  A quadratic approximation of neural network loss landscapes has been
extensively used to study the optimization process of these networks. Though,
it usually holds in a very small neighborhood of the minimum, it cannot explain
many phenomena observed during the optimization process. In this work, we study
the structure of neural network loss functions and its implication on
optimization in a region beyond the reach of a good quadratic approximation.
Numerically, we observe that neural network loss functions possesses a
multiscale structure, manifested in two ways: (1) in a neighborhood of minima,
the loss mixes a continuum of scales and grows subquadratically, and (2) in a
larger region, the loss shows several separate scales clearly. Using the
subquadratic growth, we are able to explain the Edge of Stability phenomenon
[5] observed for the gradient descent (GD) method. Using the separate scales,
we explain the working mechanism of learning rate decay by simple examples.
Finally, we study the origin of the multiscale structure and propose that the
non-convexity of the models and the non-uniformity of training data is one of
the causes. By constructing a two-layer neural network problem we show that
training data with different magnitudes give rise to different scales of the
loss function, producing subquadratic growth and multiple separate scales.
