  Stochastic gradient algorithms are often unstable when applied to functions
that do not have Lipschitz-continuous and/or bounded gradients. Gradient
clipping is a simple and effective technique to stabilize the training process
for problems that are prone to the exploding gradient problem. Despite its
widespread popularity, the convergence properties of the gradient clipping
heuristic are poorly understood, especially for stochastic problems. This paper
establishes both qualitative and quantitative convergence results of the
clipped stochastic (sub)gradient method (SGD) for non-smooth convex functions
with rapidly growing subgradients. Our analyses show that clipping enhances the
stability of SGD and that the clipped SGD algorithm enjoys finite convergence
rates in many cases. We also study the convergence of a clipped method with
momentum, which includes clipped SGD as a special case, for weakly convex
problems under standard assumptions. With a novel Lyapunov analysis, we show
that the proposed method achieves the best-known rate for the considered class
of problems, demonstrating the effectiveness of clipped methods also in this
regime. Numerical results confirm our theoretical developments.
