Regularized Gradient Clipping Provably Trains Wide and Deep Neural Networks

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 (Liu et al., 2020), to be true for various neural networks for any depth within a neighbourhood of the initialisation. In various disciplines, ranging from control theory to machine learning theory there has been a long history of trying to understand the nature of convergence on non-convex objectives for first order optimization algorithms i.e algorithms which only have access to an (estimate of) the gradient of the objective Maryak & Chin (2001); Fang et al. (1997).