Variational Symplectic Accelerated Optimization on Lie Groups

TL;DR

This paper introduces a novel variational discretization method for accelerated optimization on Lie groups, leveraging differential equations and Bregman dynamics to improve convergence in large-scale problems.

Contribution

It develops a Lie group variational discretization framework based on Bregman Lagrangian and Hamiltonian dynamics, extending accelerated optimization techniques to Lie groups.

Findings

Demonstrates the method on attitude determination.

Applies to vision-based localization tasks.

Shows improved convergence properties.

Abstract

There has been significant interest in generalizations of the Nesterov accelerated gradient descent algorithm due to its improved performance guarantee compared to the standard gradient descent algorithm, and its applicability to large scale optimization problems arising in deep learning. A particularly fruitful approach is based on numerical discretizations of differential equations that describe the continuous time limit of the Nesterov algorithm, and a generalization involving time-dependent Bregman Lagrangian and Hamiltonian dynamics that converges at an arbitrarily fast rate to the minimum. We develop a Lie group variational discretization based on an extended path space formulation of the Bregman Lagrangian on Lie groups, and analyze its computational properties with two examples in attitude determination and vision-based localization.