  It has repeatedly been observed that loss minimization by stochastic gradient
descent (SGD) leads to heavy-tailed distributions of neural network parameters.
Here, we analyze a continuous diffusion approximation of SGD, called
homogenized stochastic gradient descent, show that it behaves asymptotically
heavy-tailed, and give explicit upper and lower bounds on its tail-index. We
validate these bounds in numerical experiments and show that they are typically
close approximations to the empirical tail-index of SGD iterates. In addition,
their explicit form enables us to quantify the interplay between optimization
parameters and the tail-index. Doing so, we contribute to the ongoing
discussion on links between heavy tails and the generalization performance of
neural networks as well as the ability of SGD to avoid suboptimal local minima.
