  It is well-known that overparametrized neural networks trained using
gradient-based methods quickly achieve small training error with appropriate
hyperparameter settings. Recent papers have proved this statement theoretically
for highly overparametrized networks under reasonable assumptions. These
results either assume that the activation function is ReLU or they crucially
depend on the minimum eigenvalue of a certain Gram matrix depending on the
data, random initialization and the activation function. In the later case,
existing works only prove that this minimum eigenvalue is non-zero and do not
provide quantitative bounds. On the empirical side, a contemporary line of
investigations has proposed a number of alternative activation functions which
tend to perform better than ReLU at least in some settings but no clear
understanding has emerged. This state of affairs underscores the importance of
theoretically understanding the impact of activation functions on training. In
the present paper, we provide theoretical results about the effect of
activation function on the training of highly overparametrized 2-layer neural
networks. A crucial property that governs the performance of an activation is
whether or not it is smooth. For non-smooth activations such as ReLU, SELU and
ELU, all eigenvalues of the associated Gram matrix are large under minimal
assumptions on the data. For smooth activations such as tanh, swish and
polynomials, the situation is more complex. If the subspace spanned by the data
has small dimension then the minimum eigenvalue of the Gram matrix can be small
leading to slow training. But if the dimension is large and the data satisfies
another mild condition, then the eigenvalues are large. If we allow deep
networks, then the small data dimension is not a limitation provided that the
depth is sufficient. We discuss a number of extensions and applications of
these results.
