009545
  • Grant Number: AC05-84OR21400
  • DOI: 10.2172/217721 | External Link
  • Office of Scientific & Technical Information Report Number: 217721
  • Archival Resource Key: ark:/67531/metadc669928

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  • April 1, 1996

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  • June 29, 2015, 9:42 p.m.

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  • Jan. 19, 2016, 8:44 p.m.

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Rao, N.S.V. & Protopopescu, V.A. On PAC learning of functions with smoothness properties using feedforward sigmoidal networks, report, April 1, 1996; Tennessee. (digital.library.unt.edu/ark:/67531/metadc669928/: accessed October 20, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.