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

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 ${\mu }_x,{\mu }_y\ge 0$. 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.