Numerous applications in machine learning and data analytics can be formulated as equilibrium computation over Riemannian manifolds. Despite the extensive investigation of their Euclidean counterparts, the performance of Riemannian gradient-based algorithms remain opaque and poorly understood. We revisit the original scheme of Riemannian gradient descent (RGD) and analyze it under a geodesic monotonicity assumption, which includes the well-studied geodesically convex-concave min-max optimization problem as a special case. Our main contribution is to show that, despite the phenomenon of distance distortion, the RGD scheme, with a step size that is agnostic to the manifold's curvature, achieves a curvature-independent and linear last-iterate convergence rate in the geodesically strongly monotone setting. To the best of our knowledge, the possibility of curvature-independent rates and/or last-iterate convergence in the Riemannian setting has not been considered before.

翻译：机器学习和数据分析中的众多应用可表述为黎曼流形上的均衡计算问题。尽管欧几里得对应问题已被广泛研究，但基于黎曼梯度的算法性能仍不明确且理解有限。本文重新审视了黎曼梯度下降（RGD）的原始方案，并在测地单调性假设下对其进行分析，该假设将广泛研究的测地凸-凹极小极大优化问题作为特例包含在内。我们的主要贡献在于证明：尽管存在距离畸变现象，但采用与流形曲率无关的步长时，RGD方案在测地强单调设定下可实现与曲率无关的线性最终迭代收敛速率。据我们所知，在黎曼设定中考虑曲率无关收敛速率和/或最终迭代收敛的可能性此前尚未被探讨过。