将镜面下降法拓展到黎曼流形，解决大规模流形优化问题。
Mirror Descent on Riemannian Manifolds
- 通过重参数化构建黎曼流形上的镜面下降框架
- 给出非渐近收敛保证，适用于确定与随机情形
- 在Stiefel流形上还原经典方法并扩展为随机版本
镜面下降（MD）是一种广泛应用于大规模优化的高效一阶方法，常见于图像处理、策略优化和神经网络训练。本文将MD推广至黎曼流形优化，通过重参数化建立黎曼镜面下降（RMD）框架，并提出其随机变体。同时，本文建立了RMD与随机RMD的非渐近收敛性理论。作为在Stiefel流形上的应用，所提框架退化为文献[26]提出的曲线梯度下降（CGD）方法；进一步地，将随机RMD应用于该设置，得到一种有效的随机扩展版本，可解决大规模流形优化问题。
原文摘要 · Abstract (English)
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.
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