  Gradient clipping is commonly used in training deep neural networks partly
due to its practicability in relieving the exploding gradient problem.
Recently, \citet{zhang2019gradient} show that clipped (stochastic) Gradient
Descent (GD) converges faster than vanilla GD/SGD via introducing a new
assumption called $(L_0, L_1)$-smoothness, which characterizes the violent
fluctuation of gradients typically encountered in deep neural networks.
However, their iteration complexities on the problem-dependent parameters are
rather pessimistic, and theoretical justification of clipping combined with
other crucial techniques, e.g. momentum acceleration, are still lacking. In
this paper, we bridge the gap by presenting a general framework to study the
clipping algorithms, which also takes momentum methods into consideration. We
provide convergence analysis of the framework in both deterministic and
stochastic setting, and demonstrate the tightness of our results by comparing
them with existing lower bounds. Our results imply that the efficiency of
clipping methods will not degenerate even in highly non-smooth regions of the
landscape. Experiments confirm the superiority of clipping-based methods in
deep learning tasks.
