Iteration-complexity of gradient, subgradient and proximal point methods on Riemannian manifolds

TL;DR

This paper analyzes the iteration complexity of gradient, subgradient, and proximal point methods for optimization on Riemannian manifolds, providing improved bounds and extending results to Hadamard manifolds.

Contribution

It offers new iteration-complexity bounds for these methods on Riemannian manifolds with non-negative curvature and Hadamard manifolds, advancing theoretical understanding.

Findings

Derived new iteration-complexity bounds for gradient and subgradient methods.

Extended complexity analysis to the proximal point method on Hadamard manifolds.

Improved upon existing results in Riemannian convex optimization.

Abstract

This paper considers optimization problems on Riemannian manifolds and analyzes iteration-complexity for gradient and subgradient methods on manifolds with non-negative curvature. By using tools from the Riemannian convex analysis and exploring directly the tangent space of the manifold, we obtain different iteration-complexity bounds for the aforementioned methods, complementing and improving related results. Moreover, we also establish iteration-complexity bound for the proximal point method on Hadamard manifolds.