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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.

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Are you asking about a line-search part of a gradient descent method, or something else? Perhaps about interval arithmetic? – hardmath Mar 16 '12 at 18:51
    
Yes, this algorithm should use interval arithmetic. – James Rohal Mar 16 '12 at 21:15

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