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Probabilistic lower bounds for approximation by shallow perceptron networks.

Věra Kůrková1, Marcello Sanguineti2

  • 1Institute of Computer Science, Czech Academy of Sciences, Pod Vodárenskou věží, 2 - 18207 Prague, Czech Republic.

Neural Networks : the Official Journal of the International Neural Network Society
|May 9, 2017
PubMed
Summary
This summary is machine-generated.

Shallow perceptron networks struggle to approximate many functions. Achieving good approximation requires a large number of network units, exceeding polynomial bounds relative to the domain size.

Keywords:
Chernoff–Hoeffding boundsLower bounds on approximation ratesModel complexityPerceptronsShallow networks

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Area of Science:

  • Computer Science
  • Machine Learning
  • Artificial Intelligence

Background:

  • Shallow perceptron networks are fundamental neural network architectures.
  • Understanding their approximation capabilities is crucial for designing efficient AI models.
  • Previous research has explored limitations, but precise bounds for general functions remain an active area.

Purpose of the Study:

  • To investigate the approximation limitations of shallow perceptron networks.
  • To derive lower bounds on approximation errors for binary-valued functions.
  • To determine the necessary network size for effective function approximation.

Main Methods:

  • Derivation of lower bounds on approximation errors.
  • Application of probabilistic Chernoff-Hoeffding bounds.
  • Estimation of function set sizes computable by shallow networks.

Main Results:

  • A significant lower bound on approximation error is established for shallow networks.
  • It is proven that a large number of network units is required for good approximation.
  • This requirement scales beyond polynomial in the logarithm of the domain size for random functions.

Conclusions:

  • Shallow perceptron networks have inherent limitations in approximating arbitrary functions.
  • Effective approximation necessitates a substantial increase in network complexity.
  • The findings provide theoretical insights into the capacity of shallow neural networks.