Implicit Riemannian Optimism with Applications to Min-Max Problems

TL;DR

This paper introduces a Riemannian optimistic online learning algorithm for Hadamard manifolds that handles in-manifold constraints and achieves regret bounds comparable to Euclidean settings, with applications to min-max problems.

Contribution

It presents a novel Riemannian optimistic online learning algorithm with inexact implicit updates, extending prior Euclidean methods to Hadamard manifolds and addressing min-max problems.

Findings

Matches Euclidean regret bounds without geometric dependence

Develops algorithms for g-convex, g-concave min-max problems on manifolds

Nearly matches gradient oracle complexity lower bounds for Euclidean problems

Abstract

We introduce a Riemannian optimistic online learning algorithm for Hadamard manifolds based on inexact implicit updates. Unlike prior work, our method can handle in-manifold constraints, and matches the best known regret bounds in the Euclidean setting with no dependence on geometric constants, like the minimum curvature. Building on this, we develop algorithms for g-convex, g-concave smooth min-max problems on Hadamard manifolds. Notably, one method nearly matches the gradient oracle complexity of the lower bound for Euclidean problems, for the first time.