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Neural Networks and Rational Functions

2020-03-09 | |  52 |   37 |   0

Abstract

Neural networks and rational functions efficiently approximate each other. In more detail, it is shown here that for any ReLU network, there exists a rational function of degree O(poly log(1/ε)) which is -close, and similarly for any rational function there exists a ReLU network of size O(poly log(1/ε)) which is -close. By contrast, polynomials need degree Ω(poly(1/ε)) to approximate even a single ReLU. When converting a ReLU network to a rational function as above, the hidden constants depend exponentially on the number of layers, which is shown to be tight; in other words, a compositional representation can be beneficial even for rational functions.

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