  Current theoretical results on optimization trajectories of neural networks
trained by gradient descent typically have the form of rigorous but potentially
loose bounds on the loss values. In the present work we take a different
approach and show that the learning trajectory can be characterized by an
explicit asymptotic at large training times. Specifically, the leading term in
the asymptotic expansion of the loss behaves as a power law $L(t) \sim
t^{-\xi}$ with exponent $\xi$ expressed only through the data dimension, the
smoothness of the activation function, and the class of function being
approximated. Our results are based on spectral analysis of the integral
operator representing the linearized evolution of a large network trained on
the expected loss. Importantly, the techniques we employ do not require
specific form of a data distribution, for example Gaussian, thus making our
findings sufficiently universal.
