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

Recent years have witnessed exciting progress in the study of stochastic
variance reduced gradient methods (e.g., SVRG, SAGA), their accelerated
variants (e.g, Katyusha) and their extensions in many different settings (e.g.,
online, sparse, asynchronous, distributed). Among them, accelerated methods
enjoy improved convergence rates but have complex coupling structures, which
makes them hard to be extended to more settings (e.g., sparse and asynchronous)
due to the existence of perturbation. In this paper, we introduce a simple
stochastic variance reduced algorithm (MiG), which enjoys the best-known
convergence rates for both strongly convex and non-strongly convex problems.
Moreover, we also present its efficient sparse and asynchronous variants, and
theoretically analyze its convergence rates in these settings. Finally,
extensive experiments for various machine learning problems such as logistic
regression are given to illustrate the practical improvement in both serial and
asynchronous settings.

本篇论文介绍了一种简单的随机方差减小 (MiG) 算法及其在强凸和非强凸问题中最佳的收敛速率，并在稀疏和异步情况下介绍了其有效的变体并在这些情况下理论化分析其收敛速率。最后，我们进行了大量的实验，如逻辑回归等，以证明在串行和异步设置中的实际改进。