First-order Methods for Geodesically Convex Optimization

TL;DR

This paper extends first-order optimization methods to geodesically convex functions on Hadamard manifolds, analyzing their convergence and complexity while highlighting the influence of manifold curvature.

Contribution

It provides the first global complexity analysis of first-order algorithms for general geodesically convex optimization on Hadamard manifolds, considering smooth and nonsmooth cases.

Findings

Upper bounds for deterministic and stochastic gradient methods.

Convergence rates depend on sectional curvature.

Analysis applies to both smooth and nonsmooth g-convex functions.

Abstract

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.