  Recent research on accelerated gradient methods of use in optimization has
demonstrated that these methods can be derived as discretizations of dynamical
systems. This, in turn, has provided a basis for more systematic
investigations, especially into the geometric structure of those dynamical
systems and their structure--preserving discretizations. In this work, we
introduce dynamical systems defined through a contact geometry which are not
only naturally suited to the optimization goal but also subsume all previous
methods based on geometric dynamical systems. As a consequence, all the
deterministic flows used in optimization share an extremely interesting
geometric property: they are invariant under contact transformations. In our
main result, we exploit this observation to show that the celebrated Bregman
Hamiltonian system can always be transformed into an equivalent but separable
Hamiltonian by means of a contact transformation. This in turn enables the
development of fast and robust discretizations through geometric contact
splitting integrators. As an illustration, we propose the Relativistic Bregman
algorithm, and show in some paradigmatic examples that it compares favorably
with respect to standard optimization algorithms such as classical momentum and
Nesterov's accelerated gradient.

Optimization and Control;Machine Learning