Riemannian Dueling Optimization

TL;DR

This paper extends dueling optimization to Riemannian manifolds, proposing new algorithms with theoretical guarantees and demonstrating their effectiveness on synthetic and real-world problems.

Contribution

It introduces Riemannian dueling optimization algorithms, RDNGD and RDFW, with convergence analysis and applicability to manifold-constrained problems.

Findings

RDNGD achieves convergence under geodesic smoothness and convexity.

RDFW provides a projection-free alternative with complexity guarantees.

Numerical experiments validate the algorithms' effectiveness.

Abstract

Dueling optimization considers optimizing an objective with access to only a comparison oracle of the objective function. It finds important applications in emerging fields such as recommendation systems and robotics. Existing works on dueling optimization mainly focused on unconstrained problems in the Euclidean space. In this work, we study dueling optimization over Riemannian manifolds, which covers important applications that cannot be solved by existing dueling optimization algorithms. In particular, we propose a Riemannian Dueling Normalized Gradient Descent (RDNGD) method and establish its iteration complexity when the objective function is geodesically L-smooth or geodesically (strongly) convex. We also propose a projection-free algorithm, named Riemannian Dueling Frank-Wolfe (RDFW) method, to deal with the situation where projection is prohibited. We establish the iteration and oracle complexities for RDFW. We illustrate the effectiveness of the proposed algorithms through numerical experiments on both synthetic and real applications.