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-Łojasiewicz inequality which was recently proven to be true for various neural networks for any depth within a neighbourhood of the initialisation.