# Mirror Descent on Riemannian Manifolds

arXiv: https://arxiv.org/abs/2603.17527 · HTML: https://mlsanity.com/paper/2603.17527/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 31 references that are in the map.
- First posted on arXiv 18 Mar 2026.

## Builds on (most cited first, 16 of 16)

- [Unitary Evolution Recurrent Neural Networks](https://mlsanity.com/paper/1511.06464/) (2015)
- [First-order Methods for Geodesically Convex Optimization](https://mlsanity.com/paper/1602.06053/) (2016)
- [Policy Mirror Descent for Regularized Reinforcement Learning: A Generalized Framework with Linear Convergence](https://mlsanity.com/paper/2105.11066/) (2021)
- [Riemannian SVRG: Fast Stochastic Optimization on Riemannian Manifolds](https://mlsanity.com/paper/1605.07147/) (2016)
- [The Information Geometry of Mirror Descent](https://mlsanity.com/paper/1310.7780/) (2013)
- [Averaging Stochastic Gradient Descent on Riemannian Manifolds](https://mlsanity.com/paper/1802.09128/) (2018)
- [Implicit Bias of Gradient Descent on Reparametrized Models: On Equivalence to Mirror Descent](https://mlsanity.com/paper/2207.04036/) (2022)
- [Mirror Descent Maximizes Generalized Margin and Can Be Implemented Efficiently](https://mlsanity.com/paper/2205.12808/) (2022)
- [Riemannian Dictionary Learning and Sparse Coding for Positive Definite Matrices](https://mlsanity.com/paper/1507.02772/) (2015)
- [Policy Optimization with Stochastic Mirror Descent](https://mlsanity.com/paper/1906.10462/) (2019)
- [Mirrorless Mirror Descent: A Natural Derivation of Mirror Descent](https://mlsanity.com/paper/2004.01025/) (2020)
- [Negative curvature obstructs acceleration for strongly geodesically convex optimization, even with exact first-order oracles](https://mlsanity.com/paper/2111.13263/) (2021)
- [Convergence of Online Mirror Descent](https://mlsanity.com/paper/1802.06357/) (2018)
- [Mirror descent of Hopfield model](https://mlsanity.com/paper/2211.15880/) (2022)
- [Sufficient conditions for non-asymptotic convergence of Riemannian optimisation methods](https://mlsanity.com/paper/2212.05972/) (2022)
- [Revisit First-order Methods for Geodesically Convex Optimization](https://mlsanity.com/paper/2504.06814/) (2025)

## Abstract (arXiv)

Mirror Descent (MD) is a scalable first-order method widely used in large-scale optimization, with applications in image processing, policy optimization, and neural network training. This paper generalizes MD to optimization on Riemannian manifolds. In particular, we develop a Riemannian Mirror Descent (RMD) framework via reparameterization and further propose a stochastic variant of RMD. We also establish non-asymptotic convergence guarantees for both RMD and stochastic RMD. As an application to the Stiefel manifold, our RMD framework reduces to the Curvilinear Gradient Descent (CGD) method proposed in [26]. Moreover, when specializing the stochastic RMD framework to the Stiefel setting, we obtain a stochastic extension of CGD, which effectively addresses large-scale manifold optimization problems.