Regularized Gradient Clipping Provably Trains Wide and Deep Neural
  Networks
Matteo Tucat Anirbit Mukherjee
In this work, we instantiate a regularized form of the gradient clipping
algorithm and prove that it can converge to the global minima of deep neural
network loss functions provided that the net is of sufficient width. We present
empirical evidence that our theoretically founded regularized gradient clipping
algorithm is also competitive with the state-of-the-art deep-learning
heuristics. Hence the algorithm presented here constitutes a new approach to
rigorous deep learning.
  The modification we do to standard gradient clipping is designed to leverage
the PL* condition, a variant of the Polyak-Lojasiewicz inequality which was
recently proven to be true for various neural networks for any depth within a
neighborhood of the initialisation.