Time-adaptive Lagrangian Variational Integrators for Accelerated Optimization on Manifolds

TL;DR

This paper develops a framework for time-adaptive Lagrangian variational integrators that improve computational efficiency in accelerated optimization on manifolds, extending existing methods to more general geometric spaces.

Contribution

It introduces a novel approach for time-adaptivity in Lagrangian variational integrators applicable to Riemannian manifolds and Lie groups, addressing limitations of Hamiltonian methods.

Findings

Enhanced efficiency in optimization algorithms on manifolds.

Framework applicable to normed vector spaces and Lie groups.

Addresses energy preservation issues with variable time-steps.

Abstract

A variational framework for accelerated optimization was recently introduced on normed vector spaces and Riemannian manifolds in Wibisono et al. (2016) and Duruisseaux and Leok (2021). It was observed that a careful combination of timeadaptivity and symplecticity in the numerical integration can result in a significant gain in computational efficiency. It is however well known that symplectic integrators lose their near energy preservation properties when variable time-steps are used. The most common approach to circumvent this problem involves the Poincare transformation on the Hamiltonian side, and was used in Duruisseaux et al. (2021) to construct efficient explicit algorithms for symplectic accelerated optimization. However, the current formulations of Hamiltonian variational integrators do not make intrinsic sense on more general spaces such as Riemannian manifolds and Lie groups. In contrast, Lagrangian variational integrators are well-defined on manifolds, so we develop here a framework for time-adaptivity in Lagrangian variational integrators and use the resulting geometric integrators to solve optimization problems on normed vector spaces and Lie groups.