Are there any references discussing an interval algorithm for the vanilla gradient descent method given a function $f \colon \mathbb{R}^n \to \mathbb{R}$?
Edit: In particular, I am searching for an interval algorithm that bounds the path $\phi\bigl([0,t^*]\bigr)$ where $\phi$ is the solution to the ODE \phi'(t) = -\nabla f\bigl(\phi(t)\bigr), \quad \phi(0)=x_0. There are plenty of interval ODE solvers but that is not sufficient for my purpose. I need more precise control over how big the intervals bounding the solution become and how they step.