Mirror Descent on Riemannian Manifolds
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866917351471972352 |
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| author | Jiang, Jiaxin Shi, Lei Tan, Jiyuan |
| author_facet | Jiang, Jiaxin Shi, Lei Tan, Jiyuan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_17527 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Mirror Descent on Riemannian Manifolds Jiang, Jiaxin Shi, Lei Tan, Jiyuan Machine Learning Optimization and Control 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. |
| title | Mirror Descent on Riemannian Manifolds |
| topic | Machine Learning Optimization and Control |
| url | https://arxiv.org/abs/2603.17527 |