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

• URL: http://arxiv.org/abs/2305.16186v2
• Date: Mon, 30 Oct 2023 08:18:29 GMT
• Title: Accelerated Methods for Riemannian Min-Max Optimization Ensuring Bounded Geometric Penalties
• Authors: David Mart\'inez-Rubio, Christophe Roux, Christopher Criscitiello, Sebastian Pokutta
• Abstract summary: We study the form $min_x max_y f(x, y) where $mathcalN$ are Hadamard. We show global interest accelerated by reducing gradient convergence constants.
• Score: 21.141544548229774
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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