Region-Preserving Accelerated Gradient Methods for Geodesically Convex Optimization

Abstract

Accelerated Riemannian optimization methods often require a priori assumptions that their iterates remain in a region where geometric bounds hold. We develop a parameterized inertial dynamical system and an explicit accelerated gradient method for smooth geodesically convex optimization on Riemannian manifolds with locally bounded sectional curvature. Our setting allows both positive and negative curvature without requiring completeness of the ambient manifold. Starting from a curvature-corrected continuous model, we construct the discrete updates by eliminating indefinite cross terms in the Lyapunov energy difference and derive verifiable conditions for a fixed step size. The resulting energy analysis establishes preservation of the initial geodesic ball without assuming iterate containment in advance. We further establish accelerated convergence rates for both geodesically convex and strongly geodesically convex objectives, without requiring restarts.