# Towards Riemannian Accelerated Gradient Methods

arXiv: https://arxiv.org/abs/1806.02812 · HTML: https://mlsanity.com/paper/1806.02812/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Jun 2018.

## Built on by (most cited first, 12 of 12)

- [Fast and accurate optimization on the orthogonal manifold without retraction](https://mlsanity.com/paper/2102.07432/) (2021)
- [A Survey of Geometric Optimization for Deep Learning: From Euclidean Space to Riemannian Manifold](https://mlsanity.com/paper/2302.08210/) (2023)
- [Convergence of Kinetic Langevin Monte Carlo on Lie groups](https://mlsanity.com/paper/2403.12012/) (2024)
- [Accelerated Markov Chain Monte Carlo Algorithms on Discrete States](https://mlsanity.com/paper/2505.12599/) (2025)
- [Natural gradient descent with momentum](https://mlsanity.com/paper/2604.15554/) (2026)
- [Riemannian Stochastic Gradient Method for Nested Composition Optimization](https://mlsanity.com/paper/2207.09350/) (2022)
- [Quantitative Convergences of Lie Group Momentum Optimizers](https://mlsanity.com/paper/2405.20390/) (2024)
- [Learning-Rate-Free Stochastic Optimization over Riemannian Manifolds](https://mlsanity.com/paper/2406.02296/) (2024)
- [Local Linear Convergence of Infeasible Optimization with Orthogonal Constraints](https://mlsanity.com/paper/2412.05689/) (2024)
- [Finite sample bounds for barycenter estimation in geodesic spaces](https://mlsanity.com/paper/2502.14069/) (2025)
- [Accelerated Multiplicative Weights Update Avoids Saddle Points almost always](https://mlsanity.com/paper/2204.11407/) (2022)
- [Continuous-time Riemannian SGD and SVRG Flows on Wasserstein Probabilistic Space](https://mlsanity.com/paper/2401.13530/) (2024)

## Abstract (arXiv)

We propose a Riemannian version of Nesterov's Accelerated Gradient algorithm (RAGD), and show that for geodesically smooth and strongly convex problems, within a neighborhood of the minimizer whose radius depends on the condition number as well as the sectional curvature of the manifold, RAGD converges to the minimizer with acceleration. Unlike the algorithm in (Liu et al., 2017) that requires the exact solution to a nonlinear equation which in turn may be intractable, our algorithm is constructive and computationally tractable. Our proof exploits a new estimate sequence and a novel bound on the nonlinear metric distortion, both ideas may be of independent interest.