  Geodesic convexity generalizes the notion of (vector space) convexity to
nonlinear metric spaces. But unlike convex optimization, geodesically convex
(g-convex) optimization is much less developed. In this paper we contribute to
the understanding of g-convex optimization by developing iteration complexity
analysis for several first-order algorithms on Hadamard manifolds.
Specifically, we prove upper bounds for the global complexity of deterministic
and stochastic (sub)gradient methods for optimizing smooth and nonsmooth
g-convex functions, both with and without strong g-convexity. Our analysis also
reveals how the manifold geometry, especially \emph{sectional curvature},
impacts convergence rates. To the best of our knowledge, our work is the first
to provide global complexity analysis for first-order algorithms for general
g-convex optimization.
