  An influential line of recent work has focused on the generalization
properties of unregularized gradient-based learning procedures applied to
separable linear classification with exponentially-tailed loss functions. The
ability of such methods to generalize well has been attributed to the their
implicit bias towards large margin predictors, both asymptotically as well as
in finite time. We give an additional unified explanation for this
generalization and relate it to two simple properties of the optimization
objective, that we refer to as realizability and self-boundedness. We introduce
a general setting of unconstrained stochastic convex optimization with these
properties, and analyze generalization of gradient methods through the lens of
algorithmic stability. In this broader setting, we obtain sharp stability
bounds for gradient descent and stochastic gradient descent which apply even
for a very large number of gradient steps, and use them to derive general
generalization bounds for these algorithms. Finally, as direct applications of
the general bounds, we return to the setting of linear classification with
separable data and establish several novel test loss and test accuracy bounds
for gradient descent and stochastic gradient descent for a variety of loss
functions with different tail decay rates. In some of these cases, our bounds
significantly improve upon the existing generalization error bounds in the
literature.
