NeuroCOLT: Neural Networks and Computational Learning Theory

NeuroCOLT

Neural Networks and Computational Learning Theory

1994	1995
1996	1997
1998	1999
2000	2001

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NeuroCOLT Technical Report NC-TR-95-004

Degree of Approximation Results for Feedforward Networks Approximating Unknown Mappings and Their Derivatives

Kurt Hornik, Technische Universität
Wien,

Maxwell Stinchcombe, University of California,
San Diego,

Halbert White, University of California,
San Diego,

Peter Auer, Technische Universität
Graz

Abstract
Barron (1993) has given rates for hidden layer feedforward networks with sigmoid activation functions approximating a class of functions satisfying a certain smoothness condition. These rates do not depend on the dimension of the input space. We extend Barron's results to feedforward networks with possibly non-sigmoid activation functions approximating mappings and their derivatives simultaneously. Our conditions are similar but not identical to Barron's, but we obtain the same rates of approximation, showing that the approximation error decreases at rates as fast as $n^{-\frac{1}{2}}$, where $n$ is the number of hidden units. The dimension of the input space appears only in the constants of our bounds.

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