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.

本文研究的是优化问题，其中函数在乘积 Riemann 流形上是几何上凸的，设计了加速的方法，改进了现有的方法，并在 Riemann 最小 - 最大的情况下证明了全局线性收敛。

保证几何惩罚有界的黎曼极小 - 极大优化的加速方法

Accelerated Methods for Riemannian Min-Max Optimization Ensuring Bounded  Geometric Penalties

Several first order stochastic optimization methods commonly used in the
Euclidean domain such as stochastic gradient descent (SGD), accelerated
gradient descent or variance reduced methods have already been adapted to
certain Riemannian settings. However, some of the most popular of these
optimization tools - namely Adam , Adagrad and the more recent Amsgrad - remain
to be generalized to Riemannian manifolds. We discuss the difficulty of
generalizing such adaptive schemes to the most agnostic Riemannian setting, and
then provide algorithms and convergence proofs for geodesically convex
objectives in the particular case of a product of Riemannian manifolds, in
which adaptivity is implemented across manifolds in the cartesian product. Our
generalization is tight in the sense that choosing the Euclidean space as
Riemannian manifold yields the same algorithms and regret bounds as those that
were already known for the standard algorithms. Experimentally, we show faster
convergence and to a lower train loss value for Riemannian adaptive methods
over their corresponding baselines on the realistic task of embedding the
WordNet taxonomy in the Poincare ball.

将 Adam、Adagrad 和 Amsgrad 等流行的自适应随机优化方法扩展到里曼流形上面的困难以及基于里曼流形的优化算法和渐进结果的提出，同时在实验中证明该算法比原算法更快且表现更好。