# Global convergence of the gradient method for functions definable in o-minimal structures

arXiv: https://arxiv.org/abs/2303.03534 · HTML: https://mlsanity.com/paper/2303.03534/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Mar 2023.

## Built on by (most cited first, 5 of 5)

- [SAD Neural Networks: Divergent Gradient Flows and Asymptotic Optimality via o-minimal Structures](https://mlsanity.com/paper/2505.09572/) (2025)
- [Taming Binarized Neural Networks and Mixed-Integer Programs](https://mlsanity.com/paper/2310.04469/) (2023)
- [Piecewise Polynomial Regression of Tame Functions via Integer Programming](https://mlsanity.com/paper/2311.13544/) (2023)
- [Global convergence of gradient descent for phase retrieval](https://mlsanity.com/paper/2410.09990/) (2024)
- [Modeling high dimensional point clouds with the spherical cluster model](https://mlsanity.com/paper/2512.21960/) (2025)

## Abstract (arXiv)

We consider the gradient method with variable step size for minimizing functions that are definable in o-minimal structures on the real field and differentiable with locally Lipschitz gradients. We prove that global convergence holds if continuous gradient trajectories are bounded, with the minimum gradient norm vanishing at the rate o(1/k) if the step sizes are greater than a positive constant. If additionally the gradient is continuously differentiable, all saddle points are strict, and the step sizes are constant, then convergence to a local minimum holds almost surely over any bounded set of initial points.