# From Nesterov's Estimate Sequence to Riemannian Acceleration

arXiv: https://arxiv.org/abs/2001.08876 · HTML: https://mlsanity.com/paper/2001.08876/

## Key facts

- Cited by 24 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Jan 2020.

## Built on by (most cited first, 24 of 24)

- [A Survey of Geometric Optimization for Deep Learning: From Euclidean Space to Riemannian Manifold](https://mlsanity.com/paper/2302.08210/) (2023)
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- [Momentum Stiefel Optimizer, with Applications to Suitably-Orthogonal Attention, and Optimal Transport](https://mlsanity.com/paper/2205.14173/) (2022)
- [Simplifying Momentum-based Positive-definite Submanifold Optimization with Applications to Deep Learning](https://mlsanity.com/paper/2302.09738/) (2023)
- [Sion's Minimax Theorem in Geodesic Metric Spaces and a Riemannian Extragradient Algorithm](https://mlsanity.com/paper/2202.06950/) (2022)
- [Optimization without Retraction on the Random Generalized Stiefel Manifold](https://mlsanity.com/paper/2405.01702/) (2024)
- [Optimization on manifolds: A symplectic approach](https://mlsanity.com/paper/2107.11231/) (2021)
- [Minimizing Dynamic Regret on Geodesic Metric Spaces](https://mlsanity.com/paper/2302.08652/) (2023)
- [Riemannian accelerated gradient methods via extrapolation](https://mlsanity.com/paper/2208.06619/) (2022)
- [No-go Theorem for Acceleration in the Hyperbolic Plane](https://mlsanity.com/paper/2101.05657/) (2021)
- [Accelerated Riemannian Optimization: Handling Constraints with a Prox to Bound Geometric Penalties](https://mlsanity.com/paper/2211.14645/) (2022)
- [Riemannian Projection-free Online Learning](https://mlsanity.com/paper/2305.19349/) (2023)
- [Decentralized Online Riemannian Optimization Beyond Hadamard Manifolds](https://mlsanity.com/paper/2509.07779/) (2025)
- [Understanding Riemannian Acceleration via a Proximal Extragradient Framework](https://mlsanity.com/paper/2111.02763/) (2021)
- [Efficient Optimization with Orthogonality Constraint: a Randomized Riemannian Submanifold Method](https://mlsanity.com/paper/2505.12378/) (2025)
- [Low-complexity subspace-descent over symmetric positive definite manifold](https://mlsanity.com/paper/2305.02041/) (2023)
- [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)
- [Rieoptax: Riemannian Optimization in JAX](https://mlsanity.com/paper/2210.04840/) (2022)
- [Extragradient Type Methods for Riemannian Variational Inequality Problems](https://mlsanity.com/paper/2309.14155/) (2023)
- [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)
- [Quantum Optimization via Gradient-Based Hamiltonian Descent](https://mlsanity.com/paper/2505.14670/) (2025)
- [Riemannian Zeroth-Order Gradient Estimation with Structure-Preserving Metrics for Geodesically Incomplete Manifolds](https://mlsanity.com/paper/2601.08039/) (2026)

## Abstract (arXiv)

We propose the first global accelerated gradient method for Riemannian manifolds. Toward establishing our result we revisit Nesterov's estimate sequence technique and develop an alternative analysis for it that may also be of independent interest. Then, we extend this analysis to the Riemannian setting, localizing the key difficulty due to non-Euclidean structure into a certain ``metric distortion.'' We control this distortion by developing a novel geometric inequality, which permits us to propose and analyze a Riemannian counterpart to Nesterov's accelerated gradient method.