Nonsmooth Riemannian optimization with inexact manifold primitives via bundle methods

TL;DR

This paper introduces a bundle method for nonsmooth geodesically convex optimization on Hadamard manifolds, providing the first complexity bounds using inexact primitives and subgradients.

Contribution

It develops a proximal bundle method with non-asymptotic complexity bounds that only require inexact manifold primitives and subgradients.

Findings

First oracle-complexity bounds for inexact primitives on Hadamard manifolds.

Achieves sublinear convergence for general objectives.

Attains optimal linear convergence under sharp function growth.

Abstract

Optimization on Hadamard manifolds -- the natural Riemannian setting for globally geodesically convex problems -- relies on exponential maps to retract tangent vectors and parallel transport to connect tangent spaces across the manifold. These primitives are often computationally expensive, leading software packages to rely on approximations: first-order retractions and vector transports. However, existing results for optimization on Hadamard manifolds either require exact primitives or lack non-asymptotic rates. We bridge this gap by introducing a proximal bundle method for nonsmooth geodesically convex optimization and establishing the first oracle-complexity bounds that rely only on subgradients and inexact primitives. We obtain sublinear rates for general objectives and optimal linear convergence under sharp function growth.