  We establish a data-dependent notion of algorithmic stability for Stochastic
Gradient Descent (SGD), and employ it to develop novel generalization bounds.
This is in contrast to previous distribution-free algorithmic stability results
for SGD which depend on the worst-case constants. By virtue of the
data-dependent argument, our bounds provide new insights into learning with SGD
on convex and non-convex problems. In the convex case, we show that the bound
on the generalization error depends on the risk at the initialization point. In
the non-convex case, we prove that the expected curvature of the objective
function around the initialization point has crucial influence on the
generalization error. In both cases, our results suggest a simple data-driven
strategy to stabilize SGD by pre-screening its initialization. As a corollary,
our results allow us to show optimistic generalization bounds that exhibit fast
convergence rates for SGD subject to a vanishing empirical risk and low noise
of stochastic gradient.
