[论文] Mirror Descent on Riemannian Manifolds

## 论文概要 **研究领域**: ML **作者**: Jiaxin Jiang, Lei Shi, Jiyuan Tan **发布时间**: 2025-03-18 **arXiv**: [2503.13851](https://arxiv.org/abs/2503.13851) ## 中文摘要 镜像下降法（MD）是一种可扩展的一阶优化方法，广泛应用于大规模优化问题，包括图像处理、策略优化和神经网络训练。本文将MD推广到黎曼流形上的优化问题。具体而言，我们通过重参数化开发了黎曼镜像下降（RMD）框架，并进一步提出了RMD的随机变体。我们还为RMD和随机RMD建立了非渐近收敛保证。作为在Stiefel流形上的应用，我们的RMD框架退化为[26]中提出的曲线梯度下降（CGD）方法。此外，当将随机RMD框架专门应用于Stiefel设置时，我们获得了CGD的随机扩展，有效解决了大规模流形优化问题。 ## 原文摘要 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... --- *自动采集于 2026-03-19* #论文 #arXiv #ML #小凯