Hamiltonian Deep Neural Networks Guaranteeing Nonvanishing Gradients by Design

Deep neural networks (DNNs) training can be difficult due to vanishing and exploding gradients during weight optimization through backpropagation. To address this problem, we propose a general class of Hamiltonian DNNs (H-DNNs) that stem from the discretization of continuous-time Hamiltonian systems and include several existing DNN architectures based on ordinary differential equations. Our main result is that a broad set of H-DNNs ensures nonvanishing gradients by design for an arbitrary network depth. This is obtained by proving that, using a semi-implicit Euler discretization scheme, the backward sensitivity matrices involved in gradient computations are symplectic. We also provide an upper bound to the magnitude of sensitivity matrices and show that exploding gradients can be controlled through regularization. The good performance of H-DNNs is demonstrated on benchmark classification problems, including image classification with the MNIST dataset.