Iteration-Complexity of the Subgradient Method on Riemannian Manifolds with Lower Bounded Curvature

TL;DR

This paper analyzes the iteration complexity of the subgradient method for convex optimization on Riemannian manifolds with lower bounded curvature, providing bounds for different step-size strategies.

Contribution

It establishes and improves iteration-complexity bounds for the subgradient method on Riemannian manifolds, covering exogenous and Polyak's step-size rules.

Findings

Derived iteration-complexity bounds for exogenous step-size

Derived iteration-complexity bounds for Polyak's step-size

Completes and improves recent theoretical results

Abstract

The subgradient method for convex optimization problems on complete Riemannian manifolds with lower bounded sectional curvature is analyzed in this paper. Iteration-complexity bounds of the subgradient method with exogenous step-size and Polyak's step-size are stablished, completing and improving recent results on the subject.