  We study the relationship between the frequency of a function and the speed
at which a neural network learns it. We build on recent results that show that
the dynamics of overparameterized neural networks trained with gradient descent
can be well approximated by a linear system. When normalized training data is
uniformly distributed on a hypersphere, the eigenfunctions of this linear
system are spherical harmonic functions. We derive the corresponding
eigenvalues for each frequency after introducing a bias term in the model. This
bias term had been omitted from the linear network model without significantly
affecting previous theoretical results. However, we show theoretically and
experimentally that a shallow neural network without bias cannot represent or
learn simple, low frequency functions with odd frequencies. Our results lead to
specific predictions of the time it will take a network to learn functions of
varying frequency. These predictions match the empirical behavior of both
shallow and deep networks.
