Here we study structure-preserving discretizations for a certain class of dissipative (conformal) Hamiltonian systems, allowing us to analyze the symplectic structure of both Nesterov and heavy ball, besides providing several new insights into these methods. Moreover, we propose a new algorithm based on a dissipative relativistic system that normalizes the momentum and may result in more stable/faster optimization.

In this appendix, we include all the material missing from the main paper. Moreover, we restate a key result which connects random sampling and submodular maximization. The original version of the theorem was due to Feige et al. In fact, in what follows we exclusively use S and O for their final versions. Before stating the next lemma, let us introduce some notation for the sake of readability.