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

TL;DR

This paper develops accelerated optimization algorithms for min-max problems on Riemannian manifolds, achieving faster convergence and removing previous assumptions about iterate bounds.

Contribution

It introduces new g-convex optimization results, improves existing accelerated methods by reducing geometric constants, and removes assumptions on iterate boundedness in Riemannian min-max problems.

Findings

Global linear convergence for metric-projected Riemannian gradient descent.

Improved accelerated methods with reduced geometric constants.

Removed previous assumptions about iterates remaining in compact sets.

Abstract

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.