Initialization of ReLUs for Dynamical Isometry

Deep learning relies on good initialization schemes and hyperparameter choices prior to training a neural network. Random weight initializations induce random network ensembles, which give rise to the trainability, training speed, and sometimes also generalization ability of an instance. In addition, such ensembles provide theoretical insights into the space of candidate models of which one is selected during training. The results obtained so far rely on mean field approximations that assume infinite layer width and that study average squared signals. We derive the joint signal output distribution exactly, without mean field assumptions, for fully-connected networks with Gaussian weights and biases, and analyze deviations from the mean field results. For rectified linear units, we further discuss limitations of the standard initialization scheme, such as its lack of dynamical isometry, and propose a simple alternative that overcomes these by initial parameter sharing.