Open-Loop Riemannian Frank--Wolfe: Fast Rates under Error Bounds and Scaling Inequalities

What the math gives to ML

The paper gives a non-adaptive step-size principle for Frank–Wolfe updates on curved feasible sets: setting the step solely from the iteration index can achieve polynomial rates under geometric error bounds, without line searches or gradient-based feedback. This is potentially useful for neural layers whose parameters lie on spheres, geodesic balls, or other Riemannian constraint sets, where projection or retraction can be expensive or distort the constraint. The most practical transfer is a Riemannian Frank–Wolfe optimizer for unit-norm or spherical parameter blocks, using the schedule \(\eta_k=a/(k+a)\) and testing whether recent-window Frank–Wolfe gaps exhibit the predicted decay. The guarantee is conditional on a local Hölderian error bound and directional scaling inequality, so these assumptions should be treated as experimentally testable diagnostics rather than claimed universally.