In this work, we study optimization problems of the form $\min_x \max_y f(x, y)$, where $f(x, y)$ is defined on a product Riemannian manifold $\mathcal{M} \times \mathcal{N}$ and is $\mu_x$-strongly geodesically convex (g-convex) in $x$ and $\mu_y$-strongly g-concave in $y$, for $\mu_x, \mu_y \geq 0$. We design accelerated methods when $f$ is $(L_x, L_y, L_{xy})$-smooth and $\mathcal{M}$, $\mathcal{N}$ are Hadamard. To that aim we introduce new g-convex optimization results, of independent interest: we show global linear convergence for metric-projected Riemannian gradient descent and improve existing accelerated methods by reducing geometric constants. Additionally, we complete the analysis of two previous works applying to the Riemannian min-max case by removing an assumption about iterates staying in a pre-specified compact set.

翻译：本文研究如下形式的优化问题：$\min_x \max_y f(x, y)$，其中 $f(x, y)$ 定义在乘积黎曼流形 $\mathcal{M} \times \mathcal{N}$ 上，且对于 $\mu_x, \mu_y \geq 0$，$f$ 在 $x$ 上是 $\mu_x$-强测地线凸（g-凸）的，在 $y$ 上是 $\mu_y$-强测地线凹（g-凹）的。我们设计加速方法，适用于 $f$ 为 $(L_x, L_y, L_{xy})$-光滑且 $\mathcal{M}$、$\mathcal{N}$ 为 Hadamard 流形的情形。为此，我们引入了新的 g-凸优化结果，这些结果本身具有独立意义：我们证明了度量投影黎曼梯度下降的全局线性收敛性，并通过减少几何常数改进了现有加速方法。此外，我们完善了两项先前工作对黎曼极小极大情形的分析，去除了关于迭代点始终落在预设紧集内的假设。