Error Bounds in Riemannian Optimization

[5amohtaerg]

From my understanding, people are generally more interested in the convergence bounds regarding Riemannian optimization. However, I am wondering if there are any ways to study the error bounds in this context.
Specifically, I'm considering a simple scenario:

The error bound of single-task regression on the Stiefel Manifold using Riemannian optimization.

My questions are:

1. Are there established methods or recent developments in studying error bounds for Riemannian optimization problems, particularly on the Stiefel manifold?
2. For the case of single-task regression on the Stiefel manifold, is it possible to derive an error bound? If so, what form might it take?
3. How does the geometry of the Stiefel manifold affect the error bound analysis compared to optimization in Euclidean space?

Any insights, relevant literature, or suggestions for approaching this problem would be greatly appreciated. Thank you!