  Recently there are a considerable amount of work devoted to the study of the
algorithmic stability and generalization for stochastic gradient descent (SGD).
However, the existing stability analysis requires to impose restrictive
assumptions on the boundedness of gradients, strong smoothness and convexity of
loss functions. In this paper, we provide a fine-grained analysis of stability
and generalization for SGD by substantially relaxing these assumptions.
Firstly, we establish stability and generalization for SGD by removing the
existing bounded gradient assumptions. The key idea is the introduction of a
new stability measure called on-average model stability, for which we develop
novel bounds controlled by the risks of SGD iterates. This yields
generalization bounds depending on the behavior of the best model, and leads to
the first-ever-known fast bounds in the low-noise setting using stability
approach. Secondly, the smoothness assumption is relaxed by considering loss
functions with Holder continuous (sub)gradients for which we show that optimal
bounds are still achieved by balancing computation and stability. To our best
knowledge, this gives the first-ever-known stability and generalization bounds
for SGD with even non-differentiable loss functions. Finally, we study learning
problems with (strongly) convex objectives but non-convex loss functions.
