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.

翻译：镜像下降法（MD）是一种可扩展的一阶优化方法，广泛应用于图像处理、策略优化和神经网络训练等大规模优化问题。本文将其推广至黎曼流形上的优化问题。具体而言，我们通过重参数化技术建立了黎曼镜像下降法（RMD）框架，并进一步提出了RMD的随机变体。同时，我们为RMD及其随机版本建立了非渐近收敛性保证。作为在Stiefel流形上的应用实例，我们的RMD框架可简化为文献[26]提出的曲率梯度下降法（CGD）。此外，将随机RMD框架特化到Stiefel流形场景时，我们得到了CGD的随机扩展版本，该方法能有效解决大规模流形优化问题。