First-Order Optimization as Minimum-Time Control

What the math gives to ML

The paper recasts first-order optimization as a minimum-time control problem: the iterate is the state, while coefficients multiplying previously observed gradients are controls. Its transferable asset is an instance-specific reachability diagnostic, based on whether a small-gradient target can be reached inside the accumulated gradient span, rather than only using worst-case smoothness constants. On quadratic objectives, this span is Krylov-like and the control conditions recover conjugate-gradient structure. A practical neural-network adaptation is a safeguarded gradient-span extrapolator that periodically searches for a lower-gradient point using a locally fitted secant curvature model, while falling back to the baseline optimizer when the prediction is unreliable.