  In recent years, various notions of capacity and complexity have been
proposed for characterizing the generalization properties of stochastic
gradient descent (SGD) in deep learning. Some of the popular notions that
correlate well with the performance on unseen data are (i) the `flatness' of
the local minimum found by SGD, which is related to the eigenvalues of the
Hessian, (ii) the ratio of the stepsize $\eta$ to the batch-size $b$, which
essentially controls the magnitude of the stochastic gradient noise, and (iii)
the `tail-index', which measures the heaviness of the tails of the network
weights at convergence. In this paper, we argue that these three seemingly
unrelated perspectives for generalization are deeply linked to each other. We
claim that depending on the structure of the Hessian of the loss at the
minimum, and the choices of the algorithm parameters $\eta$ and $b$, the SGD
iterates will converge to a \emph{heavy-tailed} stationary distribution. We
rigorously prove this claim in the setting of quadratic optimization: we show
that even in a simple linear regression problem with independent and
identically distributed data whose distribution has finite moments of all
order, the iterates can be heavy-tailed with infinite variance. We further
characterize the behavior of the tails with respect to algorithm parameters,
the dimension, and the curvature. We then translate our results into insights
about the behavior of SGD in deep learning. We support our theory with
experiments conducted on synthetic data, fully connected, and convolutional
neural networks.
