# Accelerated Optimization on Riemannian Manifolds via Projected Variational Integrators

arXiv: https://arxiv.org/abs/2201.02904 · HTML: https://mlsanity.com/paper/2201.02904/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Jan 2022.

## Built on by (most cited first, 3 of 3)

- [Tutorial on amortized optimization](https://mlsanity.com/paper/2202.00665/) (2022)
- [Projected Neural Differential Equations for Learning Constrained Dynamics](https://mlsanity.com/paper/2410.23667/) (2024)
- [Warped geometric information on the optimisation of Euclidean functions](https://mlsanity.com/paper/2308.08305/) (2023)

## Abstract (arXiv)

A variational formulation of accelerated optimization on normed spaces was recently introduced by considering a specific family of time-dependent Bregman Lagrangian and Hamiltonian systems whose corresponding trajectories converge to the minimizer of the given convex function at an arbitrary accelerated rate of O(1/tp). This framework has been exploited using time-adaptive geometric integrators to design efficient explicit algorithms for symplectic accelerated optimization. It was observed that geometric discretizations were substantially less prone to stability issues, and were therefore more robust, reliable, and computationally efficient. More recently, this variational framework has been extended to the Riemannian manifold setting by considering a more general family of time-dependent Bregman Lagrangian and Hamiltonian systems on Riemannian manifolds. It is thus natural to develop time-adaptive Hamiltonian variational integrators for accelerated optimization on Riemannian manifolds. In the past, Hamiltonian variational integrators have been constructed with holonomic constraints, but the resulting algorithms were implicit in nature, which significantly increased their cost per iteration. In this paper, we will test the performance of explicit methods based on Hamiltonian variational integrators combined with projections that constrain the numerical solution to remain on the constraint manifold.