Conformal Symplectic and Relativistic Optimization

TL;DR

This paper explores structure-preserving discretizations of dissipative Hamiltonian systems to analyze popular optimization methods and introduces a new relativistic algorithm that generalizes existing methods with potential stability and speed benefits.

Contribution

It provides a novel framework connecting Nesterov and heavy ball methods through dissipative relativistic systems, offering new insights and a generalized optimization algorithm.

Findings

Analyzes symplectic structure of Nesterov and heavy ball methods.

Proposes a new relativistic optimization algorithm.

Shows the new method generalizes existing methods with potential stability advantages.

Abstract

Arguably, the two most popular accelerated or momentum-based optimization methods in machine learning are Nesterov's accelerated gradient and Polyaks's heavy ball, both corresponding to different discretizations of a particular second order differential equation with friction. Such connections with continuous-time dynamical systems have been instrumental in demystifying acceleration phenomena in optimization. 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. Importantly, such a method generalizes both Nesterov and heavy ball, each being recovered as distinct limiting cases, and has potential advantages at no additional cost.