Year: 2026
Type: preprint
Abstract: Projection-free methods replace projections by linear minimization oracles, making their convergence fundamentally sensitive to the geometry of the feasible region. We develop a geometric framework for this dependence in Riemannian Frank--Wolfe optimization. A power-type double-geodesic modulus quantifies the intrinsic geometry of the feasible set, and an exact transfer principle shows that this modulus induces a Frank--Wolfe scaling margin modulated by the differential of the Riemannian logarithm map. The resulting factorization separates the contributions of feasible-set geometry and ambient Riemannian distortion. Rauch comparison provides explicit curvature-dependent certificates for this distortion, while on the positive-definite cone equipped with the affine-invariant metric the logarithm differential admits an exact spectral characterization in terms of relative eigenvalue spread. Combined with a quadratic error bound, the power-type geometry yields the convergence rate $O(t^{-p/(p-1)})$ for the gap-feedback short-step policy. For every fixed $p>3$, a compact feasible set in a hyperbolic plane attains the matching rate $Θ(t^{-p/(p-1)})$ under the same policy. The geometry-derived exponent $p/(p-1)$ is therefore sharp for the gap-feedback short-step policy over the geometry-controlled class considered here.
Source: arXiv (Cornell University)
Author: Nguyen, Binh T.
Institution: Vietnam National University Ho Chi Minh City
Language: English
Subfield: Numerical Analysis
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7220711266