Neural Networks and Rational Functions

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/epsilon) ) which is 6-close, and similarly for any rational function there exists a ReLU network of size O (poly log(1/epsilon) ) which is epsilon-close. By contrast, polynomials need degree Omega (poly (1 /epsilon) ) 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.