Bregman dynamics, contact transformations and convex optimization
This provides a unified geometric framework for optimization algorithms, potentially benefiting researchers and practitioners in machine learning and numerical optimization, though it appears to be an incremental theoretical advancement building on existing dynamical systems approaches.
The authors tackled the problem of developing more systematic geometric foundations for accelerated gradient optimization methods by introducing dynamical systems defined through contact geometry that subsume previous geometric approaches. They showed that Bregman Hamiltonian systems can always be transformed into separable Hamiltonians via contact transformations, enabling the development of fast and robust discretizations, and demonstrated that their proposed Relativistic Bregman algorithm performs favorably compared to standard methods like classical momentum and Nesterov's accelerated gradient in paradigmatic examples.
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.