Bregman dynamics, contact transformations and convex optimization

TL;DR

This paper introduces a novel geometric framework for optimization using contact transformations, enabling the transformation of Bregman Hamiltonian systems into separable forms and leading to the development of the Relativistic Bregman algorithm with improved performance.

Contribution

It presents a new contact geometric approach to dynamical systems in optimization, unifying previous methods and enabling the creation of robust, fast discretizations like the Relativistic Bregman algorithm.

Findings

Relativistic Bregman algorithm outperforms classical momentum methods.

All deterministic optimization flows are invariant under contact transformations.

Bregman Hamiltonian systems can be transformed into separable Hamiltonians.

Abstract

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.